Hypothèse bourdieusienne sur l’Université

[SHB]

L’Université n’est pas exclusivement l’apanage des fils de la petite bourgeoisie a fort capital social et culturel. Je crois qu’elle est majoritairement le fait des familles de classe moyenne a bon ou fort capital culturel.
.
L’hypothèse est la suivante : l’Université n’est-elle pas le lieu ou la classe moyenne a fort capital culturel cherche a faire fructifier ledit capital culturel en capital social a travers le diplôme académique qui offre une certaine ascension dans la hiérarchie de classe?
.
Autrement, on peut dire que la culture est le moyen que trouves la classe moyenne pour espérer grimper vers la petite et grande bourgeoisie.
.
De ce fait, peut-on dire que l’investissement du milieu de la culture par la classe moyenne et haute classe moyenne est ainsi l’une des composantes de la lutte des classes, composante qui permet à cette dernière classe d’espérer atteindre la bourgeoisie?
.
Le mot « culture » est ici entendu au sens de culture légitime.

[françois bégaudeau]

S’il y a bien deux domaines où le bourdieusisme demande urgemment à être remis à jour, c’est bien celui de la culture – quelle est la culture légitime aujourd’hui?- et celui de l’université, devenue la poubelle des études après le bac.

[SHB]

je définirais la culture légitime par l’anti-rap encore bien présent (exemple avec les levées de bouclier contre aya nakamura avec son côté rap et « racaille ») + la promotion des auteurs dits légitimes en contradiction avec des sciences sociale dites « amateures » + encore un mépris d’internet bien présent chez les élites + promotion du « beau » en contradiction avec le « n’importe quoi » qui représente tous les aspects de la contre-culture artistique, etc…..

[SHB]

globalement en science sociales la culture intellectuelle légitime se traduit par un mépris pour les auteurs communistes ou anarchistes vus comme engagés et « militants », en opposition a un monde universitaire libéral se vivant tout en nuance et en pondération.

[SHB]

pour aller vite, de tout temps, la culture légitime a été la culture de la bourgeoisie. Il suffit donc de voir ce que la bourgeoisie exècre pour connaitre la « sous culture » et voir ce que la bourgeoisie fétichise pour voir la culture légitime

[SHB]

En gros tu allume CNews une journée et les débats te donne un peu le ton des cultures méprisées et des cultures adorées.

[françois bégaudeau]

Trois signes d’une réponse fébrile :
-Tu multiplies les réponses.
-tu mélanges deux voire trois conceptions de la culture légitime, dont une que tu inventes.
-Tu es très abstrait.

Dans le détail :
-« je définirais la culture légitime par l’anti-rap encore bien présent (exemple avec les levées de bouclier contre aya nakamura avec son côté rap et « racaille »)
C’est une réponse par la négative, et en outre elle est très très très discutable. Tu convoques une anecdote (les JO) que tu lis à l’envers. Car sur la question de la légitimité, ce qui importe ici c’est que Nakamura soit désignée comme chanteuse par les instances… légitimes. Cette désignation est une légitimation pur jus. Qu’il y ait ensuite des gens pour la contester, c’est secondaire. Surtout qu’en l’occurrence l’illégitimité qu’ils proclament n’est pas de type bourdieusien (distinction de gout) mais de type national-raciste.
Plus généralement le rap est un très mauvais exemple. C’est LA musique qui a été légitimée depuis trente ans.  » La culture légitime a été la culture de la bourgeoisie », dis-tu (merci de m’en informer, ca me donne envie d’un livre). Mais précisément tout un pan de la bourgeoisie… écoute du rap, et le glorifie. Et ce depuis les années 90.
Pose toi maintenant la question de ce que serait un gout musical spécifique à la bourgeoisie, et donc par là légitime. Mozart? 1% de la bourgeoisie écoute Mozart.
– » encore un mépris d’internet bien présent chez les élites ». 1 Internet n’est pas un gout culturel. Hors sujet. 2 les « élites » peuvent dire ce qu’elles veulent d’Internet, elles en sont des usageres frénétiques. 3 « les élites » ne cessent de financer des programmes de développement d’Internet – fibre, 5G, numérisation de tous les services, numérisation de l école. Tu planes.
-« promotion du « beau » en contradiction avec le « n’importe quoi » qui représente tous les aspects de la contre-culture artistique, etc….. » Inutile de répondre à cet argument que son abstraction invalide.
-« globalement en science sociales la culture intellectuelle légitime se traduit par un mépris pour les auteurs communistes ou anarchistes vus comme engagés et « militants », en opposition a un monde universitaire libéral se vivant tout en nuance et en pondération. » Même dévoiement de la notion de culture légitime que sur Nakamura. Ici la ligne de front est idéologique, ce qui n’a rien à voir avec la légitimité – Sartre le communiste était détesté par la bourgeoisie mais nul n’aurait nié sa légitimité d’écrivain. On trouve d’ailleurs de nombreux contre-exemples de ce que tu dis : par exemple la prix Goncourt de Vuillard, à l’occasion d’un livre qui expose l’entente entre capital et nazisme ; le succès de Despentes perçue comme une autrice de gauche radicale, etc.
-« En gros tu allume CNews une journée et les débats te donne un peu le ton des cultures méprisées et des cultures adorées. » Il est précisément très peu question de culture sur Cnews. Signe que décidément la bourgeoisie ne se positionne plus dans la culture. Tout juste entendra-t-on les gens de C news déblatérer abstraitement contre le « monde de la culture », perçu comme dominant et à gauche. Encore un renversement total de ton hypothèse : ce que pourfend Praud, c’est la culture… légitime.
– » et voir ce que la bourgeoisie fétichise pour voir la culture légitime » C’était précisément l’objet de ma question. A laquelle tu n’as pas répondu.

[V]

D’autant que Bourdieu a donné les outils pour le mettre à jour. Ayant porté un structuralisme dit génétique, il fait un distinguo entre le caractère transhistorique des structures (mécanismes de reproduction, de distinction…) et les choses sur lesquelles ces dernières vont porter (exemplairement des pratiques culturelles ou certains cursus académiques…). Ainsi, pour répondre aux critiques (notamment des marxistes) visant les structuralistes (au sens large) concernant l’absence d’historicité, Bourdieu stipule que des changements phénoménaux existent là où les structures demeurent. Il faut voir cela dans la perspective de ce qu’il appelle une « économie générale des pratiques », à savoir qu’une chose sociale repose sur une appréciation/dépréciation symbolique et non intrinsèque – ce qui laisse place à des évolutions, voire des inversions.

Pour prendre des exemples concrets, et revenir spécifiquement au sujet abordé : la hausse du niveau d’études dans la population et la perte de prestige de certaines filières universitaires. Ces points sont analysés dans un chapitre de « La Distinction » dans lequel Bourdieu parle de « translation vers le haut ».
Sur la culture, Bourdieu évoque des standards musicaux (jazz ou classique) qui ont perdu de leur valeur (symbolique) en raison d’une trop large diffusion. Ils perdent, par conséquent, leur caractère distinctif pour un pan de la bourgeoisie dont la raison sociale réside dans la détention d’un capital culturel.
Aujourd’hui, je crois que ce pan devient de plus en plus minoritaire d’une part et que sa quête de distinction (ce qui est immuable, donc) porte sur un mélange de kitsch, culture rétro et appropriation – parfois sur le mode de la distanciation ironique – de la culture populaire ou de masse.

[Charles]

Justement il faudrait peut-être remettre en question la distinction. On est d’accord qu’elle ne prend plus du tout la même forme qu’avant, au moins dans les objets et pratiques culturels. Mais je me demande même si elle passe encore par la culture. Faute d’une culture légitime très délimitée, est-ce qu’elle peut opérer avec la même efficacité qu’avant? Par exemple, un de mes cinéastes vivants préférés est Hong Sang-Soo, cinéaste sud coréen bien connu de ce site qui fait des films d’1h-1h30 max avec deux mouvements de caméra, zéro intrigue ou presque et dont le dernier film est presque entièrement flou. Ses films font max 40.000 entrées mais sont sélectionnés dans les festivals internationaux où ils remportent souvent des prix.
On pourrait dire que ça se pose là en termes de goût distinctif. Sauf qu’en dehors de quelques cinéphiles personne ne le connaît et que parmi les cinéphiles la moitié le déteste et vous prend pour un snob tandis que l’autre moitié l’adore. C’est donc au sein d’un petit milieu que cette distinction peut éventuellement jouer. C’est très différent de quelqu’un qui affirme dans les années 70 n’écouter que Bach et Mahler car tout le monde ou presque connait les deux ou sait à peu près ce dont on parle. Quand il n’y a plus de référent commun, même approximatif, et que toute la culture est morcelée, la distinction par la culture a beaucoup moins de sens. Non?

[V]

Je ne crois pas. Aimer Jul ou Aya Nakamura en tant que bourgeois est éminemment distinctif (surtout quand ça s’accompagne d’un mépris latent). Comme je le disais à la fin de mon post, la distinction bourgeoise passe aujourd’hui soit par une distanciation ironique de façon à ne pas trop s’assimiler à l’objet (ex : jeunes bourgeois qui rigolent en soirée sur les paroles sexistes d’un morceau de rap), soit par une réappropriation et une intellectualisation par le prisme de ses valeurs (ex : le monde universitaire sur les séries, le livre sur Timothée Chalamet présenté comme symbole d’une nouvelle masculinité ou encore les Popstars devenues des icônes féministes et/ou queer). On parle ici de référents parfaitement communs. Simplement, ce qui caractérise le bourgeois, qu’il écoute Bach ou le dernier morceau du rappeur SDM, c’est de ne jamais aimer les choses pour ce qu’elles sont. Vis-à-vis du paysage culturel, il se positionne toujours par rapport à ce que ça dit d’eux, à la manière dont ils se perçoivent et dont ils veulent être perçus. Même quand ils consomment un pur divertissement, ils en passent par certaines justifications. Je pense que la hiérarchie se joue précisément à ce niveau.
Et il faut faire attention à un point : si les véritables esthètes sont bien souvent issus de la bourgeoisie, ils sont par définition minoritaires (car fondamentalement improductifs). C’est d’autant plus le cas aujourd’hui où cette caractéristique est de moins en moins reconnue socialement pour les raisons susdites. En aimant des réalisateurs du type de Hong Sang-Soo, tu es extrêmement minoritaire par rapport à des époques antérieures où c’était déjà minoritaire.

Par ailleurs, quant au manque de référents communs, ce que tu avances ressemble à la thèse de l’archipelisation : il y aurait moins une hiérarchisation symbolique, qu’un éclatement et une étanchéité des objets et pratiques culturels. Encore une fois, il faut opérer le distinguo entre les changements phénoménaux et la permanence de la structure : si l’offre culturelle change du fait de nouveaux supports (ex : les plateformes de streaming), les œuvres sont toujours, de manière hétéronome, tributaires des supports de diffusion (qui sont éminemment matériels). Et ça, ça reste commun. Peu importe qu’on regarde deux contenus différents sur Netflix, on regarde Netflix. Et alors, on revient à ce qui est énoncé plus-haut.

[Demi Habile]

and also the definition of the unpolarized cross section to write
X
spins
Z
|M12→34|
2
(2π)
4
δ
4
(p1 + p2 − p3 − p4)
d
3p3
(2π)
32E3
d
3p4
(2π)
32E4
=
4F g1g2 σ12→34, (1.31)
where F ≡ [(p1 · p2)
2 − m2
1m2
2
]
1/2
and the spin factors g1, g2 come from the average
over initial spins. This way, the collision term (1.29) is written in a more compact form
g1
Z
C[f1]
d
3p1
(2π)
3
= −
Z
σvMøl (dn1dn2 − dn
eq
1 dn
eq
2
), (1.32)
where σ =
P
(all f)
σ12→f is the total annihilation cross section summed over all the
possible final states and vMøl ≡
F
E1E2
. The so called Møller velocity, vMøl, is defined in
such a way that the product vMøln1n2 is invariant under Lorentz transformations and,
in terms of particle velocities ~v1 and ~v2, it is given by the expression
vMøl =
h
~v2
1 − ~v2
2

2
− |~v1 × ~v2|
2
i1/2
. (1.33)
Due to symmetry considerations, the distributions in kinetic equilibrium are proportional to those in chemical equilibrium, with a proportionality factor independent of
the momentum. Therefore, the collision term (1.32), both before and after decoupling,
can be written in the form
g1
Z
C[f1]
d
3p1
(2π)
3
= −hσvMøli(n1n2 − n
eq
1 n
eq
2
), (1.34)
where the thermal averaged total annihilation cross section times the Møller velocity
has been defined by the expression
hσvMøli =
R
σvMøldn
eq
1 dn
eq
2
R
dn
eq
1 dn
eq
2
. (1.35)
We will come back to the thermal averaged cross section in the next subsection.
We are, now, able to write the full integrated Boltzmann equation, using the expressions (1.28), (1.34) that we have derived for the Liouville and the collision term,
respectively. In the simplified but interesting case of identical particles 1 and 2, the
Boltzmann equation is, finally, written as
n˙ + 3Hn = −hσvMøli(n
2 − n
2
eq). (1.36)
18 Dark Matter
However, instead of using n, it is more convenient to take the expansion of the universe
into account and calculate the number density per comoving volume Y , which can be
defined as the ratio of the number and entropy densities: Y ≡ n/s. The total entropy
density S = R3
s (R is the scale factor) remains constant, hence we can obtain a
differential equation for Y by dividing (1.36) by S. Before we write the final form
of the Boltzmann equation that it is used for the relic density calculations, we have
to change the variable that parametrizes the comoving density. In practice, the time
variable t is not convenient and the temperature of the Universe (actually the photon
temperature, since the photons were the last particles that went out of equilibrium) is
used instead. However, it proves even more useful to use as time variable the quantity
defined by x ≡ m/T with m the DM mass, so that Eq. (1.36) transforms into
dY
dx
=
1
3H
ds
dx
hσvMøli

Y
2 − Y
2
eq
. (1.37)
Last, using the Hubble parameter (1.2) for a radiation dominated Universe and the
expressions (1.20), (1.21) for the energy and entropy density, the Boltzmann equation
is written in its final form
dY
dx
= −
r
45GN
π
g
1/2
∗ m
x
2
hσvMøli

Y
2 − Y
2
eq
, (1.38)
where the effective degrees of freedom g
1/2
∗ have been defined by
g
1/2
∗ ≡
heff
g
1/2
eff

1 +
1
3
T
heff
dheff
dT

. (1.39)
The equilibrium density per comoving volume Yeq ≡ neq/s can be expressed as
Yeq(x) = 45g
4π
4
x
2K2(x)
heff(m/x)
, (1.40)
with K2 the modified Bessel function of second kind.
1.4.3 Thermal average of the annihilation cross section
We are going to derive a simple formula that one can use to calculate the thermal
average of the cross section times velocity, based again on the analysis of [38]. We will
use the assumption that equilibrium functions follow the Maxwell-Boltzmann distribution, instead of the actual Bose-Einstein or Fermi-Dirac. This is a well established
assumption if the freeze out occurs after T ≃ m/3 or for x >∼ 3, which is actually the
case for WIMPs. Under this assumption, the expression (1.35) gives, in the cosmic
comoving frame,
hσvMøli =
R
vMøle
−E1/T e
−E2/T d
3p1d
3p2
R
e
−E1/T e
−E2/T d
3p1d
3p2
. (1.4
1.4.3 Thermal average of the annihilation cross section 19
The volume element can be written as d3p1d
3p2 = 4πp1dE14πp2dE2
1
2
cos θ, with θ the
angle between ~p1 and ~p2. After changing the integration variables to E+, E−, s given
by
E+ = E1 + E2, E− = E1 − E2, s = 2m2 + 2E1E2 − 2p1p2 cos θ, (1.42)
(with s = −(p1 − p2)
2 one of the Mandelstam variables,) the volume element becomes
d
3p1d
3p2 = 2π
2E1E2dE+dE−ds and the initial integration region
{E1 > m, E2 > m, | cos θ| ≤ 1i
transforms into
|E−| ≤
1 −
4m2
s
1/2
(E
2
+ − s)
1/2
, E+ ≥
√
s, s ≥ 4m2
. (1.43)
After some algebraic calculations, it can be found that the quantity hσvMøliE1E2
depends only on s, specifically vMølE1E2 =
1
2
p
s(s − 4m2
). Hence, the numerator of the expression (1.41), which after changing the integration variables reads
2π
2
R
dE+
R
dE−
R
dsσvMølE1E2e
−E+/T , can be written, eventually, as
Z
vMøle
−E1/T e
−E2/T = 2π
2
Z ∞
4m2
dsσ(s − 4m2
)
Z
dE+e
−E+/T (E
2
+ − s)
1/2
. (1.44)
The integral over E+ can be written with the help of the modified Bessel function of
the first kind K1 as √
s T K1(
√
s/T). The denominator of (1.41) can be treated in a
similar way, so that the thermal average is, finally, given by the expression
hσvMøli =
1
8m4TK2
2
(x)
Z ∞
4m2
ds σ(s)(s − 4m2
)
√
s K1(
√
s/T). (1.45)
Eqs. (1.38)–(1.40) along with this last Eq. (1.45) are all we need in order to calculate
the relic density of a WIMP, if its total annihilation cross section in terms of the
Mandelstam variable s is known.
In many cases, in order to avoid the numerical integration in Eq. (1.45), an approximation for hσvMøli can be used. The thermal average is expanded in powers of x
−1
(or, equivalently, in powers of the squared WIMP velocity):
hσvMøli = a + bx−1 + . . . . (1.46)
(The coefficient a corresponds to the s-wave contribution to the cross section, the
coefficient b to the p-wave contribution, and so on.) This partial wave expansion gives
a quite good approximation, provided there are no s-channel resonances and thresholds
for the final states [39].
In [40], it was shown that, after expanding the integrands of Eq. (1.41) in powers
of x
−1
, all the integrations can be performed analytically. As we saw, the expression
20 Dark Matter
vMølE1E2 depends on momenta only through s. Therefore, one can form the Lorentz
invariant quantity
w(s) ≡ σ(s)vMølE1E2 =
1
2
σ(s)
p
s(s − 4m2
). (1.47)
The integration involves the Taylor expansion of this quantity w around s/4m2 = 1
and the general formula for the partial wave expansion of the thermal average is [40]
hσvMøli =
1
m2

w −
3
2
(2w − w
′
)x
−1 +
3
8
(16w − 8w
′ + 5w
′′)x
−2
−
5
16
(30w − 15w
′ + 3w
′′ − 7x
′′′)x
−3 + O(x
−4
)

s/4m2=1
, (1.48)
where primes denote derivatives with respect to s/4m2 and all quantities have to be
evaluated at s = 4m2
.
1.5 Direct Detection of DM
Since the beginning of 1980s, it has been realized that besides the numerous facts showing evidence for the existence of these new dark particles, it is also possible to detect
them directly. Already in 1985, two pioneering articles [41, 42] appeared, describing
the detection methods for WIMPs. Since WIMPs are expected to cluster gravitationally together with ordinary stars in the Milky Way halo, they would pass also through
Earth and, in principle, they can be detected through scattering with the nuclei in a
detector’s material. In practice, one has to measure the recoil energy deposited by this
scattering.
However, although one can deduce from rotation curves that DM dominates the
dark halo in the outer parts of our galaxy, it is not so obvious from direct measurements
whether there is any substantial amount of DM inside the solar radius R0 ≃ 8 kpc.
Using indirect methods (involving the determination of the gravitational potential,
through the measuring of the kinematics of stars, both near the mid-plane of the
galactic disk and at heights several times the disk thickness), it is almost certain
that the DM is also present in the solar system, with a local density ρ0 = (0.3 ±
0.1) GeV cm−3
[43].
This value for the local density implies that for a WIMP mass of order ∼ 100 GeV,
the local number density is n0 ∼ 10−3
cm−3
. It is also expected that the WIMPs
velocity is similar to the velocity with which the Sun orbits around the galactic center
(v0 ≃ 220 km s−1
), since they are both moving under the same gravitational potential.
These two quantities allow to estimate the order of magnitude of the incident flux
of WIMPs on the Earth: J0 = n0v0 ∼ 105
cm−2
s
−1
. This value is manifestly large,
but the very weak interactions of the DM particles with ordinary matter makes their
detection a difficult, although in principle feasible, task. In order to compensate for
the very low WIMP-nucleus scattering cross section, very large detectors are required.
1.5.1 Elastic scattering event rate 21
1.5.1 Elastic scattering event rate
In the following, we will confine ourselves to the elastic scattering with nuclei. Although
inelastic scattering of WIMPs off nuclei in a detector or off orbital electrons producing
an excited state is possible, the event rate of these processes is quite suppressed. In
contrast, during an elastic scattering the nucleus recoils as a whole.
The direct detection experiments measure the number of events per day and per
kilogram of the detector material, as a function of the amount of energy Q deposited
in the detector. This event rate would be given by R = nWIMP nnuclei σv in a simplified
model with WIMPs moving with a constant velocity v. The number density of WIMPs
is nWIMP = ρ0/mX and the number density of nuclei is just the ratio of the detector’s
mass over the nuclear mass mN .
For accurate calculations, one should take into account that the WIMPs move in the
halo not with a uniform velocity, but rather following a velocity distribution f(v). The
Earth’s motion in the solar system should be included into this distribution function.
The scattering cross section σ also depends on the velocity. Actually, the cross section
can be parametrized by a nuclear form factor F(Q) as
dσ =
σ
4m2
r
v
2
F
2
(Q)d|~q|
2
, (1.49)
where |~q|
2 = 2m2
r
v
2
(1 − cos θ) is the momentum transferred during the scattering,
mr =
mXmN
mX+mN
is the reduced mass of the WIMP – nucleus system and θ is the scattering
angle in the center of momentum frame. Therefore, one can write a general expression
for the differential event rate per unit detector mass as
dR =
ρ0
mX
1
mN
σF2
(Q)d|~q|
2
4m2
r
v
2
vf(v)dv. (1.50)
The energy deposited in the detector (transferred to the nucleus through one elastic
scattering) is
Q =
|~q|
2
2mN
=
m2
r
v
2
mN
(1 − cos θ). (1.51)
Therefore, the differential event rate over deposited energy can be written, using the
equations (1.50) and (1.51), as
dR
dQ
=
σρ0
√
πv0mXm2
r
F
2
(Q)T(Q), (1.52)
where, following [37], we have defined the dimensionless quantity T(Q) as
T(Q) ≡
√
π
2
v0
Z ∞
vmin
f(v)
v
dv, (1.53)
with the minimum velocity given by vmin =
qQmN
2m2
r
, obtained by Eq. (1.51). Finally,
the event rate R can be calculated by integrating (1.52) over the energy
R =
Z ∞
ET
dR
dQ
dQ. (1.54)
22 Dark Matter
The integration is performed for energies larger than the threshold energy ET of the
detector, below which it is insensitive to WIMP-nucleus recoils.
Using Eqs. (1.54) and (1.52), one can derive the scattering cross section from the
event rate. The experimental collaborations prefer to give their results already in terms
of the scattering cross section as a function of the WIMP mass. To be more precise,
the WIMP-nucleus total cross section consists of two parts: the spin-dependent (SD)
cross section and the spin-independent (SI) one. The former comes from axial current
couplings, whereas the latter comes from scalar-scalar and vector-vector couplings.
The SD cross section is much suppressed compared to the SI one in the case of heavy
nuclei targets and it vanishes if the nucleus contains an even number of nucleons, since
in this case the total nuclear spin is zero.
We see that two uncertainties enter the above calculation: the exact value of the
local density ρ0 and the exact form of the velocity distribution f(v). To these, one
has to include one more. The cross section σ that appears in the previous expressions
concerns the WIMP-nucleon cross section. The couplings of a WIMP with the various
quarks that constitute the nucleon are not the same and the WIMP-nucleon cross
section depends strongly on the exact quark content of the nucleon. To be more
precise, the largest uncertainty lies on the strange content of the nucleon, but we shall
return to this point when we will calculate the cross section in a specific particle theory,
the Next-to-Minimal Supersymmetric Standard Model, in Sec. 3.5.1.
1.5.2 Experimental status
The situation of the experimental results from direct DM searches is a bit confusing. The null observations in most of the experiments led them to set upper limits
on the WIMP-nucleon cross section. These bounds are quite stringent for the spinindependent cross section7
, especially in the regime of WIMP masses of the order of
100 GeV. However, some collaborations have already reported possible DM signals,
mainly in the low mass regime. The preferred regions of these experiments do not
coincide, while some of them have been already excluded by other experiments. The
present picture, for WIMP masses ranging from 5 to 1000 GeV, is summarized in Fig.
1.5, 1.6.
Fig. 1.5 mainly presents upper bounds coming from XENON100 [44]. XENON100
[46] is an experiment located at the Gran Sasso underground laboratory in Italy. It
contains in total 165 kg of liquid Xenon, with 65 kg acting as target mass and the
rest shielding the detector from background radiation. For these upper limits, 225
live days of data were used. The minimum value for the predicted upper bounds on
the cross section is 2 · 10−45 cm2
for WIMP mass ∼ 55 GeV (at 90% confidence level),
almost one order of magnitude lower than the previously released limits [47] by the
same collaboration, using 100 live days of data.
The stringent upper bounds up-to-date (at least for WIMP mass larger than about
7 GeV) come from the first results of the LUX experiment (see Fig. 1.6), after the first
7For the spin-dependent scattering, the exclusion limits are quite relaxed. Hence, we will focus on
the SI cross sections.
1.5.2 Experimental status 23
Figure 1.5: The XENON100 exclusion limit (thick blue line), along with the expected
sensitivity in green (1σ) and yellow (2σ) band. Other upper bounds are also shown as
well as detection claims. From [44].
85.3 live-days of its operation [45]. LUX [53] is a detector containing liquid Xenon, as
XENON100, but in larger quantity, with total mass 370 kg. Its operation started on
April 2013 with a goal to clearly detect or exclude WIMPs with a spin independent
cross section ∼ 2 · 10−46 cm2
.
In Fig. 1.5, except of the XENON100 bounds and other experimental limits on larger
WIMP-nucleon cross section, some detection claims also appear. These come from
DAMA [48,49], CoGeNT [50] and CRESST-II [51] experiments. The first positive result
came from DAMA [52], back in 2000. Since then, the experiment has accumulated 1.17
ton-yr of data over 13 years of operation. DAMA consists of 250 kg of radio pure NaI
scintillator and looks for the annual modulation of the WIMP flux in order to reduce
the influence of the background.
The annual modulation of the DM flux (see [54] for a recent review) is due to the
Earth’s orbital motion relative to the rotation of the galactic disk. The galactic disk
rotation through an essentially non-rotating DM halo, creates an effective DM wind in
the solar frame. During the earth’s heliocentric orbit, this wind reaches a maximum
when the Earth is moving fastest in the direction of the disk rotation (this happens
in the beginning of June) and a minimum when it is moving fastest in the opposite
direction (beginning of December).
DAMA claims an 8.9σ annual modulation with a minimum flux on May 26±7 days,
consistent with the expectation. Since the detector’s target consists of two different
nuclei and the experiment cannot distinguish between sodium and iodine recoils, there
24 Dark Matter
Figure 1.6: The LUX 90% confidence exclusion limit (blue line) with the 1σ range
(shaded area). The XENON100 upper bound is represented by the red line. The inset
shows also preferred regions by CoGeNT (shaded light red), CDMS II silicon detector
(shaded green), CRESST II (shaded yellow) and DAMA (shaded gray). From [45].
is no model independent way to determine the exact region in the cross section versus
WIMP mass plane to which the observed modulation corresponds. However, one can
assume two cases: one that the WIMP scattering off the sodium nucleus dominates the
recoil energy and the other with the iodine recoils dominating. The former corresponds
[55] to a light WIMP (∼ 10 GeV) and quite large scattering cross section and the latter
to a heavier WIMP (∼ 50 to 100 GeV) with smaller cross section (see Fig. 1.5).
The positive result of DAMA was followed many years later by the ones of CoGeNT
and CRESST-II, and more recently by the silicon detector of CDMS [56] (Fig. 1.7).
The discrepancy of the results raised a lot of debates among the experiments (for
example, [64–67]) and by some the positive results are regarded as controversial. On
the other hand, it also raised an effort to find a physical explanation behind this
inconsistency (see, for example, [68–71]).
1.6 Indirect Methods for DM Detection
The same annihilation processes that determined the DM relic abundance in the early
Universe also occur today in galactic regions where the DM concentration is higher.
This fact rises the possibility of detecting potential WIMP pair annihilations indirectly
through their imprints on the cosmic rays. Therefore, the indirect DM searches aim
at the detection of an excess over the known astrophysical background of charged
particles, photons or neutrinos.
Charged particles – electrons, protons and their antiparticles – may originate from
direct products (pair of SM particles) of WIMP annihilations, after their decay and
1.6 Indirect Methods for DM Detection 25
Figure 1.7: The blue contours represent preferred regions for a possible signal at 68%
and 90% C.L. using the silicon detector of CMDS [56]. The blue dotted line represents
the upper limit obtained by the same analysis and the blue solid line is the combined
limit with the silicon CDMS data set reported in [57]. Other limits also appear:
from the CMDS standard germanium detector (light and dark red dashed line, for
standard [58] and low threshold analysis [59], respectively), EDELWEISS [60] (dashed
orange), XENON10 [61] (dash-dotted green) and XENON100 [44] (long-dash-dotted
green). The filled regions identify possible signal regions associated with data from
CoGeNT [62] (dashed yellow, 90% C.L.), DAMA [49,55] (dotted tan, 99.7% C.L.) and
CRESST-II [51, 63] (dash-dotted pink, 95.45% C.L.) experiments. Taken from [56].
through the process of showering and hadronization. Although the exact shape of the
resulting spectrum would depend on the specific process, it is expected to show a steep
cutoff at the WIMP mass. Once produced in the DM halo, the charged particles have
to travel to the point of detection through the turbulent galactic field, which will cause
diffusion. Apart from that, a lot of processes disturb the propagation of the charged
particles, such as bremsstrahlung, inverse Compton scattering with CMB photons and
many others. Therefore, the uncertainties that enter the propagation of the charged
flux until it reaches the telescope are important (contrary to the case of photons and
neutrinos that propagate almost unperturbed through the galaxy).
As in the case of direct detection, the experimental status of charged particle detection concerning the DM is confusing. After some hints from HEAT [72] and AMS01 [73] (the former a far-infrared telescope in Antarctica, the latter a spectrometer,
prototype for AMS-02 mounted on the International Space Station [74]), the PAMELA
satellite observed [75, 76] a steep increase in the energy spectrum of positron fraction
e
+/(e
+ + e
−)
8
. Later FERMI satellite [77] and AMS-02 [78] confirmed the results up
8The searches for charged particles focus on the antiparticles in order to have a reduced background,
26 Dark Matter
Figure 1.8: A compilation of data of charged cosmic rays, together with plausible but
uncertain astrophysical backgrounds, taken from [79]. Left: Positron flux. Center:
Antiproton flux. Right: Sum of electrons and positrons.
to energies of ∼ 200 GeV. However, the excess of positrons is not followed by an excess
of antiprotons, whose flux seems to coincide with the predicted background [75]. In
Fig. 1.8, three plots summarizing the situation are shown [79].
The observed excess is very difficult to explain in terms of DM [79]. To begin with,
the annihilation cross section required to reproduce the excess is quite large, many
orders of magnitude larger than the thermal cross section. Moreover, an “ordinary”
WIMP with large annihilation cross section giving rise to charged leptons is expected
to give, additionally, a large number of antiprotons, a fact in contradiction with the
observations. Although a lot of work has been done to fit a DM particle to the observed
pattern, it is quite possible that the excesses come from a yet unknown astrophysical
source. We are not going to discuss further this matter, but we end with a comment.
If this excess is due to a source other than DM, then a possible DM positron excess
would be lost under this formidable background.
A last hint for DM came from the detection of highly energetic photons. However,
we will interrupt this discussion, since this signal and a possible explanation is the
subject of Ch. 4. There, we will also see the upper bounds on the annihilation cross
section being set due to the absence of excesses in diffuse γ radiation.
since they are much less abundant than the corresponding particles.
CHAPTER 2
PARTICLE PHYSICS
Since the DM comprises of particles, it should be explained by a general particle physics
theory. We start in the following section by describing the Standard Model (SM) of
particle physics. Although the SM describes so far the fundamental particles and their
interactions quite accurately, it cannot provide a DM candidate. Besides, the SM
suffers from some theoretical problems, which we discuss in Sec. 2.2. We will see that
these problems can be solved if one introduces a new symmetry, the supersymmetry,
which we describe in Sec. 2.3. We finish this chapter by briefly describing in Sec. 2.4 a
supersymmetric extension of the SM with the minimal additional particle content, the
Minimal Supersymmetric Standard Model (MSSM).
2.1 The Standard Model of Particle Physics
The Standard Model (SM) of particle physics1
consists of two well developed theories,
the quantum chromodynamics (QCD) and the electroweak (EW) theory. The former
describes the strong interactions among the quarks, whereas the latter describes the
electroweak interactions (the weak and electromagnetic interactions in a unified context) between fermions. The EW theory took its final form in the late 1960s by the
introduction by S. Weinberg [85] and A. Salam [86] of the Higgs mechanism that gives
masses to the SM particles, which followed the unification of electromagnetic and weak
interactions [87,88]. At the same time, the EW model preserves the gauge invariance,
making the theory renormalizable, as shown later by ’t Hooft [89]. On the other hand,
QCD obtained its final form some years later, after the confirmation of the existence
of quarks. Of course, the history of the SM is much longer and it can be traced back to
1920s with the formulation of a theoretical basis for a Quantum Field Theory (QFT).
Since then, the SM had many successes. The SM particle content was completed with
the discovery of the heaviest of the quarks, the top quark [90,91], in 1995 and, recently,
with the discovery of the Higgs boson [92, 93].
1There are many good textbooks on the SM and Quantum Field Theory, e.g. [80–84].
28 Particle Physics
The key concept within the SM, as in every QFT, is that of symmetries. Each
interaction respects a gauge symmetry, based on a Lie algebra. The strong interaction is
described by an SU(3)c symmetry, where the subscript c stands for color, the conserved
charge of strong interactions. The EW interactions, on the other hand, are based on
a SU(2)L × U(1)Y Lie algebra. Here, as we will subsequently see, L refers to the
left-handed fermions and Y is the hypercharge, the conserved charge under the U(1).
SU(2)L conserves a quantity known as weak isospin I. Therefore, the SM contains the
internal symmetries of the unitary product group
SU(2)L × U(1)Y × SU(3)c. (2.1)
2.1.1 The particle content of the SM
We mention for completeness that particles are divided into two main classes according
to the statistics they follow. The bosons are particles with integer spin and follow the
Bose-Einstein distribution, whereas fermions have half-integer spin and follow the
Dirac-Einstein statistics, obeying the Pauli exclusion principle. In the SM, all the
fermions have spin 1/2, whereas the bosons have spin 1 with only exception the Higgs
boson, which is a scalar (spin zero). We begin the description of the SM particles with
the fermions.
Each fermion is classified in irreducible representations of each individual Lie algebra, according to the conserved quantum numbers, i.e. the color C, the weak isospin
I and the hypercharge Y . A first classification of fermions can be done into leptons
and quarks, which transform differently under the SU(3)c. Leptons are singlets under
this transformation, while quarks act as triplets (the fundamental representation of
this group). The EW interactions violate maximally the parity symmetry and SU(2)L
acts only on states with negative chirality (left-handed). A Dirac spinor Ψ can be
decomposed into left and right chirality components using, respectively, the projection
operators PL =
1
2
(1 − γ5) and PR =
1
2
(1 + γ5):
ΨL = PLΨ and ΨR = PRΨ. (2.2)
Left-handed fermions have I = 1/2, with a third component of the isospin I3 = ±1/2.
Fermions with positive I3 are called up-type fermions and those with negative are
called down-type. These behave the same way under SU(2)L and form doublets with
one fermion of each type. On the other hand, right-handed fermions have I = 0 and
form singlets that do not undergo weak interactions. The hypercharge is written in
terms of the electric charge Q and the third component of the isospin I3 through the
Gell-Mann–Nishijima relation:
Q = I3 + Y/2. (2.3)
Therefore, left- and right-handed components transform differently under the U(1)Y ,
since they have different hypercharge.
The fermionic sector of the SM comprises three generations of fermions, transforming as spinors under Lorentz transformations. Each generation has the same structure.
For leptons, it is an SU(2)L doublet with components consisting of one left-handed
2.1.2 The SM Lagrangian 29
charged lepton and one neutrino (neutrinos are only left-handed in the SM), along
with a gauge singlet right-handed charged lepton. The quark doublet consists of an
up- (u) and a down-type (d) (left-handed) quark and the pattern is completed by the
two corresponding SU(2)L singlet right-handed quarks. We write these representations
as
Quarks: Q ≡

u
i
L
d
i
L
!
, ui
R, di
R Leptons: L ≡

ν
i
L
e
i
L
!
, ei
R, (2.4)
with i = 1, 2, 3 the generation index.
Having briefly described the fermionic sector, we turn to the bosonic sector of
the SM. It consists of the gauge bosons that mediate the interactions and the Higgs
boson that gives masses to the particles through a spontaneous symmetry breaking,
the electroweak symmetry breaking (EWSB) [94–98], which we shall describe in Sec.
2.1.3. Before the EWSB, these bosons are
• three Wa
µ
(a = 1, 2, 3) weak bosons, associated with the generators of SU(2)L,
• one neutral Bµ boson, associated with the generator of U(1)Y ,
• eight gluons Ga
µ
(a = 1, . . . , 8), associated with the generators of SU(3)c, and
• the complex scalar Higgs doublet Φ =
φ
+
φ
0
!
.
After the EWSB, the EW boson states mix and give the two W± bosons, the neutral
Z boson and the massless photon γ. From the symmetry breaking, one scalar degree of
freedom remains which is the famous (neutral) Higgs boson [97–99]. We will return to
the mixed physical states, after describing the Higgs mechanism for symmetry breaking.
A complete list of the SM particles (the physical states after EWSB) is shown in Table
2.1.
2.1.2 The SM Lagrangian
The gauge bosons are responsible for the mediation of the interactions and are associated with the generators of the corresponding symmetry. The EW gauge bosons Bµ
and Wa
µ
are associated, respectively, with the generator Y of the U(1)Y and the three
generators T
a
2
of the SU(2)L. The latter are defined as half of the Pauli matrices τ
a
(T
a
2 =
1
2
τ
a
) and they obey the algebra

T
a
2
, Tb
2

= iǫabcT
c
2
, (2.5)
where ǫ
abc is the fully antisymmetric Levi-Civita tensor. The eight gluons are associated
with an equal number of generators T
a
3
(Gell-Mann matrices) of SU(3)c and obey the
Lie algebra

T
a
3
, Tb
3

= if abcT
c
3
, with Tr
T
a
3 T
b
3

=
1
2
δ
ab
, (2.6)
30 Particle Physics
Name symbol mass charge (|e|) spin
Leptons
electron e 0.511 MeV −1 1/2
electron neutrino νe 0 (<2 eV) 0 1/2
muon µ 105.7 MeV −1 1/2
muon neutrino νµ 0 (<2 eV) 0 1/2
tau τ 1.777 GeV −1 1/2
tau neutrino ντ 0 (<2 eV) 0 1/2
Quarks
up u 2.7
+0.7
−0.5 MeV 2/3 1/2
down d 4.8
+0.7
−0.3 MeV −1/3 1/2
strange s (95 ± 5) MeV −1/3 1/2
charm c (1.275 ± 0.025) GeV 2/3 1/2
bottom b (4.18 ± 0.03) GeV −1/3 1/2
top t (173.5 ± 0.6 ± 0.8) GeV 2/3 1/2
Bosons
photon γ 0 (<10−18 eV) 0 (<10−35) 1
W boson W± (80.385 ± 0.015) GeV ±1 1
Z boson Z (91.1876 ± 0.0021) GeV 0 1
gluon g 0 (.O(1) MeV) 0 1
Higgs H
(125.3 ± 0.4 ± 0.5) GeV
0 0
(126.0 ± 0.4 ± 0.4) GeV
Table 2.1: The particle content of the SM. All values are those given in [100], except of
the Higgs mass that is taken from [92, 93] (up and down row, respectively), assuming
that the observed excess corresponds to the SM Higgs. The u, d and s quark masses
are estimates of so-called “current-quark masses” in a mass-independent subtraction
scheme as MS at a scale ∼ 2 GeV. The c and b quark masses are the running masses
in the MS scheme. The values in the parenthesis are the current experimental limits.
with f
abc the structure constants of the group.
Using the structure constants of the corresponding groups, we define the field
strengths for the gauge bosons as
Bµν ≡ ∂µBν − ∂νBµ, (2.7a)
Wµν ≡ ∂µWa
ν − ∂νWa
µ + g2ǫ
abcWb
µWc
ν
(2.7b)
and
G
a
µν ≡ ∂µG
a
ν − ∂νG
a
µ + g3f
abcG
b
µG
c
ν
. (2.7c)
2.1.2 The SM Lagrangian 31
We use the notation g1, g2 and g3 for the coupling constants of U(1)Y , SU(2)L and
SU(3)c, respectively. As in any Yang-Mills theory, the non-abelian gauge groups lead
to self-interactions, which is not the case for the abelian U(1)Y group.
Before we finally write the full Lagrangian, we have to introduce the covariant
derivative for fermions, which in a general form can be written as
DµΨ =
∂µ − ig1
1
2
Y Bµ − ig2T
a
2 Wa
µ − ig3T
a
3 G
a
µ

Ψ. (2.8)
This form has to be understood as that, depending on Ψ, only the relevant terms
apply, hence for SU(2)L singlet leptons only the two first terms inside the parenthesis
are relevant, for doublet leptons the three first terms and for the corresponding quark
singlets and doublets the last term also participates. We also have to notice that in
order to retain the gauge symmetry, mass terms are forbidden in the Lagrangian. For
example, the mass term mψψ¯ = m

ψ¯
LψR + ψ¯
RψL

(with ψ¯ ≡ ψ
†γ
0
) is not invariant
under SU(2)L. This paradox is solved by the introduction of the Higgs scalar field
(see next subsection). The SM Lagrangian can be now written2
, split for simplicity in
three parts, each describing the gauge bosons, the fermions and the scalar sector,
LSM = Lgauge + Lfermion + Lscalar, (2.9)
with
Lgauge = −
1
4
G
a
µνG
µν
a −
1
4
Wa
µνWµν
a −
1
4
BµνB
µν
, (2.10a)
Lfermion = iL¯Dµγ
µL + ie¯RDµγµeR
+ iQ¯Dµγ
µQ + iu¯RDµγ
µuR + i
¯dRDµγ
µ
dR
−

heL¯ΦeR + hdQ¯ΦdR + huQ¯ΦeuR + h.c.

(2.10b)
and
Lscalar = (DµΦ)†
(DµΦ) − V (Φ†Φ), (2.10c)
where
V (Φ†Φ) = µ
2Φ
†Φ + λ

Φ
†Φ
2
(2.11)
is the scalar Higgs potential. Φ is the conjugate of Φ, related to the charge conjugate e
by Φ =e iτ2Φ
⋆
, with τi the Pauli matrices. The covariant derivative acting on the Higgs
scalar field gives
DµΦ =
∂µ − ig1
1
2
Y Bµ − ig2T
a
2 Wa
µ

Φ. (2.12)
Before we proceed to the description of the Higgs mechanism, a last comment concerning the SM Lagrangian is in order. If we restore the generation indices, we see that
2For simplicity, from now on we are going to omit the generations indice
32 Particle Physics
the Yukawa couplings h are 3 × 3, in general complex, matrices. As any complex matrix, they can be diagonalized with the help of two unitary matrices VL and VR, which
are related by VR = U
†VL with U again a unitary matrix. The diagonalization in the
quark sector to the mass eigenstates induces a mixing among the flavors (generations),
described by the Cabibbo–Kobayashi–Maskawa (CKM) matrix [101, 102]. The CKM
matrix is defined by
VCKM ≡ V
u
L
†
V
d
L
†
, (2.13)
where V
u
L
, V
d
L
are the unitary matrices that diagonalize the Yukawa couplings Hu
, Hd
,
respectively. This product of the two matrices appears in the charged current when it
is expressed in terms of the observable mass eigenstates.
2.1.3 Mass generation through the Higgs mechanism
We will start by examining the scalar potential (2.11). The vacuum expectation value
(vev) of the Higgs field hΦi ≡ h0|Φ|0i is given by the minimum of the potential. For
µ
2 > 0, the potential is always non-negative and Φ has a zero vev. The hypothesis of
the Higgs mechanism is that µ
2 < 0. In this case, the field Φ will acquire a vev
hΦi =
1
2

0
v
!
with v =
r
−
µ2
λ
. (2.14)
Since the charged component of Φ still has a zero vev, the U(1)Q symmetry of quantum
electrodynamics (QED) remains unbroken.
We expand the field Φ around the minima v in terms of real fields, and at leading
order we have
Φ(x) =
θ2(x) + iθ1(x)
√
1
2
(v + H(x)) − iθ3(x)
!
=
1
√
2
e
iθa(x)τ
a

0
v + H(x)
!
. (2.15)
We can eliminate the unphysical degrees of freedom θa, using the fact that the theory
remains gauge invariant. Therefore, we perform the following SU(2)L gauge transformation on Φ (unitary gauge)
Φ(x) → e
−iθa(x)τ
a
Φ(x), (2.16)
so that
Φ(x) = 1
√
2

0
v + H(x)
!
. (2.17)
We are going to use the following definitions for the gauge fields
W±
µ ≡
1
2

W1
µ ∓ iW2
µ

, (2.18a)
Zµ ≡
1
p
g
2
1 + g
2
2

g2W3
µ − g1Bµ

, (2.18b)
Aµ ≡
1
p
g
2
1 + g
2
2

g1W3
µ + g2Bµ

, (2.1
2.2 Limits of the SM and the emergence of supersymmetry 33
Then, the kinetic term for Φ (see Eq. (2.10c)) can be written in the unitary gauge as
(DµΦ)†
(D
µΦ) = 1
2
(∂µH)
2 + M2
W W+
µ W−µ +
1
2
M2
ZZµZ
µ
, (2.19)
with
MW ≡
1
2
g2v and MZ ≡
1
2
q
g
2
1 + g
2
2
v. (2.20)
We see that the definitions (2.18) correspond to the physical states of the gauge bosons
that have acquired masses due to the non-zero Higgs vev, given by (2.20). The photon
has remained massless, which reflects the fact that after the spontaneous breakdown of
SU(2)L × U(1)Y the U(1)Q remained unbroken. Among the initial degrees of freedom
of the complex scalar field Φ, three were absorbed by W± and Z and one remained as
the neutral Higgs particle with squared mass
m2
H = 2λv2
. (2.21)
We note that λ should be positive so that the scalar potential (2.11) is bounded from
below.
Fermions also acquire masses due to the Higgs mechanism. The Yukawa terms in
the fermionic part (2.10b) of the SM Lagrangian are written, after expanding around
the vev in the unitary gauge,
LY = −
1
√
2
hee¯L(v + H)eR −
1
√
2
hd
¯dL(v + H)dR −
1
√
2
huu¯L(v + H)uR + h.c. . (2.22)
Therefore, we can identify the masses of the fermions as
me
i =
h
i
e
v
√
2
, md
i =
h
i
d
v
√
2
, mui =
h
i
u
v
√
2
, (2.23)
where we have written explicitly the generation indices.
2.2 Limits of the SM and the emergence of supersymmetry
2.2.1 General discussion of the SM problems
The SM has been proven extremely successful and has been tested in high precision
in many different experiments. It has predicted many new particles before their final
discovery and also explained how the particles gain their masses. Its last triumph was
of course the discovery of a boson that seems to be very similar to the Higgs boson of
the SM. However, it is generally accepted that the SM cannot be the ultimate theory. It
is not only observed phenomena that the SM does not explain; SM also faces important
theoretical issues.
The most prominent among the inconsistencies of the SM with observations is the
oscillations among neutrinos of different generations. In order for the oscillations to
34 Particle Physics
φ φ
k
Figure 2.1: The scalar one-loop diagram giving rise to quadratic divergences.
occur, neutrinos should have non-zero masses. However, minimal modifications of the
SM are able to fit with the data of neutrino physics. Another issue that a more complete theory has to face is the matter asymmetry, the observed dominance of matter
over antimatter in the Universe. In addition, in order to comply with the standard
cosmological model, it has to provide the appropriate particle(s) that drove the inflation. Last, but not least, we saw that in order to explain the DM that dominates the
Universe, a massive, stable weakly interacting particle must exist. Such a particle is
not present in the SM.
On the other hand, the SM also suffers from a theoretical perspective. For example,
the SM counts 19 free parameters; one expects that a fundamental theory would have
a much smaller number of free parameters. Simple modifications of the SM have been
proposed relating some of these parameters. Grand unified theories (GUTs) unify
the gauge couplings at a high scale ∼ 1016 GeV. However, this unification is only
approximate unless the GUT is embedded in a supersymmetric framework. Another
serious problem of the SM is that of naturalness. This will be the topic of the following
subsection.
2.2.2 The naturalness problem of the SM
The presence of fundamental scalar fields, like the Higgs, gives rise to quadratic divergences. The diagram of Fig. 2.1 contributes to the squared mass of the scalar
δm2 = λ
Z Λ
d
4k
(2π)
4
k
−2
. (2.24)
This contribution is approximated by δm2 ∼ λΛ
2/(16π
2
), quadratic in a cut-off Λ,
which should be finite. For the case of the Higgs scalar field, one has to include its
couplings to the gauge fields and the top quark3
. Therefore,
δm2
H =
3Λ2
8π
2v
2

4m2
t − 2M2
W − M2
Z − m2
H

+ O(ln Λ
µ
)

, (2.25)
where we have used Eq. (2.21) and m2
H ≡ m2
0 + δm2
H.
3Since the contribution to the squared mass correction are quadratic in the Yukawa couplings (or
quark masses), the lighter quarks can be neglected
2.2.3 A way out 35
Taking Λ as a fundamental scale Λ ∼ MP l ∼ 1019 GeV we have
m2
0 = m2
H −
3Λ2
8π
2v
2

4m2
t − 2M2
W − M2
Z − m2
H

(2.26)
and we can see that m2
0 has to be adjusted to a precision of about 30 orders of magnitude
in order to achieve an EW scale Higgs mass. This is considered as an intolerable finetuning, which is against the general belief that the observable properties of a theory
have to be stable under small variations of the fundamental (bare) parameters. It is
exactly the above behavior that is considered as unnatural. Although the SM could
be self-consistent without imposing a large scale, grand unification of the parameters
introduce a hierarchy problem between the different scales.
A more strict definition of naturalness comes from ’t Hooft [103], which we rewrite
here:
At an energy scale µ, a physical parameter or set of physical parameters
αi(µ) is allowed to be very small only if the replacement αi(µ) = 0 would
increase the symmetry of the system.
Clearly, this is not the case here. Although mH is small compared to the fundamental
scale Λ, it is not protected by any symmetry and a fine-tuning is necessary.
2.2.3 A way out
The naturalness in the ’t Hooft sense is inspired by quantum electrodynamics, which is
the archetype for a natural theory. For example, the corrections to the electron mass
me are themselves proportional to me, with a dimensionless proportionality factor that
behaves like ∼ ln Λ. In general, fermion masses are protected by the chiral symmetry; small values (compared to the fundamental scale) of these masses enhances the
symmetry.
If a new symmetry exists in nature, relating fermion fields to scalar fields, then each
scalar mass would be related somehow to the corresponding fermion mass. Therefore,
the scalar mass itself can be naturally small compared to Λ, since this would mean
that the fermion mass is small, which enhances the chiral symmetry. Such a symmetry,
relating bosons to fermions and vice versa, is known as supersymmetry [104, 105].
Actually, as we will see later, if this new symmetry remains unbroken, the masses of
the conjugate bosons and fermions would have to be equal.
In order to make the above statement more concrete, we consider a toy model with
two additional complex scalar fields feL and feR. We will discuss only the quadratic
divergences that come from corrections to the Higgs mass due to a fermion. The
generalization for the contributions from the gauge bosons or the self-interaction is
straightforward. The interactions in this toy model of the new scalar fields with the
Higgs are described by the Lagrangian
Lfefφe = λfe|φ|
2

|feL|
2 + |feR|
2

. (2.27
36 Particle Physics
It can be easily checked that the quadratic divergence coming from a fermion at one
loop is exactly canceled, as long as the new quartic coupling λfe obeys the relation
λfe = −λ
2
f
(λf is the Yukawa coupling for the fermion f).
2.3 A brief summary of Supersymmetry
Supersymmetry (SUSY) is a symmetry relating fermions and bosons. The supersymmetry transformation should turn a boson state into a fermion state and vice versa. If
Q is the operator that generates such transformations, then
Q |bosoni = |fermioni Q |fermioni = |bosoni. (2.28)
Due to commutation and anticommutation rules of bosons and fermions, Q has to
be an anticommuting spinor operator, carrying spin angular momentum 1/2. Since
spinors are complex objects, the hermitian conjugate Q†
is also a symmetry operator4
.
There is a no-go theorem, the Coleman-Mandula theorem [106], that restricts the
conserved charges which transform as tensors under the Lorentz group to the generators
of translations Pµ and the generators of Lorentz transformations Mµν. Although this
theorem can be evaded in the case of supersymmetry due to the anticommutation
properties of Q, Q†
[107], it restricts the underlying algebra of supersymmetry [108].
Therefore, the basic supersymmetric algebra can be written as5
{Q, Q†
} = P
µ
, (2.29a)
{Q, Q} = {Q
†
, Q†
} = 0, (2.29b)
[P
µ
, Q] = [P
µ
, Q] = 0. (2.29c)
In the following, we summarize the basic conclusions derived from this algebra.
• The single-particle states of a supersymmetric theory fall into irreducible representations of the SUSY algebra, called supermultiplets. A supermultiplet contains
both fermion and boson states, called superpartners.
• Superpartners must have equal masses: Consider |Ωi and |Ω
′
i as the superpartners, |Ω
′
i should be proportional to some combination of the Q and Q† operators
acting on |Ωi, up to a space-time translation or rotation. Since −P
2
commutes
with Q, Q† and all space-time translation and rotation operators, |Ωi, |Ω
′
i will
have equal eigenvalues of −P
2 and thus equal masses.
• Superpartners must be in the same representation of gauge groups, since Q, Q†
commute with the generators of gauge transformations. This means that they
have equal charges, weak isospin and color degrees of freedom.
4We will confine ourselves to the phenomenologically more interesting case of N = 1 supersymmetry, with N referring to the number of distinct copies of Q, Q†
.
5We present a simplified version, omitting spinor indices in Q and Q†
.
2.3 A brief summary of Supersymmetry 37
• Each supermultiplet contains an equal number of fermion and boson degrees of
freedom (nF and nB, respectively): Consider the operator (−1)2s
, with s the spin
angular momentum, and the states |ii that have the same eigenvalue p
µ of P
µ
.
Then, using the SUSY algebra (2.29) and the completeness relation P
i
|ii hi| =
1, we have P
i
hi|(−1)2sP
µ
|ii = 0. On the other hand, P
i
hi|(−1)2sP
µ
|ii =
p
µTr [(−1)2s
] ∝ nB − nF . Therefore, nF = nB.
As addendum to the last point, we see that two kind of supermultiplets are possible
(neglecting gravity):
• A chiral (or matter or scalar ) supermultiplet, which consists of a single Weyl
fermion (with two spin helicity states, nF = 2) and two real scalars (each with
nB = 1), which can be replaced by a single complex scalar field.
• A gauge (or vector ) supermultiplet, which consists of a massless spin 1 boson
(two helicity states, nB = 2) and a massless spin 1/2 fermion (nF = 2).
Other combinations either are reduced to combinations of the above supermultiplets
or lead to non-renormalizable interactions.
It is possible to study supersymmetry in a geometric approach, using a space-time
manifold extended by four fermionic (Grassmann) coordinates. This manifold is called
superspace. The fields, in turn, expressed in terms of the extended set of coordinates
are called superfields. We are not going to discuss the technical details of this topic
(the interested reader may refer to the rich bibliography, for example [109–111]).
However, it is important to mention a very useful function of the superfields, the
superpotential. A generic form of a (renormalizable) superpotential in terms of the
superfields Φ is the following b
W =
1
2
MijΦbiΦbj +
1
6
y
ijkΦbiΦbjΦbk. (2.30)
The Lagrangian density can always be written according to the superpotential. The
superpotential has also to fulfill some requirements. In order for the Lagrangian to
be supersymmetric invariant, W has to be holomorphic in the complex scalar fields
(it does not involve hermitian conjugates Φb† of the superfields). Conventionally, W
involves only left chiral superfields. Instead of the SU(2)L singlet right chiral fermion
fields, one can use their left chiral charge conjugates.
As we mentioned before, the members of a supermultiplet have equal masses. This
contradicts our experience, since the partners of the light SM particles would have been
detected long time ago. Hence, the supersymmetry should be broken at a large energy
scale. The common approach is that SUSY is broken in a hidden sector, very weakly
coupled to the visible sector. Then, one has to explain how the SUSY breaking mediated to the visible sector. The two most popular scenarios are the gravity mediation
scenario [112–114] and the Gauge-Mediated SUSY Breaking (GSMB) [113, 115–117],
where the mediation occurs through gauge interactions.
There are two approaches with which one can address the SUSY breaking. In the
first approach, one refers to a GUT unification and determines the supersymmetric
38 Particle Physics
breaking parameters at low energies through the renormalization group equations.
This approach results in a small number of free parameters. In the second approach,
the starting point is the low energy scale. In this case, the SUSY breaking has to be
parametrized by the addition of breaking terms to the low energy Lagrangian. This
results in a larger set of free parameters. These terms should not reintroduce quadratic
divergences to the scalar masses, since the cancellation of these divergences was the
main motivation for SUSY. Then, one talks about soft breaking terms.
2.4 The Minimal Supersymmetric Standard Model
One can construct a supersymmetric version of the standard model with a minimal
content of particles. This model is known as the Minimal Supersymmetric Standard
Model (MSSM). In a SUSY extension of the SM, each of the SM particles is either in a
chiral or in a gauge supermultiplet, and should have a superpartner with spin differing
by 1/2.
The spin-0 partners of quarks and leptons are called squarks and sleptons, respectively (or collectively sfermions), and they have to reside in chiral supermultiplets.
The left- and right-handed components of fermions are distinct 2-component Weyl
fermions with different gauge transformations in the SM, so that each must have its
own complex scalar superpartner. The gauge bosons of the SM reside in gauge supermultiplets, along with their spin-1/2 superpartners, which are called gauginos. Every
gaugino field, like its gauge boson partner, transforms as the adjoint representation of
the corresponding gauge group. They have left- and right-handed components which
are charge conjugates of each other: (λeL)
c = λeR.
The Higgs boson, since it is a spin-0 particle, should reside in a chiral supermultiplet. However, we saw (in the fermionic part of the SM Lagrangian, Eq. (2.10b))
that the Y = 1/2 Higgs in the SM can give mass to both up- and down-type quarks,
only if the conjugate Higgs field with Y = −1/2 is involved. Since in the superpotential there are no conjugate fields, two Higgs doublets have to be introduced. Each
Higgs supermultiplet would have hypercharge Y = +1/2 or Y = −1/2. The Higgs
with the negative hypercharge gives mass to the down-type fermions and it is called
down-type Higgs (Hd, or H1 in the SLHA convention [118]) and the other one gives
mass to up-type fermions and it is called up-type Higgs (Hu, or H2).
The MSSM respects a discrete Z2 symmetry, the R-parity. If one writes the most
general terms in the supersymmetric Lagrangian (still gauge-invariant and holomorphic), some of them would lead to non-observed processes. The most obvious constraint
comes from the non-observed proton decay, which arises from a term that violates both
lepton and baryon numbers (L and B, respectively) by one unit. In order to avoid these
terms, R-parity, a multiplicative conserved quantum number, is introduced, defined as
PR = (−1)3(B−L)+2s
, (2.31)
with s the spin of the particle.
The R even particles are the SM particles, whereas the R odd are the new particles
introduced by the MSSM and are called supersymmetric particles. Due to R-parity,
2.4 The Minimal Supersymmetric Standard Model 39
if it is exactly conserved, there can be no mixing among odd and even particles and,
additionally, each interaction vertex in the theory can only involve an even number of
supersymmetric particles. The phenomenological consequences are quite important.
First, the lightest among the odd-parity particles is stable. This particle is known
as the lightest supersymmetric particle (LSP). Second, in collider experiments, supersymmetric particles can only be produced in pairs. The first of these consequences
was a breakthrough for the incorporation of DM into a general theory. If the LSP is
electrically neutral, it interacts only weakly and it consists an attractive candidate for
DM.
We are not going to enter further into the details of the MSSM6
. Although MSSM
offers a possible DM candidate, there is a strong theoretical reason to move from the
minimal model. This reason is the so-called µ-problem of the MSSM, with which we
begin the discussion of the next chapter, where we shall describe more thoroughly the
Next-to-Minimal Supersymmetric Standard Model.
6We refer to [110] for an excellent and detailed description of MSSM.
40 Particle Physics
Part II
Dark Matter in the
Next-to-Minimal Supersymmetric
Standard Model

CHAPTER 3
THE NEXT-TO-MINIMAL
SUPERSYMMETRIC STANDARD
MODEL
The Next-to-Minimal Supersymmetric Standard Model (NMSSM) is an extension of
the MSSM by a chiral, SU(2)L singlet superfield Sb (see [119, 120] for reviews). The
introduction of this field solves the µ-problem1
from which the MSSM suffers, but
also leads to a different phenomenology from that of the minimal model. The scalar
component of the additional field mixes with the scalar Higgs doublets, leading to three
CP-even mass eigenstates and two CP-odd eigenstates (as in the MSSM a doublet-like
pair of charged Higgs also exists). On the other hand, the fermionic component of the
singlet (singlino) mixes with gauginos and higgsinos, forming five neutral states, the
neutralinos.
Concerning the CP-even sector, a new possibility opens. The lightest Higgs mass
eigenstate may have evaded the detection due to a sizeable singlet component. Besides,
the SM-like Higgs is naturally heavier than in the MSSM [123–126]. Therefore, a SMlike Higgs mass ∼ 125 GeV is much easier to explain [127–141]. The singlet component
of the CP-odd Higgs also allows for a potentially very light pseudoscalar with suppressed couplings to SM particles, with various consequences, especially on low energy
observables (for example, [142–145]). The singlino component of the neutralino may
also play an important role for both collider phenomenology and DM. This is the case
when the neutralino is the LSP and the lightest neutralino has a significant singlino
component.
We start the discussion about the NMSSM by describing the µ-problem and how
this is solved in the context of the NMSSM. In Sec. 3.2 we introduce the NMSSM
Lagrangian and we write the mass matrices of the Higgs sector particles and the su1However, historically, the introduction of a singlet field preceded the µ-problem, e.g. [104, 105,
121, 122].
44 The Next-to-Minimal Supersymmetric Standard Model
persymmetric particles, at tree level. We continue by examining, in Sec. 3.3, the DM
candidates in the NMSSM and particularly the neutralino. The processes which determine the neutralino relic density are described in Sec. 3.4. The detection possibilities
of a potential NMSSM neutralino as DM are discussed in (Sec. 3.5). We close this
chapter (Sec. 3.6) by examining possible ways to include non-zero neutrino masses and
the additional DM candidates that are introduced.
3.1 Motivation – The µ-problem of the MSSM
As we saw, the minimal extension of the SM, the MSSM, contains two Higgs SU(2)L
doublets Hu and Hd. The Lagrangian of the MSSM should contain a supersymmetric
mass term, µHuHd, for these two doublets. There are several reasons, which we will
subsequently review, that require the existence of such a term. On the other hand,
the fact that |µ| cannot be very large, actually it should be of the order of the EW
scale, brings back the problem of naturalness. A parameter of the model should be
much smaller than the “natural” scale (the GUT or the Planck scale) before the EW
symmetry breaking. This leads to the so-called µ-problem of the MSSM [146].
The reasons that such a term should exist in the Lagrangian of the MSSM are
mainly phenomenological. The doublets Hu and Hd are components of chiral superfields that also contain fermionic SU(2)L doublets. Their electrically charged components mix with the superpartners of the W± bosons, forming two charged Dirac
fermions, the charginos. The unsuccessful searches for charginos in LEP have excluded
charginos with masses almost up to its kinetic limit (∼ 104 GeV) [147]. Since the µ term
determines the mass of the charginos, µ cannot be zero and actually |µ| >∼ 100 GeV,
independently of the other free parameters of the model. Moreover, µ = 0 would result
in a Peccei-Quinn symmetry of the Higgs sector and an undesirable massless axion.
Finally, there is one more reason for µ 6= 0 related to the mass generation by the Higgs
mechanism. The term µHuHd will be accompanied by a soft SUSY breaking term
BµHuHd. This term is necessary so that both neutral components of Hu and Hd are
non-vanishing at the minimum of the potential.
The Higgs mechanism also requires that µ is not too large. In order to generate
the EW symmetry breaking, the Higgs potential has to be unstable at its origin Hu =
Hd = 0. Soft SUSY breaking terms for Hu and Hd of the order of the SUSY breaking
scale generate such an instability. However, the µ induced squared masses for Hu,
Hd are always positive and would destroy the instability in case they dominate the
negative soft mass terms.
The NMSSM is able to solve the µ-problem by dynamically generating the mass
µ. This is achieved by the introduction of an SU(2)L singlet scalar field S. When S
acquires a vev, a mass term for the Hu and Hd emerges with an effective mass µeff of
the correct order, as long as the vev is of the order of the SUSY breaking scale. This
can be obtained in a more “natural” way through the soft SUSY breaking terms.
3.2 The NMSSM Lagrangian 45
3.2 The NMSSM Lagrangian
All the necessary information for the Lagrangian of the NMSSM can be extracted from
the superpotential and the soft SUSY breaking Lagrangian, containing the soft gaugino and scalar masses, and the trilinear couplings. We begin with the superpotential,
writing all the interactions of the NMSSM superfields, which include the MSSM superfields and the additional gauge singlet chiral superfield2 Sb. Hence, the superpotential
reads
W = λSbHbu · Hbd +
1
3
κSb3
+ huQb · HbuUbc
R + hdHbd · QbDbc
R + heHbd · LbEbc
R.
(3.1)
The couplings to quarks and leptons have to be understood as 3 × 3 matrices and the
quark and lepton fields as vectors in the flavor space. The SU(2)L doublet superfields
are given (as in the MSSM) by
Qb =

UbL
DbL
!
, Lb =

νb
EbL
!
, Hbu =

Hb +
u
Hb0
u
!
, Hbd =

Hb0
d
Hb −
d
!
(3.2)
and the product of two doublets is given by, for example, Qb · Hbu = UbLHb0
u − Hb +
u DbL.
An important fact to note is that the superpotential given by (3.1) does not include all possible renormalizable couplings (which respect R-parity). The most general
superpotential would also include the terms
W ⊃ µHbu · Hbd +
1
2
µ
′Sb2 + ξF s, b (3.3)
with the first two terms corresponding to supersymmetric masses and the third one,
with ξF of dimension mass2
, to a tadpole term. However, the above dimensionful
parameters µ, µ
′ and ξF should be of the order of the SUSY breaking scale, a fact
that contradicts the motivation behind the NMSSM. Here, we omit these terms and
we will work with the scale invariant superpotential (3.1). The Lagrangian of a scale
invariant superpotential possesses an accidental Z3 symmetry, which corresponds to a
multiplication of all the components of all chiral fields by a phase ei2π/3
.
The corresponding soft SUSY breaking masses and couplings are
−Lsof t = m2
Hu
|Hu|
2 + m2
Hd
|Hd|
2 + m2
S
|S|
2
+ m2
Q|Q|
2 + m2
D|DR|
2 + m2
U
|UR|
2 + m2
L
|L|
2 + m2
E|ER|
2
+

huAuQ · HuU
c
R − hdAdQ · HdD
c
R − heAeL · HdE
c
R
+λAλHu · HdS +
1
3
κAκS
3 + h.c.

+
1
2
M1λ1λ1 +
1
2
M2λ
i
2λ
i
2 +
1
2
M3λ
a
3λ
a
3
,
(3.4)
2Here, the hatted capital letters denote chiral superfields, whereas the corresponding unhatted
ones indicate their complex scalar components.
46 The Next-to-Minimal Supersymmetric Standard Model
where we have also included the soft breaking masses for the gauginos. λ1 is the U(1)Y
gaugino (bino), λ
i
2 with i = 1, 2, 3 is the SU(2)L gaugino (winos) and, finally, the λ
a
3
with a = 1, . . . , 8 denotes the SU(3)c gaugino (gluinos).
The scalar potential, expressed by the so-called D and F terms, can be written
explicitly using the general formula
V =
1
2

D
aD
a + D
′2

+ F
⋆
i Fi
, (3.5)
where
D
a = g2Φ
∗
i T
a
ijΦj (3.6a)
D
′ =
1
2
g1YiΦ
∗
i Φi (3.6b)
Fi =
∂W
∂Φi
. (3.6c)
We remind that T
a are the SU(2)L generators and Yi the hypercharge of the scalar
field Φi
. The Yukawa interactions and fermion mass terms are given by the general
Lagrangian
LY ukawa = −
1
2
∂
2W
∂Φi∂Φj
ψiψj + h.c.
, (3.7)
using the superpotential (3.1). The two-component spinor ψi
is the superpartner of
the scalar Φi
.
3.2.1 Higgs sector
Using the general form of the scalar potential, the following Higgs potential is derived
VHiggs =

λ

H
+
u H
−
d − H
0
uH
0
d

+ κS2

2
+

m2
Hu + |λS|
2

H
0
u

2
+

H
+
u

2

+

m2
Hd + |λS|
2

H
0
d

2
+

H
−
d

2

+
1
8

g
2
1 + g
2
2

H
0
u

2
+

H
+
u

2
−

H
0
d

2
−

H
−
d

2
2
+
1
2
g
2
2

H
+
u H
0
d
⋆
+ H
0
uH
−
d
⋆

2
+ m2
S
|S|
2 +

λAλ

H
+
u H
−
d − H
0
uH
0
d

S +
1
3
κAκS
3 + h.c.

.
(3.8)
The neutral physical Higgs states are defined through the relations
H
0
u = vu +
1
√
2
(HuR + iHuI ), H0
d = vd +
1
√
2
(HdR + iHdI ),
S = s +
1
√
2
(SR + iSI ),
3.2.1 Higgs sector 47
where vu, vd and s are, respectively, the real vevs of Hu, Hd and S, which have to be
obtained from the minima of the scalar potential (3.8), after expanding the fields using
Eq. (3.9). We notice that when S acquires a vev, a term µeffHbu · Hbd appears in the
superpotential, with
µeff = λs, (3.10)
solving the µ-problem.
Therefore, the Higgs sector of the NMSSM is characterized by the seven parameters
λ, κ, m2
Hu
, m2
Hd
, m2
S
, Aλ and Aκ. One can express the three soft masses by the three
vevs using the minimization equations of the Higgs potential (3.8), which are given by
vu

m2
Hu + µ
2
eff + λ
2
v
2
d +
1
2
g
2

v
2
u − v
2
d

− vdµeff(Aλ + κs) = 0
vd

m2
Hd + µ
2
eff + λ
2
v
2
u +
1
2
g
2

v
2
d − v
2
u

− vuµeff(Aλ + κs) = 0
s

m2
S + κAκs + 2κ
2σ
2 + λ
2

v
2
u + v
2
d

− 2λκvuvd

− λAλvuvd = 0,
(3.11)
where we have defined
g
2 ≡
1
2

g
2
1 + g
2
2

. (3.12)
One can also define the β angle by
tan β =
vu
vd
. (3.13)
The Z boson mass is given by MZ = gv with v
2 = v
2
u + v
2
d ≃ (174 GeV)2
. Hence, with
MZ given, the set of parameters that describes the Higgs sector of the NMSSM can be
chosen to be the following
λ, κ, Aλ, Aκ, tan b and µeff. (3.14)
CP-even Higgs masses
One can obtain the Higgs mass matrices at tree level by expanding the Higgs potential
(3.8) around the vevs, using Eq. (3.9). We begin by writing3
the squared mass matrix
M2
S
of the scalar Higgses in the basis (HdR, HuR, SR):
M2
S =


g
2
v
2
d + µ tan βBeff (2λ
2 − g
2
) vuvd − µBeff 2λµvd − λ (Aλ + 2κs) vu
g
2
v
2
u +
µ
tan βBeff 2λµvu − λ (Aλ + 2κs) vd
λAλ
vuvd
s + κAκs + (2κs)
2

 ,
(3.15)
where we have defined Beff ≡ Aλ + κs (it plays the role of the B parameter of the
MSSM).
3For economy of space, we omit in this expression the subscript from µ
48 The Next-to-Minimal Supersymmetric Standard Model
Although an analytical diagonalization of the above 3 × 3 mass matrix is lengthy,
there is a crucial conclusion that comes from the approximate diagonalization of the
upper 2 × 2 submatrix. If it is rotated by an angle β, one of its diagonal elements
is M2
Z
(cos2 2β +
λ
2
g
2 sin2
2β) which is an upper bound for its lightest eigenvalue. The
first term is the same one as in the MSSM. The conclusion is that in the NMSSM
the lightest CP-even Higgs can be heavier than the corresponding of the MSSM, as
long as λ is large and tan β relatively small. Therefore, it is much easier to explain
the observed mass of the SM-like Higgs. However, λ is bounded from above in order
to avoid the appearance of the Landau pole below the GUT scale. Depending on the
other free parameters, λ should obey λ <∼ 0.7.
CP-odd Higgs masses
For the pseudoscalar case, the squared mass matrix in the basis (HdI , HuI , SI ) is
M2
P =


µeff (Aλ + κs) tan β µeff (Aλ + κs) λvu (Aλ − 2κs)
µeff
tan β
(Aλ + κs) λvd (Aλ − 2κs)
λ (Aλ + 4κs)
vuvd
s − 3κAκs

 . (3.16)
One eigenstate of this matrix corresponds to an unphysical massless Goldstone
boson G. In order to drop the Goldstone boson, we write the matrix in the basis
(A, G, SI ) by rotating the upper 2 × 2 submatrix by an angle β. After dropping the
massless mode, the 2 × 2 squared mass matrix turns out to be
M2
P =
2µeff
sin 2β
(Aλ + κs) λ (Aλ − 2κs) v
λ (Aλ + 4κs)
vuvd
s − 3Aκs
!
. (3.17)
Charged Higgs mass
The charged Higgs squared mass matrix is given, in the basis (H+
u
, H−
d
⋆
), by
M2
± =

µeff (Aλ + κs) + vuvd

1
2
g
2
2 − λ

cot β 1
1 tan β
!
, (3.18)
which contains one Goldstone boson and one physical mass eigenstate H± with eigenvalue
m2
± =
2µeff
sin 2β
(Aλ + κs) + v
2

1
2
g
2
2 − λ

. (3.19)
3.2.2 Sfermion sector
The mass matrix for the up-type squarks is given in the basis (ueR, ueL) by
Mu =

m2
u + h
2
u
v
2
u −
1
3
(v
2
u − v
2
d
) g
2
1 hu (Auvu − µeffvd)
hu (Auvu − µeffvd) m2
Q + h
2
u
v
2
u +
1
12 (v
2
u − v
2
d
) (g
2
1 − 3g
2
2
)
!
, (3.20)
3.2.3 Gaugino and higgsino sector 49
whereas for down-type squarks the mass matrix reads in the basis (deR, deL)
Md =

m2
d + h
2
d
v
2
d −
1
6
(v
2
u − v
2
d
) g
2
1 hd (Advd − µeffvu)
hd (Advd − µeffvu) m2
Q + h
2
d
v
2
d +
1
12 (v
2
u − v
2
d
) (g
2
1 − 3g
2
2
)
!
. (3.21)
The off-diagonal terms are proportional to the Yukawa coupling hu for the up-type
squarks and hd for the down-type ones. Therefore, the two lightest generations remain
approximately unmixed. For the third generation, the mass matrices are diagonalized
by a rotation by an angle θT and θB, respectively, for the stop and sbottom. The mass
eigenstates are, then, given by
et1 = cos θT
etL + sin θT
etR, et2 = cos θT
etL − sin θT
etR, (3.22)
eb1 = cos θB
ebL + sin θB
ebR, eb2 = cos θB
ebL − sin θB
ebR. (3.23)
In the slepton sector, for a similar reason, only the left- and right-handed staus are
mixed and their mass matrix
Mτ =

m2
E3 + h
2
τ
v
2
d −
1
2
(v
2
u − v
2
d
) g
2
1 hτ (Aτ vd − µeffvu)
hτ (Aτ vd − µeffvu) m2
L3 + h
2
τ
v
2
d −
1
4
(v
2
u − v
2
d
) (g
2
1 − g
2
2
)
!
(3.24)
is diagonalized after a rotation by an angle θτ . Their mass eigenstates are given by
τe1 = cos θτ τeL + sin θτ τeR, τe2 = cos θτ τeL − sin θτ τeR. (3.25)
Finally, the sneutrino masses are
mνe = m2
L −
1
4

v
2
u − v
2
d
g
2
1 + g
2
2

. (3.26)
3.2.3 Gaugino and higgsino sector
The gauginos λ1 and λ
3
2 mix with the neutral higgsinos ψ
0
d
, ψ
0
u
and ψS to form neutral
particles, the neutralinos. The 5 × 5 mass matrix of the neutralinos is written in the
basis
(−iλ1, −iλ3
2
, ψ0
d
, ψ0
u
, ψS) ≡ (B, e W , f He0
d
, He0
u
, Se) (3.27)
as
M0 =


M1 0 − √
1
2
g1vd √
1
2
g1vu 0
M2 √
1
2
g2vd − √
1
2
g2vu 0
0 −µeff −λvu
0 −λvd
2κs


. (3.28)
The mass matrix (3.28) is diagonalized by an orthogonal matrix Nij . The mass eigenstates of the neutralinos are usually denoted by χ
0
i
, with i = 1, . . . , 5, with increasing
masses (i = 1 corresponds to the lightest neutralino). These are given by
χ
0
i = Ni1Be + Ni2Wf + Ni3He0
d + Ni4He0
u + Ni5S. e (3.2
50 The Next-to-Minimal Supersymmetric Standard Model
We use the convention of a real matrix Nij , so that the physical masses mχ
0
i
are real,
but not necessarily positive.
In the charged sector, the SU(2)L charged gauginos λ
− = √
1
2
(λ
1
2 + iλ2
2
), λ
+ =
√
1
2
(λ
1
2 − iλ2
2
) mix with the charged higgsinos ψ
−
d
and ψ
+
u
, forming the charginos ψ
±:
ψ
± =

−iλ±
ψ
±
u
!
. (3.30)
The chargino mass matrix in the basis (ψ
−, ψ+) is
M± =

M2 g2vu
g2vd µeff !
. (3.31)
Since it is not symmetric, the diagonalization requires different rotations of ψ
− and
ψ
+. We denote these rotations by U and V , respectively, so that the mass eigenstates
are obtained by
χ
− = Uψ−, χ+ = V ψ+. (3.32)
3.3 DM Candidates in the NMSSM
Let us first review the characteristics that a DM candidate particle should have. First,
it should be massive in order to account for the missing mass in the galaxies. Second,
it must be electrically and color neutral. Otherwise, it would have condensed with
baryonic matter, forming anomalous heavy isotopes. Such isotopes are absent in nature. Finally, it should be stable and interact only weakly, in order to fit the observed
relic density.
In the NMSSM there are two possible candidates. Both can be stable particles if
they are the LSPs of the supersymmetric spectrum. The first one is the sneutrino (see
[148,149] for early discussions on MSSM sneutrino LSP). However, although sneutrinos
are WIMPs, their large coupling to the Z bosons result in a too large annihilation cross
section. Hence, if they were the DM particles, their relic density would have been very
small compared to the observed value. Exceptions are very massive sneutrinos, heavier
than about 5 TeV [150]. Furthermore, the same coupling result in a large scattering
cross section off the nuclei of the detectors. Therefore, sneutrinos are also excluded by
direct detection experiments.
The other possibility is the lightest neutralino. Neutralinos fulfill successfully, at
least in principle, all the requirements for a DM candidate. However, the resulting
relic density, although weakly interacting, may vary over many orders of magnitude as
a function of the free parameters of the theory. In the next sections we will investigate
further the properties of the lightest neutralino as the DM particle. We begin by
studying its annihilation that determines the DM relic density.
3.4 Neutralino relic density 51
3.4 Neutralino relic density
We remind that the neutralinos are mixed states composed of bino, wino, higgsinos
and the singlino. The exact content of the lightest neutralino determines its pair
annihilation channels and, therefore, its relic density (for detailed analyses, we refer
to [151–154]). Subsequently, we will briefly describe the neutralino pair annihilation
in various scenarios. We classify these scenarios with respect to the lightest neutralino
content.
Before we proceed, we should mention another mechanism that affects the neutralino LSP relic density. If there is a supersymmetric particle with mass close to the
LSP (but always larger), it would be abundant during the freeze-out and LSP coannihilations with this particle would contribute to the total annihilation cross section.
This particle, which is the Next-to-Lightest Supersymmetric Particle (NLSP), is most
commonly a stau or a stop. In the above sense, coannihilations refer not only to the
LSP–NLSP coannihilations, but also to the NLSP–NLSP annihilations, since the latter
reduce the number density of the NLSPs [155].
• Bino-like LSP
In principle, if the lightest neutralino is mostly bino-like, the total annihilation
cross section is expected to be small. Therefore, a bino-like neutralino LSP would
have been overabundant. The reason for this is that there is only one available
annihilation channel via t-channel sfermion exchange, since all couplings to gauge
bosons require a higgsino component. The cross section is even more reduced
when the sfermion mass is large.
However, there are still two ways to achieve the correct relic density. The first one
is using the coannihilation effect: if there is a sfermion with a mass slightly larger
(some GeV) than the LSP mass, their coannihilations can be proved to reduce
efficiently the relic density to the desired value. The second one concerns a binolike LSP, with a very small but non-negligible higgsino component. In this case,
if in addition the lightest CP-odd Higgs A1 is light enough, the annihilation to a
pair A1A1 (through an s-channel CP-even Higgs Hi exchange) can be enhanced
via Hi resonances. In this scenario a fine-tuning of the masses is necessary.
• Higgsino-like LSP
A mostly higgsino LSP is as well problematic. The strong couplings of the higgsinos to the gauge bosons lead to very large annihilation cross section. Therefore,
a possible higgsino LSP would have a very small relic density.
• Mixed bino–higgsino LSP
In this case, as it was probably expected, one can easily fit the relic density to
the observed value. This kind of LSP annihilates to W+W−, ZZ, W±H∓, ZHi
,
HiAj
, b
¯b and τ
+τ
− through s-channel Z or Higgs boson exchange or t-channel
neutralino or chargino exchange. The last two channels are the dominant ones
when the Higgs coupling to down-type fermions is enhanced, which occurs more
commonly in the regime of relatively large tan β. The annihilation channel to a
52 The Next-to-Minimal Supersymmetric Standard Model
pair of top quarks also contributes to the total cross section, if it is kinematically
allowed. However, in order to achieve the correct relic density, the higgsino
component cannot be very large.
• Singlino-like LSP
Since a mostly singlino LSP has small couplings to SM particles, the resulting relic
density is expected to be large. However, there are some annihilation channels
that can be enhanced in order to reduce the relic density. These include the
s-channel (scalar or pseudoscalar) Higgs exchange and the t-channel neutralino
exchange.
For the former, any Higgs with sufficient large singlet component gives an important contribution to the cross section, increasing with the parameter κ (since
the singlino-singlino-singlet coupling is proportional to κ). Concerning the latter
annihilation, in order to enhance it, one needs large values of the parameter λ.
In this case, the neutralino-neutralino-singlet coupling, which is proportional to
λ, is large and the annihilation proceeds giving a pair of scalar HsHs or a pair
of pseudoscalar AsAs singlet like Higgs.
As in the case of bino-like LSP, one can also use the effect of s-channel resonances
or coannihilations. In the latter case, an efficient NLSP can be the neutralino
χ
0
2
or the lightest stau τe1. In the case that the neutralino NLSP is higgsinolike, the LSP-NLSP coannihilation through a (doublet-like) Higgs exchange can
be proved very efficient. A stau NLSP reduces the relic density through NLSPNLSP annihilations, which is the only possibility in the case that both parameters
κ and λ are small. We refer to [156,157] for further discussion on this possibility.
Assuming universality conditions the wino mass M2 has to be larger than the bino
mass M1 (M2 ∼ 2M1). This is the reason that we have not considered a wino-like LSP.
3.5 Detection of neutralino DM
3.5.1 Direct detection
Since neutralinos are Majorana fermions, the effective Lagrangian describing their
elastic scattering with a quark in a nucleon can be written, in the Dirac fermion
notation, as [158]
Leff = a
SI
i χ¯
0
1χ
0
1
q¯iqi + a
SD
i χ¯
0
1γ5γµχ
0
1
q¯iγ5γ
µ
qi
, (3.33)
with i = u, d corresponding to up- and down-type quarks, respectively. The Lagrangian has to be understood as summing over the quark generations.
In this expression, we have omitted terms containing the operator ψγ¯
5ψ or a combination of ψγ¯
5γµψ and ψγ¯
µψ (with ψ = χ, q). This is a well qualified assumption:
Due to the small velocity of WIMPs, the momentum transfer ~q is very small compared
3.5.1 Direct detection 53
to the reduced mass of the WIMP-nucleus system. In the extreme limit of zero momentum transfer, the above operators vanish4
. Hence, we are left with the Lagrangian
(3.33) consisting of two terms, the first one corresponding to spin-independent (SI)
interactions and the second to spin-dependent (SD) ones. In the following, we will
focus again only to SI scattering, since the detector sensitivity to SD scattering is low,
as it has been already mentioned in Sec. 1.5.1.
The SI cross section for the neutralino-nucleus scattering can be written as [158]
(see, also, [159])
σ
SI
tot =
4m2
r
π
[Zfp + (A − Z)fn]
2
. (3.34)
mr is the neutralino-nucleus reduced mass mr =
mχmN
mχ+mN
, and Z, A are the atomic and
the nucleon number, respectively. It is more common, however, to use an expression
for the cross section normalized to the nucleon. In this case, on has for the neutralinoproton scattering
σ
SI
p =
4
π

mpmχ
0
1
mp + mχ
0
1
!2
f
2
p ≃
4m2
χ
0
1
π
f
2
p
, (3.35)
with a similar expression for the neutron.
The form factor fp is related to the couplings a to quarks through the expression
(omitting the “SI” superscripts)
fp
mp
=
X
q=u,d,s
f
p
T q
aq
mq
+
2
27
fT G X
q=c,b,t
aq
mq
. (3.36)
A similar expression may be obtained for the neutron form factor fn, by the replacement
p → n in the previous expression (henceforth, we focus to neutralino-proton scattering).
The parameters fT q are defined by the quark mass matrix elements
hp| mqqq¯ |pi = mpfT q, (3.37)
which corresponds to the contribution of the quark q to the proton mass and the
parameter fT G is related to them by
fT G = 1 −
X
q=u,d,s
fT q. (3.38)
The above parameters can be obtained by the following quantities
σπN =
1
2
(mu + md)(Bu + Bd) and σ0 =
1
2
(mu + md)(Bu + Bd − 2Bs,) (3.39)
with Bq = hN| qq¯ |Ni, which are obtained by chiral perturbation theory [160] or by
lattice simulations. Unfortunately, the uncertainties on the values of these quantities
are large (see [161], for more recent values and error bars).
4While there are possible circumstances in which the operators of (3.33) are also suppressed and,
therefore, comparable to the operators omitted, they are not phenomenologically interesting.
54 The Next-to-Minimal Supersymmetric Standard Model
χ
0
1
χ
0
1
χ
0
1 χ
0
1
qe
q q
q q
Hi
Figure 3.1: Feynman diagrams contributing to the elastic neutralino-quark scalar scattering amplitude at tree level.
The SI neutralino-nucleon interactions arise from t-channel Higgs exchange and
s-channel squark exchange at tree level (see Fig. 3.1), with one-loop contributions from
neutralino-gluon interactions. In practice, the s-channel Higgs exchange contribution
to the scattering amplitude dominates, especially due to the large masses of squarks.
In this case, the effective couplings a are given by
a
SI
d =
X
3
i=1
1
m2
Hi
C
1
i Cχ
0
1χ
0
1Hi
, aSI
u =
X
3
i=1
1
m2
Hi
C
2
i Cχ
0
1χ
0
1Hi
. (3.40)
C
1
i
and C
2
i
are the Higgs Hi couplings to down- and up-type quarks, respectively, given
by
C
1
i =
g2md
2MW cos β
Si1, C2
i =
g2mu
2MW sin β
Si2, (3.41)
with S the mixing matrix of the CP-even Higgs mass eigenstates and md, mu the
corresponding quark mass. We see from Eqs. (3.36) and (3.41) that the final cross
section (3.35) is independent of each quark mass. We write for completeness the
neutralino-neutralino-Higgs coupling Cχ
0
1χ
0
1Hi
:
Cχ
0
1χ
0
1Hi =
√
2λ (Si1N14N15 + Si2N13N15 + Si3N13N14) −
√
2κSi3N
2
+ g1 (Si1N11N13 − Si2N11N14) − g2 (Si1N12N13 − Si2N12N14), (3.42)
with N the neutralino mixing matrix given in (3.29).
The resulting cross section is proportional to m−4
Hi
. In the NMSSM, it is possible
for the lightest scalar Higgs eigenstate to be quite light, evading detection due to its
singlet nature. This scenario can give rise to large values of SI scattering cross section,
provided that the doublet components of th

[Charles]

@ V : Très bien donc dans ce cas ce ne sont pas plus les oeuvres qui sont distinctives mais les raisons derrière leur consommation, ce qui me semble être une évolution significative, pour ne pas une régression, de la distinction si celle-ci ne touche plus les oeuvres en elles-mêmes. (Je nuance un peu tout de suite car il me semble que Bourdieu faisait la différence entre les bourgeois et les prolos au musée en expliquant que seuls les premiers savaient que devant un tableau il n’y avait rien à dire, que l’appréciation devait être muette ou presque.) Mais parmi les amateurs de rap tu as beaucoup de bourgeois qui l’aiment au premier degré, c’est même la majorité. Personne n’écoute Orelsan comme quelqu’un qui regarde la télé-réalité pour « se moquer parce qu’ils sont vraiment débiles ». Au contraire, la mode est plutôt d’assumer tous les types de divertissements et refuser la hiérarchie.
Sur les séries, c’est un peu plus compliqué car celles-ci n’ont pas simplement été récupérées et intellectualisées par la bourgeoisie mais elles se sont elles-mêmes mises à singer le cinéma et une forme d’auteurisme, elles sont aussi auto-légitimées.

[Demi Habile]

and also the definition of the unpolarized cross section to write
X
spins
Z
|M12→34|
2
(2π)
4
δ
4
(p1 + p2 − p3 − p4)
d
3p3
(2π)
32E3
d
3p4
(2π)
32E4
=
4F g1g2 σ12→34, (1.31)
where F ≡ [(p1 · p2)
2 − m2
1m2
2
]
1/2
and the spin factors g1, g2 come from the average
over initial spins. This way, the collision term (1.29) is written in a more compact form
g1
Z
C[f1]
d
3p1
(2π)
3
= −
Z
σvMøl (dn1dn2 − dn
eq
1 dn
eq
2
), (1.32)
where σ =
P
(all f)
σ12→f is the total annihilation cross section summed over all the
possible final states and vMøl ≡
F
E1E2
. The so called Møller velocity, vMøl, is defined in
such a way that the product vMøln1n2 is invariant under Lorentz transformations and,
in terms of particle velocities ~v1 and ~v2, it is given by the expression
vMøl =
h
~v2
1 − ~v2
2

2
− |~v1 × ~v2|
2
i1/2
. (1.33)
Due to symmetry considerations, the distributions in kinetic equilibrium are proportional to those in chemical equilibrium, with a proportionality factor independent of
the momentum. Therefore, the collision term (1.32), both before and after decoupling,
can be written in the form
g1
Z
C[f1]
d
3p1
(2π)
3
= −hσvMøli(n1n2 − n
eq
1 n
eq
2
), (1.34)
where the thermal averaged total annihilation cross section times the Møller velocity
has been defined by the expression
hσvMøli =
R
σvMøldn
eq
1 dn
eq
2
R
dn
eq
1 dn
eq
2
. (1.35)
We will come back to the thermal averaged cross section in the next subsection.
We are, now, able to write the full integrated Boltzmann equation, using the expressions (1.28), (1.34) that we have derived for the Liouville and the collision term,
respectively. In the simplified but interesting case of identical particles 1 and 2, the
Boltzmann equation is, finally, written as
n˙ + 3Hn = −hσvMøli(n
2 − n
2
eq). (1.36)
18 Dark Matter
However, instead of using n, it is more convenient to take the expansion of the universe
into account and calculate the number density per comoving volume Y , which can be
defined as the ratio of the number and entropy densities: Y ≡ n/s. The total entropy
density S = R3
s (R is the scale factor) remains constant, hence we can obtain a
differential equation for Y by dividing (1.36) by S. Before we write the final form
of the Boltzmann equation that it is used for the relic density calculations, we have
to change the variable that parametrizes the comoving density. In practice, the time
variable t is not convenient and the temperature of the Universe (actually the photon
temperature, since the photons were the last particles that went out of equilibrium) is
used instead. However, it proves even more useful to use as time variable the quantity
defined by x ≡ m/T with m the DM mass, so that Eq. (1.36) transforms into
dY
dx
=
1
3H
ds
dx
hσvMøli

Y
2 − Y
2
eq
. (1.37)
Last, using the Hubble parameter (1.2) for a radiation dominated Universe and the
expressions (1.20), (1.21) for the energy and entropy density, the Boltzmann equation
is written in its final form
dY
dx
= −
r
45GN
π
g
1/2
∗ m
x
2
hσvMøli

Y
2 − Y
2
eq
, (1.38)
where the effective degrees of freedom g
1/2
∗ have been defined by
g
1/2
∗ ≡
heff
g
1/2
eff

1 +
1
3
T
heff
dheff
dT

. (1.39)
The equilibrium density per comoving volume Yeq ≡ neq/s can be expressed as
Yeq(x) = 45g
4π
4
x
2K2(x)
heff(m/x)
, (1.40)
with K2 the modified Bessel function of second kind.
1.4.3 Thermal average of the annihilation cross section
We are going to derive a simple formula that one can use to calculate the thermal
average of the cross section times velocity, based again on the analysis of [38]. We will
use the assumption that equilibrium functions follow the Maxwell-Boltzmann distribution, instead of the actual Bose-Einstein or Fermi-Dirac. This is a well established
assumption if the freeze out occurs after T ≃ m/3 or for x >∼ 3, which is actually the
case for WIMPs. Under this assumption, the expression (1.35) gives, in the cosmic
comoving frame,
hσvMøli =
R
vMøle
−E1/T e
−E2/T d
3p1d
3p2
R
e
−E1/T e
−E2/T d
3p1d
3p2
. (1.4
1.4.3 Thermal average of the annihilation cross section 19
The volume element can be written as d3p1d
3p2 = 4πp1dE14πp2dE2
1
2
cos θ, with θ the
angle between ~p1 and ~p2. After changing the integration variables to E+, E−, s given
by
E+ = E1 + E2, E− = E1 − E2, s = 2m2 + 2E1E2 − 2p1p2 cos θ, (1.42)
(with s = −(p1 − p2)
2 one of the Mandelstam variables,) the volume element becomes
d
3p1d
3p2 = 2π
2E1E2dE+dE−ds and the initial integration region
{E1 > m, E2 > m, | cos θ| ≤ 1i
transforms into
|E−| ≤
1 −
4m2
s
1/2
(E
2
+ − s)
1/2
, E+ ≥
√
s, s ≥ 4m2
. (1.43)
After some algebraic calculations, it can be found that the quantity hσvMøliE1E2
depends only on s, specifically vMølE1E2 =
1
2
p
s(s − 4m2
). Hence, the numerator of the expression (1.41), which after changing the integration variables reads
2π
2
R
dE+
R
dE−
R
dsσvMølE1E2e
−E+/T , can be written, eventually, as
Z
vMøle
−E1/T e
−E2/T = 2π
2
Z ∞
4m2
dsσ(s − 4m2
)
Z
dE+e
−E+/T (E
2
+ − s)
1/2
. (1.44)
The integral over E+ can be written with the help of the modified Bessel function of
the first kind K1 as √
s T K1(
√
s/T). The denominator of (1.41) can be treated in a
similar way, so that the thermal average is, finally, given by the expression
hσvMøli =
1
8m4TK2
2
(x)
Z ∞
4m2
ds σ(s)(s − 4m2
)
√
s K1(
√
s/T). (1.45)
Eqs. (1.38)–(1.40) along with this last Eq. (1.45) are all we need in order to calculate
the relic density of a WIMP, if its total annihilation cross section in terms of the
Mandelstam variable s is known.
In many cases, in order to avoid the numerical integration in Eq. (1.45), an approximation for hσvMøli can be used. The thermal average is expanded in powers of x
−1
(or, equivalently, in powers of the squared WIMP velocity):
hσvMøli = a + bx−1 + . . . . (1.46)
(The coefficient a corresponds to the s-wave contribution to the cross section, the
coefficient b to the p-wave contribution, and so on.) This partial wave expansion gives
a quite good approximation, provided there are no s-channel resonances and thresholds
for the final states [39].
In [40], it was shown that, after expanding the integrands of Eq. (1.41) in powers
of x
−1
, all the integrations can be performed analytically. As we saw, the expression
20 Dark Matter
vMølE1E2 depends on momenta only through s. Therefore, one can form the Lorentz
invariant quantity
w(s) ≡ σ(s)vMølE1E2 =
1
2
σ(s)
p
s(s − 4m2
). (1.47)
The integration involves the Taylor expansion of this quantity w around s/4m2 = 1
and the general formula for the partial wave expansion of the thermal average is [40]
hσvMøli =
1
m2

w −
3
2
(2w − w
′
)x
−1 +
3
8
(16w − 8w
′ + 5w
′′)x
−2
−
5
16
(30w − 15w
′ + 3w
′′ − 7x
′′′)x
−3 + O(x
−4
)

s/4m2=1
, (1.48)
where primes denote derivatives with respect to s/4m2 and all quantities have to be
evaluated at s = 4m2
.
1.5 Direct Detection of DM
Since the beginning of 1980s, it has been realized that besides the numerous facts showing evidence for the existence of these new dark particles, it is also possible to detect
them directly. Already in 1985, two pioneering articles [41, 42] appeared, describing
the detection methods for WIMPs. Since WIMPs are expected to cluster gravitationally together with ordinary stars in the Milky Way halo, they would pass also through
Earth and, in principle, they can be detected through scattering with the nuclei in a
detector’s material. In practice, one has to measure the recoil energy deposited by this
scattering.
However, although one can deduce from rotation curves that DM dominates the
dark halo in the outer parts of our galaxy, it is not so obvious from direct measurements
whether there is any substantial amount of DM inside the solar radius R0 ≃ 8 kpc.
Using indirect methods (involving the determination of the gravitational potential,
through the measuring of the kinematics of stars, both near the mid-plane of the
galactic disk and at heights several times the disk thickness), it is almost certain
that the DM is also present in the solar system, with a local density ρ0 = (0.3 ±
0.1) GeV cm−3
[43].
This value for the local density implies that for a WIMP mass of order ∼ 100 GeV,
the local number density is n0 ∼ 10−3
cm−3
. It is also expected that the WIMPs
velocity is similar to the velocity with which the Sun orbits around the galactic center
(v0 ≃ 220 km s−1
), since they are both moving under the same gravitational potential.
These two quantities allow to estimate the order of magnitude of the incident flux
of WIMPs on the Earth: J0 = n0v0 ∼ 105
cm−2
s
−1
. This value is manifestly large,
but the very weak interactions of the DM particles with ordinary matter makes their
detection a difficult, although in principle feasible, task. In order to compensate for
the very low WIMP-nucleus scattering cross section, very large detectors are required.
1.5.1 Elastic scattering event rate 21
1.5.1 Elastic scattering event rate
In the following, we will confine ourselves to the elastic scattering with nuclei. Although
inelastic scattering of WIMPs off nuclei in a detector or off orbital electrons producing
an excited state is possible, the event rate of these processes is quite suppressed. In
contrast, during an elastic scattering the nucleus recoils as a whole.
The direct detection experiments measure the number of events per day and per
kilogram of the detector material, as a function of the amount of energy Q deposited
in the detector. This event rate would be given by R = nWIMP nnuclei σv in a simplified
model with WIMPs moving with a constant velocity v. The number density of WIMPs
is nWIMP = ρ0/mX and the number density of nuclei is just the ratio of the detector’s
mass over the nuclear mass mN .
For accurate calculations, one should take into account that the WIMPs move in the
halo not with a uniform velocity, but rather following a velocity distribution f(v). The
Earth’s motion in the solar system should be included into this distribution function.
The scattering cross section σ also depends on the velocity. Actually, the cross section
can be parametrized by a nuclear form factor F(Q) as
dσ =
σ
4m2
r
v
2
F
2
(Q)d|~q|
2
, (1.49)
where |~q|
2 = 2m2
r
v
2
(1 − cos θ) is the momentum transferred during the scattering,
mr =
mXmN
mX+mN
is the reduced mass of the WIMP – nucleus system and θ is the scattering
angle in the center of momentum frame. Therefore, one can write a general expression
for the differential event rate per unit detector mass as
dR =
ρ0
mX
1
mN
σF2
(Q)d|~q|
2
4m2
r
v
2
vf(v)dv. (1.50)
The energy deposited in the detector (transferred to the nucleus through one elastic
scattering) is
Q =
|~q|
2
2mN
=
m2
r
v
2
mN
(1 − cos θ). (1.51)
Therefore, the differential event rate over deposited energy can be written, using the
equations (1.50) and (1.51), as
dR
dQ
=
σρ0
√
πv0mXm2
r
F
2
(Q)T(Q), (1.52)
where, following [37], we have defined the dimensionless quantity T(Q) as
T(Q) ≡
√
π
2
v0
Z ∞
vmin
f(v)
v
dv, (1.53)
with the minimum velocity given by vmin =
qQmN
2m2
r
, obtained by Eq. (1.51). Finally,
the event rate R can be calculated by integrating (1.52) over the energy
R =
Z ∞
ET
dR
dQ
dQ. (1.54)
22 Dark Matter
The integration is performed for energies larger than the threshold energy ET of the
detector, below which it is insensitive to WIMP-nucleus recoils.
Using Eqs. (1.54) and (1.52), one can derive the scattering cross section from the
event rate. The experimental collaborations prefer to give their results already in terms
of the scattering cross section as a function of the WIMP mass. To be more precise,
the WIMP-nucleus total cross section consists of two parts: the spin-dependent (SD)
cross section and the spin-independent (SI) one. The former comes from axial current
couplings, whereas the latter comes from scalar-scalar and vector-vector couplings.
The SD cross section is much suppressed compared to the SI one in the case of heavy
nuclei targets and it vanishes if the nucleus contains an even number of nucleons, since
in this case the total nuclear spin is zero.
We see that two uncertainties enter the above calculation: the exact value of the
local density ρ0 and the exact form of the velocity distribution f(v). To these, one
has to include one more. The cross section σ that appears in the previous expressions
concerns the WIMP-nucleon cross section. The couplings of a WIMP with the various
quarks that constitute the nucleon are not the same and the WIMP-nucleon cross
section depends strongly on the exact quark content of the nucleon. To be more
precise, the largest uncertainty lies on the strange content of the nucleon, but we shall
return to this point when we will calculate the cross section in a specific particle theory,
the Next-to-Minimal Supersymmetric Standard Model, in Sec. 3.5.1.
1.5.2 Experimental status
The situation of the experimental results from direct DM searches is a bit confusing. The null observations in most of the experiments led them to set upper limits
on the WIMP-nucleon cross section. These bounds are quite stringent for the spinindependent cross section7
, especially in the regime of WIMP masses of the order of
100 GeV. However, some collaborations have already reported possible DM signals,
mainly in the low mass regime. The preferred regions of these experiments do not
coincide, while some of them have been already excluded by other experiments. The
present picture, for WIMP masses ranging from 5 to 1000 GeV, is summarized in Fig.
1.5, 1.6.
Fig. 1.5 mainly presents upper bounds coming from XENON100 [44]. XENON100
[46] is an experiment located at the Gran Sasso underground laboratory in Italy. It
contains in total 165 kg of liquid Xenon, with 65 kg acting as target mass and the
rest shielding the detector from background radiation. For these upper limits, 225
live days of data were used. The minimum value for the predicted upper bounds on
the cross section is 2 · 10−45 cm2
for WIMP mass ∼ 55 GeV (at 90% confidence level),
almost one order of magnitude lower than the previously released limits [47] by the
same collaboration, using 100 live days of data.
The stringent upper bounds up-to-date (at least for WIMP mass larger than about
7 GeV) come from the first results of the LUX experiment (see Fig. 1.6), after the first
7For the spin-dependent scattering, the exclusion limits are quite relaxed. Hence, we will focus on
the SI cross sections.
1.5.2 Experimental status 23
Figure 1.5: The XENON100 exclusion limit (thick blue line), along with the expected
sensitivity in green (1σ) and yellow (2σ) band. Other upper bounds are also shown as
well as detection claims. From [44].
85.3 live-days of its operation [45]. LUX [53] is a detector containing liquid Xenon, as
XENON100, but in larger quantity, with total mass 370 kg. Its operation started on
April 2013 with a goal to clearly detect or exclude WIMPs with a spin independent
cross section ∼ 2 · 10−46 cm2
.
In Fig. 1.5, except of the XENON100 bounds and other experimental limits on larger
WIMP-nucleon cross section, some detection claims also appear. These come from
DAMA [48,49], CoGeNT [50] and CRESST-II [51] experiments. The first positive result
came from DAMA [52], back in 2000. Since then, the experiment has accumulated 1.17
ton-yr of data over 13 years of operation. DAMA consists of 250 kg of radio pure NaI
scintillator and looks for the annual modulation of the WIMP flux in order to reduce
the influence of the background.
The annual modulation of the DM flux (see [54] for a recent review) is due to the
Earth’s orbital motion relative to the rotation of the galactic disk. The galactic disk
rotation through an essentially non-rotating DM halo, creates an effective DM wind in
the solar frame. During the earth’s heliocentric orbit, this wind reaches a maximum
when the Earth is moving fastest in the direction of the disk rotation (this happens
in the beginning of June) and a minimum when it is moving fastest in the opposite
direction (beginning of December).
DAMA claims an 8.9σ annual modulation with a minimum flux on May 26±7 days,
consistent with the expectation. Since the detector’s target consists of two different
nuclei and the experiment cannot distinguish between sodium and iodine recoils, there
24 Dark Matter
Figure 1.6: The LUX 90% confidence exclusion limit (blue line) with the 1σ range
(shaded area). The XENON100 upper bound is represented by the red line. The inset
shows also preferred regions by CoGeNT (shaded light red), CDMS II silicon detector
(shaded green), CRESST II (shaded yellow) and DAMA (shaded gray). From [45].
is no model independent way to determine the exact region in the cross section versus
WIMP mass plane to which the observed modulation corresponds. However, one can
assume two cases: one that the WIMP scattering off the sodium nucleus dominates the
recoil energy and the other with the iodine recoils dominating. The former corresponds
[55] to a light WIMP (∼ 10 GeV) and quite large scattering cross section and the latter
to a heavier WIMP (∼ 50 to 100 GeV) with smaller cross section (see Fig. 1.5).
The positive result of DAMA was followed many years later by the ones of CoGeNT
and CRESST-II, and more recently by the silicon detector of CDMS [56] (Fig. 1.7).
The discrepancy of the results raised a lot of debates among the experiments (for
example, [64–67]) and by some the positive results are regarded as controversial. On
the other hand, it also raised an effort to find a physical explanation behind this
inconsistency (see, for example, [68–71]).
1.6 Indirect Methods for DM Detection
The same annihilation processes that determined the DM relic abundance in the early
Universe also occur today in galactic regions where the DM concentration is higher.
This fact rises the possibility of detecting potential WIMP pair annihilations indirectly
through their imprints on the cosmic rays. Therefore, the indirect DM searches aim
at the detection of an excess over the known astrophysical background of charged
particles, photons or neutrinos.
Charged particles – electrons, protons and their antiparticles – may originate from
direct products (pair of SM particles) of WIMP annihilations, after their decay and
1.6 Indirect Methods for DM Detection 25
Figure 1.7: The blue contours represent preferred regions for a possible signal at 68%
and 90% C.L. using the silicon detector of CMDS [56]. The blue dotted line represents
the upper limit obtained by the same analysis and the blue solid line is the combined
limit with the silicon CDMS data set reported in [57]. Other limits also appear:
from the CMDS standard germanium detector (light and dark red dashed line, for
standard [58] and low threshold analysis [59], respectively), EDELWEISS [60] (dashed
orange), XENON10 [61] (dash-dotted green) and XENON100 [44] (long-dash-dotted
green). The filled regions identify possible signal regions associated with data from
CoGeNT [62] (dashed yellow, 90% C.L.), DAMA [49,55] (dotted tan, 99.7% C.L.) and
CRESST-II [51, 63] (dash-dotted pink, 95.45% C.L.) experiments. Taken from [56].
through the process of showering and hadronization. Although the exact shape of the
resulting spectrum would depend on the specific process, it is expected to show a steep
cutoff at the WIMP mass. Once produced in the DM halo, the charged particles have
to travel to the point of detection through the turbulent galactic field, which will cause
diffusion. Apart from that, a lot of processes disturb the propagation of the charged
particles, such as bremsstrahlung, inverse Compton scattering with CMB photons and
many others. Therefore, the uncertainties that enter the propagation of the charged
flux until it reaches the telescope are important (contrary to the case of photons and
neutrinos that propagate almost unperturbed through the galaxy).
As in the case of direct detection, the experimental status of charged particle detection concerning the DM is confusing. After some hints from HEAT [72] and AMS01 [73] (the former a far-infrared telescope in Antarctica, the latter a spectrometer,
prototype for AMS-02 mounted on the International Space Station [74]), the PAMELA
satellite observed [75, 76] a steep increase in the energy spectrum of positron fraction
e
+/(e
+ + e
−)
8
. Later FERMI satellite [77] and AMS-02 [78] confirmed the results up
8The searches for charged particles focus on the antiparticles in order to have a reduced background,
26 Dark Matter
Figure 1.8: A compilation of data of charged cosmic rays, together with plausible but
uncertain astrophysical backgrounds, taken from [79]. Left: Positron flux. Center:
Antiproton flux. Right: Sum of electrons and positrons.
to energies of ∼ 200 GeV. However, the excess of positrons is not followed by an excess
of antiprotons, whose flux seems to coincide with the predicted background [75]. In
Fig. 1.8, three plots summarizing the situation are shown [79].
The observed excess is very difficult to explain in terms of DM [79]. To begin with,
the annihilation cross section required to reproduce the excess is quite large, many
orders of magnitude larger than the thermal cross section. Moreover, an “ordinary”
WIMP with large annihilation cross section giving rise to charged leptons is expected
to give, additionally, a large number of antiprotons, a fact in contradiction with the
observations. Although a lot of work has been done to fit a DM particle to the observed
pattern, it is quite possible that the excesses come from a yet unknown astrophysical
source. We are not going to discuss further this matter, but we end with a comment.
If this excess is due to a source other than DM, then a possible DM positron excess
would be lost under this formidable background.
A last hint for DM came from the detection of highly energetic photons. However,
we will interrupt this discussion, since this signal and a possible explanation is the
subject of Ch. 4. There, we will also see the upper bounds on the annihilation cross
section being set due to the absence of excesses in diffuse γ radiation.
since they are much less abundant than the corresponding particles.
CHAPTER 2
PARTICLE PHYSICS
Since the DM comprises of particles, it should be explained by a general particle physics
theory. We start in the following section by describing the Standard Model (SM) of
particle physics. Although the SM describes so far the fundamental particles and their
interactions quite accurately, it cannot provide a DM candidate. Besides, the SM
suffers from some theoretical problems, which we discuss in Sec. 2.2. We will see that
these problems can be solved if one introduces a new symmetry, the supersymmetry,
which we describe in Sec. 2.3. We finish this chapter by briefly describing in Sec. 2.4 a
supersymmetric extension of the SM with the minimal additional particle content, the
Minimal Supersymmetric Standard Model (MSSM).
2.1 The Standard Model of Particle Physics
The Standard Model (SM) of particle physics1
consists of two well developed theories,
the quantum chromodynamics (QCD) and the electroweak (EW) theory. The former
describes the strong interactions among the quarks, whereas the latter describes the
electroweak interactions (the weak and electromagnetic interactions in a unified context) between fermions. The EW theory took its final form in the late 1960s by the
introduction by S. Weinberg [85] and A. Salam [86] of the Higgs mechanism that gives
masses to the SM particles, which followed the unification of electromagnetic and weak
interactions [87,88]. At the same time, the EW model preserves the gauge invariance,
making the theory renormalizable, as shown later by ’t Hooft [89]. On the other hand,
QCD obtained its final form some years later, after the confirmation of the existence
of quarks. Of course, the history of the SM is much longer and it can be traced back to
1920s with the formulation of a theoretical basis for a Quantum Field Theory (QFT).
Since then, the SM had many successes. The SM particle content was completed with
the discovery of the heaviest of the quarks, the top quark [90,91], in 1995 and, recently,
with the discovery of the Higgs boson [92, 93].
1There are many good textbooks on the SM and Quantum Field Theory, e.g. [80–84].
28 Particle Physics
The key concept within the SM, as in every QFT, is that of symmetries. Each
interaction respects a gauge symmetry, based on a Lie algebra. The strong interaction is
described by an SU(3)c symmetry, where the subscript c stands for color, the conserved
charge of strong interactions. The EW interactions, on the other hand, are based on
a SU(2)L × U(1)Y Lie algebra. Here, as we will subsequently see, L refers to the
left-handed fermions and Y is the hypercharge, the conserved charge under the U(1).
SU(2)L conserves a quantity known as weak isospin I. Therefore, the SM contains the
internal symmetries of the unitary product group
SU(2)L × U(1)Y × SU(3)c. (2.1)
2.1.1 The particle content of the SM
We mention for completeness that particles are divided into two main classes according
to the statistics they follow. The bosons are particles with integer spin and follow the
Bose-Einstein distribution, whereas fermions have half-integer spin and follow the
Dirac-Einstein statistics, obeying the Pauli exclusion principle. In the SM, all the
fermions have spin 1/2, whereas the bosons have spin 1 with only exception the Higgs
boson, which is a scalar (spin zero). We begin the description of the SM particles with
the fermions.
Each fermion is classified in irreducible representations of each individual Lie algebra, according to the conserved quantum numbers, i.e. the color C, the weak isospin
I and the hypercharge Y . A first classification of fermions can be done into leptons
and quarks, which transform differently under the SU(3)c. Leptons are singlets under
this transformation, while quarks act as triplets (the fundamental representation of
this group). The EW interactions violate maximally the parity symmetry and SU(2)L
acts only on states with negative chirality (left-handed). A Dirac spinor Ψ can be
decomposed into left and right chirality components using, respectively, the projection
operators PL =
1
2
(1 − γ5) and PR =
1
2
(1 + γ5):
ΨL = PLΨ and ΨR = PRΨ. (2.2)
Left-handed fermions have I = 1/2, with a third component of the isospin I3 = ±1/2.
Fermions with positive I3 are called up-type fermions and those with negative are
called down-type. These behave the same way under SU(2)L and form doublets with
one fermion of each type. On the other hand, right-handed fermions have I = 0 and
form singlets that do not undergo weak interactions. The hypercharge is written in
terms of the electric charge Q and the third component of the isospin I3 through the
Gell-Mann–Nishijima relation:
Q = I3 + Y/2. (2.3)
Therefore, left- and right-handed components transform differently under the U(1)Y ,
since they have different hypercharge.
The fermionic sector of the SM comprises three generations of fermions, transforming as spinors under Lorentz transformations. Each generation has the same structure.
For leptons, it is an SU(2)L doublet with components consisting of one left-handed
2.1.2 The SM Lagrangian 29
charged lepton and one neutrino (neutrinos are only left-handed in the SM), along
with a gauge singlet right-handed charged lepton. The quark doublet consists of an
up- (u) and a down-type (d) (left-handed) quark and the pattern is completed by the
two corresponding SU(2)L singlet right-handed quarks. We write these representations
as
Quarks: Q ≡

u
i
L
d
i
L
!
, ui
R, di
R Leptons: L ≡

ν
i
L
e
i
L
!
, ei
R, (2.4)
with i = 1, 2, 3 the generation index.
Having briefly described the fermionic sector, we turn to the bosonic sector of
the SM. It consists of the gauge bosons that mediate the interactions and the Higgs
boson that gives masses to the particles through a spontaneous symmetry breaking,
the electroweak symmetry breaking (EWSB) [94–98], which we shall describe in Sec.
2.1.3. Before the EWSB, these bosons are
• three Wa
µ
(a = 1, 2, 3) weak bosons, associated with the generators of SU(2)L,
• one neutral Bµ boson, associated with the generator of U(1)Y ,
• eight gluons Ga
µ
(a = 1, . . . , 8), associated with the generators of SU(3)c, and
• the complex scalar Higgs doublet Φ =
φ
+
φ
0
!
.
After the EWSB, the EW boson states mix and give the two W± bosons, the neutral
Z boson and the massless photon γ. From the symmetry breaking, one scalar degree of
freedom remains which is the famous (neutral) Higgs boson [97–99]. We will return to
the mixed physical states, after describing the Higgs mechanism for symmetry breaking.
A complete list of the SM particles (the physical states after EWSB) is shown in Table
2.1.
2.1.2 The SM Lagrangian
The gauge bosons are responsible for the mediation of the interactions and are associated with the generators of the corresponding symmetry. The EW gauge bosons Bµ
and Wa
µ
are associated, respectively, with the generator Y of the U(1)Y and the three
generators T
a
2
of the SU(2)L. The latter are defined as half of the Pauli matrices τ
a
(T
a
2 =
1
2
τ
a
) and they obey the algebra

T
a
2
, Tb
2

= iǫabcT
c
2
, (2.5)
where ǫ
abc is the fully antisymmetric Levi-Civita tensor. The eight gluons are associated
with an equal number of generators T
a
3
(Gell-Mann matrices) of SU(3)c and obey the
Lie algebra

T
a
3
, Tb
3

= if abcT
c
3
, with Tr
T
a
3 T
b
3

=
1
2
δ
ab
, (2.6)
30 Particle Physics
Name symbol mass charge (|e|) spin
Leptons
electron e 0.511 MeV −1 1/2
electron neutrino νe 0 (<2 eV) 0 1/2
muon µ 105.7 MeV −1 1/2
muon neutrino νµ 0 (<2 eV) 0 1/2
tau τ 1.777 GeV −1 1/2
tau neutrino ντ 0 (<2 eV) 0 1/2
Quarks
up u 2.7
+0.7
−0.5 MeV 2/3 1/2
down d 4.8
+0.7
−0.3 MeV −1/3 1/2
strange s (95 ± 5) MeV −1/3 1/2
charm c (1.275 ± 0.025) GeV 2/3 1/2
bottom b (4.18 ± 0.03) GeV −1/3 1/2
top t (173.5 ± 0.6 ± 0.8) GeV 2/3 1/2
Bosons
photon γ 0 (<10−18 eV) 0 (<10−35) 1
W boson W± (80.385 ± 0.015) GeV ±1 1
Z boson Z (91.1876 ± 0.0021) GeV 0 1
gluon g 0 (.O(1) MeV) 0 1
Higgs H
(125.3 ± 0.4 ± 0.5) GeV
0 0
(126.0 ± 0.4 ± 0.4) GeV
Table 2.1: The particle content of the SM. All values are those given in [100], except of
the Higgs mass that is taken from [92, 93] (up and down row, respectively), assuming
that the observed excess corresponds to the SM Higgs. The u, d and s quark masses
are estimates of so-called “current-quark masses” in a mass-independent subtraction
scheme as MS at a scale ∼ 2 GeV. The c and b quark masses are the running masses
in the MS scheme. The values in the parenthesis are the current experimental limits.
with f
abc the structure constants of the group.
Using the structure constants of the corresponding groups, we define the field
strengths for the gauge bosons as
Bµν ≡ ∂µBν − ∂νBµ, (2.7a)
Wµν ≡ ∂µWa
ν − ∂νWa
µ + g2ǫ
abcWb
µWc
ν
(2.7b)
and
G
a
µν ≡ ∂µG
a
ν − ∂νG
a
µ + g3f
abcG
b
µG
c
ν
. (2.7c)
2.1.2 The SM Lagrangian 31
We use the notation g1, g2 and g3 for the coupling constants of U(1)Y , SU(2)L and
SU(3)c, respectively. As in any Yang-Mills theory, the non-abelian gauge groups lead
to self-interactions, which is not the case for the abelian U(1)Y group.
Before we finally write the full Lagrangian, we have to introduce the covariant
derivative for fermions, which in a general form can be written as
DµΨ =
∂µ − ig1
1
2
Y Bµ − ig2T
a
2 Wa
µ − ig3T
a
3 G
a
µ

Ψ. (2.8)
This form has to be understood as that, depending on Ψ, only the relevant terms
apply, hence for SU(2)L singlet leptons only the two first terms inside the parenthesis
are relevant, for doublet leptons the three first terms and for the corresponding quark
singlets and doublets the last term also participates. We also have to notice that in
order to retain the gauge symmetry, mass terms are forbidden in the Lagrangian. For
example, the mass term mψψ¯ = m

ψ¯
LψR + ψ¯
RψL

(with ψ¯ ≡ ψ
†γ
0
) is not invariant
under SU(2)L. This paradox is solved by the introduction of the Higgs scalar field
(see next subsection). The SM Lagrangian can be now written2
, split for simplicity in
three parts, each describing the gauge bosons, the fermions and the scalar sector,
LSM = Lgauge + Lfermion + Lscalar, (2.9)
with
Lgauge = −
1
4
G
a
µνG
µν
a −
1
4
Wa
µνWµν
a −
1
4
BµνB
µν
, (2.10a)
Lfermion = iL¯Dµγ
µL + ie¯RDµγµeR
+ iQ¯Dµγ
µQ + iu¯RDµγ
µuR + i
¯dRDµγ
µ
dR
−

heL¯ΦeR + hdQ¯ΦdR + huQ¯ΦeuR + h.c.

(2.10b)
and
Lscalar = (DµΦ)†
(DµΦ) − V (Φ†Φ), (2.10c)
where
V (Φ†Φ) = µ
2Φ
†Φ + λ

Φ
†Φ
2
(2.11)
is the scalar Higgs potential. Φ is the conjugate of Φ, related to the charge conjugate e
by Φ =e iτ2Φ
⋆
, with τi the Pauli matrices. The covariant derivative acting on the Higgs
scalar field gives
DµΦ =
∂µ − ig1
1
2
Y Bµ − ig2T
a
2 Wa
µ

Φ. (2.12)
Before we proceed to the description of the Higgs mechanism, a last comment concerning the SM Lagrangian is in order. If we restore the generation indices, we see that
2For simplicity, from now on we are going to omit the generations indice
32 Particle Physics
the Yukawa couplings h are 3 × 3, in general complex, matrices. As any complex matrix, they can be diagonalized with the help of two unitary matrices VL and VR, which
are related by VR = U
†VL with U again a unitary matrix. The diagonalization in the
quark sector to the mass eigenstates induces a mixing among the flavors (generations),
described by the Cabibbo–Kobayashi–Maskawa (CKM) matrix [101, 102]. The CKM
matrix is defined by
VCKM ≡ V
u
L
†
V
d
L
†
, (2.13)
where V
u
L
, V
d
L
are the unitary matrices that diagonalize the Yukawa couplings Hu
, Hd
,
respectively. This product of the two matrices appears in the charged current when it
is expressed in terms of the observable mass eigenstates.
2.1.3 Mass generation through the Higgs mechanism
We will start by examining the scalar potential (2.11). The vacuum expectation value
(vev) of the Higgs field hΦi ≡ h0|Φ|0i is given by the minimum of the potential. For
µ
2 > 0, the potential is always non-negative and Φ has a zero vev. The hypothesis of
the Higgs mechanism is that µ
2 < 0. In this case, the field Φ will acquire a vev
hΦi =
1
2

0
v
!
with v =
r
−
µ2
λ
. (2.14)
Since the charged component of Φ still has a zero vev, the U(1)Q symmetry of quantum
electrodynamics (QED) remains unbroken.
We expand the field Φ around the minima v in terms of real fields, and at leading
order we have
Φ(x) =
θ2(x) + iθ1(x)
√
1
2
(v + H(x)) − iθ3(x)
!
=
1
√
2
e
iθa(x)τ
a

0
v + H(x)
!
. (2.15)
We can eliminate the unphysical degrees of freedom θa, using the fact that the theory
remains gauge invariant. Therefore, we perform the following SU(2)L gauge transformation on Φ (unitary gauge)
Φ(x) → e
−iθa(x)τ
a
Φ(x), (2.16)
so that
Φ(x) = 1
√
2

0
v + H(x)
!
. (2.17)
We are going to use the following definitions for the gauge fields
W±
µ ≡
1
2

W1
µ ∓ iW2
µ

, (2.18a)
Zµ ≡
1
p
g
2
1 + g
2
2

g2W3
µ − g1Bµ

, (2.18b)
Aµ ≡
1
p
g
2
1 + g
2
2

g1W3
µ + g2Bµ

, (2.1
2.2 Limits of the SM and the emergence of supersymmetry 33
Then, the kinetic term for Φ (see Eq. (2.10c)) can be written in the unitary gauge as
(DµΦ)†
(D
µΦ) = 1
2
(∂µH)
2 + M2
W W+
µ W−µ +
1
2
M2
ZZµZ
µ
, (2.19)
with
MW ≡
1
2
g2v and MZ ≡
1
2
q
g
2
1 + g
2
2
v. (2.20)
We see that the definitions (2.18) correspond to the physical states of the gauge bosons
that have acquired masses due to the non-zero Higgs vev, given by (2.20). The photon
has remained massless, which reflects the fact that after the spontaneous breakdown of
SU(2)L × U(1)Y the U(1)Q remained unbroken. Among the initial degrees of freedom
of the complex scalar field Φ, three were absorbed by W± and Z and one remained as
the neutral Higgs particle with squared mass
m2
H = 2λv2
. (2.21)
We note that λ should be positive so that the scalar potential (2.11) is bounded from
below.
Fermions also acquire masses due to the Higgs mechanism. The Yukawa terms in
the fermionic part (2.10b) of the SM Lagrangian are written, after expanding around
the vev in the unitary gauge,
LY = −
1
√
2
hee¯L(v + H)eR −
1
√
2
hd
¯dL(v + H)dR −
1
√
2
huu¯L(v + H)uR + h.c. . (2.22)
Therefore, we can identify the masses of the fermions as
me
i =
h
i
e
v
√
2
, md
i =
h
i
d
v
√
2
, mui =
h
i
u
v
√
2
, (2.23)
where we have written explicitly the generation indices.
2.2 Limits of the SM and the emergence of supersymmetry
2.2.1 General discussion of the SM problems
The SM has been proven extremely successful and has been tested in high precision
in many different experiments. It has predicted many new particles before their final
discovery and also explained how the particles gain their masses. Its last triumph was
of course the discovery of a boson that seems to be very similar to the Higgs boson of
the SM. However, it is generally accepted that the SM cannot be the ultimate theory. It
is not only observed phenomena that the SM does not explain; SM also faces important
theoretical issues.
The most prominent among the inconsistencies of the SM with observations is the
oscillations among neutrinos of different generations. In order for the oscillations to
34 Particle Physics
φ φ
k
Figure 2.1: The scalar one-loop diagram giving rise to quadratic divergences.
occur, neutrinos should have non-zero masses. However, minimal modifications of the
SM are able to fit with the data of neutrino physics. Another issue that a more complete theory has to face is the matter asymmetry, the observed dominance of matter
over antimatter in the Universe. In addition, in order to comply with the standard
cosmological model, it has to provide the appropriate particle(s) that drove the inflation. Last, but not least, we saw that in order to explain the DM that dominates the
Universe, a massive, stable weakly interacting particle must exist. Such a particle is
not present in the SM.
On the other hand, the SM also suffers from a theoretical perspective. For example,
the SM counts 19 free parameters; one expects that a fundamental theory would have
a much smaller number of free parameters. Simple modifications of the SM have been
proposed relating some of these parameters. Grand unified theories (GUTs) unify
the gauge couplings at a high scale ∼ 1016 GeV. However, this unification is only
approximate unless the GUT is embedded in a supersymmetric framework. Another
serious problem of the SM is that of naturalness. This will be the topic of the following
subsection.
2.2.2 The naturalness problem of the SM
The presence of fundamental scalar fields, like the Higgs, gives rise to quadratic divergences. The diagram of Fig. 2.1 contributes to the squared mass of the scalar
δm2 = λ
Z Λ
d
4k
(2π)
4
k
−2
. (2.24)
This contribution is approximated by δm2 ∼ λΛ
2/(16π
2
), quadratic in a cut-off Λ,
which should be finite. For the case of the Higgs scalar field, one has to include its
couplings to the gauge fields and the top quark3
. Therefore,
δm2
H =
3Λ2
8π
2v
2

4m2
t − 2M2
W − M2
Z − m2
H

+ O(ln Λ
µ
)

, (2.25)
where we have used Eq. (2.21) and m2
H ≡ m2
0 + δm2
H.
3Since the contribution to the squared mass correction are quadratic in the Yukawa couplings (or
quark masses), the lighter quarks can be neglected
2.2.3 A way out 35
Taking Λ as a fundamental scale Λ ∼ MP l ∼ 1019 GeV we have
m2
0 = m2
H −
3Λ2
8π
2v
2

4m2
t − 2M2
W − M2
Z − m2
H

(2.26)
and we can see that m2
0 has to be adjusted to a precision of about 30 orders of magnitude
in order to achieve an EW scale Higgs mass. This is considered as an intolerable finetuning, which is against the general belief that the observable properties of a theory
have to be stable under small variations of the fundamental (bare) parameters. It is
exactly the above behavior that is considered as unnatural. Although the SM could
be self-consistent without imposing a large scale, grand unification of the parameters
introduce a hierarchy problem between the different scales.
A more strict definition of naturalness comes from ’t Hooft [103], which we rewrite
here:
At an energy scale µ, a physical parameter or set of physical parameters
αi(µ) is allowed to be very small only if the replacement αi(µ) = 0 would
increase the symmetry of the system.
Clearly, this is not the case here. Although mH is small compared to the fundamental
scale Λ, it is not protected by any symmetry and a fine-tuning is necessary.
2.2.3 A way out
The naturalness in the ’t Hooft sense is inspired by quantum electrodynamics, which is
the archetype for a natural theory. For example, the corrections to the electron mass
me are themselves proportional to me, with a dimensionless proportionality factor that
behaves like ∼ ln Λ. In general, fermion masses are protected by the chiral symmetry; small values (compared to the fundamental scale) of these masses enhances the
symmetry.
If a new symmetry exists in nature, relating fermion fields to scalar fields, then each
scalar mass would be related somehow to the corresponding fermion mass. Therefore,
the scalar mass itself can be naturally small compared to Λ, since this would mean
that the fermion mass is small, which enhances the chiral symmetry. Such a symmetry,
relating bosons to fermions and vice versa, is known as supersymmetry [104, 105].
Actually, as we will see later, if this new symmetry remains unbroken, the masses of
the conjugate bosons and fermions would have to be equal.
In order to make the above statement more concrete, we consider a toy model with
two additional complex scalar fields feL and feR. We will discuss only the quadratic
divergences that come from corrections to the Higgs mass due to a fermion. The
generalization for the contributions from the gauge bosons or the self-interaction is
straightforward. The interactions in this toy model of the new scalar fields with the
Higgs are described by the Lagrangian
Lfefφe = λfe|φ|
2

|feL|
2 + |feR|
2

. (2.27
36 Particle Physics
It can be easily checked that the quadratic divergence coming from a fermion at one
loop is exactly canceled, as long as the new quartic coupling λfe obeys the relation
λfe = −λ
2
f
(λf is the Yukawa coupling for the fermion f).
2.3 A brief summary of Supersymmetry
Supersymmetry (SUSY) is a symmetry relating fermions and bosons. The supersymmetry transformation should turn a boson state into a fermion state and vice versa. If
Q is the operator that generates such transformations, then
Q |bosoni = |fermioni Q |fermioni = |bosoni. (2.28)
Due to commutation and anticommutation rules of bosons and fermions, Q has to
be an anticommuting spinor operator, carrying spin angular momentum 1/2. Since
spinors are complex objects, the hermitian conjugate Q†
is also a symmetry operator4
.
There is a no-go theorem, the Coleman-Mandula theorem [106], that restricts the
conserved charges which transform as tensors under the Lorentz group to the generators
of translations Pµ and the generators of Lorentz transformations Mµν. Although this
theorem can be evaded in the case of supersymmetry due to the anticommutation
properties of Q, Q†
[107], it restricts the underlying algebra of supersymmetry [108].
Therefore, the basic supersymmetric algebra can be written as5
{Q, Q†
} = P
µ
, (2.29a)
{Q, Q} = {Q
†
, Q†
} = 0, (2.29b)
[P
µ
, Q] = [P
µ
, Q] = 0. (2.29c)
In the following, we summarize the basic conclusions derived from this algebra.
• The single-particle states of a supersymmetric theory fall into irreducible representations of the SUSY algebra, called supermultiplets. A supermultiplet contains
both fermion and boson states, called superpartners.
• Superpartners must have equal masses: Consider |Ωi and |Ω
′
i as the superpartners, |Ω
′
i should be proportional to some combination of the Q and Q† operators
acting on |Ωi, up to a space-time translation or rotation. Since −P
2
commutes
with Q, Q† and all space-time translation and rotation operators, |Ωi, |Ω
′
i will
have equal eigenvalues of −P
2 and thus equal masses.
• Superpartners must be in the same representation of gauge groups, since Q, Q†
commute with the generators of gauge transformations. This means that they
have equal charges, weak isospin and color degrees of freedom.
4We will confine ourselves to the phenomenologically more interesting case of N = 1 supersymmetry, with N referring to the number of distinct copies of Q, Q†
.
5We present a simplified version, omitting spinor indices in Q and Q†
.
2.3 A brief summary of Supersymmetry 37
• Each supermultiplet contains an equal number of fermion and boson degrees of
freedom (nF and nB, respectively): Consider the operator (−1)2s
, with s the spin
angular momentum, and the states |ii that have the same eigenvalue p
µ of P
µ
.
Then, using the SUSY algebra (2.29) and the completeness relation P
i
|ii hi| =
1, we have P
i
hi|(−1)2sP
µ
|ii = 0. On the other hand, P
i
hi|(−1)2sP
µ
|ii =
p
µTr [(−1)2s
] ∝ nB − nF . Therefore, nF = nB.
As addendum to the last point, we see that two kind of supermultiplets are possible
(neglecting gravity):
• A chiral (or matter or scalar ) supermultiplet, which consists of a single Weyl
fermion (with two spin helicity states, nF = 2) and two real scalars (each with
nB = 1), which can be replaced by a single complex scalar field.
• A gauge (or vector ) supermultiplet, which consists of a massless spin 1 boson
(two helicity states, nB = 2) and a massless spin 1/2 fermion (nF = 2).
Other combinations either are reduced to combinations of the above supermultiplets
or lead to non-renormalizable interactions.
It is possible to study supersymmetry in a geometric approach, using a space-time
manifold extended by four fermionic (Grassmann) coordinates. This manifold is called
superspace. The fields, in turn, expressed in terms of the extended set of coordinates
are called superfields. We are not going to discuss the technical details of this topic
(the interested reader may refer to the rich bibliography, for example [109–111]).
However, it is important to mention a very useful function of the superfields, the
superpotential. A generic form of a (renormalizable) superpotential in terms of the
superfields Φ is the following b
W =
1
2
MijΦbiΦbj +
1
6
y
ijkΦbiΦbjΦbk. (2.30)
The Lagrangian density can always be written according to the superpotential. The
superpotential has also to fulfill some requirements. In order for the Lagrangian to
be supersymmetric invariant, W has to be holomorphic in the complex scalar fields
(it does not involve hermitian conjugates Φb† of the superfields). Conventionally, W
involves only left chiral superfields. Instead of the SU(2)L singlet right chiral fermion
fields, one can use their left chiral charge conjugates.
As we mentioned before, the members of a supermultiplet have equal masses. This
contradicts our experience, since the partners of the light SM particles would have been
detected long time ago. Hence, the supersymmetry should be broken at a large energy
scale. The common approach is that SUSY is broken in a hidden sector, very weakly
coupled to the visible sector. Then, one has to explain how the SUSY breaking mediated to the visible sector. The two most popular scenarios are the gravity mediation
scenario [112–114] and the Gauge-Mediated SUSY Breaking (GSMB) [113, 115–117],
where the mediation occurs through gauge interactions.
There are two approaches with which one can address the SUSY breaking. In the
first approach, one refers to a GUT unification and determines the supersymmetric
38 Particle Physics
breaking parameters at low energies through the renormalization group equations.
This approach results in a small number of free parameters. In the second approach,
the starting point is the low energy scale. In this case, the SUSY breaking has to be
parametrized by the addition of breaking terms to the low energy Lagrangian. This
results in a larger set of free parameters. These terms should not reintroduce quadratic
divergences to the scalar masses, since the cancellation of these divergences was the
main motivation for SUSY. Then, one talks about soft breaking terms.
2.4 The Minimal Supersymmetric Standard Model
One can construct a supersymmetric version of the standard model with a minimal
content of particles. This model is known as the Minimal Supersymmetric Standard
Model (MSSM). In a SUSY extension of the SM, each of the SM particles is either in a
chiral or in a gauge supermultiplet, and should have a superpartner with spin differing
by 1/2.
The spin-0 partners of quarks and leptons are called squarks and sleptons, respectively (or collectively sfermions), and they have to reside in chiral supermultiplets.
The left- and right-handed components of fermions are distinct 2-component Weyl
fermions with different gauge transformations in the SM, so that each must have its
own complex scalar superpartner. The gauge bosons of the SM reside in gauge supermultiplets, along with their spin-1/2 superpartners, which are called gauginos. Every
gaugino field, like its gauge boson partner, transforms as the adjoint representation of
the corresponding gauge group. They have left- and right-handed components which
are charge conjugates of each other: (λeL)
c = λeR.
The Higgs boson, since it is a spin-0 particle, should reside in a chiral supermultiplet. However, we saw (in the fermionic part of the SM Lagrangian, Eq. (2.10b))
that the Y = 1/2 Higgs in the SM can give mass to both up- and down-type quarks,
only if the conjugate Higgs field with Y = −1/2 is involved. Since in the superpotential there are no conjugate fields, two Higgs doublets have to be introduced. Each
Higgs supermultiplet would have hypercharge Y = +1/2 or Y = −1/2. The Higgs
with the negative hypercharge gives mass to the down-type fermions and it is called
down-type Higgs (Hd, or H1 in the SLHA convention [118]) and the other one gives
mass to up-type fermions and it is called up-type Higgs (Hu, or H2).
The MSSM respects a discrete Z2 symmetry, the R-parity. If one writes the most
general terms in the supersymmetric Lagrangian (still gauge-invariant and holomorphic), some of them would lead to non-observed processes. The most obvious constraint
comes from the non-observed proton decay, which arises from a term that violates both
lepton and baryon numbers (L and B, respectively) by one unit. In order to avoid these
terms, R-parity, a multiplicative conserved quantum number, is introduced, defined as
PR = (−1)3(B−L)+2s
, (2.31)
with s the spin of the particle.
The R even particles are the SM particles, whereas the R odd are the new particles
introduced by the MSSM and are called supersymmetric particles. Due to R-parity,
2.4 The Minimal Supersymmetric Standard Model 39
if it is exactly conserved, there can be no mixing among odd and even particles and,
additionally, each interaction vertex in the theory can only involve an even number of
supersymmetric particles. The phenomenological consequences are quite important.
First, the lightest among the odd-parity particles is stable. This particle is known
as the lightest supersymmetric particle (LSP). Second, in collider experiments, supersymmetric particles can only be produced in pairs. The first of these consequences
was a breakthrough for the incorporation of DM into a general theory. If the LSP is
electrically neutral, it interacts only weakly and it consists an attractive candidate for
DM.
We are not going to enter further into the details of the MSSM6
. Although MSSM
offers a possible DM candidate, there is a strong theoretical reason to move from the
minimal model. This reason is the so-called µ-problem of the MSSM, with which we
begin the discussion of the next chapter, where we shall describe more thoroughly the
Next-to-Minimal Supersymmetric Standard Model.
6We refer to [110] for an excellent and detailed description of MSSM.
40 Particle Physics
Part II
Dark Matter in the
Next-to-Minimal Supersymmetric
Standard Model

CHAPTER 3
THE NEXT-TO-MINIMAL
SUPERSYMMETRIC STANDARD
MODEL
The Next-to-Minimal Supersymmetric Standard Model (NMSSM) is an extension of
the MSSM by a chiral, SU(2)L singlet superfield Sb (see [119, 120] for reviews). The
introduction of this field solves the µ-problem1
from which the MSSM suffers, but
also leads to a different phenomenology from that of the minimal model. The scalar
component of the additional field mixes with the scalar Higgs doublets, leading to three
CP-even mass eigenstates and two CP-odd eigenstates (as in the MSSM a doublet-like
pair of charged Higgs also exists). On the other hand, the fermionic component of the
singlet (singlino) mixes with gauginos and higgsinos, forming five neutral states, the
neutralinos.
Concerning the CP-even sector, a new possibility opens. The lightest Higgs mass
eigenstate may have evaded the detection due to a sizeable singlet component. Besides,
the SM-like Higgs is naturally heavier than in the MSSM [123–126]. Therefore, a SMlike Higgs mass ∼ 125 GeV is much easier to explain [127–141]. The singlet component
of the CP-odd Higgs also allows for a potentially very light pseudoscalar with suppressed couplings to SM particles, with various consequences, especially on low energy
observables (for example, [142–145]). The singlino component of the neutralino may
also play an important role for both collider phenomenology and DM. This is the case
when the neutralino is the LSP and the lightest neutralino has a significant singlino
component.
We start the discussion about the NMSSM by describing the µ-problem and how
this is solved in the context of the NMSSM. In Sec. 3.2 we introduce the NMSSM
Lagrangian and we write the mass matrices of the Higgs sector particles and the su1However, historically, the introduction of a singlet field preceded the µ-problem, e.g. [104, 105,
121, 122].
44 The Next-to-Minimal Supersymmetric Standard Model
persymmetric particles, at tree level. We continue by examining, in Sec. 3.3, the DM
candidates in the NMSSM and particularly the neutralino. The processes which determine the neutralino relic density are described in Sec. 3.4. The detection possibilities
of a potential NMSSM neutralino as DM are discussed in (Sec. 3.5). We close this
chapter (Sec. 3.6) by examining possible ways to include non-zero neutrino masses and
the additional DM candidates that are introduced.
3.1 Motivation – The µ-problem of the MSSM
As we saw, the minimal extension of the SM, the MSSM, contains two Higgs SU(2)L
doublets Hu and Hd. The Lagrangian of the MSSM should contain a supersymmetric
mass term, µHuHd, for these two doublets. There are several reasons, which we will
subsequently review, that require the existence of such a term. On the other hand,
the fact that |µ| cannot be very large, actually it should be of the order of the EW
scale, brings back the problem of naturalness. A parameter of the model should be
much smaller than the “natural” scale (the GUT or the Planck scale) before the EW
symmetry breaking. This leads to the so-called µ-problem of the MSSM [146].
The reasons that such a term should exist in the Lagrangian of the MSSM are
mainly phenomenological. The doublets Hu and Hd are components of chiral superfields that also contain fermionic SU(2)L doublets. Their electrically charged components mix with the superpartners of the W± bosons, forming two charged Dirac
fermions, the charginos. The unsuccessful searches for charginos in LEP have excluded
charginos with masses almost up to its kinetic limit (∼ 104 GeV) [147]. Since the µ term
determines the mass of the charginos, µ cannot be zero and actually |µ| >∼ 100 GeV,
independently of the other free parameters of the model. Moreover, µ = 0 would result
in a Peccei-Quinn symmetry of the Higgs sector and an undesirable massless axion.
Finally, there is one more reason for µ 6= 0 related to the mass generation by the Higgs
mechanism. The term µHuHd will be accompanied by a soft SUSY breaking term
BµHuHd. This term is necessary so that both neutral components of Hu and Hd are
non-vanishing at the minimum of the potential.
The Higgs mechanism also requires that µ is not too large. In order to generate
the EW symmetry breaking, the Higgs potential has to be unstable at its origin Hu =
Hd = 0. Soft SUSY breaking terms for Hu and Hd of the order of the SUSY breaking
scale generate such an instability. However, the µ induced squared masses for Hu,
Hd are always positive and would destroy the instability in case they dominate the
negative soft mass terms.
The NMSSM is able to solve the µ-problem by dynamically generating the mass
µ. This is achieved by the introduction of an SU(2)L singlet scalar field S. When S
acquires a vev, a mass term for the Hu and Hd emerges with an effective mass µeff of
the correct order, as long as the vev is of the order of the SUSY breaking scale. This
can be obtained in a more “natural” way through the soft SUSY breaking terms.
3.2 The NMSSM Lagrangian 45
3.2 The NMSSM Lagrangian
All the necessary information for the Lagrangian of the NMSSM can be extracted from
the superpotential and the soft SUSY breaking Lagrangian, containing the soft gaugino and scalar masses, and the trilinear couplings. We begin with the superpotential,
writing all the interactions of the NMSSM superfields, which include the MSSM superfields and the additional gauge singlet chiral superfield2 Sb. Hence, the superpotential
reads
W = λSbHbu · Hbd +
1
3
κSb3
+ huQb · HbuUbc
R + hdHbd · QbDbc
R + heHbd · LbEbc
R.
(3.1)
The couplings to quarks and leptons have to be understood as 3 × 3 matrices and the
quark and lepton fields as vectors in the flavor space. The SU(2)L doublet superfields
are given (as in the MSSM) by
Qb =

UbL
DbL
!
, Lb =

νb
EbL
!
, Hbu =

Hb +
u
Hb0
u
!
, Hbd =

Hb0
d
Hb −
d
!
(3.2)
and the product of two doublets is given by, for example, Qb · Hbu = UbLHb0
u − Hb +
u DbL.
An important fact to note is that the superpotential given by (3.1) does not include all possible renormalizable couplings (which respect R-parity). The most general
superpotential would also include the terms
W ⊃ µHbu · Hbd +
1
2
µ
′Sb2 + ξF s, b (3.3)
with the first two terms corresponding to supersymmetric masses and the third one,
with ξF of dimension mass2
, to a tadpole term. However, the above dimensionful
parameters µ, µ
′ and ξF should be of the order of the SUSY breaking scale, a fact
that contradicts the motivation behind the NMSSM. Here, we omit these terms and
we will work with the scale invariant superpotential (3.1). The Lagrangian of a scale
invariant superpotential possesses an accidental Z3 symmetry, which corresponds to a
multiplication of all the components of all chiral fields by a phase ei2π/3
.
The corresponding soft SUSY breaking masses and couplings are
−Lsof t = m2
Hu
|Hu|
2 + m2
Hd
|Hd|
2 + m2
S
|S|
2
+ m2
Q|Q|
2 + m2
D|DR|
2 + m2
U
|UR|
2 + m2
L
|L|
2 + m2
E|ER|
2
+

huAuQ · HuU
c
R − hdAdQ · HdD
c
R − heAeL · HdE
c
R
+λAλHu · HdS +
1
3
κAκS
3 + h.c.

+
1
2
M1λ1λ1 +
1
2
M2λ
i
2λ
i
2 +
1
2
M3λ
a
3λ
a
3
,
(3.4)
2Here, the hatted capital letters denote chiral superfields, whereas the corresponding unhatted
ones indicate their complex scalar components.
46 The Next-to-Minimal Supersymmetric Standard Model
where we have also included the soft breaking masses for the gauginos. λ1 is the U(1)Y
gaugino (bino), λ
i
2 with i = 1, 2, 3 is the SU(2)L gaugino (winos) and, finally, the λ
a
3
with a = 1, . . . , 8 denotes the SU(3)c gaugino (gluinos).
The scalar potential, expressed by the so-called D and F terms, can be written
explicitly using the general formula
V =
1
2

D
aD
a + D
′2

+ F
⋆
i Fi
, (3.5)
where
D
a = g2Φ
∗
i T
a
ijΦj (3.6a)
D
′ =
1
2
g1YiΦ
∗
i Φi (3.6b)
Fi =
∂W
∂Φi
. (3.6c)
We remind that T
a are the SU(2)L generators and Yi the hypercharge of the scalar
field Φi
. The Yukawa interactions and fermion mass terms are given by the general
Lagrangian
LY ukawa = −
1
2
∂
2W
∂Φi∂Φj
ψiψj + h.c.
, (3.7)
using the superpotential (3.1). The two-component spinor ψi
is the superpartner of
the scalar Φi
.
3.2.1 Higgs sector
Using the general form of the scalar potential, the following Higgs potential is derived
VHiggs =

λ

H
+
u H
−
d − H
0
uH
0
d

+ κS2

2
+

m2
Hu + |λS|
2

H
0
u

2
+

H
+
u

2

+

m2
Hd + |λS|
2

H
0
d

2
+

H
−
d

2

+
1
8

g
2
1 + g
2
2

H
0
u

2
+

H
+
u

2
−

H
0
d

2
−

H
−
d

2
2
+
1
2
g
2
2

H
+
u H
0
d
⋆
+ H
0
uH
−
d
⋆

2
+ m2
S
|S|
2 +

λAλ

H
+
u H
−
d − H
0
uH
0
d

S +
1
3
κAκS
3 + h.c.

.
(3.8)
The neutral physical Higgs states are defined through the relations
H
0
u = vu +
1
√
2
(HuR + iHuI ), H0
d = vd +
1
√
2
(HdR + iHdI ),
S = s +
1
√
2
(SR + iSI ),
3.2.1 Higgs sector 47
where vu, vd and s are, respectively, the real vevs of Hu, Hd and S, which have to be
obtained from the minima of the scalar potential (3.8), after expanding the fields using
Eq. (3.9). We notice that when S acquires a vev, a term µeffHbu · Hbd appears in the
superpotential, with
µeff = λs, (3.10)
solving the µ-problem.
Therefore, the Higgs sector of the NMSSM is characterized by the seven parameters
λ, κ, m2
Hu
, m2
Hd
, m2
S
, Aλ and Aκ. One can express the three soft masses by the three
vevs using the minimization equations of the Higgs potential (3.8), which are given by
vu

m2
Hu + µ
2
eff + λ
2
v
2
d +
1
2
g
2

v
2
u − v
2
d

− vdµeff(Aλ + κs) = 0
vd

m2
Hd + µ
2
eff + λ
2
v
2
u +
1
2
g
2

v
2
d − v
2
u

− vuµeff(Aλ + κs) = 0
s

m2
S + κAκs + 2κ
2σ
2 + λ
2

v
2
u + v
2
d

− 2λκvuvd

− λAλvuvd = 0,
(3.11)
where we have defined
g
2 ≡
1
2

g
2
1 + g
2
2

. (3.12)
One can also define the β angle by
tan β =
vu
vd
. (3.13)
The Z boson mass is given by MZ = gv with v
2 = v
2
u + v
2
d ≃ (174 GeV)2
. Hence, with
MZ given, the set of parameters that describes the Higgs sector of the NMSSM can be
chosen to be the following
λ, κ, Aλ, Aκ, tan b and µeff. (3.14)
CP-even Higgs masses
One can obtain the Higgs mass matrices at tree level by expanding the Higgs potential
(3.8) around the vevs, using Eq. (3.9). We begin by writing3
the squared mass matrix
M2
S
of the scalar Higgses in the basis (HdR, HuR, SR):
M2
S =


g
2
v
2
d + µ tan βBeff (2λ
2 − g
2
) vuvd − µBeff 2λµvd − λ (Aλ + 2κs) vu
g
2
v
2
u +
µ
tan βBeff 2λµvu − λ (Aλ + 2κs) vd
λAλ
vuvd
s + κAκs + (2κs)
2

 ,
(3.15)
where we have defined Beff ≡ Aλ + κs (it plays the role of the B parameter of the
MSSM).
3For economy of space, we omit in this expression the subscript from µ
48 The Next-to-Minimal Supersymmetric Standard Model
Although an analytical diagonalization of the above 3 × 3 mass matrix is lengthy,
there is a crucial conclusion that comes from the approximate diagonalization of the
upper 2 × 2 submatrix. If it is rotated by an angle β, one of its diagonal elements
is M2
Z
(cos2 2β +
λ
2
g
2 sin2
2β) which is an upper bound for its lightest eigenvalue. The
first term is the same one as in the MSSM. The conclusion is that in the NMSSM
the lightest CP-even Higgs can be heavier than the corresponding of the MSSM, as
long as λ is large and tan β relatively small. Therefore, it is much easier to explain
the observed mass of the SM-like Higgs. However, λ is bounded from above in order
to avoid the appearance of the Landau pole below the GUT scale. Depending on the
other free parameters, λ should obey λ <∼ 0.7.
CP-odd Higgs masses
For the pseudoscalar case, the squared mass matrix in the basis (HdI , HuI , SI ) is
M2
P =


µeff (Aλ + κs) tan β µeff (Aλ + κs) λvu (Aλ − 2κs)
µeff
tan β
(Aλ + κs) λvd (Aλ − 2κs)
λ (Aλ + 4κs)
vuvd
s − 3κAκs

 . (3.16)
One eigenstate of this matrix corresponds to an unphysical massless Goldstone
boson G. In order to drop the Goldstone boson, we write the matrix in the basis
(A, G, SI ) by rotating the upper 2 × 2 submatrix by an angle β. After dropping the
massless mode, the 2 × 2 squared mass matrix turns out to be
M2
P =
2µeff
sin 2β
(Aλ + κs) λ (Aλ − 2κs) v
λ (Aλ + 4κs)
vuvd
s − 3Aκs
!
. (3.17)
Charged Higgs mass
The charged Higgs squared mass matrix is given, in the basis (H+
u
, H−
d
⋆
), by
M2
± =

µeff (Aλ + κs) + vuvd

1
2
g
2
2 − λ

cot β 1
1 tan β
!
, (3.18)
which contains one Goldstone boson and one physical mass eigenstate H± with eigenvalue
m2
± =
2µeff
sin 2β
(Aλ + κs) + v
2

1
2
g
2
2 − λ

. (3.19)
3.2.2 Sfermion sector
The mass matrix for the up-type squarks is given in the basis (ueR, ueL) by
Mu =

m2
u + h
2
u
v
2
u −
1
3
(v
2
u − v
2
d
) g
2
1 hu (Auvu − µeffvd)
hu (Auvu − µeffvd) m2
Q + h
2
u
v
2
u +
1
12 (v
2
u − v
2
d
) (g
2
1 − 3g
2
2
)
!
, (3.20)
3.2.3 Gaugino and higgsino sector 49
whereas for down-type squarks the mass matrix reads in the basis (deR, deL)
Md =

m2
d + h
2
d
v
2
d −
1
6
(v
2
u − v
2
d
) g
2
1 hd (Advd − µeffvu)
hd (Advd − µeffvu) m2
Q + h
2
d
v
2
d +
1
12 (v
2
u − v
2
d
) (g
2
1 − 3g
2
2
)
!
. (3.21)
The off-diagonal terms are proportional to the Yukawa coupling hu for the up-type
squarks and hd for the down-type ones. Therefore, the two lightest generations remain
approximately unmixed. For the third generation, the mass matrices are diagonalized
by a rotation by an angle θT and θB, respectively, for the stop and sbottom. The mass
eigenstates are, then, given by
et1 = cos θT
etL + sin θT
etR, et2 = cos θT
etL − sin θT
etR, (3.22)
eb1 = cos θB
ebL + sin θB
ebR, eb2 = cos θB
ebL − sin θB
ebR. (3.23)
In the slepton sector, for a similar reason, only the left- and right-handed staus are
mixed and their mass matrix
Mτ =

m2
E3 + h
2
τ
v
2
d −
1
2
(v
2
u − v
2
d
) g
2
1 hτ (Aτ vd − µeffvu)
hτ (Aτ vd − µeffvu) m2
L3 + h
2
τ
v
2
d −
1
4
(v
2
u − v
2
d
) (g
2
1 − g
2
2
)
!
(3.24)
is diagonalized after a rotation by an angle θτ . Their mass eigenstates are given by
τe1 = cos θτ τeL + sin θτ τeR, τe2 = cos θτ τeL − sin θτ τeR. (3.25)
Finally, the sneutrino masses are
mνe = m2
L −
1
4

v
2
u − v
2
d
g
2
1 + g
2
2

. (3.26)
3.2.3 Gaugino and higgsino sector
The gauginos λ1 and λ
3
2 mix with the neutral higgsinos ψ
0
d
, ψ
0
u
and ψS to form neutral
particles, the neutralinos. The 5 × 5 mass matrix of the neutralinos is written in the
basis
(−iλ1, −iλ3
2
, ψ0
d
, ψ0
u
, ψS) ≡ (B, e W , f He0
d
, He0
u
, Se) (3.27)
as
M0 =


M1 0 − √
1
2
g1vd √
1
2
g1vu 0
M2 √
1
2
g2vd − √
1
2
g2vu 0
0 −µeff −λvu
0 −λvd
2κs


. (3.28)
The mass matrix (3.28) is diagonalized by an orthogonal matrix Nij . The mass eigenstates of the neutralinos are usually denoted by χ
0
i
, with i = 1, . . . , 5, with increasing
masses (i = 1 corresponds to the lightest neutralino). These are given by
χ
0
i = Ni1Be + Ni2Wf + Ni3He0
d + Ni4He0
u + Ni5S. e (3.2
50 The Next-to-Minimal Supersymmetric Standard Model
We use the convention of a real matrix Nij , so that the physical masses mχ
0
i
are real,
but not necessarily positive.
In the charged sector, the SU(2)L charged gauginos λ
− = √
1
2
(λ
1
2 + iλ2
2
), λ
+ =
√
1
2
(λ
1
2 − iλ2
2
) mix with the charged higgsinos ψ
−
d
and ψ
+
u
, forming the charginos ψ
±:
ψ
± =

−iλ±
ψ
±
u
!
. (3.30)
The chargino mass matrix in the basis (ψ
−, ψ+) is
M± =

M2 g2vu
g2vd µeff !
. (3.31)
Since it is not symmetric, the diagonalization requires different rotations of ψ
− and
ψ
+. We denote these rotations by U and V , respectively, so that the mass eigenstates
are obtained by
χ
− = Uψ−, χ+ = V ψ+. (3.32)
3.3 DM Candidates in the NMSSM
Let us first review the characteristics that a DM candidate particle should have. First,
it should be massive in order to account for the missing mass in the galaxies. Second,
it must be electrically and color neutral. Otherwise, it would have condensed with
baryonic matter, forming anomalous heavy isotopes. Such isotopes are absent in nature. Finally, it should be stable and interact only weakly, in order to fit the observed
relic density.
In the NMSSM there are two possible candidates. Both can be stable particles if
they are the LSPs of the supersymmetric spectrum. The first one is the sneutrino (see
[148,149] for early discussions on MSSM sneutrino LSP). However, although sneutrinos
are WIMPs, their large coupling to the Z bosons result in a too large annihilation cross
section. Hence, if they were the DM particles, their relic density would have been very
small compared to the observed value. Exceptions are very massive sneutrinos, heavier
than about 5 TeV [150]. Furthermore, the same coupling result in a large scattering
cross section off the nuclei of the detectors. Therefore, sneutrinos are also excluded by
direct detection experiments.
The other possibility is the lightest neutralino. Neutralinos fulfill successfully, at
least in principle, all the requirements for a DM candidate. However, the resulting
relic density, although weakly interacting, may vary over many orders of magnitude as
a function of the free parameters of the theory. In the next sections we will investigate
further the properties of the lightest neutralino as the DM particle. We begin by
studying its annihilation that determines the DM relic density.
3.4 Neutralino relic density 51
3.4 Neutralino relic density
We remind that the neutralinos are mixed states composed of bino, wino, higgsinos
and the singlino. The exact content of the lightest neutralino determines its pair
annihilation channels and, therefore, its relic density (for detailed analyses, we refer
to [151–154]). Subsequently, we will briefly describe the neutralino pair annihilation
in various scenarios. We classify these scenarios with respect to the lightest neutralino
content.
Before we proceed, we should mention another mechanism that affects the neutralino LSP relic density. If there is a supersymmetric particle with mass close to the
LSP (but always larger), it would be abundant during the freeze-out and LSP coannihilations with this particle would contribute to the total annihilation cross section.
This particle, which is the Next-to-Lightest Supersymmetric Particle (NLSP), is most
commonly a stau or a stop. In the above sense, coannihilations refer not only to the
LSP–NLSP coannihilations, but also to the NLSP–NLSP annihilations, since the latter
reduce the number density of the NLSPs [155].
• Bino-like LSP
In principle, if the lightest neutralino is mostly bino-like, the total annihilation
cross section is expected to be small. Therefore, a bino-like neutralino LSP would
have been overabundant. The reason for this is that there is only one available
annihilation channel via t-channel sfermion exchange, since all couplings to gauge
bosons require a higgsino component. The cross section is even more reduced
when the sfermion mass is large.
However, there are still two ways to achieve the correct relic density. The first one
is using the coannihilation effect: if there is a sfermion with a mass slightly larger
(some GeV) than the LSP mass, their coannihilations can be proved to reduce
efficiently the relic density to the desired value. The second one concerns a binolike LSP, with a very small but non-negligible higgsino component. In this case,
if in addition the lightest CP-odd Higgs A1 is light enough, the annihilation to a
pair A1A1 (through an s-channel CP-even Higgs Hi exchange) can be enhanced
via Hi resonances. In this scenario a fine-tuning of the masses is necessary.
• Higgsino-like LSP
A mostly higgsino LSP is as well problematic. The strong couplings of the higgsinos to the gauge bosons lead to very large annihilation cross section. Therefore,
a possible higgsino LSP would have a very small relic density.
• Mixed bino–higgsino LSP
In this case, as it was probably expected, one can easily fit the relic density to
the observed value. This kind of LSP annihilates to W+W−, ZZ, W±H∓, ZHi
,
HiAj
, b
¯b and τ
+τ
− through s-channel Z or Higgs boson exchange or t-channel
neutralino or chargino exchange. The last two channels are the dominant ones
when the Higgs coupling to down-type fermions is enhanced, which occurs more
commonly in the regime of relatively large tan β. The annihilation channel to a
52 The Next-to-Minimal Supersymmetric Standard Model
pair of top quarks also contributes to the total cross section, if it is kinematically
allowed. However, in order to achieve the correct relic density, the higgsino
component cannot be very large.
• Singlino-like LSP
Since a mostly singlino LSP has small couplings to SM particles, the resulting relic
density is expected to be large. However, there are some annihilation channels
that can be enhanced in order to reduce the relic density. These include the
s-channel (scalar or pseudoscalar) Higgs exchange and the t-channel neutralino
exchange.
For the former, any Higgs with sufficient large singlet component gives an important contribution to the cross section, increasing with the parameter κ (since
the singlino-singlino-singlet coupling is proportional to κ). Concerning the latter
annihilation, in order to enhance it, one needs large values of the parameter λ.
In this case, the neutralino-neutralino-singlet coupling, which is proportional to
λ, is large and the annihilation proceeds giving a pair of scalar HsHs or a pair
of pseudoscalar AsAs singlet like Higgs.
As in the case of bino-like LSP, one can also use the effect of s-channel resonances
or coannihilations. In the latter case, an efficient NLSP can be the neutralino
χ
0
2
or the lightest stau τe1. In the case that the neutralino NLSP is higgsinolike, the LSP-NLSP coannihilation through a (doublet-like) Higgs exchange can
be proved very efficient. A stau NLSP reduces the relic density through NLSPNLSP annihilations, which is the only possibility in the case that both parameters
κ and λ are small. We refer to [156,157] for further discussion on this possibility.
Assuming universality conditions the wino mass M2 has to be larger than the bino
mass M1 (M2 ∼ 2M1). This is the reason that we have not considered a wino-like LSP.
3.5 Detection of neutralino DM
3.5.1 Direct detection
Since neutralinos are Majorana fermions, the effective Lagrangian describing their
elastic scattering with a quark in a nucleon can be written, in the Dirac fermion
notation, as [158]
Leff = a
SI
i χ¯
0
1χ
0
1
q¯iqi + a
SD
i χ¯
0
1γ5γµχ
0
1
q¯iγ5γ
µ
qi
, (3.33)
with i = u, d corresponding to up- and down-type quarks, respectively. The Lagrangian has to be understood as summing over the quark generations.
In this expression, we have omitted terms containing the operator ψγ¯
5ψ or a combination of ψγ¯
5γµψ and ψγ¯
µψ (with ψ = χ, q). This is a well qualified assumption:
Due to the small velocity of WIMPs, the momentum transfer ~q is very small compared
3.5.1 Direct detection 53
to the reduced mass of the WIMP-nucleus system. In the extreme limit of zero momentum transfer, the above operators vanish4
. Hence, we are left with the Lagrangian
(3.33) consisting of two terms, the first one corresponding to spin-independent (SI)
interactions and the second to spin-dependent (SD) ones. In the following, we will
focus again only to SI scattering, since the detector sensitivity to SD scattering is low,
as it has been already mentioned in Sec. 1.5.1.
The SI cross section for the neutralino-nucleus scattering can be written as [158]
(see, also, [159])
σ
SI
tot =
4m2
r
π
[Zfp + (A − Z)fn]
2
. (3.34)
mr is the neutralino-nucleus reduced mass mr =
mχmN
mχ+mN
, and Z, A are the atomic and
the nucleon number, respectively. It is more common, however, to use an expression
for the cross section normalized to the nucleon. In this case, on has for the neutralinoproton scattering
σ
SI
p =
4
π

mpmχ
0
1
mp + mχ
0
1
!2
f
2
p ≃
4m2
χ
0
1
π
f
2
p
, (3.35)
with a similar expression for the neutron.
The form factor fp is related to the couplings a to quarks through the expression
(omitting the “SI” superscripts)
fp
mp
=
X
q=u,d,s
f
p
T q
aq
mq
+
2
27
fT G X
q=c,b,t
aq
mq
. (3.36)
A similar expression may be obtained for the neutron form factor fn, by the replacement
p → n in the previous expression (henceforth, we focus to neutralino-proton scattering).
The parameters fT q are defined by the quark mass matrix elements
hp| mqqq¯ |pi = mpfT q, (3.37)
which corresponds to the contribution of the quark q to the proton mass and the
parameter fT G is related to them by
fT G = 1 −
X
q=u,d,s
fT q. (3.38)
The above parameters can be obtained by the following quantities
σπN =
1
2
(mu + md)(Bu + Bd) and σ0 =
1
2
(mu + md)(Bu + Bd − 2Bs,) (3.39)
with Bq = hN| qq¯ |Ni, which are obtained by chiral perturbation theory [160] or by
lattice simulations. Unfortunately, the uncertainties on the values of these quantities
are large (see [161], for more recent values and error bars).
4While there are possible circumstances in which the operators of (3.33) are also suppressed and,
therefore, comparable to the operators omitted, they are not phenomenologically interesting.
54 The Next-to-Minimal Supersymmetric Standard Model
χ
0
1
χ
0
1
χ
0
1 χ
0
1
qe
q q
q q
Hi
Figure 3.1: Feynman diagrams contributing to the elastic neutralino-quark scalar scattering amplitude at tree level.
The SI neutralino-nucleon interactions arise from t-channel Higgs exchange and
s-channel squark exchange at tree level (see Fig. 3.1), with one-loop contributions from
neutralino-gluon interactions. In practice, the s-channel Higgs exchange contribution
to the scattering amplitude dominates, especially due to the large masses of squarks.
In this case, the effective couplings a are given by
a
SI
d =
X
3
i=1
1
m2
Hi
C
1
i Cχ
0
1χ
0
1Hi
, aSI
u =
X
3
i=1
1
m2
Hi
C
2
i Cχ
0
1χ
0
1Hi
. (3.40)
C
1
i
and C
2
i
are the Higgs Hi couplings to down- and up-type quarks, respectively, given
by
C
1
i =
g2md
2MW cos β
Si1, C2
i =
g2mu
2MW sin β
Si2, (3.41)
with S the mixing matrix of the CP-even Higgs mass eigenstates and md, mu the
corresponding quark mass. We see from Eqs. (3.36) and (3.41) that the final cross
section (3.35) is independent of each quark mass. We write for completeness the
neutralino-neutralino-Higgs coupling Cχ
0
1χ
0
1Hi
:
Cχ
0
1χ
0
1Hi =
√
2λ (Si1N14N15 + Si2N13N15 + Si3N13N14) −
√
2κSi3N
2

+ g1 (Si1N11N13 − Si2N11N14) − g2 (Si1N12N13 − Si2N12N14), (3.42)
with N the neutralino mixing matrix given in (3.29).
The resulting cross section is proportional to m−4
Hi
. In the NMSSM, it is possible
for the lightest scalar Higgs eigenstate to be quite light, evading detection due to its
singlet nature. This scenario can give rise to large values of SI scattering cross section,
provided that the doublet components of th

[V]

Oui, c’est soit le détachement ironique, SOIT une certaine intellectualisation. Concernant Orelsan, on est davantage sur la deuxième modalité (preuve en est des livres, articles ou vidéos qui « analysent » ses textes de manière politique ou littéraire). Booba, par exemple, c’est l’un ou l’autre (selon les morceaux et les auditeurs). Pour caricaturer, certains rigolent sur son côté bourrin, d’autres le comparent à Céline ou Proust. De plus, il ne faut pas oublier qu’il y a des logiques de distinction au sein même des champs culturels non-académiques (rap, jeux vidéos, cinéma de genre…). Ces luttes symboliques de légitimité sont internes et externes, ce qui rend les choses très mouvantes, par-delà les œuvres et ce qu’elles recèlent.

Enfin, je ne sais pas si on peut parler de régression, à moins de tomber dans une lecture décliniste et, surtout, morale de la distinction. Les œuvres n’ont jamais été distinctives en soi, contrairement à ce que l’on y projette. Et ta nuance est bienvenue : dire à une époque que t’aimais Van Gogh, parce que le jaune se trouvait être ta couleur préférée, c’était se distinguer par la négative. Or, dans l’absolu, il s’agissait là d’une réception profane et naïve ni plus ni moins valable qu’une lecture plus érudite ou cérébrale. Rien de nouveau sous le soleil, donc.
En fait, Bourdieu n’entre jamais dans des considérations esthétiques. Son ontologie s’inscrit justement dans ce que Lordon appelle la « condition anarchique ». Tout comme les prolos, les bourgeois ont toujours ignorer les raisons sociales qui guident leurs différentes appréciations en société. Ils mobilisent des jugements de valeurs intrinsèques pour dissimuler des mobiles extrinsèques ; tant est si bien que les hiérarchies n’existent pas a priori.
Maintenant, je rapporte ici une lecture sociologique et non esthétique qui n’est aucunement réductible à son objectivation sociale. La distinction ne rend pas le goût moins sincère, puisqu’il s’agit in fine de l’intérioriser. A ce titre, FB a raison : aimons les choses pour ce qu’elles nous font (ex : sa passion du rock par rapport à la critique de Clouscard).

[Demi Habile]

and also the definition of the unpolarized cross section to write
X
spin
Z
|M12→34|
2
(2π)
4
δ
4
(p1 + p2 − p3 − p4)
d
3p3
(2π)
32E3
d
3p4
(2π)
32E4
=
4F g1g2 σ12→34, (1.31)
where F ≡ [(p1 · p2)
2 − m2
1m2
2
]
1/2
and the spin factors g1, g2 come from the average
over initial spins. This way, the collision term (1.29) is written in a more compact form
g1
Z
C[f1]
d
3p1
(2π)
3
= −
Z
σvMøl (dn1dn2 − dn
eq
1 dn
eq
2
), (1.32)
where σ =
P
(all f)
σ12→f is the total annihilation cross section summed over all the
possible final states and vMøl ≡
F
E1E2
. The so called Møller velocity, vMøl, is defined in
such a way that the product vMøln1n2 is invariant under Lorentz transformations and,
in terms of particle velocities ~v1 and ~v2, it is given by the expression
vMøl =
h
~v2
1 − ~v2
2

2
− |~v1 × ~v2|
2
i1/2
. (1.33)
Due to symmetry considerations, the distributions in kinetic equilibrium are proportional to those in chemical equilibrium, with a proportionality factor independent of
the momentum. Therefore, the collision term (1.32), both before and after decoupling,
can be written in the form
g1
Z
C[f1]
d
3p1
(2π)
3
= −hσvMøli(n1n2 − n
eq
1 n
eq
2
), (1.34)
where the thermal averaged total annihilation cross section times the Møller velocity
has been defined by the expression
hσvMøli =
R
σvMøldn
eq
1 dn
eq
2
R
dn
eq
1 dn
eq
2
. (1.35)
We will come back to the thermal averaged cross section in the next subsection.
We are, now, able to write the full integrated Boltzmann equation, using the expressions (1.28), (1.34) that we have derived for the Liouville and the collision term,
respectively. In the simplified but interesting case of identical particles 1 and 2, the
Boltzmann equation is, finally, written as
n˙ + 3Hn = −hσvMøli(n
2 − n
2
eq). (1.36)
18 Dark Matter
However, instead of using n, it is more convenient to take the expansion of the universe
into account and calculate the number density per comoving volume Y , which can be
defined as the ratio of the number and entropy densities: Y ≡ n/s. The total entropy
density S = R3
s (R is the scale factor) remains constant, hence we can obtain a
differential equation for Y by dividing (1.36) by S. Before we write the final form
of the Boltzmann equation that it is used for the relic density calculations, we have
to change the variable that parametrizes the comoving density. In practice, the time
variable t is not convenient and the temperature of the Universe (actually the photon
temperature, since the photons were the last particles that went out of equilibrium) is
used instead. However, it proves even more useful to use as time variable the quantity
defined by x ≡ m/T with m the DM mass, so that Eq. (1.36) transforms into
dY
dx
=
1
3H
ds
dx
hσvMøli

Y
2 − Y
2
eq
. (1.37)
Last, using the Hubble parameter (1.2) for a radiation dominated Universe and the
expressions (1.20), (1.21) for the energy and entropy density, the Boltzmann equation
is written in its final form
dY
dx
= −
r
45GN
π
g
1/2
∗ m
x
2
hσvMøli

Y
2 − Y
2
eq
, (1.38)
where the effective degrees of freedom g
1/2
∗ have been defined by
g
1/2
∗ ≡
heff
g
1/2
eff

1 +
1
3
T
heff
dheff
dT

. (1.39)
The equilibrium density per comoving volume Yeq ≡ neq/s can be expressed as
Yeq(x) = 45g
4π
4
x
2K2(x)
heff(m/x)
, (1.40)
with K2 the modified Bessel function of second kind.
1.4.3 Thermal average of the annihilation cross section
We are going to derive a simple formula that one can use to calculate the thermal
average of the cross section times velocity, based again on the analysis of [38]. We will
use the assumption that equilibrium functions follow the Maxwell-Boltzmann distribution, instead of the actual Bose-Einstein or Fermi-Dirac. This is a well established
assumption if the freeze out occurs after T ≃ m/3 or for x >∼ 3, which is actually the
case for WIMPs. Under this assumption, the expression (1.35) gives, in the cosmic
comoving frame,
hσvMøli =
R
vMøle
−E1/T e
−E2/T d
3p1d
3p2
R
e
−E1/T e
−E2/T d
3p1d
3p2
. (1.4
1.4.3 Thermal average of the annihilation cross section 19
The volume element can be written as d3p1d
3p2 = 4πp1dE14πp2dE2
1
2
cos θ, with θ the
angle between ~p1 and ~p2. After changing the integration variables to E+, E−, s given
by
E+ = E1 + E2, E− = E1 − E2, s = 2m2 + 2E1E2 − 2p1p2 cos θ, (1.42)
(with s = −(p1 − p2)
2 one of the Mandelstam variables,) the volume element becomes
d
3p1d
3p2 = 2π
2E1E2dE+dE−ds and the initial integration region
{E1 > m, E2 > m, | cos θ| ≤ 1i
transforms into
|E−| ≤
1 −
4m2
s
1/2
(E
2
+ − s)
1/2
, E+ ≥
√
s, s ≥ 4m2
. (1.43)
After some algebraic calculations, it can be found that the quantity hσvMøliE1E2
depends only on s, specifically vMølE1E2 =
1
2
p
s(s − 4m2
). Hence, the numerator of the expression (1.41), which after changing the integration variables reads
2π
2
R
dE+
R
dE−
R
dsσvMølE1E2e
−E+/T , can be written, eventually, as
Z
vMøle
−E1/T e
−E2/T = 2π
2
Z ∞
4m2
dsσ(s − 4m2
)
Z
dE+e
−E+/T (E
2
+ − s)
1/2
. (1.44)
The integral over E+ can be written with the help of the modified Bessel function of
the first kind K1 as √
s T K1(
√
s/T). The denominator of (1.41) can be treated in a
similar way, so that the thermal average is, finally, given by the expression
hσvMøli =
1
8m4TK2
2
(x)
Z ∞
4m2
ds σ(s)(s − 4m2
)
√
s K1(
√
s/T). (1.45)
Eqs. (1.38)–(1.40) along with this last Eq. (1.45) are all we need in order to calculate
the relic density of a WIMP, if its total annihilation cross section in terms of the
Mandelstam variable s is known.
In many cases, in order to avoid the numerical integration in Eq. (1.45), an approximation for hσvMøli can be used. The thermal average is expanded in powers of x
−1
(or, equivalently, in powers of the squared WIMP velocity):
hσvMøli = a + bx−1 + . . . . (1.46)
(The coefficient a corresponds to the s-wave contribution to the cross section, the
coefficient b to the p-wave contribution, and so on.) This partial wave expansion gives
a quite good approximation, provided there are no s-channel resonances and thresholds
for the final states [39].
In [40], it was shown that, after expanding the integrands of Eq. (1.41) in powers
of x
−1
, all the integrations can be performed analytically. As we saw, the expression
20 Dark Matter
vMølE1E2 depends on momenta only through s. Therefore, one can form the Lorentz
invariant quantity
w(s) ≡ σ(s)vMølE1E2 =
1
2
σ(s)
p
s(s − 4m2
). (1.47)
The integration involves the Taylor expansion of this quantity w around s/4m2 = 1
and the general formula for the partial wave expansion of the thermal average is [40]
hσvMøli =
1
m2

w −
3
2
(2w − w
′
)x
−1 +
3
8
(16w − 8w
′ + 5w
′′)x
−2
−
5
16
(30w − 15w
′ + 3w
′′ − 7x
′′′)x
−3 + O(x
−4
)

s/4m2=1
, (1.48)
where primes denote derivatives with respect to s/4m2 and all quantities have to be
evaluated at s = 4m2
.
1.5 Direct Detection of DM
Since the beginning of 1980s, it has been realized that besides the numerous facts showing evidence for the existence of these new dark particles, it is also possible to detect
them directly. Already in 1985, two pioneering articles [41, 42] appeared, describing
the detection methods for WIMPs. Since WIMPs are expected to cluster gravitationally together with ordinary stars in the Milky Way halo, they would pass also through
Earth and, in principle, they can be detected through scattering with the nuclei in a
detector’s material. In practice, one has to measure the recoil energy deposited by this
scattering.
However, although one can deduce from rotation curves that DM dominates the
dark halo in the outer parts of our galaxy, it is not so obvious from direct measurements
whether there is any substantial amount of DM inside the solar radius R0 ≃ 8 kpc.
Using indirect methods (involving the determination of the gravitational potential,
through the measuring of the kinematics of stars, both near the mid-plane of the
galactic disk and at heights several times the disk thickness), it is almost certain
that the DM is also present in the solar system, with a local density ρ0 = (0.3 ±
0.1) GeV cm−3
[43].
This value for the local density implies that for a WIMP mass of order ∼ 100 GeV,
the local number density is n0 ∼ 10−3
cm−3
. It is also expected that the WIMPs
velocity is similar to the velocity with which the Sun orbits around the galactic center
(v0 ≃ 220 km s−1
), since they are both moving under the same gravitational potential.
These two quantities allow to estimate the order of magnitude of the incident flux
of WIMPs on the Earth: J0 = n0v0 ∼ 105
cm−2
s
−1
. This value is manifestly large,
but the very weak interactions of the DM particles with ordinary matter makes their
detection a difficult, although in principle feasible, task. In order to compensate for
the very low WIMP-nucleus scattering cross section, very large detectors are required.
1.5.1 Elastic scattering event rate 21
1.5.1 Elastic scattering event rate
In the following, we will confine ourselves to the elastic scattering with nuclei. Although
inelastic scattering of WIMPs off nuclei in a detector or off orbital electrons producing
an excited state is possible, the event rate of these processes is quite suppressed. In
contrast, during an elastic scattering the nucleus recoils as a whole.
The direct detection experiments measure the number of events per day and per
kilogram of the detector material, as a function of the amount of energy Q deposited
in the detector. This event rate would be given by R = nWIMP nnuclei σv in a simplified
model with WIMPs moving with a constant velocity v. The number density of WIMPs
is nWIMP = ρ0/mX and the number density of nuclei is just the ratio of the detector’s
mass over the nuclear mass mN .
For accurate calculations, one should take into account that the WIMPs move in the
halo not with a uniform velocity, but rather following a velocity distribution f(v). The
Earth’s motion in the solar system should be included into this distribution function.
The scattering cross section σ also depends on the velocity. Actually, the cross section
can be parametrized by a nuclear form factor F(Q) as
dσ =
σ
4m2
r
v
2
F
2
(Q)d|~q|
2
, (1.49)
where |~q|
2 = 2m2
r
v
2
(1 − cos θ) is the momentum transferred during the scattering,
mr =
mXmN
mX+mN
is the reduced mass of the WIMP – nucleus system and θ is the scattering
angle in the center of momentum frame. Therefore, one can write a general expression
for the differential event rate per unit detector mass as
dR =
ρ0
mX
1
mN
σF2
(Q)d|~q|
2
4m2
r
v
2
vf(v)dv. (1.50)
The energy deposited in the detector (transferred to the nucleus through one elastic
scattering) is
Q =
|~q|
2
2mN
=
m2
r
v
2
mN
(1 − cos θ). (1.51)
Therefore, the differential event rate over deposited energy can be written, using the
equations (1.50) and (1.51), as
dR
dQ
=
σρ0
√
πv0mXm2
r
F
2
(Q)T(Q), (1.52)
where, following [37], we have defined the dimensionless quantity T(Q) as
T(Q) ≡
√
π
2
v0
Z ∞
vmin
f(v)
v
dv, (1.53)
with the minimum velocity given by vmin =
qQmN
2m2
r
, obtained by Eq. (1.51). Finally,
the event rate R can be calculated by integrating (1.52) over the energy
R =
Z ∞
ET
dR
dQ
dQ. (1.54)
22 Dark Matter
The integration is performed for energies larger than the threshold energy ET of the
detector, below which it is insensitive to WIMP-nucleus recoils.
Using Eqs. (1.54) and (1.52), one can derive the scattering cross section from the
event rate. The experimental collaborations prefer to give their results already in terms
of the scattering cross section as a function of the WIMP mass. To be more precise,
the WIMP-nucleus total cross section consists of two parts: the spin-dependent (SD)
cross section and the spin-independent (SI) one. The former comes from axial current
couplings, whereas the latter comes from scalar-scalar and vector-vector couplings.
The SD cross section is much suppressed compared to the SI one in the case of heavy
nuclei targets and it vanishes if the nucleus contains an even number of nucleons, since
in this case the total nuclear spin is zero.
We see that two uncertainties enter the above calculation: the exact value of the
local density ρ0 and the exact form of the velocity distribution f(v). To these, one
has to include one more. The cross section σ that appears in the previous expressions
concerns the WIMP-nucleon cross section. The couplings of a WIMP with the various
quarks that constitute the nucleon are not the same and the WIMP-nucleon cross
section depends strongly on the exact quark content of the nucleon. To be more
precise, the largest uncertainty lies on the strange content of the nucleon, but we shall
return to this point when we will calculate the cross section in a specific particle theory,
the Next-to-Minimal Supersymmetric Standard Model, in Sec. 3.5.1.
1.5.2 Experimental status
The situation of the experimental results from direct DM searches is a bit confusing. The null observations in most of the experiments led them to set upper limits
on the WIMP-nucleon cross section. These bounds are quite stringent for the spinindependent cross section7
, especially in the regime of WIMP masses of the order of
100 GeV. However, some collaborations have already reported possible DM signals,
mainly in the low mass regime. The preferred regions of these experiments do not
coincide, while some of them have been already excluded by other experiments. The
present picture, for WIMP masses ranging from 5 to 1000 GeV, is summarized in Fig.
1.5, 1.6.
Fig. 1.5 mainly presents upper bounds coming from XENON100 [44]. XENON100
[46] is an experiment located at the Gran Sasso underground laboratory in Italy. It
contains in total 165 kg of liquid Xenon, with 65 kg acting as target mass and the
rest shielding the detector from background radiation. For these upper limits, 225
live days of data were used. The minimum value for the predicted upper bounds on
the cross section is 2 · 10−45 cm2
for WIMP mass ∼ 55 GeV (at 90% confidence level),
almost one order of magnitude lower than the previously released limits [47] by the
same collaboration, using 100 live days of data.
The stringent upper bounds up-to-date (at least for WIMP mass larger than about
7 GeV) come from the first results of the LUX experiment (see Fig. 1.6), after the first
7For the spin-dependent scattering, the exclusion limits are quite relaxed. Hence, we will focus on
the SI cross sections.
1.5.2 Experimental status 23
Figure 1.5: The XENON100 exclusion limit (thick blue line), along with the expected
sensitivity in green (1σ) and yellow (2σ) band. Other upper bounds are also shown as
well as detection claims. From [44].
85.3 live-days of its operation [45]. LUX [53] is a detector containing liquid Xenon, as
XENON100, but in larger quantity, with total mass 370 kg. Its operation started on
April 2013 with a goal to clearly detect or exclude WIMPs with a spin independent
cross section ∼ 2 · 10−46 cm2
.
In Fig. 1.5, except of the XENON100 bounds and other experimental limits on larger
WIMP-nucleon cross section, some detection claims also appear. These come from
DAMA [48,49], CoGeNT [50] and CRESST-II [51] experiments. The first positive result
came from DAMA [52], back in 2000. Since then, the experiment has accumulated 1.17
ton-yr of data over 13 years of operation. DAMA consists of 250 kg of radio pure NaI
scintillator and looks for the annual modulation of the WIMP flux in order to reduce
the influence of the background.
The annual modulation of the DM flux (see [54] for a recent review) is due to the
Earth’s orbital motion relative to the rotation of the galactic disk. The galactic disk
rotation through an essentially non-rotating DM halo, creates an effective DM wind in
the solar frame. During the earth’s heliocentric orbit, this wind reaches a maximum
when the Earth is moving fastest in the direction of the disk rotation (this happens
in the beginning of June) and a minimum when it is moving fastest in the opposite
direction (beginning of December).
DAMA claims an 8.9σ annual modulation with a minimum flux on May 26±7 days,
consistent with the expectation. Since the detector’s target consists of two different
nuclei and the experiment cannot distinguish between sodium and iodine recoils, there
24 Dark Matter
Figure 1.6: The LUX 90% confidence exclusion limit (blue line) with the 1σ range
(shaded area). The XENON100 upper bound is represented by the red line. The inset
shows also preferred regions by CoGeNT (shaded light red), CDMS II silicon detector
(shaded green), CRESST II (shaded yellow) and DAMA (shaded gray). From [45].
is no model independent way to determine the exact region in the cross section versus
WIMP mass plane to which the observed modulation corresponds. However, one can
assume two cases: one that the WIMP scattering off the sodium nucleus dominates the
recoil energy and the other with the iodine recoils dominating. The former corresponds
[55] to a light WIMP (∼ 10 GeV) and quite large scattering cross section and the latter
to a heavier WIMP (∼ 50 to 100 GeV) with smaller cross section (see Fig. 1.5).
The positive result of DAMA was followed many years later by the ones of CoGeNT
and CRESST-II, and more recently by the silicon detector of CDMS [56] (Fig. 1.7).
The discrepancy of the results raised a lot of debates among the experiments (for
example, [64–67]) and by some the positive results are regarded as controversial. On
the other hand, it also raised an effort to find a physical explanation behind this
inconsistency (see, for example, [68–71]).
1.6 Indirect Methods for DM Detection
The same annihilation processes that determined the DM relic abundance in the early
Universe also occur today in galactic regions where the DM concentration is higher.
This fact rises the possibility of detecting potential WIMP pair annihilations indirectly
through their imprints on the cosmic rays. Therefore, the indirect DM searches aim
at the detection of an excess over the known astrophysical background of charged
particles, photons or neutrinos.
Charged particles – electrons, protons and their antiparticles – may originate from
direct products (pair of SM particles) of WIMP annihilations, after their decay and
1.6 Indirect Methods for DM Detection 25
Figure 1.7: The blue contours represent preferred regions for a possible signal at 68%
and 90% C.L. using the silicon detector of CMDS [56]. The blue dotted line represents
the upper limit obtained by the same analysis and the blue solid line is the combined
limit with the silicon CDMS data set reported in [57]. Other limits also appear:
from the CMDS standard germanium detector (light and dark red dashed line, for
standard [58] and low threshold analysis [59], respectively), EDELWEISS [60] (dashed
orange), XENON10 [61] (dash-dotted green) and XENON100 [44] (long-dash-dotted
green). The filled regions identify possible signal regions associated with data from
CoGeNT [62] (dashed yellow, 90% C.L.), DAMA [49,55] (dotted tan, 99.7% C.L.) and
CRESST-II [51, 63] (dash-dotted pink, 95.45% C.L.) experiments. Taken from [56].
through the process of showering and hadronization. Although the exact shape of the
resulting spectrum would depend on the specific process, it is expected to show a steep
cutoff at the WIMP mass. Once produced in the DM halo, the charged particles have
to travel to the point of detection through the turbulent galactic field, which will cause
diffusion. Apart from that, a lot of processes disturb the propagation of the charged
particles, such as bremsstrahlung, inverse Compton scattering with CMB photons and
many others. Therefore, the uncertainties that enter the propagation of the charged
flux until it reaches the telescope are important (contrary to the case of photons and
neutrinos that propagate almost unperturbed through the galaxy).
As in the case of direct detection, the experimental status of charged particle detection concerning the DM is confusing. After some hints from HEAT [72] and AMS01 [73] (the former a far-infrared telescope in Antarctica, the latter a spectrometer,
prototype for AMS-02 mounted on the International Space Station [74]), the PAMELA
satellite observed [75, 76] a steep increase in the energy spectrum of positron fraction
e
+/(e
+ + e
−)
8
. Later FERMI satellite [77] and AMS-02 [78] confirmed the results up
8The searches for charged particles focus on the antiparticles in order to have a reduced background,
26 Dark Matter
Figure 1.8: A compilation of data of charged cosmic rays, together with plausible but
uncertain astrophysical backgrounds, taken from [79]. Left: Positron flux. Center:
Antiproton flux. Right: Sum of electrons and positrons.
to energies of ∼ 200 GeV. However, the excess of positrons is not followed by an excess
of antiprotons, whose flux seems to coincide with the predicted background [75]. In
Fig. 1.8, three plots summarizing the situation are shown [79].
The observed excess is very difficult to explain in terms of DM [79]. To begin with,
the annihilation cross section required to reproduce the excess is quite large, many
orders of magnitude larger than the thermal cross section. Moreover, an “ordinary”
WIMP with large annihilation cross section giving rise to charged leptons is expected
to give, additionally, a large number of antiprotons, a fact in contradiction with the
observations. Although a lot of work has been done to fit a DM particle to the observed
pattern, it is quite possible that the excesses come from a yet unknown astrophysical
source. We are not going to discuss further this matter, but we end with a comment.
If this excess is due to a source other than DM, then a possible DM positron excess
would be lost under this formidable background.
A last hint for DM came from the detection of highly energetic photons. However,
we will interrupt this discussion, since this signal and a possible explanation is the
subject of Ch. 4. There, we will also see the upper bounds on the annihilation cross
section being set due to the absence of excesses in diffuse γ radiation.
since they are much less abundant than the corresponding particles.
CHAPTER 2
PARTICLE PHYSICS
Since the DM comprises of particles, it should be explained by a general particle physics
theory. We start in the following section by describing the Standard Model (SM) of
particle physics. Although the SM describes so far the fundamental particles and their
interactions quite accurately, it cannot provide a DM candidate. Besides, the SM
suffers from some theoretical problems, which we discuss in Sec. 2.2. We will see that
these problems can be solved if one introduces a new symmetry, the supersymmetry,
which we describe in Sec. 2.3. We finish this chapter by briefly describing in Sec. 2.4 a
supersymmetric extension of the SM with the minimal additional particle content, the
Minimal Supersymmetric Standard Model (MSSM).
2.1 The Standard Model of Particle Physics
The Standard Model (SM) of particle physics1
consists of two well developed theories,
the quantum chromodynamics (QCD) and the electroweak (EW) theory. The former
describes the strong interactions among the quarks, whereas the latter describes the
electroweak interactions (the weak and electromagnetic interactions in a unified context) between fermions. The EW theory took its final form in the late 1960s by the
introduction by S. Weinberg [85] and A. Salam [86] of the Higgs mechanism that gives
masses to the SM particles, which followed the unification of electromagnetic and weak
interactions [87,88]. At the same time, the EW model preserves the gauge invariance,
making the theory renormalizable, as shown later by ’t Hooft [89]. On the other hand,
QCD obtained its final form some years later, after the confirmation of the existence
of quarks. Of course, the history of the SM is much longer and it can be traced back to
1920s with the formulation of a theoretical basis for a Quantum Field Theory (QFT).
Since then, the SM had many successes. The SM particle content was completed with
the discovery of the heaviest of the quarks, the top quark [90,91], in 1995 and, recently,
with the discovery of the Higgs boson [92, 93].
1There are many good textbooks on the SM and Quantum Field Theory, e.g. [80–84].
28 Particle Physics
The key concept within the SM, as in every QFT, is that of symmetries. Each
interaction respects a gauge symmetry, based on a Lie algebra. The strong interaction is
described by an SU(3)c symmetry, where the subscript c stands for color, the conserved
charge of strong interactions. The EW interactions, on the other hand, are based on
a SU(2)L × U(1)Y Lie algebra. Here, as we will subsequently see, L refers to the
left-handed fermions and Y is the hypercharge, the conserved charge under the U(1).
SU(2)L conserves a quantity known as weak isospin I. Therefore, the SM contains the
internal symmetries of the unitary product group
SU(2)L × U(1)Y × SU(3)c. (2.1)
2.1.1 The particle content of the SM
We mention for completeness that particles are divided into two main classes according
to the statistics they follow. The bosons are particles with integer spin and follow the
Bose-Einstein distribution, whereas fermions have half-integer spin and follow the
Dirac-Einstein statistics, obeying the Pauli exclusion principle. In the SM, all the
fermions have spin 1/2, whereas the bosons have spin 1 with only exception the Higgs
boson, which is a scalar (spin zero). We begin the description of the SM particles with
the fermions.
Each fermion is classified in irreducible representations of each individual Lie algebra, according to the conserved quantum numbers, i.e. the color C, the weak isospin
I and the hypercharge Y . A first classification of fermions can be done into leptons
and quarks, which transform differently under the SU(3)c. Leptons are singlets under
this transformation, while quarks act as triplets (the fundamental representation of
this group). The EW interactions violate maximally the parity symmetry and SU(2)L
acts only on states with negative chirality (left-handed). A Dirac spinor Ψ can be
decomposed into left and right chirality components using, respectively, the projection
operators PL =
1
2
(1 − γ5) and PR =
1
2
(1 + γ5):
ΨL = PLΨ and ΨR = PRΨ. (2.2)
Left-handed fermions have I = 1/2, with a third component of the isospin I3 = ±1/2.
Fermions with positive I3 are called up-type fermions and those with negative are
called down-type. These behave the same way under SU(2)L and form doublets with
one fermion of each type. On the other hand, right-handed fermions have I = 0 and
form singlets that do not undergo weak interactions. The hypercharge is written in
terms of the electric charge Q and the third component of the isospin I3 through the
Gell-Mann–Nishijima relation:
Q = I3 + Y/2. (2.3)
Therefore, left- and right-handed components transform differently under the U(1)Y ,
since they have different hypercharge.
The fermionic sector of the SM comprises three generations of fermions, transforming as spinors under Lorentz transformations. Each generation has the same structure.
For leptons, it is an SU(2)L doublet with components consisting of one left-handed
2.1.2 The SM Lagrangian 29
charged lepton and one neutrino (neutrinos are only left-handed in the SM), along
with a gauge singlet right-handed charged lepton. The quark doublet consists of an
up- (u) and a down-type (d) (left-handed) quark and the pattern is completed by the
two corresponding SU(2)L singlet right-handed quarks. We write these representations
as
Quarks: Q ≡

u
i
L
d
i
L
!
, ui
R, di
R Leptons: L ≡

ν
i
L
e
i
L
!
, ei
R, (2.4)
with i = 1, 2, 3 the generation index.
Having briefly described the fermionic sector, we turn to the bosonic sector of
the SM. It consists of the gauge bosons that mediate the interactions and the Higgs
boson that gives masses to the particles through a spontaneous symmetry breaking,
the electroweak symmetry breaking (EWSB) [94–98], which we shall describe in Sec.
2.1.3. Before the EWSB, these bosons are
• three Wa
µ
(a = 1, 2, 3) weak bosons, associated with the generators of SU(2)L,
• one neutral Bµ boson, associated with the generator of U(1)Y ,
• eight gluons Ga
µ
(a = 1, . . . , 8), associated with the generators of SU(3)c, and
• the complex scalar Higgs doublet Φ =
φ
+
φ
0
!
.
After the EWSB, the EW boson states mix and give the two W± bosons, the neutral
Z boson and the massless photon γ. From the symmetry breaking, one scalar degree of
freedom remains which is the famous (neutral) Higgs boson [97–99]. We will return to
the mixed physical states, after describing the Higgs mechanism for symmetry breaking.
A complete list of the SM particles (the physical states after EWSB) is shown in Table
2.1.
2.1.2 The SM Lagrangian
The gauge bosons are responsible for the mediation of the interactions and are associated with the generators of the corresponding symmetry. The EW gauge bosons Bµ
and Wa
µ
are associated, respectively, with the generator Y of the U(1)Y and the three
generators T
a
2
of the SU(2)L. The latter are defined as half of the Pauli matrices τ
a
(T
a
2 =
1
2
τ
a
) and they obey the algebra

T
a
2
, Tb
2

= iǫabcT
c
2
, (2.5)
where ǫ
abc is the fully antisymmetric Levi-Civita tensor. The eight gluons are associated
with an equal number of generators T
a
3
(Gell-Mann matrices) of SU(3)c and obey the
Lie algebra

T
a
3
, Tb
3

= if abcT
c
3
, with Tr
T
a
3 T
b
3

=
1
2
δ
ab
, (2.6)
30 Particle Physics
Name symbol mass charge (|e|) spin
Leptons
electron e 0.511 MeV −1 1/2
electron neutrino νe 0 (<2 eV) 0 1/2
muon µ 105.7 MeV −1 1/2
muon neutrino νµ 0 (<2 eV) 0 1/2
tau τ 1.777 GeV −1 1/2
tau neutrino ντ 0 (<2 eV) 0 1/2
Quarks
up u 2.7
+0.7
−0.5 MeV 2/3 1/2
down d 4.8
+0.7
−0.3 MeV −1/3 1/2
strange s (95 ± 5) MeV −1/3 1/2
charm c (1.275 ± 0.025) GeV 2/3 1/2
bottom b (4.18 ± 0.03) GeV −1/3 1/2
top t (173.5 ± 0.6 ± 0.8) GeV 2/3 1/2
Bosons
photon γ 0 (<10−18 eV) 0 (<10−35) 1
W boson W± (80.385 ± 0.015) GeV ±1 1
Z boson Z (91.1876 ± 0.0021) GeV 0 1
gluon g 0 (.O(1) MeV) 0 1
Higgs H
(125.3 ± 0.4 ± 0.5) GeV
0 0
(126.0 ± 0.4 ± 0.4) GeV
Table 2.1: The particle content of the SM. All values are those given in [100], except of
the Higgs mass that is taken from [92, 93] (up and down row, respectively), assuming
that the observed excess corresponds to the SM Higgs. The u, d and s quark masses
are estimates of so-called “current-quark masses” in a mass-independent subtraction
scheme as MS at a scale ∼ 2 GeV. The c and b quark masses are the running masses
in the MS scheme. The values in the parenthesis are the current experimental limits.
with f
abc the structure constants of the group.
Using the structure constants of the corresponding groups, we define the field
strengths for the gauge bosons as
Bµν ≡ ∂µBν − ∂νBµ, (2.7a)
Wµν ≡ ∂µWa
ν − ∂νWa
µ + g2ǫ
abcWb
µWc
ν
(2.7b)
and
G
a
µν ≡ ∂µG
a
ν − ∂νG
a
µ + g3f
abcG
b
µG
c
ν
. (2.7c)
2.1.2 The SM Lagrangian 31
We use the notation g1, g2 and g3 for the coupling constants of U(1)Y , SU(2)L and
SU(3)c, respectively. As in any Yang-Mills theory, the non-abelian gauge groups lead
to self-interactions, which is not the case for the abelian U(1)Y group.
Before we finally write the full Lagrangian, we have to introduce the covariant
derivative for fermions, which in a general form can be written as
DµΨ =
∂µ − ig1
1
2
Y Bµ − ig2T
a
2 Wa
µ − ig3T
a
3 G
a
µ

Ψ. (2.8)
This form has to be understood as that, depending on Ψ, only the relevant terms
apply, hence for SU(2)L singlet leptons only the two first terms inside the parenthesis
are relevant, for doublet leptons the three first terms and for the corresponding quark
singlets and doublets the last term also participates. We also have to notice that in
order to retain the gauge symmetry, mass terms are forbidden in the Lagrangian. For
example, the mass term mψψ¯ = m

ψ¯
LψR + ψ¯
RψL

(with ψ¯ ≡ ψ
†γ
0
) is not invariant
under SU(2)L. This paradox is solved by the introduction of the Higgs scalar field
(see next subsection). The SM Lagrangian can be now written2
, split for simplicity in
three parts, each describing the gauge bosons, the fermions and the scalar sector,
LSM = Lgauge + Lfermion + Lscalar, (2.9)
with
Lgauge = −
1
4
G
a
µνG
µν
a −
1
4
Wa
µνWµν
a −
1
4
BµνB
µν
, (2.10a)
Lfermion = iL¯Dµγ
µL + ie¯RDµγµeR
+ iQ¯Dµγ
µQ + iu¯RDµγ
µuR + i
¯dRDµγ
µ
dR
−

heL¯ΦeR + hdQ¯ΦdR + huQ¯ΦeuR + h.c.

(2.10b)
and
Lscalar = (DµΦ)†
(DµΦ) − V (Φ†Φ), (2.10c)
where
V (Φ†Φ) = µ
2Φ
†Φ + λ

Φ
†Φ
2
(2.11)
is the scalar Higgs potential. Φ is the conjugate of Φ, related to the charge conjugate e
by Φ =e iτ2Φ
⋆
, with τi the Pauli matrices. The covariant derivative acting on the Higgs
scalar field gives
DµΦ =
∂µ − ig1
1
2
Y Bµ − ig2T
a
2 Wa
µ

Φ. (2.12)
Before we proceed to the description of the Higgs mechanism, a last comment concerning the SM Lagrangian is in order. If we restore the generation indices, we see that
2For simplicity, from now on we are going to omit the generations indice
32 Particle Physics
the Yukawa couplings h are 3 × 3, in general complex, matrices. As any complex matrix, they can be diagonalized with the help of two unitary matrices VL and VR, which
are related by VR = U
†VL with U again a unitary matrix. The diagonalization in the
quark sector to the mass eigenstates induces a mixing among the flavors (generations),
described by the Cabibbo–Kobayashi–Maskawa (CKM) matrix [101, 102]. The CKM
matrix is defined by
VCKM ≡ V
u
L
†
V
d
L
†
, (2.13)
where V
u
L
, V
d
L
are the unitary matrices that diagonalize the Yukawa couplings Hu
, Hd
,
respectively. This product of the two matrices appears in the charged current when it
is expressed in terms of the observable mass eigenstates.
2.1.3 Mass generation through the Higgs mechanism
We will start by examining the scalar potential (2.11). The vacuum expectation value
(vev) of the Higgs field hΦi ≡ h0|Φ|0i is given by the minimum of the potential. For
µ
2 > 0, the potential is always non-negative and Φ has a zero vev. The hypothesis of
the Higgs mechanism is that µ
2 < 0. In this case, the field Φ will acquire a vev
hΦi =
1
2

0
v
!
with v =
r
−
µ2
λ
. (2.14)
Since the charged component of Φ still has a zero vev, the U(1)Q symmetry of quantum
electrodynamics (QED) remains unbroken.
We expand the field Φ around the minima v in terms of real fields, and at leading
order we have
Φ(x) =
θ2(x) + iθ1(x)
√
1
2
(v + H(x)) − iθ3(x)
!
=
1
√
2
e
iθa(x)τ
a

0
v + H(x)
!
. (2.15)
We can eliminate the unphysical degrees of freedom θa, using the fact that the theory
remains gauge invariant. Therefore, we perform the following SU(2)L gauge transformation on Φ (unitary gauge)
Φ(x) → e
−iθa(x)τ
a
Φ(x), (2.16)
so that
Φ(x) = 1
√
2

0
v + H(x)
!
. (2.17)
We are going to use the following definitions for the gauge fields
W±
µ ≡
1
2

W1
µ ∓ iW2
µ

, (2.18a)
Zµ ≡
1
p
g
2
1 + g
2
2

g2W3
µ − g1Bµ

, (2.18b)
Aµ ≡
1
p
g
2
1 + g
2
2

g1W3
µ + g2Bµ

, (2.1
2.2 Limits of the SM and the emergence of supersymmetry 33
Then, the kinetic term for Φ (see Eq. (2.10c)) can be written in the unitary gauge as
(DµΦ)†
(D
µΦ) = 1
2
(∂µH)
2 + M2
W W+
µ W−µ +
1
2
M2
ZZµZ
µ
, (2.19)
with
MW ≡
1
2
g2v and MZ ≡
1
2
q
g
2
1 + g
2
2
v. (2.20)
We see that the definitions (2.18) correspond to the physical states of the gauge bosons
that have acquired masses due to the non-zero Higgs vev, given by (2.20). The photon
has remained massless, which reflects the fact that after the spontaneous breakdown of
SU(2)L × U(1)Y the U(1)Q remained unbroken. Among the initial degrees of freedom
of the complex scalar field Φ, three were absorbed by W± and Z and one remained as
the neutral Higgs particle with squared mass
m2
H = 2λv2
. (2.21)
We note that λ should be positive so that the scalar potential (2.11) is bounded from
below.
Fermions also acquire masses due to the Higgs mechanism. The Yukawa terms in
the fermionic part (2.10b) of the SM Lagrangian are written, after expanding around
the vev in the unitary gauge,
LY = −
1
√
2
hee¯L(v + H)eR −
1
√
2
hd
¯dL(v + H)dR −
1
√
2
huu¯L(v + H)uR + h.c. . (2.22)
Therefore, we can identify the masses of the fermions as
me
i =
h
i
e
v
√
2
, md
i =
h
i
d
v
√
2
, mui =
h
i
u
v
√
2
, (2.23)
where we have written explicitly the generation indices.
2.2 Limits of the SM and the emergence of supersymmetry
2.2.1 General discussion of the SM problems
The SM has been proven extremely successful and has been tested in high precision
in many different experiments. It has predicted many new particles before their final
discovery and also explained how the particles gain their masses. Its last triumph was
of course the discovery of a boson that seems to be very similar to the Higgs boson of
the SM. However, it is generally accepted that the SM cannot be the ultimate theory. It
is not only observed phenomena that the SM does not explain; SM also faces important
theoretical issues.
The most prominent among the inconsistencies of the SM with observations is the
oscillations among neutrinos of different generations. In order for the oscillations to
34 Particle Physics
φ φ
k
Figure 2.1: The scalar one-loop diagram giving rise to quadratic divergences.
occur, neutrinos should have non-zero masses. However, minimal modifications of the
SM are able to fit with the data of neutrino physics. Another issue that a more complete theory has to face is the matter asymmetry, the observed dominance of matter
over antimatter in the Universe. In addition, in order to comply with the standard
cosmological model, it has to provide the appropriate particle(s) that drove the inflation. Last, but not least, we saw that in order to explain the DM that dominates the
Universe, a massive, stable weakly interacting particle must exist. Such a particle is
not present in the SM.
On the other hand, the SM also suffers from a theoretical perspective. For example,
the SM counts 19 free parameters; one expects that a fundamental theory would have
a much smaller number of free parameters. Simple modifications of the SM have been
proposed relating some of these parameters. Grand unified theories (GUTs) unify
the gauge couplings at a high scale ∼ 1016 GeV. However, this unification is only
approximate unless the GUT is embedded in a supersymmetric framework. Another
serious problem of the SM is that of naturalness. This will be the topic of the following
subsection.
2.2.2 The naturalness problem of the SM
The presence of fundamental scalar fields, like the Higgs, gives rise to quadratic divergences. The diagram of Fig. 2.1 contributes to the squared mass of the scalar
δm2 = λ
Z Λ
d
4k
(2π)
4
k
−2
. (2.24)
This contribution is approximated by δm2 ∼ λΛ
2/(16π
2
), quadratic in a cut-off Λ,
which should be finite. For the case of the Higgs scalar field, one has to include its
couplings to the gauge fields and the top quark3
. Therefore,
δm2
H =
3Λ2
8π
2v
2

4m2
t − 2M2
W − M2
Z − m2
H

+ O(ln Λ
µ
)

, (2.25)
where we have used Eq. (2.21) and m2
H ≡ m2
0 + δm2
H.
3Since the contribution to the squared mass correction are quadratic in the Yukawa couplings (or
quark masses), the lighter quarks can be neglected
2.2.3 A way out 35
Taking Λ as a fundamental scale Λ ∼ MP l ∼ 1019 GeV we have
m2
0 = m2
H −
3Λ2
8π
2v
2

4m2
t − 2M2
W − M2
Z − m2
H

(2.26)
and we can see that m2
0 has to be adjusted to a precision of about 30 orders of magnitude
in order to achieve an EW scale Higgs mass. This is considered as an intolerable finetuning, which is against the general belief that the observable properties of a theory
have to be stable under small variations of the fundamental (bare) parameters. It is
exactly the above behavior that is considered as unnatural. Although the SM could
be self-consistent without imposing a large scale, grand unification of the parameters
introduce a hierarchy problem between the different scales.
A more strict definition of naturalness comes from ’t Hooft [103], which we rewrite
here:
At an energy scale µ, a physical parameter or set of physical parameters
αi(µ) is allowed to be very small only if the replacement αi(µ) = 0 would
increase the symmetry of the system.
Clearly, this is not the case here. Although mH is small compared to the fundamental
scale Λ, it is not protected by any symmetry and a fine-tuning is necessary.
2.2.3 A way out
The naturalness in the ’t Hooft sense is inspired by quantum electrodynamics, which is
the archetype for a natural theory. For example, the corrections to the electron mass
me are themselves proportional to me, with a dimensionless proportionality factor that
behaves like ∼ ln Λ. In general, fermion masses are protected by the chiral symmetry; small values (compared to the fundamental scale) of these masses enhances the
symmetry.
If a new symmetry exists in nature, relating fermion fields to scalar fields, then each
scalar mass would be related somehow to the corresponding fermion mass. Therefore,
the scalar mass itself can be naturally small compared to Λ, since this would mean
that the fermion mass is small, which enhances the chiral symmetry. Such a symmetry,
relating bosons to fermions and vice versa, is known as supersymmetry [104, 105].
Actually, as we will see later, if this new symmetry remains unbroken, the masses of
the conjugate bosons and fermions would have to be equal.
In order to make the above statement more concrete, we consider a toy model with
two additional complex scalar fields feL and feR. We will discuss only the quadratic
divergences that come from corrections to the Higgs mass due to a fermion. The
generalization for the contributions from the gauge bosons or the self-interaction is
straightforward. The interactions in this toy model of the new scalar fields with the
Higgs are described by the Lagrangian
Lfefφe = λfe|φ|
2

|feL|
2 + |feR|
2

. (2.27
36 Particle Physics
It can be easily checked that the quadratic divergence coming from a fermion at one
loop is exactly canceled, as long as the new quartic coupling λfe obeys the relation
λfe = −λ
2
f
(λf is the Yukawa coupling for the fermion f).
2.3 A brief summary of Supersymmetry
Supersymmetry (SUSY) is a symmetry relating fermions and bosons. The supersymmetry transformation should turn a boson state into a fermion state and vice versa. If
Q is the operator that generates such transformations, then
Q |bosoni = |fermioni Q |fermioni = |bosoni. (2.28)
Due to commutation and anticommutation rules of bosons and fermions, Q has to
be an anticommuting spinor operator, carrying spin angular momentum 1/2. Since
spinors are complex objects, the hermitian conjugate Q†
is also a symmetry operator4
.
There is a no-go theorem, the Coleman-Mandula theorem [106], that restricts the
conserved charges which transform as tensors under the Lorentz group to the generators
of translations Pµ and the generators of Lorentz transformations Mµν. Although this
theorem can be evaded in the case of supersymmetry due to the anticommutation
properties of Q, Q†
[107], it restricts the underlying algebra of supersymmetry [108].
Therefore, the basic supersymmetric algebra can be written as5
{Q, Q†
} = P
µ
, (2.29a)
{Q, Q} = {Q
†
, Q†
} = 0, (2.29b)
[P
µ
, Q] = [P
µ
, Q] = 0. (2.29c)
In the following, we summarize the basic conclusions derived from this algebra.
• The single-particle states of a supersymmetric theory fall into irreducible representations of the SUSY algebra, called supermultiplets. A supermultiplet contains
both fermion and boson states, called superpartners.
• Superpartners must have equal masses: Consider |Ωi and |Ω
′
i as the superpartners, |Ω
′
i should be proportional to some combination of the Q and Q† operators
acting on |Ωi, up to a space-time translation or rotation. Since −P
2
commutes
with Q, Q† and all space-time translation and rotation operators, |Ωi, |Ω
′
i will
have equal eigenvalues of −P
2 and thus equal masses.
• Superpartners must be in the same representation of gauge groups, since Q, Q†
commute with the generators of gauge transformations. This means that they
have equal charges, weak isospin and color degrees of freedom.
4We will confine ourselves to the phenomenologically more interesting case of N = 1 supersymmetry, with N referring to the number of distinct copies of Q, Q†
.
5We present a simplified version, omitting spinor indices in Q and Q†
.
2.3 A brief summary of Supersymmetry 37
• Each supermultiplet contains an equal number of fermion and boson degrees of
freedom (nF and nB, respectively): Consider the operator (−1)2s
, with s the spin
angular momentum, and the states |ii that have the same eigenvalue p
µ of P
µ
.
Then, using the SUSY algebra (2.29) and the completeness relation P
i
|ii hi| =
1, we have P
i
hi|(−1)2sP
µ
|ii = 0. On the other hand, P
i
hi|(−1)2sP
µ
|ii =
p
µTr [(−1)2s
] ∝ nB − nF . Therefore, nF = nB.
As addendum to the last point, we see that two kind of supermultiplets are possible
(neglecting gravity):
• A chiral (or matter or scalar ) supermultiplet, which consists of a single Weyl
fermion (with two spin helicity states, nF = 2) and two real scalars (each with
nB = 1), which can be replaced by a single complex scalar field.
• A gauge (or vector ) supermultiplet, which consists of a massless spin 1 boson
(two helicity states, nB = 2) and a massless spin 1/2 fermion (nF = 2).
Other combinations either are reduced to combinations of the above supermultiplets
or lead to non-renormalizable interactions.
It is possible to study supersymmetry in a geometric approach, using a space-time
manifold extended by four fermionic (Grassmann) coordinates. This manifold is called
superspace. The fields, in turn, expressed in terms of the extended set of coordinates
are called superfields. We are not going to discuss the technical details of this topic
(the interested reader may refer to the rich bibliography, for example [109–111]).
However, it is important to mention a very useful function of the superfields, the
superpotential. A generic form of a (renormalizable) superpotential in terms of the
superfields Φ is the following b
W =
1
2
MijΦbiΦbj +
1
6
y
ijkΦbiΦbjΦbk. (2.30)
The Lagrangian density can always be written according to the superpotential. The
superpotential has also to fulfill some requirements. In order for the Lagrangian to
be supersymmetric invariant, W has to be holomorphic in the complex scalar fields
(it does not involve hermitian conjugates Φb† of the superfields). Conventionally, W
involves only left chiral superfields. Instead of the SU(2)L singlet right chiral fermion
fields, one can use their left chiral charge conjugates.
As we mentioned before, the members of a supermultiplet have equal masses. This
contradicts our experience, since the partners of the light SM particles would have been
detected long time ago. Hence, the supersymmetry should be broken at a large energy
scale. The common approach is that SUSY is broken in a hidden sector, very weakly
coupled to the visible sector. Then, one has to explain how the SUSY breaking mediated to the visible sector. The two most popular scenarios are the gravity mediation
scenario [112–114] and the Gauge-Mediated SUSY Breaking (GSMB) [113, 115–117],
where the mediation occurs through gauge interactions.
There are two approaches with which one can address the SUSY breaking. In the
first approach, one refers to a GUT unification and determines the supersymmetric
38 Particle Physics
breaking parameters at low energies through the renormalization group equations.
This approach results in a small number of free parameters. In the second approach,
the starting point is the low energy scale. In this case, the SUSY breaking has to be
parametrized by the addition of breaking terms to the low energy Lagrangian. This
results in a larger set of free parameters. These terms should not reintroduce quadratic
divergences to the scalar masses, since the cancellation of these divergences was the
main motivation for SUSY. Then, one talks about soft breaking terms.
2.4 The Minimal Supersymmetric Standard Model
One can construct a supersymmetric version of the standard model with a minimal
content of particles. This model is known as the Minimal Supersymmetric Standard
Model (MSSM). In a SUSY extension of the SM, each of the SM particles is either in a
chiral or in a gauge supermultiplet, and should have a superpartner with spin differing
by 1/2.
The spin-0 partners of quarks and leptons are called squarks and sleptons, respectively (or collectively sfermions), and they have to reside in chiral supermultiplets.
The left- and right-handed components of fermions are distinct 2-component Weyl
fermions with different gauge transformations in the SM, so that each must have its
own complex scalar superpartner. The gauge bosons of the SM reside in gauge supermultiplets, along with their spin-1/2 superpartners, which are called gauginos. Every
gaugino field, like its gauge boson partner, transforms as the adjoint representation of
the corresponding gauge group. They have left- and right-handed components which
are charge conjugates of each other: (λeL)
c = λeR.
The Higgs boson, since it is a spin-0 particle, should reside in a chiral supermultiplet. However, we saw (in the fermionic part of the SM Lagrangian, Eq. (2.10b))
that the Y = 1/2 Higgs in the SM can give mass to both up- and down-type quarks,
only if the conjugate Higgs field with Y = −1/2 is involved. Since in the superpotential there are no conjugate fields, two Higgs doublets have to be introduced. Each
Higgs supermultiplet would have hypercharge Y = +1/2 or Y = −1/2. The Higgs
with the negative hypercharge gives mass to the down-type fermions and it is called
down-type Higgs (Hd, or H1 in the SLHA convention [118]) and the other one gives
mass to up-type fermions and it is called up-type Higgs (Hu, or H2).
The MSSM respects a discrete Z2 symmetry, the R-parity. If one writes the most
general terms in the supersymmetric Lagrangian (still gauge-invariant and holomorphic), some of them would lead to non-observed processes. The most obvious constraint
comes from the non-observed proton decay, which arises from a term that violates both
lepton and baryon numbers (L and B, respectively) by one unit. In order to avoid these
terms, R-parity, a multiplicative conserved quantum number, is introduced, defined as
PR = (−1)3(B−L)+2s
, (2.31)
with s the spin of the particle.
The R even particles are the SM particles, whereas the R odd are the new particles
introduced by the MSSM and are called supersymmetric particles. Due to R-parity,
2.4 The Minimal Supersymmetric Standard Model 39
if it is exactly conserved, there can be no mixing among odd and even particles and,
additionally, each interaction vertex in the theory can only involve an even number of
supersymmetric particles. The phenomenological consequences are quite important.
First, the lightest among the odd-parity particles is stable. This particle is known
as the lightest supersymmetric particle (LSP). Second, in collider experiments, supersymmetric particles can only be produced in pairs. The first of these consequences
was a breakthrough for the incorporation of DM into a general theory. If the LSP is
electrically neutral, it interacts only weakly and it consists an attractive candidate for
DM.
We are not going to enter further into the details of the MSSM6
. Although MSSM
offers a possible DM candidate, there is a strong theoretical reason to move from the
minimal model. This reason is the so-called µ-problem of the MSSM, with which we
begin the discussion of the next chapter, where we shall describe more thoroughly the
Next-to-Minimal Supersymmetric Standard Model.
6We refer to [110] for an excellent and detailed description of MSSM.
40 Particle Physics
Part II
Dark Matter in the
Next-to-Minimal Supersymmetric
Standard Model

CHAPTER 3
THE NEXT-TO-MINIMAL
SUPERSYMMETRIC STANDARD
MODEL
The Next-to-Minimal Supersymmetric Standard Model (NMSSM) is an extension of
the MSSM by a chiral, SU(2)L singlet superfield Sb (see [119, 120] for reviews). The
introduction of this field solves the µ-problem1
from which the MSSM suffers, but
also leads to a different phenomenology from that of the minimal model. The scalar
component of the additional field mixes with the scalar Higgs doublets, leading to three
CP-even mass eigenstates and two CP-odd eigenstates (as in the MSSM a doublet-like
pair of charged Higgs also exists). On the other hand, the fermionic component of the
singlet (singlino) mixes with gauginos and higgsinos, forming five neutral states, the
neutralinos.
Concerning the CP-even sector, a new possibility opens. The lightest Higgs mass
eigenstate may have evaded the detection due to a sizeable singlet component. Besides,
the SM-like Higgs is naturally heavier than in the MSSM [123–126]. Therefore, a SMlike Higgs mass ∼ 125 GeV is much easier to explain [127–141]. The singlet component
of the CP-odd Higgs also allows for a potentially very light pseudoscalar with suppressed couplings to SM particles, with various consequences, especially on low energy
observables (for example, [142–145]). The singlino component of the neutralino may
also play an important role for both collider phenomenology and DM. This is the case
when the neutralino is the LSP and the lightest neutralino has a significant singlino
component.
We start the discussion about the NMSSM by describing the µ-problem and how
this is solved in the context of the NMSSM. In Sec. 3.2 we introduce the NMSSM
Lagrangian and we write the mass matrices of the Higgs sector particles and the su1However, historically, the introduction of a singlet field preceded the µ-problem, e.g. [104, 105,
121, 122].
44 The Next-to-Minimal Supersymmetric Standard Model
persymmetric particles, at tree level. We continue by examining, in Sec. 3.3, the DM
candidates in the NMSSM and particularly the neutralino. The processes which determine the neutralino relic density are described in Sec. 3.4. The detection possibilities
of a potential NMSSM neutralino as DM are discussed in (Sec. 3.5). We close this
chapter (Sec. 3.6) by examining possible ways to include non-zero neutrino masses and
the additional DM candidates that are introduced.
3.1 Motivation – The µ-problem of the MSSM
As we saw, the minimal extension of the SM, the MSSM, contains two Higgs SU(2)L
doublets Hu and Hd. The Lagrangian of the MSSM should contain a supersymmetric
mass term, µHuHd, for these two doublets. There are several reasons, which we will
subsequently review, that require the existence of such a term. On the other hand,
the fact that |µ| cannot be very large, actually it should be of the order of the EW
scale, brings back the problem of naturalness. A parameter of the model should be
much smaller than the “natural” scale (the GUT or the Planck scale) before the EW
symmetry breaking. This leads to the so-called µ-problem of the MSSM [146].
The reasons that such a term should exist in the Lagrangian of the MSSM are
mainly phenomenological. The doublets Hu and Hd are components of chiral superfields that also contain fermionic SU(2)L doublets. Their electrically charged components mix with the superpartners of the W± bosons, forming two charged Dirac
fermions, the charginos. The unsuccessful searches for charginos in LEP have excluded
charginos with masses almost up to its kinetic limit (∼ 104 GeV) [147]. Since the µ term
determines the mass of the charginos, µ cannot be zero and actually |µ| >∼ 100 GeV,
independently of the other free parameters of the model. Moreover, µ = 0 would result
in a Peccei-Quinn symmetry of the Higgs sector and an undesirable massless axion.
Finally, there is one more reason for µ 6= 0 related to the mass generation by the Higgs
mechanism. The term µHuHd will be accompanied by a soft SUSY breaking term
BµHuHd. This term is necessary so that both neutral components of Hu and Hd are
non-vanishing at the minimum of the potential.
The Higgs mechanism also requires that µ is not too large. In order to generate
the EW symmetry breaking, the Higgs potential has to be unstable at its origin Hu =
Hd = 0. Soft SUSY breaking terms for Hu and Hd of the order of the SUSY breaking
scale generate such an instability. However, the µ induced squared masses for Hu,
Hd are always positive and would destroy the instability in case they dominate the
negative soft mass terms.
The NMSSM is able to solve the µ-problem by dynamically generating the mass
µ. This is achieved by the introduction of an SU(2)L singlet scalar field S. When S
acquires a vev, a mass term for the Hu and Hd emerges with an effective mass µeff of
the correct order, as long as the vev is of the order of the SUSY breaking scale. This
can be obtained in a more “natural” way through the soft SUSY breaking terms.
3.2 The NMSSM Lagrangian 45
3.2 The NMSSM Lagrangian
All the necessary information for the Lagrangian of the NMSSM can be extracted from
the superpotential and the soft SUSY breaking Lagrangian, containing the soft gaugino and scalar masses, and the trilinear couplings. We begin with the superpotential,
writing all the interactions of the NMSSM superfields, which include the MSSM superfields and the additional gauge singlet chiral superfield2 Sb. Hence, the superpotential
reads
W = λSbHbu · Hbd +
1
3
κSb3
+ huQb · HbuUbc
R + hdHbd · QbDbc
R + heHbd · LbEbc
R.
(3.1)
The couplings to quarks and leptons have to be understood as 3 × 3 matrices and the
quark and lepton fields as vectors in the flavor space. The SU(2)L doublet superfields
are given (as in the MSSM) by
Qb =

UbL
DbL
!
, Lb =

νb
EbL
!
, Hbu =

Hb +
u
Hb0
u
!
, Hbd =

Hb0
d
Hb −
d
!
(3.2)
and the product of two doublets is given by, for example, Qb · Hbu = UbLHb0
u − Hb +
u DbL.
An important fact to note is that the superpotential given by (3.1) does not include all possible renormalizable couplings (which respect R-parity). The most general
superpotential would also include the terms
W ⊃ µHbu · Hbd +
1
2
µ
′Sb2 + ξF s, b (3.3)
with the first two terms corresponding to supersymmetric masses and the third one,
with ξF of dimension mass2
, to a tadpole term. However, the above dimensionful
parameters µ, µ
′ and ξF should be of the order of the SUSY breaking scale, a fact
that contradicts the motivation behind the NMSSM. Here, we omit these terms and
we will work with the scale invariant superpotential (3.1). The Lagrangian of a scale
invariant superpotential possesses an accidental Z3 symmetry, which corresponds to a
multiplication of all the components of all chiral fields by a phase ei2π/3
.
The corresponding soft SUSY breaking masses and couplings are
−Lsof t = m2
Hu
|Hu|
2 + m2
Hd
|Hd|
2 + m2
S
|S|
2
+ m2
Q|Q|
2 + m2
D|DR|
2 + m2
U
|UR|
2 + m2
L
|L|
2 + m2
E|ER|
2
+

huAuQ · HuU
c
R − hdAdQ · HdD
c
R − heAeL · HdE
c
R
+λAλHu · HdS +
1
3
κAκS
3 + h.c.

+
1
2
M1λ1λ1 +
1
2
M2λ
i
2λ
i
2 +
1
2
M3λ
a
3λ
a
3
,
(3.4)
2Here, the hatted capital letters denote chiral superfields, whereas the corresponding unhatted
ones indicate their complex scalar components.
46 The Next-to-Minimal Supersymmetric Standard Model
where we have also included the soft breaking masses for the gauginos. λ1 is the U(1)Y
gaugino (bino), λ
i
2 with i = 1, 2, 3 is the SU(2)L gaugino (winos) and, finally, the λ
a
3
with a = 1, . . . , 8 denotes the SU(3)c gaugino (gluinos).
The scalar potential, expressed by the so-called D and F terms, can be written
explicitly using the general formula
V =
1
2

D
aD
a + D
′2

+ F
⋆
i Fi
, (3.5)
where
D
a = g2Φ
∗
i T
a
ijΦj (3.6a)
D
′ =
1
2
g1YiΦ
∗
i Φi (3.6b)
Fi =
∂W
∂Φi
. (3.6c)
We remind that T
a are the SU(2)L generators and Yi the hypercharge of the scalar
field Φi
. The Yukawa interactions and fermion mass terms are given by the general
Lagrangian
LY ukawa = −
1
2
∂
2W
∂Φi∂Φj
ψiψj + h.c.
, (3.7)
using the superpotential (3.1). The two-component spinor ψi
is the superpartner of
the scalar Φi
.
3.2.1 Higgs sector
Using the general form of the scalar potential, the following Higgs potential is derived
VHiggs =

λ

H
+
u H
−
d − H
0
uH
0
d

+ κS2

2
+

m2
Hu + |λS|
2

H
0
u

2
+

H
+
u

2

+

m2
Hd + |λS|
2

H
0
d

2
+

H
−
d

2

+
1
8

g
2
1 + g
2
2

H
0
u

2
+

H
+
u

2
−

H
0
d

2
−

H
−
d

2
2
+
1
2
g
2
2

H
+
u H
0
d
⋆
+ H
0
uH
−
d
⋆

2
+ m2
S
|S|
2 +

λAλ

H
+
u H
−
d − H
0
uH
0
d

S +
1
3
κAκS
3 + h.c.

.
(3.8)
The neutral physical Higgs states are defined through the relations
H
0
u = vu +
1
√
2
(HuR + iHuI ), H0
d = vd +
1
√
2
(HdR + iHdI ),
S = s +
1
√
2
(SR + iSI ),
3.2.1 Higgs sector 47
where vu, vd and s are, respectively, the real vevs of Hu, Hd and S, which have to be
obtained from the minima of the scalar potential (3.8), after expanding the fields using
Eq. (3.9). We notice that when S acquires a vev, a term µeffHbu · Hbd appears in the
superpotential, with
µeff = λs, (3.10)
solving the µ-problem.
Therefore, the Higgs sector of the NMSSM is characterized by the seven parameters
λ, κ, m2
Hu
, m2
Hd
, m2
S
, Aλ and Aκ. One can express the three soft masses by the three
vevs using the minimization equations of the Higgs potential (3.8), which are given by
vu

m2
Hu + µ
2
eff + λ
2
v
2
d +
1
2
g
2

v
2
u − v
2
d

− vdµeff(Aλ + κs) = 0
vd

m2
Hd + µ
2
eff + λ
2
v
2
u +
1
2
g
2

v
2
d − v
2
u

− vuµeff(Aλ + κs) = 0
s

m2
S + κAκs + 2κ
2σ
2 + λ
2

v
2
u + v
2
d

− 2λκvuvd

− λAλvuvd = 0,
(3.11)
where we have defined
g
2 ≡
1
2

g
2
1 + g
2
2

. (3.12)
One can also define the β angle by
tan β =
vu
vd
. (3.13)
The Z boson mass is given by MZ = gv with v
2 = v
2
u + v
2
d ≃ (174 GeV)2
. Hence, with
MZ given, the set of parameters that describes the Higgs sector of the NMSSM can be
chosen to be the following
λ, κ, Aλ, Aκ, tan b and µeff. (3.14)
CP-even Higgs masses
One can obtain the Higgs mass matrices at tree level by expanding the Higgs potential
(3.8) around the vevs, using Eq. (3.9). We begin by writing3
the squared mass matrix
M2
S
of the scalar Higgses in the basis (HdR, HuR, SR):
M2
S =


g
2
v
2
d + µ tan βBeff (2λ
2 − g
2
) vuvd − µBeff 2λµvd − λ (Aλ + 2κs) vu
g
2
v
2
u +
µ
tan βBeff 2λµvu − λ (Aλ + 2κs) vd
λAλ
vuvd
s + κAκs + (2κs)
2

 ,
(3.15)
where we have defined Beff ≡ Aλ + κs (it plays the role of the B parameter of the
MSSM).
3For economy of space, we omit in this expression the subscript from µ
48 The Next-to-Minimal Supersymmetric Standard Model
Although an analytical diagonalization of the above 3 × 3 mass matrix is lengthy,
there is a crucial conclusion that comes from the approximate diagonalization of the
upper 2 × 2 submatrix. If it is rotated by an angle β, one of its diagonal elements
is M2
Z
(cos2 2β +
λ
2
g
2 sin2
2β) which is an upper bound for its lightest eigenvalue. The
first term is the same one as in the MSSM. The conclusion is that in the NMSSM
the lightest CP-even Higgs can be heavier than the corresponding of the MSSM, as
long as λ is large and tan β relatively small. Therefore, it is much easier to explain
the observed mass of the SM-like Higgs. However, λ is bounded from above in order
to avoid the appearance of the Landau pole below the GUT scale. Depending on the
other free parameters, λ should obey λ <∼ 0.7.
CP-odd Higgs masses
For the pseudoscalar case, the squared mass matrix in the basis (HdI , HuI , SI ) is
M2
P =


µeff (Aλ + κs) tan β µeff (Aλ + κs) λvu (Aλ − 2κs)
µeff
tan β
(Aλ + κs) λvd (Aλ − 2κs)
λ (Aλ + 4κs)
vuvd
s − 3κAκs

 . (3.16)
One eigenstate of this matrix corresponds to an unphysical massless Goldstone
boson G. In order to drop the Goldstone boson, we write the matrix in the basis
(A, G, SI ) by rotating the upper 2 × 2 submatrix by an angle β. After dropping the
massless mode, the 2 × 2 squared mass matrix turns out to be
M2
P =
2µeff
sin 2β
(Aλ + κs) λ (Aλ − 2κs) v
λ (Aλ + 4κs)
vuvd
s − 3Aκs
!
. (3.17)
Charged Higgs mass
The charged Higgs squared mass matrix is given, in the basis (H+
u
, H−
d
⋆
), by
M2
± =

µeff (Aλ + κs) + vuvd

1
2
g
2
2 − λ

cot β 1
1 tan β
!
, (3.18)
which contains one Goldstone boson and one physical mass eigenstate H± with eigenvalue
m2
± =
2µeff
sin 2β
(Aλ + κs) + v
2

1
2
g
2
2 − λ

. (3.19)
3.2.2 Sfermion sector
The mass matrix for the up-type squarks is given in the basis (ueR, ueL) by
Mu =

m2
u + h
2
u
v
2
u −
1
3
(v
2
u − v
2
d
) g
2
1 hu (Auvu − µeffvd)
hu (Auvu − µeffvd) m2
Q + h
2
u
v
2
u +
1
12 (v
2
u − v
2
d
) (g
2
1 − 3g
2
2
)
!
, (3.20)
3.2.3 Gaugino and higgsino sector 49
whereas for down-type squarks the mass matrix reads in the basis (deR, deL)
Md =

m2
d + h
2
d
v
2
d −
1
6
(v
2
u − v
2
d
) g
2
1 hd (Advd − µeffvu)
hd (Advd − µeffvu) m2
Q + h
2
d
v
2
d +
1
12 (v
2
u − v
2
d
) (g
2
1 − 3g
2
2
)
!
. (3.21)
The off-diagonal terms are proportional to the Yukawa coupling hu for the up-type
squarks and hd for the down-type ones. Therefore, the two lightest generations remain
approximately unmixed. For the third generation, the mass matrices are diagonalized
by a rotation by an angle θT and θB, respectively, for the stop and sbottom. The mass
eigenstates are, then, given by
et1 = cos θT
etL + sin θT
etR, et2 = cos θT
etL − sin θT
etR, (3.22)
eb1 = cos θB
ebL + sin θB
ebR, eb2 = cos θB
ebL − sin θB
ebR. (3.23)
In the slepton sector, for a similar reason, only the left- and right-handed staus are
mixed and their mass matrix
Mτ =

m2
E3 + h
2
τ
v
2
d −
1
2
(v
2
u − v
2
d
) g
2
1 hτ (Aτ vd − µeffvu)
hτ (Aτ vd − µeffvu) m2
L3 + h
2
τ
v
2
d −
1
4
(v
2
u − v
2
d
) (g
2
1 − g
2
2
)
!
(3.24)
is diagonalized after a rotation by an angle θτ . Their mass eigenstates are given by
τe1 = cos θτ τeL + sin θτ τeR, τe2 = cos θτ τeL − sin θτ τeR. (3.25)
Finally, the sneutrino masses are
mνe = m2
L −
1
4

v
2
u − v
2
d
g
2
1 + g
2
2

. (3.26)
3.2.3 Gaugino and higgsino sector
The gauginos λ1 and λ
3
2 mix with the neutral higgsinos ψ
0
d
, ψ
0
u
and ψS to form neutral
particles, the neutralinos. The 5 × 5 mass matrix of the neutralinos is written in the
basis
(−iλ1, −iλ3
2
, ψ0
d
, ψ0
u
, ψS) ≡ (B, e W , f He0
d
, He0
u
, Se) (3.27)
as
M0 =


M1 0 − √
1
2
g1vd √
1
2
g1vu 0
M2 √
1
2
g2vd − √
1
2
g2vu 0
0 −µeff −λvu
0 −λvd
2κs


. (3.28)
The mass matrix (3.28) is diagonalized by an orthogonal matrix Nij . The mass eigenstates of the neutralinos are usually denoted by χ
0
i
, with i = 1, . . . , 5, with increasing
masses (i = 1 corresponds to the lightest neutralino). These are given by
χ
0
i = Ni1Be + Ni2Wf + Ni3He0
d + Ni4He0
u + Ni5S. e (3.2
50 The Next-to-Minimal Supersymmetric Standard Model
We use the convention of a real matrix Nij , so that the physical masses mχ
0
i
are real,
but not necessarily positive.
In the charged sector, the SU(2)L charged gauginos λ
− = √
1
2
(λ
1
2 + iλ2
2
), λ
+ =
√
1
2
(λ
1
2 − iλ2
2
) mix with the charged higgsinos ψ
−
d
and ψ
+
u
, forming the charginos ψ
±:
ψ
± =

−iλ±
ψ
±
u
!
. (3.30)
The chargino mass matrix in the basis (ψ
−, ψ+) is
M± =

M2 g2vu
g2vd µeff !
. (3.31)
Since it is not symmetric, the diagonalization requires different rotations of ψ
− and
ψ
+. We denote these rotations by U and V , respectively, so that the mass eigenstates
are obtained by
χ
− = Uψ−, χ+ = V ψ+. (3.32)
3.3 DM Candidates in the NMSSM
Let us first review the characteristics that a DM candidate particle should have. First,
it should be massive in order to account for the missing mass in the galaxies. Second,
it must be electrically and color neutral. Otherwise, it would have condensed with
baryonic matter, forming anomalous heavy isotopes. Such isotopes are absent in nature. Finally, it should be stable and interact only weakly, in order to fit the observed
relic density.
In the NMSSM there are two possible candidates. Both can be stable particles if
they are the LSPs of the supersymmetric spectrum. The first one is the sneutrino (see
[148,149] for early discussions on MSSM sneutrino LSP). However, although sneutrinos
are WIMPs, their large coupling to the Z bosons result in a too large annihilation cross
section. Hence, if they were the DM particles, their relic density would have been very
small compared to the observed value. Exceptions are very massive sneutrinos, heavier
than about 5 TeV [150]. Furthermore, the same coupling result in a large scattering
cross section off the nuclei of the detectors. Therefore, sneutrinos are also excluded by
direct detection experiments.
The other possibility is the lightest neutralino. Neutralinos fulfill successfully, at
least in principle, all the requirements for a DM candidate. However, the resulting
relic density, although weakly interacting, may vary over many orders of magnitude as
a function of the free parameters of the theory. In the next sections we will investigate
further the properties of the lightest neutralino as the DM particle. We begin by
studying its annihilation that determines the DM relic density.
3.4 Neutralino relic density 51
3.4 Neutralino relic density
We remind that the neutralinos are mixed states composed of bino, wino, higgsinos
and the singlino. The exact content of the lightest neutralino determines its pair
annihilation channels and, therefore, its relic density (for detailed analyses, we refer
to [151–154]). Subsequently, we will briefly describe the neutralino pair annihilation
in various scenarios. We classify these scenarios with respect to the lightest neutralino
content.
Before we proceed, we should mention another mechanism that affects the neutralino LSP relic density. If there is a supersymmetric particle with mass close to the
LSP (but always larger), it would be abundant during the freeze-out and LSP coannihilations with this particle would contribute to the total annihilation cross section.
This particle, which is the Next-to-Lightest Supersymmetric Particle (NLSP), is most
commonly a stau or a stop. In the above sense, coannihilations refer not only to the
LSP–NLSP coannihilations, but also to the NLSP–NLSP annihilations, since the latter
reduce the number density of the NLSPs [155].
• Bino-like LSP
In principle, if the lightest neutralino is mostly bino-like, the total annihilation
cross section is expected to be small. Therefore, a bino-like neutralino LSP would
have been overabundant. The reason for this is that there is only one available
annihilation channel via t-channel sfermion exchange, since all couplings to gauge
bosons require a higgsino component. The cross section is even more reduced
when the sfermion mass is large.
However, there are still two ways to achieve the correct relic density. The first one
is using the coannihilation effect: if there is a sfermion with a mass slightly larger
(some GeV) than the LSP mass, their coannihilations can be proved to reduce
efficiently the relic density to the desired value. The second one concerns a binolike LSP, with a very small but non-negligible higgsino component. In this case,
if in addition the lightest CP-odd Higgs A1 is light enough, the annihilation to a
pair A1A1 (through an s-channel CP-even Higgs Hi exchange) can be enhanced
via Hi resonances. In this scenario a fine-tuning of the masses is necessary.
• Higgsino-like LSP
A mostly higgsino LSP is as well problematic. The strong couplings of the higgsinos to the gauge bosons lead to very large annihilation cross section. Therefore,
a possible higgsino LSP would have a very small relic density.
• Mixed bino–higgsino LSP
In this case, as it was probably expected, one can easily fit the relic density to
the observed value. This kind of LSP annihilates to W+W−, ZZ, W±H∓, ZHi
,
HiAj
, b
¯b and τ
+τ
− through s-channel Z or Higgs boson exchange or t-channel
neutralino or chargino exchange. The last two channels are the dominant ones
when the Higgs coupling to down-type fermions is enhanced, which occurs more
commonly in the regime of relatively large tan β. The annihilation channel to a
52 The Next-to-Minimal Supersymmetric Standard Model
pair of top quarks also contributes to the total cross section, if it is kinematically
allowed. However, in order to achieve the correct relic density, the higgsino
component cannot be very large.
• Singlino-like LSP
Since a mostly singlino LSP has small couplings to SM particles, the resulting relic
density is expected to be large. However, there are some annihilation channels
that can be enhanced in order to reduce the relic density. These include the
s-channel (scalar or pseudoscalar) Higgs exchange and the t-channel neutralino
exchange.
For the former, any Higgs with sufficient large singlet component gives an important contribution to the cross section, increasing with the parameter κ (since
the singlino-singlino-singlet coupling is proportional to κ). Concerning the latter
annihilation, in order to enhance it, one needs large values of the parameter λ.
In this case, the neutralino-neutralino-singlet coupling, which is proportional to
λ, is large and the annihilation proceeds giving a pair of scalar HsHs or a pair
of pseudoscalar AsAs singlet like Higgs.
As in the case of bino-like LSP, one can also use the effect of s-channel resonances
or coannihilations. In the latter case, an efficient NLSP can be the neutralino
χ
0
2
or the lightest stau τe1. In the case that the neutralino NLSP is higgsinolike, the LSP-NLSP coannihilation through a (doublet-like) Higgs exchange can
be proved very efficient. A stau NLSP reduces the relic density through NLSPNLSP annihilations, which is the only possibility in the case that both parameters
κ and λ are small. We refer to [156,157] for further discussion on this possibility.
Assuming universality conditions the wino mass M2 has to be larger than the bino
mass M1 (M2 ∼ 2M1). This is the reason that we have not considered a wino-like LSP.
3.5 Detection of neutralino DM
3.5.1 Direct detection
Since neutralinos are Majorana fermions, the effective Lagrangian describing their
elastic scattering with a quark in a nucleon can be written, in the Dirac fermion
notation, as [158]
Leff = a
SI
i χ¯
0
1χ
0
1
q¯iqi + a
SD
i χ¯
0
1γ5γµχ
0
1
q¯iγ5γ
µ
qi
, (3.33)
with i = u, d corresponding to up- and down-type quarks, respectively. The Lagrangian has to be understood as summing over the quark generations.
In this expression, we have omitted terms containing the operator ψγ¯
5ψ or a combination of ψγ¯
5γµψ and ψγ¯
µψ (with ψ = χ, q). This is a well qualified assumption:
Due to the small velocity of WIMPs, the momentum transfer ~q is very small compared
3.5.1 Direct detection 53
to the reduced mass of the WIMP-nucleus system. In the extreme limit of zero momentum transfer, the above operators vanish4
. Hence, we are left with the Lagrangian
(3.33) consisting of two terms, the first one corresponding to spin-independent (SI)
interactions and the second to spin-dependent (SD) ones. In the following, we will
focus again only to SI scattering, since the detector sensitivity to SD scattering is low,
as it has been already mentioned in Sec. 1.5.1.
The SI cross section for the neutralino-nucleus scattering can be written as [158]
(see, also, [159])
σ
SI
tot =
4m2
r
π
[Zfp + (A − Z)fn]
2
. (3.34)
mr is the neutralino-nucleus reduced mass mr =
mχmN
mχ+mN
, and Z, A are the atomic and
the nucleon number, respectively. It is more common, however, to use an expression
for the cross section normalized to the nucleon. In this case, on has for the neutralinoproton scattering
σ
SI
p =
4
π

mpmχ
0
1
mp + mχ
0
1
!2
f
2
p ≃
4m2
χ
0
1
π
f
2
p
, (3.35)
with a similar expression for the neutron.
The form factor fp is related to the couplings a to quarks through the expression
(omitting the “SI” superscripts)
fp
mp
=
X
q=u,d,s
f
p
T q
aq
mq
+
2
27
fT G X
q=c,b,t
aq
mq
. (3.36)
A similar expression may be obtained for the neutron form factor fn, by the replacement
p → n in the previous expression (henceforth, we focus to neutralino-proton scattering).
The parameters fT q are defined by the quark mass matrix elements
hp| mqqq¯ |pi = mpfT q, (3.37)
which corresponds to the contribution of the quark q to the proton mass and the
parameter fT G is related to them by
fT G = 1 −
X
q=u,d,s
fT q. (3.38)
The above parameters can be obtained by the following quantities
σπN =
1
2
(mu + md)(Bu + Bd) and σ0 =
1
2
(mu + md)(Bu + Bd − 2Bs,) (3.39)
with Bq = hN| qq¯ |Ni, which are obtained by chiral perturbation theory [160] or by
lattice simulations. Unfortunately, the uncertainties on the values of these quantities
are large (see [161], for more recent values and error bars).
4While there are possible circumstances in which the operators of (3.33) are also suppressed and,
therefore, comparable to the operators omitted, they are not phenomenologically interesting.
54 The Next-to-Minimal Supersymmetric Standard Model
χ
0
1
χ
0
1
χ
0
1 χ
0
1
qe
q q
q q
Hi
Figure 3.1: Feynman diagrams contributing to the elastic neutralino-quark scalar scattering amplitude at tree level.
The SI neutralino-nucleon interactions arise from t-channel Higgs exchange and
s-channel squark exchange at tree level (see Fig. 3.1), with one-loop contributions from
neutralino-gluon interactions. In practice, the s-channel Higgs exchange contribution
to the scattering amplitude dominates, especially due to the large masses of squarks.
In this case, the effective couplings a are given by
a
SI
d =
X
3
i=1
1
m2
Hi
C
1
i Cχ
0
1χ
0
1Hi
, aSI
u =
X
3
i=1
1
m2
Hi
C
2
i Cχ
0
1χ
0
1Hi
. (3.40)
C
1
i
and C
2
i
are the Higgs Hi couplings to down- and up-type quarks, respectively, given
by
C
1
i =
g2md
2MW cos β
Si1, C2
i =
g2mu
2MW sin β
Si2, (3.41)
with S the mixing matrix of the CP-even Higgs mass eigenstates and md, mu the
corresponding quark mass. We see from Eqs. (3.36) and (3.41) that the final cross
section (3.35) is independent of each quark mass. We write for completeness the
neutralino-neutralino-Higgs coupling Cχ
0
1χ
0
1Hi
:
Cχ
0
1χ
0
1Hi =
√
2λ (Si1N14N15 + Si2N13N15 + Si3N13N14) −
√
2κSi3N

15
+ g1 (Si1N11N13 − Si2N11N14) − g2 (Si1N12N13 − Si2N12N14), (3.42)
with N the neutralino mixing matrix given in (3.29).
The resulting cross section is proportional to m−4
Hi
. In the NMSSM, it is possible
for the lightest scalar Higgs eigenstate to be quite light, evading detection due to its
singlet nature. This scenario can give rise to large values of SI scattering cross section,
provided that the doublet components of th

[I.G.Y.]

@V

« ce qui caractérise le bourgeois, qu’il écoute Bach ou le dernier morceau du rappeur SDM, c’est de ne jamais aimer les choses pour ce qu’elles sont ».

Puisqu’il s’agit d’une caractérisation, c’est que ledit critère est fondamental dans le raisonnement qu’on tient ici. Or ce critère qui prend le bourgeois pour un être entièrement extérieur à lui-même et qui n’a de puissance que par mimétisme ne fonctionne pas. Il y a évidemment des bourgeois qui aiment Bach non pas pour ce que Bach représente mais pour la musique qu’il a composée. On ne peut croire un seul instant que les participants à « La Tribune des critiques de disque » sur France Musique, par exemple, n’aiment pas Bach avant tout pour ce qu’il a fait (je ne sais pas si tous aiment Bach, je n’écoute cette émission que deux fois par an, mais il est facile de postuler qu’une large majorité aime Bach). Or la sociologie des participants (et des auditeurs) de cette émission est très manifestement « non-prolétarienne ».
.
Je reconnais que cette grille de lecture est tout à fait valable (que ce soit dans la bourgeoisie ou non d’ailleurs), mais ne caractérise pas le bourgeois, loin s’en faut. Là où il peut fonctionner à plein, si on prend le cas de la bourgeoisie, c’est lorsque les valeurs culturelles sont immédiatement économiques : par exemple, sur le marché de l’art contemporain bankable.

[Demi Habile]

and also the definition of the unpolarized cross section to write
X
spins
Z
|M12→34|
2
(2π)
4
δ
4
(p1 + p2 − p3 − p4)
d
3p3
(2π)
32E3
d
3p4
(2π)
32E4
=
4F g1g2 σ12→34, (1.31)
where F ≡ [(p1 · p2)
2 − m2
1m2
2
]
1/2
and the spin factors come from the average
over initial spins. This way, the collision term (1.29) is written in a more compact form
g1
Z
C[f1]
d
3p1
(2π)
3
= −
Z
σvMøl (dn1dn2 − dn
eq
1 dn
eq
2
), (1.32)
where σ =
P
(all f)
σ12→f is the total annihilation cross section summed over all the
possible final states and vMøl ≡
F
E1E2
. The so called Møller velocity, vMøl, is defined in
such a way that the product vMøln1n2 is invariant under Lorentz transformations and,
in terms of particle velocities ~v1 and ~v2, it is given by the expression
vMøl =
h
~v2
1 − ~v2
2

2
− |~v1 × ~v2|
2
i1/2
. (1.33)
Due to symmetry considerations, the distributions in kinetic equilibrium are proportional to those in chemical equilibrium, with a proportionality factor independent of
the momentum. Therefore, the collision term (1.32), both before and after decoupling,
can be written in the form
g1
Z
C[f1]
d
3p1
(2π)
3
= −hσvMøli(n1n2 − n
eq
1 n
eq
2
), (1.34)
where the thermal averaged total annihilation cross section times the Møller velocity
has been defined by the expression
hσvMøli =
R
σvMøldn
eq
1 dn
eq
2
R
dn
eq
1 dn
eq
2
. (1.35)
We will come back to the thermal averaged cross section in the next subsection.
We are, now, able to write the full integrated Boltzmann equation, using the expressions (1.28), (1.34) that we have derived for the Liouville and the collision term,
respectively. In the simplified but interesting case of identical particles 1 and 2, the
Boltzmann equation is, finally, written as
n˙ + 3Hn = −hσvMøli(n
2 − n
2
eq). (1.36)
18 Dark Matter
However, instead of using n, it is more convenient to take the expansion of the universe
into account and calculate the number density per comoving volume Y , which can be
defined as the ratio of the number and entropy densities: Y ≡ n/s. The total entropy
density S = R3
s (R is the scale factor) remains constant, hence we can obtain a
differential equation for Y by dividing (1.36) by S. Before we write the final form
of the Boltzmann equation that it is used for the relic density calculations, we have
to change the variable that parametrizes the comoving density. In practice, the time
variable t is not convenient and the temperature of the Universe (actually the photon
temperature, since the photons were the last particles that went out of equilibrium) is
used instead. However, it proves even more useful to use as time variable the quantity
defined by x ≡ m/T with m the DM mass, so that Eq. (1.36) transforms into
dY
dx
=
1
3H
ds
dx
hσvMøli

Y
2 − Y
2
eq
. (1.37)
Last, using the Hubble parameter (1.2) for a radiation dominated Universe and the
expressions (1.20), (1.21) for the energy and entropy density, the Boltzmann equation
is written in its final form
dY
dx
= −
r
45GN
π
g
1/2
∗ m
x
2
hσvMøli

Y
2 − Y
2
eq
, (1.38)
where the effective degrees of freedom g
1/2
∗ have been defined by
g
1/2
∗ ≡
heff
g
1/2
eff

1 +
1
3
T
heff
dheff
dT

. (1.39)
The equilibrium density per comoving volume Yeq ≡ neq/s can be expressed as
Yeq(x) = 45g
4π
4
x
2K2(x)
heff(m/x)
, (1.40)
with K2 the modified Bessel function of second kind.
1.4.3 Thermal average of the annihilation cross section
We are going to derive a simple formula that one can use to calculate the thermal
average of the cross section times velocity, based again on the analysis of [38]. We will
use the assumption that equilibrium functions follow the Maxwell-Boltzmann distribution, instead of the actual Bose-Einstein or Fermi-Dirac. This is a well established
assumption if the freeze out occurs after T ≃ m/3 or for x >∼ 3, which is actually the
case for WIMPs. Under this assumption, the expression (1.35) gives, in the cosmic
comoving frame,
hσvMøli =
R
vMøle
−E1/T e
−E2/T d
3p1d
3p2
R
e
−E1/T e
−E2/T d
3p1d
3p2
. (1.4
1.4.3 Thermal average of the annihilation cross section 19
The volume element can be written as d3p1d
3p2 = 4πp1dE14πp2dE2
1
2
cos θ, with θ the
angle between ~p1 and ~p2. After changing the integration variables to E+, E−, s given
by
E+ = E1 + E2, E− = E1 − E2, s = 2m2 + 2E1E2 − 2p1p2 cos θ, (1.42)
(with s = −(p1 − p2)
2 one of the Mandelstam variables,) the volume element becomes
d
3p1d
3p2 = 2π
2E1E2dE+dE−ds and the initial integration region
{E1 > m, E2 > m, | cos θ| ≤ 1i
transforms into
|E−| ≤
1 −
4m2
s
1/2
(E
2
+ − s)
1/2
, E+ ≥
√
s, s ≥ 4m2
. (1.43)
After some algebraic calculations, it can be found that the quantity hσvMøliE1E2
depends only on s, specifically vMølE1E2 =
1
2
p
s(s − 4m2
). Hence, the numerator of the expression (1.41), which after changing the integration variables reads
2π
2
R
dE+
R
dE−
R
dsσvMølE1E2e
−E+/T , can be written, eventually, as
Z
vMøle
−E1/T e
−E2/T = 2π
2
Z ∞
4m2
dsσ(s − 4m2
)
Z
dE+e
−E+/T (E
2
+ − s)
1/2
. (1.44)
The integral over E+ can be written with the help of the modified Bessel function of
the first kind K1 as √
s T K1(
√
s/T). The denominator of (1.41) can be treated in a
similar way, so that the thermal average is, finally, given by the expression
hσvMøli =
1
8m4TK2
2
(x)
Z ∞
4m2
ds σ(s)(s − 4m2
)
√
s K1(
√
s/T). (1.45)
Eqs. (1.38)–(1.40) along with this last Eq. (1.45) are all we need in order to calculate
the relic density of a WIMP, if its total annihilation cross section in terms of the
Mandelstam variable s is known.
In many cases, in order to avoid the numerical integration in Eq. (1.45), an approximation for hσvMøli can be used. The thermal average is expanded in powers of x
−1
(or, equivalently, in powers of the squared WIMP velocity):
hσvMøli = a + bx−1 + . . . . (1.46)
(The coefficient a corresponds to the s-wave contribution to the cross section, the
coefficient b to the p-wave contribution, and so on.) This partial wave expansion gives
a quite good approximation, provided there are no s-channel resonances and thresholds
for the final states [39].
In [40], it was shown that, after expanding the integrands of Eq. (1.41) in powers
of x
−1
, all the integrations can be performed analytically. As we saw, the expression
20 Dark Matter
vMølE1E2 depends on momenta only through s. Therefore, one can form the Lorentz
invariant quantity
w(s) ≡ σ(s)vMølE1E2 =
1
2
σ(s)
p
s(s − 4m2
). (1.47)
The integration involves the Taylor expansion of this quantity w around s/4m2 = 1
and the general formula for the partial wave expansion of the thermal average is [40]
hσvMøli =
1
m2

w −
3
2
(2w − w
′
)x
−1 +
3
8
(16w − 8w
′ + 5w
′′)x
−2
−
5
16
(30w − 15w
′ + 3w
′′ − 7x
′′′)x
−3 + O(x
−4
)

s/4m2=1
, (1.48)
where primes denote derivatives with respect to s/4m2 and all quantities have to be
evaluated at s = 4m2
.
1.5 Direct Detection of DM
Since the beginning of 1980s, it has been realized that besides the numerous facts showing evidence for the existence of these new dark particles, it is also possible to detect
them directly. Already in 1985, two pioneering articles [41, 42] appeared, describing
the detection methods for WIMPs. Since WIMPs are expected to cluster gravitationally together with ordinary stars in the Milky Way halo, they would pass also through
Earth and, in principle, they can be detected through scattering with the nuclei in a
detector’s material. In practice, one has to measure the recoil energy deposited by this
scattering.
However, although one can deduce from rotation curves that DM dominates the
dark halo in the outer parts of our galaxy, it is not so obvious from direct measurements
whether there is any substantial amount of DM inside the solar radius R0 ≃ 8 kpc.
Using indirect methods (involving the determination of the gravitational potential,
through the measuring of the kinematics of stars, both near the mid-plane of the
galactic disk and at heights several times the disk thickness), it is almost certain
that the DM is also present in the solar system, with a local density ρ0 = (0.3 ±
0.1) GeV cm−3
[43].
This value for the local density implies that for a WIMP mass of order ∼ 100 GeV,
the local number density is n0 ∼ 10−3
cm−3
. It is also expected that the WIMPs
velocity is similar to the velocity with which the Sun orbits around the galactic center
(v0 ≃ 220 km s−1
), since they are both moving under the same gravitational potential.
These two quantities allow to estimate the order of magnitude of the incident flux
of WIMPs on the Earth: J0 = n0v0 ∼ 105
cm−2
s
−1
. This value is manifestly large,
but the very weak interactions of the DM particles with ordinary matter makes their
detection a difficult, although in principle feasible, task. In order to compensate for
the very low WIMP-nucleus scattering cross section, very large detectors are required.
1.5.1 Elastic scattering event rate 21
1.5.1 Elastic scattering event rate
In the following, we will confine ourselves to the elastic scattering with nuclei. Although
inelastic scattering of WIMPs off nuclei in a detector or off orbital electrons producing
an excited state is possible, the event rate of these processes is quite suppressed. In
contrast, during an elastic scattering the nucleus recoils as a whole.
The direct detection experiments measure the number of events per day and per
kilogram of the detector material, as a function of the amount of energy Q deposited
in the detector. This event rate would be given by R = nWIMP nnuclei σv in a simplified
model with WIMPs moving with a constant velocity v. The number density of WIMPs
is nWIMP = ρ0/mX and the number density of nuclei is just the ratio of the detector’s
mass over the nuclear mass mN .
For accurate calculations, one should take into account that the WIMPs move in the
halo not with a uniform velocity, but rather following a velocity distribution f(v). The
Earth’s motion in the solar system should be included into this distribution function.
The scattering cross section σ also depends on the velocity. Actually, the cross section
can be parametrized by a nuclear form factor F(Q) as
dσ =
σ
4m2
r
v
2
F
2
(Q)d|~q|
2
, (1.49)
where |~q|
2 = 2m2
r
v
2
(1 − cos θ) is the momentum transferred during the scattering,
mr =
mXmN
mX+mN
is the reduced mass of the WIMP – nucleus system and θ is the scattering
angle in the center of momentum frame. Therefore, one can write a general expression
for the differential event rate per unit detector mass as
dR =
ρ0
mX
1
mN
σF2
(Q)d|~q|
2
4m2
r
v
2
vf(v)dv. (1.50)
The energy deposited in the detector (transferred to the nucleus through one elastic
scattering) is
Q =
|~q|
2
2mN
=
m2
r
v
2
mN
(1 − cos θ). (1.51)
Therefore, the differential event rate over deposited energy can be written, using the
equations (1.50) and (1.51), as
dR
dQ
=
σρ0
√
πv0mXm2
r
F
2
(Q)T(Q), (1.52)
where, following [37], we have defined the dimensionless quantity T(Q) as
T(Q) ≡
√
π
2
v0
Z ∞
vmin
f(v)
v
dv, (1.53)
with the minimum velocity given by vmin =
qQmN
2m2
r
, obtained by Eq. (1.51). Finally,
the event rate R can be calculated by integrating (1.52) over the energy
R =
Z ∞
ET
dR
dQ
dQ. (1.54)
22 Dark Matter
The integration is performed for energies larger than the threshold energy ET of the
detector, below which it is insensitive to WIMP-nucleus recoils.
Using Eqs. (1.54) and (1.52), one can derive the scattering cross section from the
event rate. The experimental collaborations prefer to give their results already in terms
of the scattering cross section as a function of the WIMP mass. To be more precise,
the WIMP-nucleus total cross section consists of two parts: the spin-dependent (SD)
cross section and the spin-independent (SI) one. The former comes from axial current
couplings, whereas the latter comes from scalar-scalar and vector-vector couplings.
The SD cross section is much suppressed compared to the SI one in the case of heavy
nuclei targets and it vanishes if the nucleus contains an even number of nucleons, since
in this case the total nuclear spin is zero.
We see that two uncertainties enter the above calculation: the exact value of the
local density ρ0 and the exact form of the velocity distribution f(v). To these, one
has to include one more. The cross section σ that appears in the previous expressions
concerns the WIMP-nucleon cross section. The couplings of a WIMP with the various
quarks that constitute the nucleon are not the same and the WIMP-nucleon cross
section depends strongly on the exact quark content of the nucleon. To be more
precise, the largest uncertainty lies on the strange content of the nucleon, but we shall
return to this point when we will calculate the cross section in a specific particle theory,
the Next-to-Minimal Supersymmetric Standard Model, in Sec. 3.5.1.
1.5.2 Experimental status
The situation of the experimental results from direct DM searches is a bit confusing. The null observations in most of the experiments led them to set upper limits
on the WIMP-nucleon cross section. These bounds are quite stringent for the spinindependent cross section7
, especially in the regime of WIMP masses of the order of
100 GeV. However, some collaborations have already reported possible DM signals,
mainly in the low mass regime. The preferred regions of these experiments do not
coincide, while some of them have been already excluded by other experiments. The
present picture, for WIMP masses ranging from 5 to 1000 GeV, is summarized in Fig.
1.5, 1.6.
Fig. 1.5 mainly presents upper bounds coming from XENON100 [44]. XENON100
[46] is an experiment located at the Gran Sasso underground laboratory in Italy. It
contains in total 165 kg of liquid Xenon, with 65 kg acting as target mass and the
rest shielding the detector from background radiation. For these upper limits, 225
live days of data were used. The minimum value for the predicted upper bounds on
the cross section is 2 · 10−45 cm2
for WIMP mass ∼ 55 GeV (at 90% confidence level),
almost one order of magnitude lower than the previously released limits [47] by the
same collaboration, using 100 live days of data.
The stringent upper bounds up-to-date (at least for WIMP mass larger than about
7 GeV) come from the first results of the LUX experiment (see Fig. 1.6), after the first
7For the spin-dependent scattering, the exclusion limits are quite relaxed. Hence, we will focus on
the SI cross sections.
1.5.2 Experimental status 23
Figure 1.5: The XENON100 exclusion limit (thick blue line), along with the expected
sensitivity in green (1σ) and yellow (2σ) band. Other upper bounds are also shown as
well as detection claims. From [44].
85.3 live-days of its operation [45]. LUX [53] is a detector containing liquid Xenon, as
XENON100, but in larger quantity, with total mass 370 kg. Its operation started on
April 2013 with a goal to clearly detect or exclude WIMPs with a spin independent
cross section ∼ 2 · 10−46 cm2
.
In Fig. 1.5, except of the XENON100 bounds and other experimental limits on larger
WIMP-nucleon cross section, some detection claims also appear. These come from
DAMA [48,49], CoGeNT [50] and CRESST-II [51] experiments. The first positive result
came from DAMA [52], back in 2000. Since then, the experiment has accumulated 1.17
ton-yr of data over 13 years of operation. DAMA consists of 250 kg of radio pure NaI
scintillator and looks for the annual modulation of the WIMP flux in order to reduce
the influence of the background.
The annual modulation of the DM flux (see [54] for a recent review) is due to the
Earth’s orbital motion relative to the rotation of the galactic disk. The galactic disk
rotation through an essentially non-rotating DM halo, creates an effective DM wind in
the solar frame. During the earth’s heliocentric orbit, this wind reaches a maximum
when the Earth is moving fastest in the direction of the disk rotation (this happens
in the beginning of June) and a minimum when it is moving fastest in the opposite
direction (beginning of December).
DAMA claims an 8.9σ annual modulation with a minimum flux on May 26±7 days,
consistent with the expectation. Since the detector’s target consists of two different
nuclei and the experiment cannot distinguish between sodium and iodine recoils, there
24 Dark Matter
Figure 1.6: The LUX 90% confidence exclusion limit (blue line) with the 1σ range
(shaded area). The XENON100 upper bound is represented by the red line. The inset
shows also preferred regions by CoGeNT (shaded light red), CDMS II silicon detector
(shaded green), CRESST II (shaded yellow) and DAMA (shaded gray). From [45].
is no model independent way to determine the exact region in the cross section versus
WIMP mass plane to which the observed modulation corresponds. However, one can
assume two cases: one that the WIMP scattering off the sodium nucleus dominates the
recoil energy and the other with the iodine recoils dominating. The former corresponds
[55] to a light WIMP (∼ 10 GeV) and quite large scattering cross section and the latter
to a heavier WIMP (∼ 50 to 100 GeV) with smaller cross section (see Fig. 1.5).
The positive result of DAMA was followed many years later by the ones of CoGeNT
and CRESST-II, and more recently by the silicon detector of CDMS [56] (Fig. 1.7).
The discrepancy of the results raised a lot of debates among the experiments (for
example, [64–67]) and by some the positive results are regarded as controversial. On
the other hand, it also raised an effort to find a physical explanation behind this
inconsistency (see, for example, [68–71]).
1.6 Indirect Methods for DM Detection
The same annihilation processes that determined the DM relic abundance in the early
Universe also occur today in galactic regions where the DM concentration is higher.
This fact rises the possibility of detecting potential WIMP pair annihilations indirectly
through their imprints on the cosmic rays. Therefore, the indirect DM searches aim
at the detection of an excess over the known astrophysical background of charged
particles, photons or neutrinos.
Charged particles – electrons, protons and their antiparticles – may originate from
direct products (pair of SM particles) of WIMP annihilations, after their decay and
1.6 Indirect Methods for DM Detection 25
Figure 1.7: The blue contours represent preferred regions for a possible signal at 68%
and 90% C.L. using the silicon detector of CMDS [56]. The blue dotted line represents
the upper limit obtained by the same analysis and the blue solid line is the combined
limit with the silicon CDMS data set reported in [57]. Other limits also appear:
from the CMDS standard germanium detector (light and dark red dashed line, for
standard [58] and low threshold analysis [59], respectively), EDELWEISS [60] (dashed
orange), XENON10 [61] (dash-dotted green) and XENON100 [44] (long-dash-dotted
green). The filled regions identify possible signal regions associated with data from
CoGeNT [62] (dashed yellow, 90% C.L.), DAMA [49,55] (dotted tan, 99.7% C.L.) and
CRESST-II [51, 63] (dash-dotted pink, 95.45% C.L.) experiments. Taken from [56].
through the process of showering and hadronization. Although the exact shape of the
resulting spectrum would depend on the specific process, it is expected to show a steep
cutoff at the WIMP mass. Once produced in the DM halo, the charged particles have
to travel to the point of detection through the turbulent galactic field, which will cause
diffusion. Apart from that, a lot of processes disturb the propagation of the charged
particles, such as bremsstrahlung, inverse Compton scattering with CMB photons and
many others. Therefore, the uncertainties that enter the propagation of the charged
flux until it reaches the telescope are important (contrary to the case of photons and
neutrinos that propagate almost unperturbed through the galaxy).
As in the case of direct detection, the experimental status of charged particle detection concerning the DM is confusing. After some hints from HEAT [72] and AMS01 [73] (the former a far-infrared telescope in Antarctica, the latter a spectrometer,
prototype for AMS-02 mounted on the International Space Station [74]), the PAMELA
satellite observed [75, 76] a steep increase in the energy spectrum of positron fraction
e
+/(e
+ + e
−)
8
. Later FERMI satellite [77] and AMS-02 [78] confirmed the results up
8The searches for charged particles focus on the antiparticles in order to have a reduced background,
26 Dark Matter
Figure 1.8: A compilation of data of charged cosmic rays, together with plausible but
uncertain astrophysical backgrounds, taken from [79]. Left: Positron flux. Center:
Antiproton flux. Right: Sum of electrons and positrons.
to energies of ∼ 200 GeV. However, the excess of positrons is not followed by an excess
of antiprotons, whose flux seems to coincide with the predicted background [75]. In
Fig. 1.8, three plots summarizing the situation are shown [79].
The observed excess is very difficult to explain in terms of DM [79]. To begin with,
the annihilation cross section required to reproduce the excess is quite large, many
orders of magnitude larger than the thermal cross section. Moreover, an “ordinary”
WIMP with large annihilation cross section giving rise to charged leptons is expected
to give, additionally, a large number of antiprotons, a fact in contradiction with the
observations. Although a lot of work has been done to fit a DM particle to the observed
pattern, it is quite possible that the excesses come from a yet unknown astrophysical
source. We are not going to discuss further this matter, but we end with a comment.
If this excess is due to a source other than DM, then a possible DM positron excess
would be lost under this formidable background.
A last hint for DM came from the detection of highly energetic photons. However,
we will interrupt this discussion, since this signal and a possible explanation is the
subject of Ch. 4. There, we will also see the upper bounds on the annihilation cross
section being set due to the absence of excesses in diffuse γ radiation.
since they are much less abundant than the corresponding particles.
CHAPTER 2
PARTICLE PHYSICS
Since the DM comprises of particles, it should be explained by a general particle physics
theory. We start in the following section by describing the Standard Model (SM) of
particle physics. Although the SM describes so far the fundamental particles and their
interactions quite accurately, it cannot provide a DM candidate. Besides, the SM
suffers from some theoretical problems, which we discuss in Sec. 2.2. We will see that
these problems can be solved if one introduces a new symmetry, the supersymmetry,
which we describe in Sec. 2.3. We finish this chapter by briefly describing in Sec. 2.4 a
supersymmetric extension of the SM with the minimal additional particle content, the
Minimal Supersymmetric Standard Model (MSSM).
2.1 The Standard Model of Particle Physics
The Standard Model (SM) of particle physics1
consists of two well developed theories,
the quantum chromodynamics (QCD) and the electroweak (EW) theory. The former
describes the strong interactions among the quarks, whereas the latter describes the
electroweak interactions (the weak and electromagnetic interactions in a unified context) between fermions. The EW theory took its final form in the late 1960s by the
introduction by S. Weinberg [85] and A. Salam [86] of the Higgs mechanism that gives
masses to the SM particles, which followed the unification of electromagnetic and weak
interactions [87,88]. At the same time, the EW model preserves the gauge invariance,
making the theory renormalizable, as shown later by ’t Hooft [89]. On the other hand,
QCD obtained its final form some years later, after the confirmation of the existence
of quarks. Of course, the history of the SM is much longer and it can be traced back to
1920s with the formulation of a theoretical basis for a Quantum Field Theory (QFT).
Since then, the SM had many successes. The SM particle content was completed with
the discovery of the heaviest of the quarks, the top quark [90,91], in 1995 and, recently,
with the discovery of the Higgs boson [92, 93].
1There are many good textbooks on the SM and Quantum Field Theory, e.g. [80–84].
28 Particle Physics
The key concept within the SM, as in every QFT, is that of symmetries. Each
interaction respects a gauge symmetry, based on a Lie algebra. The strong interaction is
described by an SU(3)c symmetry, where the subscript c stands for color, the conserved
charge of strong interactions. The EW interactions, on the other hand, are based on
a SU(2)L × U(1)Y Lie algebra. Here, as we will subsequently see, L refers to the
left-handed fermions and Y is the hypercharge, the conserved charge under the U(1).
SU(2)L conserves a quantity known as weak isospin I. Therefore, the SM contains the
internal symmetries of the unitary product group
SU(2)L × U(1)Y × SU(3)c. (2.1)
2.1.1 The particle content of the SM
We mention for completeness that particles are divided into two main classes according
to the statistics they follow. The bosons are particles with integer spin and follow the
Bose-Einstein distribution, whereas fermions have half-integer spin and follow the
Dirac-Einstein statistics, obeying the Pauli exclusion principle. In the SM, all the
fermions have spin 1/2, whereas the bosons have spin 1 with only exception the Higgs
boson, which is a scalar (spin zero). We begin the description of the SM particles with
the fermions.
Each fermion is classified in irreducible representations of each individual Lie algebra, according to the conserved quantum numbers, i.e. the color C, the weak isospin
I and the hypercharge Y . A first classification of fermions can be done into leptons
and quarks, which transform differently under the SU(3)c. Leptons are singlets under
this transformation, while quarks act as triplets (the fundamental representation of
this group). The EW interactions violate maximally the parity symmetry and SU(2)L
acts only on states with negative chirality (left-handed). A Dirac spinor Ψ can be
decomposed into left and right chirality components using, respectively, the projection
operators PL =
1
2
(1 − γ5) and PR =
1
2
(1 + γ5):
ΨL = PLΨ and ΨR = PRΨ. (2.2)
Left-handed fermions have I = 1/2, with a third component of the isospin I3 = ±1/2.
Fermions with positive I3 are called up-type fermions and those with negative are
called down-type. These behave the same way under SU(2)L and form doublets with
one fermion of each type. On the other hand, right-handed fermions have I = 0 and
form singlets that do not undergo weak interactions. The hypercharge is written in
terms of the electric charge Q and the third component of the isospin I3 through the
Gell-Mann–Nishijima relation:
Q = I3 + Y/2. (2.3)
Therefore, left- and right-handed components transform differently under the U(1)Y ,
since they have different hypercharge.
The fermionic sector of the SM comprises three generations of fermions, transforming as spinors under Lorentz transformations. Each generation has the same structure.
For leptons, it is an SU(2)L doublet with components consisting of one left-handed
2.1.2 The SM Lagrangian 29
charged lepton and one neutrino (neutrinos are only left-handed in the SM), along
with a gauge singlet right-handed charged lepton. The quark doublet consists of an
up- (u) and a down-type (d) (left-handed) quark and the pattern is completed by the
two corresponding SU(2)L singlet right-handed quarks. We write these representations
as
Quarks: Q ≡

u
i
L
d
i
L
!
, ui
R, di
R Leptons: L ≡

ν
i
L
e
i
L
!
, ei
R, (2.4)
with i = 1, 2, 3 the generation index.
Having briefly described the fermionic sector, we turn to the bosonic sector of
the SM. It consists of the gauge bosons that mediate the interactions and the Higgs
boson that gives masses to the particles through a spontaneous symmetry breaking,
the electroweak symmetry breaking (EWSB) [94–98], which we shall describe in Sec.
2.1.3. Before the EWSB, these bosons are
• three Wa
µ
(a = 1, 2, 3) weak bosons, associated with the generators of SU(2)L,
• one neutral Bµ boson, associated with the generator of U(1)Y ,
• eight gluons Ga
µ
(a = 1, . . . , 8), associated with the generators of SU(3)c, and
• the complex scalar Higgs doublet Φ =
φ
+
φ
0
!
.
After the EWSB, the EW boson states mix and give the two W± bosons, the neutral
Z boson and the massless photon γ. From the symmetry breaking, one scalar degree of
freedom remains which is the famous (neutral) Higgs boson [97–99]. We will return to
the mixed physical states, after describing the Higgs mechanism for symmetry breaking.
A complete list of the SM particles (the physical states after EWSB) is shown in Table
2.1.
2.1.2 The SM Lagrangian
The gauge bosons are responsible for the mediation of the interactions and are associated with the generators of the corresponding symmetry. The EW gauge bosons Bµ
and Wa
µ
are associated, respectively, with the generator Y of the U(1)Y and the three
generators T
a
2
of the SU(2)L. The latter are defined as half of the Pauli matrices τ
a
(T
a
2 =
1
2
τ
a
) and they obey the algebra

T
a
2
, Tb
2

= iǫabcT
c
2
, (2.5)
where ǫ
abc is the fully antisymmetric Levi-Civita tensor. The eight gluons are associated
with an equal number of generators T
a
3
(Gell-Mann matrices) of SU(3)c and obey the
Lie algebra

T
a
3
, Tb
3

= if abcT
c
3
, with Tr
T
a
3 T
b
3

=
1
2
δ
ab
, (2.6)
30 Particle Physics
Name symbol mass charge (|e|) spin
Leptons
electron e 0.511 MeV −1 1/2
electron neutrino νe 0 (<2 eV) 0 1/2
muon µ 105.7 MeV −1 1/2
muon neutrino νµ 0 (<2 eV) 0 1/2
tau τ 1.777 GeV −1 1/2
tau neutrino ντ 0 (<2 eV) 0 1/2
Quarks
up u 2.7
+0.7
−0.5 MeV 2/3 1/2
down d 4.8
+0.7
−0.3 MeV −1/3 1/2
strange s (95 ± 5) MeV −1/3 1/2
charm c (1.275 ± 0.025) GeV 2/3 1/2
bottom b (4.18 ± 0.03) GeV −1/3 1/2
top t (173.5 ± 0.6 ± 0.8) GeV 2/3 1/2
Bosons
photon γ 0 (<10−18 eV) 0 (<10−35) 1
W boson W± (80.385 ± 0.015) GeV ±1 1
Z boson Z (91.1876 ± 0.0021) GeV 0 1
gluon g 0 (.O(1) MeV) 0 1
Higgs H
(125.3 ± 0.4 ± 0.5) GeV
0 0
(126.0 ± 0.4 ± 0.4) GeV
Table 2.1: The particle content of the SM. All values are those given in [100], except of
the Higgs mass that is taken from [92, 93] (up and down row, respectively), assuming
that the observed excess corresponds to the SM Higgs. The u, d and s quark masses
are estimates of so-called “current-quark masses” in a mass-independent subtraction
scheme as MS at a scale ∼ 2 GeV. The c and b quark masses are the running masses
in the MS scheme. The values in the parenthesis are the current experimental limits.
with f
abc the structure constants of the group.
Using the structure constants of the corresponding groups, we define the field
strengths for the gauge bosons as
Bµν ≡ ∂µBν − ∂νBµ, (2.7a)
Wµν ≡ ∂µWa
ν − ∂νWa
µ + g2ǫ
abcWb
µWc
ν
(2.7b)
and
G
a
µν ≡ ∂µG
a
ν − ∂νG
a
µ + g3f
abcG
b
µG
c
ν
. (2.7c)
2.1.2 The SM Lagrangian 31
We use the notation g1, g2 and g3 for the coupling constants of U(1)Y , SU(2)L and
SU(3)c, respectively. As in any Yang-Mills theory, the non-abelian gauge groups lead
to self-interactions, which is not the case for the abelian U(1)Y group.
Before we finally write the full Lagrangian, we have to introduce the covariant
derivative for fermions, which in a general form can be written as
DµΨ =
∂µ − ig1
1
2
Y Bµ − ig2T
a
2 Wa
µ − ig3T
a
3 G
a
µ

Ψ. (2.8)
This form has to be understood as that, depending on Ψ, only the relevant terms
apply, hence for SU(2)L singlet leptons only the two first terms inside the parenthesis
are relevant, for doublet leptons the three first terms and for the corresponding quark
singlets and doublets the last term also participates. We also have to notice that in
order to retain the gauge symmetry, mass terms are forbidden in the Lagrangian. For
example, the mass term mψψ¯ = m

ψ¯
LψR + ψ¯
RψL

(with ψ¯ ≡ ψ
†γ
0
) is not invariant
under SU(2)L. This paradox is solved by the introduction of the Higgs scalar field
(see next subsection). The SM Lagrangian can be now written2
, split for simplicity in
three parts, each describing the gauge bosons, the fermions and the scalar sector,
LSM = Lgauge + Lfermion + Lscalar, (2.9)
with
Lgauge = −
1
4
G
a
µνG
µν
a −
1
4
Wa
µνWµν
a −
1
4
BµνB
µν
, (2.10a)
Lfermion = iL¯Dµγ
µL + ie¯RDµγµeR
+ iQ¯Dµγ
µQ + iu¯RDµγ
µuR + i
¯dRDµγ
µ
dR
−

heL¯ΦeR + hdQ¯ΦdR + huQ¯ΦeuR + h.c.

(2.10b)
and
Lscalar = (DµΦ)†
(DµΦ) − V (Φ†Φ), (2.10c)
where
V (Φ†Φ) = µ
2Φ
†Φ + λ

Φ
†Φ
2
(2.11)
is the scalar Higgs potential. Φ is the conjugate of Φ, related to the charge conjugate e
by Φ =e iτ2Φ
⋆
, with τi the Pauli matrices. The covariant derivative acting on the Higgs
scalar field gives
DµΦ =
∂µ − ig1
1
2
Y Bµ − ig2T
a
2 Wa
µ

Φ. (2.12)
Before we proceed to the description of the Higgs mechanism, a last comment concerning the SM Lagrangian is in order. If we restore the generation indices, we see that
2For simplicity, from now on we are going to omit the generations indice
32 Particle Physics
the Yukawa couplings h are 3 × 3, in general complex, matrices. As any complex matrix, they can be diagonalized with the help of two unitary matrices VL and VR, which
are related by VR = U
†VL with U again a unitary matrix. The diagonalization in the
quark sector to the mass eigenstates induces a mixing among the flavors (generations),
described by the Cabibbo–Kobayashi–Maskawa (CKM) matrix [101, 102]. The CKM
matrix is defined by
VCKM ≡ V
u
L
†
V
d
L
†
, (2.13)
where V
u
L
, V
d
L
are the unitary matrices that diagonalize the Yukawa couplings Hu
, Hd
,
respectively. This product of the two matrices appears in the charged current when it
is expressed in terms of the observable mass eigenstates.
2.1.3 Mass generation through the Higgs mechanism
We will start by examining the scalar potential (2.11). The vacuum expectation value
(vev) of the Higgs field hΦi ≡ h0|Φ|0i is given by the minimum of the potential. For
µ
2 > 0, the potential is always non-negative and Φ has a zero vev. The hypothesis of
the Higgs mechanism is that µ
2 < 0. In this case, the field Φ will acquire a vev
hΦi =
1
2

0
v
!
with v =
r
−
µ2
λ
. (2.14)
Since the charged component of Φ still has a zero vev, the U(1)Q symmetry of quantum
electrodynamics (QED) remains unbroken.
We expand the field Φ around the minima v in terms of real fields, and at leading
order we have
Φ(x) =
θ2(x) + iθ1(x)
√
1
2
(v + H(x)) − iθ3(x)
!
=
1
√
2
e
iθa(x)τ
a

0
v + H(x)
!
. (2.15)
We can eliminate the unphysical degrees of freedom θa, using the fact that the theory
remains gauge invariant. Therefore, we perform the following SU(2)L gauge transformation on Φ (unitary gauge)
Φ(x) → e
−iθa(x)τ
a
Φ(x), (2.16)
so that
Φ(x) = 1
√
2

0
v + H(x)
!
. (2.17)
We are going to use the following definitions for the gauge fields
W±
µ ≡
1
2

W1
µ ∓ iW2
µ

, (2.18a)
Zµ ≡
1
p
g
2
1 + g
2
2

g2W3
µ − g1Bµ

, (2.18b)
Aµ ≡
1
p
g
2
1 + g
2
2

g1W3
µ + g2Bµ

, (2.1
2.2 Limits of the SM and the emergence of supersymmetry 33
Then, the kinetic term for Φ (see Eq. (2.10c)) can be written in the unitary gauge as
(DµΦ)†
(D
µΦ) = 1
2
(∂µH)
2 + M2
W W+
µ W−µ +
1
2
M2
ZZµZ
µ
, (2.19)
with
MW ≡
1
2
g2v and MZ ≡
1
2
q
g
2
1 + g
2
2
v. (2.20)
We see that the definitions (2.18) correspond to the physical states of the gauge bosons
that have acquired masses due to the non-zero Higgs vev, given by (2.20). The photon
has remained massless, which reflects the fact that after the spontaneous breakdown of
SU(2)L × U(1)Y the U(1)Q remained unbroken. Among the initial degrees of freedom
of the complex scalar field Φ, three were absorbed by W± and Z and one remained as
the neutral Higgs particle with squared mass
m2
H = 2λv2
. (2.21)
We note that λ should be positive so that the scalar potential (2.11) is bounded from
below.
Fermions also acquire masses due to the Higgs mechanism. The Yukawa terms in
the fermionic part (2.10b) of the SM Lagrangian are written, after expanding around
the vev in the unitary gauge,
LY = −
1
√
2
hee¯L(v + H)eR −
1
√
2
hd
¯dL(v + H)dR −
1
√
2
huu¯L(v + H)uR + h.c. . (2.22)
Therefore, we can identify the masses of the fermions as
me
i =
h
i
e
v
√
2
, md
i =
h
i
d
v
√
2
, mui =
h
i
u
v
√
2
, (2.23)
where we have written explicitly the generation indices.
2.2 Limits of the SM and the emergence of supersymmetry
2.2.1 General discussion of the SM problems
The SM has been proven extremely successful and has been tested in high precision
in many different experiments. It has predicted many new particles before their final
discovery and also explained how the particles gain their masses. Its last triumph was
of course the discovery of a boson that seems to be very similar to the Higgs boson of
the SM. However, it is generally accepted that the SM cannot be the ultimate theory. It
is not only observed phenomena that the SM does not explain; SM also faces important
theoretical issues.
The most prominent among the inconsistencies of the SM with observations is the
oscillations among neutrinos of different generations. In order for the oscillations to
34 Particle Physics
φ φ
k
Figure 2.1: The scalar one-loop diagram giving rise to quadratic divergences.
occur, neutrinos should have non-zero masses. However, minimal modifications of the
SM are able to fit with the data of neutrino physics. Another issue that a more complete theory has to face is the matter asymmetry, the observed dominance of matter
over antimatter in the Universe. In addition, in order to comply with the standard
cosmological model, it has to provide the appropriate particle(s) that drove the inflation. Last, but not least, we saw that in order to explain the DM that dominates the
Universe, a massive, stable weakly interacting particle must exist. Such a particle is
not present in the SM.
On the other hand, the SM also suffers from a theoretical perspective. For example,
the SM counts 19 free parameters; one expects that a fundamental theory would have
a much smaller number of free parameters. Simple modifications of the SM have been
proposed relating some of these parameters. Grand unified theories (GUTs) unify
the gauge couplings at a high scale ∼ 1016 GeV. However, this unification is only
approximate unless the GUT is embedded in a supersymmetric framework. Another
serious problem of the SM is that of naturalness. This will be the topic of the following
subsection.
2.2.2 The naturalness problem of the SM
The presence of fundamental scalar fields, like the Higgs, gives rise to quadratic divergences. The diagram of Fig. 2.1 contributes to the squared mass of the scalar
δm2 = λ
Z Λ
d
4k
(2π)
4
k
−2
. (2.24)
This contribution is approximated by δm2 ∼ λΛ
2/(16π
2
), quadratic in a cut-off Λ,
which should be finite. For the case of the Higgs scalar field, one has to include its
couplings to the gauge fields and the top quark3
. Therefore,
δm2
H =
3Λ2
8π
2v
2

4m2
t − 2M2
W − M2
Z − m2
H

+ O(ln Λ
µ
)

, (2.25)
where we have used Eq. (2.21) and m2
H ≡ m2
0 + δm2
H.
3Since the contribution to the squared mass correction are quadratic in the Yukawa couplings (or
quark masses), the lighter quarks can be neglected
2.2.3 A way out 35
Taking Λ as a fundamental scale Λ ∼ MP l ∼ 1019 GeV we have
m2
0 = m2
H −
3Λ2
8π
2v
2

4m2
t − 2M2
W − M2
Z − m2
H

(2.26)
and we can see that m2
0 has to be adjusted to a precision of about 30 orders of magnitude
in order to achieve an EW scale Higgs mass. This is considered as an intolerable finetuning, which is against the general belief that the observable properties of a theory
have to be stable under small variations of the fundamental (bare) parameters. It is
exactly the above behavior that is considered as unnatural. Although the SM could
be self-consistent without imposing a large scale, grand unification of the parameters
introduce a hierarchy problem between the different scales.
A more strict definition of naturalness comes from ’t Hooft [103], which we rewrite
here:
At an energy scale µ, a physical parameter or set of physical parameters
αi(µ) is allowed to be very small only if the replacement αi(µ) = 0 would
increase the symmetry of the system.
Clearly, this is not the case here. Although mH is small compared to the fundamental
scale Λ, it is not protected by any symmetry and a fine-tuning is necessary.
2.2.3 A way out
The naturalness in the ’t Hooft sense is inspired by quantum electrodynamics, which is
the archetype for a natural theory. For example, the corrections to the electron mass
me are themselves proportional to me, with a dimensionless proportionality factor that
behaves like ∼ ln Λ. In general, fermion masses are protected by the chiral symmetry; small values (compared to the fundamental scale) of these masses enhances the
symmetry.
If a new symmetry exists in nature, relating fermion fields to scalar fields, then each
scalar mass would be related somehow to the corresponding fermion mass. Therefore,
the scalar mass itself can be naturally small compared to Λ, since this would mean
that the fermion mass is small, which enhances the chiral symmetry. Such a symmetry,
relating bosons to fermions and vice versa, is known as supersymmetry [104, 105].
Actually, as we will see later, if this new symmetry remains unbroken, the masses of
the conjugate bosons and fermions would have to be equal.
In order to make the above statement more concrete, we consider a toy model with
two additional complex scalar fields feL and feR. We will discuss only the quadratic
divergences that come from corrections to the Higgs mass due to a fermion. The
generalization for the contributions from the gauge bosons or the self-interaction is
straightforward. The interactions in this toy model of the new scalar fields with the
Higgs are described by the Lagrangian
Lfefφe = λfe|φ|
2

|feL|
2 + |feR|
2

. (2.27
36 Particle Physics
It can be easily checked that the quadratic divergence coming from a fermion at one
loop is exactly canceled, as long as the new quartic coupling λfe obeys the relation
λfe = −λ
2
f
(λf is the Yukawa coupling for the fermion f).
2.3 A brief summary of Supersymmetry
Supersymmetry (SUSY) is a symmetry relating fermions and bosons. The supersymmetry transformation should turn a boson state into a fermion state and vice versa. If
Q is the operator that generates such transformations, then
Q |bosoni = |fermioni Q |fermioni = |bosoni. (2.28)
Due to commutation and anticommutation rules of bosons and fermions, Q has to
be an anticommuting spinor operator, carrying spin angular momentum 1/2. Since
spinors are complex objects, the hermitian conjugate Q†
is also a symmetry operator4
.
There is a no-go theorem, the Coleman-Mandula theorem [106], that restricts the
conserved charges which transform as tensors under the Lorentz group to the generators
of translations Pµ and the generators of Lorentz transformations Mµν. Although this
theorem can be evaded in the case of supersymmetry due to the anticommutation
properties of Q, Q†
[107], it restricts the underlying algebra of supersymmetry [108].
Therefore, the basic supersymmetric algebra can be written as5
{Q, Q†
} = P
µ
, (2.29a)
{Q, Q} = {Q
†
, Q†
} = 0, (2.29b)
[P
µ
, Q] = [P
µ
, Q] = 0. (2.29c)
In the following, we summarize the basic conclusions derived from this algebra.
• The single-particle states of a supersymmetric theory fall into irreducible representations of the SUSY algebra, called supermultiplets. A supermultiplet contains
both fermion and boson states, called superpartners.
• Superpartners must have equal masses: Consider |Ωi and |Ω
′
i as the superpartners, |Ω
′
i should be proportional to some combination of the Q and Q† operators
acting on |Ωi, up to a space-time translation or rotation. Since −P
2
commutes
with Q, Q† and all space-time translation and rotation operators, |Ωi, |Ω
′
i will
have equal eigenvalues of −P
2 and thus equal masses.
• Superpartners must be in the same representation of gauge groups, since Q, Q†
commute with the generators of gauge transformations. This means that they
have equal charges, weak isospin and color degrees of freedom.
4We will confine ourselves to the phenomenologically more interesting case of N = 1 supersymmetry, with N referring to the number of distinct copies of Q, Q†
.
5We present a simplified version, omitting spinor indices in Q and Q†
.
2.3 A brief summary of Supersymmetry 37
• Each supermultiplet contains an equal number of fermion and boson degrees of
freedom (nF and nB, respectively): Consider the operator (−1)2s
, with s the spin
angular momentum, and the states |ii that have the same eigenvalue p
µ of P
µ
.
Then, using the SUSY algebra (2.29) and the completeness relation P
i
|ii hi| =
1, we have P
i
hi|(−1)2sP
µ
|ii = 0. On the other hand, P
i
hi|(−1)2sP
µ
|ii =
p
µTr [(−1)2s
] ∝ nB − nF . Therefore, nF = nB.
As addendum to the last point, we see that two kind of supermultiplets are possible
(neglecting gravity):
• A chiral (or matter or scalar ) supermultiplet, which consists of a single Weyl
fermion (with two spin helicity states, nF = 2) and two real scalars (each with
nB = 1), which can be replaced by a single complex scalar field.
• A gauge (or vector ) supermultiplet, which consists of a massless spin 1 boson
(two helicity states, nB = 2) and a massless spin 1/2 fermion (nF = 2).
Other combinations either are reduced to combinations of the above supermultiplets
or lead to non-renormalizable interactions.
It is possible to study supersymmetry in a geometric approach, using a space-time
manifold extended by four fermionic (Grassmann) coordinates. This manifold is called
superspace. The fields, in turn, expressed in terms of the extended set of coordinates
are called superfields. We are not going to discuss the technical details of this topic
(the interested reader may refer to the rich bibliography, for example [109–111]).
However, it is important to mention a very useful function of the superfields, the
superpotential. A generic form of a (renormalizable) superpotential in terms of the
superfields Φ is the following b
W =
1
2
MijΦbiΦbj +
1
6
y
ijkΦbiΦbjΦbk. (2.30)
The Lagrangian density can always be written according to the superpotential. The
superpotential has also to fulfill some requirements. In order for the Lagrangian to
be supersymmetric invariant, W has to be holomorphic in the complex scalar fields
(it does not involve hermitian conjugates Φb† of the superfields). Conventionally, W
involves only left chiral superfields. Instead of the SU(2)L singlet right chiral fermion
fields, one can use their left chiral charge conjugates.
As we mentioned before, the members of a supermultiplet have equal masses. This
contradicts our experience, since the partners of the light SM particles would have been
detected long time ago. Hence, the supersymmetry should be broken at a large energy
scale. The common approach is that SUSY is broken in a hidden sector, very weakly
coupled to the visible sector. Then, one has to explain how the SUSY breaking mediated to the visible sector. The two most popular scenarios are the gravity mediation
scenario [112–114] and the Gauge-Mediated SUSY Breaking (GSMB) [113, 115–117],
where the mediation occurs through gauge interactions.
There are two approaches with which one can address the SUSY breaking. In the
first approach, one refers to a GUT unification and determines the supersymmetric
38 Particle Physics
breaking parameters at low energies through the renormalization group equations.
This approach results in a small number of free parameters. In the second approach,
the starting point is the low energy scale. In this case, the SUSY breaking has to be
parametrized by the addition of breaking terms to the low energy Lagrangian. This
results in a larger set of free parameters. These terms should not reintroduce quadratic
divergences to the scalar masses, since the cancellation of these divergences was the
main motivation for SUSY. Then, one talks about soft breaking terms.
2.4 The Minimal Supersymmetric Standard Model
One can construct a supersymmetric version of the standard model with a minimal
content of particles. This model is known as the Minimal Supersymmetric Standard
Model (MSSM). In a SUSY extension of the SM, each of the SM particles is either in a
chiral or in a gauge supermultiplet, and should have a superpartner with spin differing
by 1/2.
The spin-0 partners of quarks and leptons are called squarks and sleptons, respectively (or collectively sfermions), and they have to reside in chiral supermultiplets.
The left- and right-handed components of fermions are distinct 2-component Weyl
fermions with different gauge transformations in the SM, so that each must have its
own complex scalar superpartner. The gauge bosons of the SM reside in gauge supermultiplets, along with their spin-1/2 superpartners, which are called gauginos. Every
gaugino field, like its gauge boson partner, transforms as the adjoint representation of
the corresponding gauge group. They have left- and right-handed components which
are charge conjugates of each other: (λeL)
c = λeR.
The Higgs boson, since it is a spin-0 particle, should reside in a chiral supermultiplet. However, we saw (in the fermionic part of the SM Lagrangian, Eq. (2.10b))
that the Y = 1/2 Higgs in the SM can give mass to both up- and down-type quarks,
only if the conjugate Higgs field with Y = −1/2 is involved. Since in the superpotential there are no conjugate fields, two Higgs doublets have to be introduced. Each
Higgs supermultiplet would have hypercharge Y = +1/2 or Y = −1/2. The Higgs
with the negative hypercharge gives mass to the down-type fermions and it is called
down-type Higgs (Hd, or H1 in the SLHA convention [118]) and the other one gives
mass to up-type fermions and it is called up-type Higgs (Hu, or H2).
The MSSM respects a discrete Z2 symmetry, the R-parity. If one writes the most
general terms in the supersymmetric Lagrangian (still gauge-invariant and holomorphic), some of them would lead to non-observed processes. The most obvious constraint
comes from the non-observed proton decay, which arises from a term that violates both
lepton and baryon numbers (L and B, respectively) by one unit. In order to avoid these
terms, R-parity, a multiplicative conserved quantum number, is introduced, defined as
PR = (−1)3(B−L)+2s
, (2.31)
with s the spin of the particle.
The R even particles are the SM particles, whereas the R odd are the new particles
introduced by the MSSM and are called supersymmetric particles. Due to R-parity,
2.4 The Minimal Supersymmetric Standard Model 39
if it is exactly conserved, there can be no mixing among odd and even particles and,
additionally, each interaction vertex in the theory can only involve an even number of
supersymmetric particles. The phenomenological consequences are quite important.
First, the lightest among the odd-parity particles is stable. This particle is known
as the lightest supersymmetric particle (LSP). Second, in collider experiments, supersymmetric particles can only be produced in pairs. The first of these consequences
was a breakthrough for the incorporation of DM into a general theory. If the LSP is
electrically neutral, it interacts only weakly and it consists an attractive candidate for
DM.
We are not going to enter further into the details of the MSSM6
. Although MSSM
offers a possible DM candidate, there is a strong theoretical reason to move from the
minimal model. This reason is the so-called µ-problem of the MSSM, with which we
begin the discussion of the next chapter, where we shall describe more thoroughly the
Next-to-Minimal Supersymmetric Standard Model.
6We refer to [110] for an excellent and detailed description of MSSM.
40 Particle Physics
Part II
Dark Matter in the
Next-to-Minimal Supersymmetric
Standard Model

CHAPTER 3
THE NEXT-TO-MINIMAL
SUPERSYMMETRIC STANDARD
MODEL
The Next-to-Minimal Supersymmetric Standard Model (NMSSM) is an extension of
the MSSM by a chiral, SU(2)L singlet superfield Sb (see [119, 120] for reviews). The
introduction of this field solves the µ-problem1
from which the MSSM suffers, but
also leads to a different phenomenology from that of the minimal model. The scalar
component of the additional field mixes with the scalar Higgs doublets, leading to three
CP-even mass eigenstates and two CP-odd eigenstates (as in the MSSM a doublet-like
pair of charged Higgs also exists). On the other hand, the fermionic component of the
singlet (singlino) mixes with gauginos and higgsinos, forming five neutral states, the
neutralinos.
Concerning the CP-even sector, a new possibility opens. The lightest Higgs mass
eigenstate may have evaded the detection due to a sizeable singlet component. Besides,
the SM-like Higgs is naturally heavier than in the MSSM [123–126]. Therefore, a SMlike Higgs mass ∼ 125 GeV is much easier to explain [127–141]. The singlet component
of the CP-odd Higgs also allows for a potentially very light pseudoscalar with suppressed couplings to SM particles, with various consequences, especially on low energy
observables (for example, [142–145]). The singlino component of the neutralino may
also play an important role for both collider phenomenology and DM. This is the case
when the neutralino is the LSP and the lightest neutralino has a significant singlino
component.
We start the discussion about the NMSSM by describing the µ-problem and how
this is solved in the context of the NMSSM. In Sec. 3.2 we introduce the NMSSM
Lagrangian and we write the mass matrices of the Higgs sector particles and the su1However, historically, the introduction of a singlet field preceded the µ-problem, e.g. [104, 105,
121, 122].
44 The Next-to-Minimal Supersymmetric Standard Model
persymmetric particles, at tree level. We continue by examining, in Sec. 3.3, the DM
candidates in the NMSSM and particularly the neutralino. The processes which determine the neutralino relic density are described in Sec. 3.4. The detection possibilities
of a potential NMSSM neutralino as DM are discussed in (Sec. 3.5). We close this
chapter (Sec. 3.6) by examining possible ways to include non-zero neutrino masses and
the additional DM candidates that are introduced.
3.1 Motivation – The µ-problem of the MSSM
As we saw, the minimal extension of the SM, the MSSM, contains two Higgs SU(2)L
doublets Hu and Hd. The Lagrangian of the MSSM should contain a supersymmetric
mass term, µHuHd, for these two doublets. There are several reasons, which we will
subsequently review, that require the existence of such a term. On the other hand,
the fact that |µ| cannot be very large, actually it should be of the order of the EW
scale, brings back the problem of naturalness. A parameter of the model should be
much smaller than the “natural” scale (the GUT or the Planck scale) before the EW
symmetry breaking. This leads to the so-called µ-problem of the MSSM [146].
The reasons that such a term should exist in the Lagrangian of the MSSM are
mainly phenomenological. The doublets Hu and Hd are components of chiral superfields that also contain fermionic SU(2)L doublets. Their electrically charged components mix with the superpartners of the W± bosons, forming two charged Dirac
fermions, the charginos. The unsuccessful searches for charginos in LEP have excluded
charginos with masses almost up to its kinetic limit (∼ 104 GeV) [147]. Since the µ term
determines the mass of the charginos, µ cannot be zero and actually |µ| >∼ 100 GeV,
independently of the other free parameters of the model. Moreover, µ = 0 would result
in a Peccei-Quinn symmetry of the Higgs sector and an undesirable massless axion.
Finally, there is one more reason for µ 6= 0 related to the mass generation by the Higgs
mechanism. The term µHuHd will be accompanied by a soft SUSY breaking term
BµHuHd. This term is necessary so that both neutral components of Hu and Hd are
non-vanishing at the minimum of the potential.
The Higgs mechanism also requires that µ is not too large. In order to generate
the EW symmetry breaking, the Higgs potential has to be unstable at its origin Hu =
Hd = 0. Soft SUSY breaking terms for Hu and Hd of the order of the SUSY breaking
scale generate such an instability. However, the µ induced squared masses for Hu,
Hd are always positive and would destroy the instability in case they dominate the
negative soft mass terms.
The NMSSM is able to solve the µ-problem by dynamically generating the mass
µ. This is achieved by the introduction of an SU(2)L singlet scalar field S. When S
acquires a vev, a mass term for the Hu and Hd emerges with an effective mass µeff of
the correct order, as long as the vev is of the order of the SUSY breaking scale. This
can be obtained in a more “natural” way through the soft SUSY breaking terms.
3.2 The NMSSM Lagrangian 45
3.2 The NMSSM Lagrangian
All the necessary information for the Lagrangian of the NMSSM can be extracted from
the superpotential and the soft SUSY breaking Lagrangian, containing the soft gaugino and scalar masses, and the trilinear couplings. We begin with the superpotential,
writing all the interactions of the NMSSM superfields, which include the MSSM superfields and the additional gauge singlet chiral superfield2 Sb. Hence, the superpotential
reads
W = λSbHbu · Hbd +
1
3
κSb3
+ huQb · HbuUbc
R + hdHbd · QbDbc
R + heHbd · LbEbc
R.
(3.1)
The couplings to quarks and leptons have to be understood as 3 × 3 matrices and the
quark and lepton fields as vectors in the flavor space. The SU(2)L doublet superfields
are given (as in the MSSM) by
Qb =

UbL
DbL
!
, Lb =

νb
EbL
!
, Hbu =

Hb +
u
Hb0
u
!
, Hbd =

Hb0
d
Hb −
d
!
(3.2)
and the product of two doublets is given by, for example, Qb · Hbu = UbLHb0
u − Hb +
u DbL.
An important fact to note is that the superpotential given by (3.1) does not include all possible renormalizable couplings (which respect R-parity). The most general
superpotential would also include the terms
W ⊃ µHbu · Hbd +
1
2
µ
′Sb2 + ξF s, b (3.3)
with the first two terms corresponding to supersymmetric masses and the third one,
with ξF of dimension mass2
, to a tadpole term. However, the above dimensionful
parameters µ, µ
′ and ξF should be of the order of the SUSY breaking scale, a fact
that contradicts the motivation behind the NMSSM. Here, we omit these terms and
we will work with the scale invariant superpotential (3.1). The Lagrangian of a scale
invariant superpotential possesses an accidental Z3 symmetry, which corresponds to a
multiplication of all the components of all chiral fields by a phase ei2π/3
.
The corresponding soft SUSY breaking masses and couplings are
−Lsof t = m2
Hu
|Hu|
2 + m2
Hd
|Hd|
2 + m2
S
|S|
2
+ m2
Q|Q|
2 + m2
D|DR|
2 + m2
U
|UR|
2 + m2
L
|L|
2 + m2
E|ER|
2
+

huAuQ · HuU
c
R − hdAdQ · HdD
c
R − heAeL · HdE
c
R
+λAλHu · HdS +
1
3
κAκS
3 + h.c.

+
1
2
M1λ1λ1 +
1
2
M2λ
i
2λ
i
2 +
1
2
M3λ
a
3λ
a
3
,
(3.4)
2Here, the hatted capital letters denote chiral superfields, whereas the corresponding unhatted
ones indicate their complex scalar components.
46 The Next-to-Minimal Supersymmetric Standard Model
where we have also included the soft breaking masses for the gauginos. λ1 is the U(1)Y
gaugino (bino), λ
i
2 with i = 1, 2, 3 is the SU(2)L gaugino (winos) and, finally, the λ
a
3
with a = 1, . . . , 8 denotes the SU(3)c gaugino (gluinos).
The scalar potential, expressed by the so-called D and F terms, can be written
explicitly using the general formula
V =
1
2

D
aD
a + D
′2

+ F
⋆
i Fi
, (3.5)
where
D
a = g2Φ
∗
i T
a
ijΦj (3.6a)
D
′ =
1
2
g1YiΦ
∗
i Φi (3.6b)
Fi =
∂W
∂Φi
. (3.6c)
We remind that T
a are the SU(2)L generators and Yi the hypercharge of the scalar
field Φi
. The Yukawa interactions and fermion mass terms are given by the general
Lagrangian
LY ukawa = −
1
2
∂
2W
∂Φi∂Φj
ψiψj + h.c.
, (3.7)
using the superpotential (3.1). The two-component spinor ψi
is the superpartner of
the scalar Φi
.
3.2.1 Higgs sector
Using the general form of the scalar potential, the following Higgs potential is derived
VHiggs =

λ

H
+
u H
−
d − H
0
uH
0
d

+ κS2

2
+

m2
Hu + |λS|
2

H
0
u

2
+

H
+
u

2

+

m2
Hd + |λS|
2

H
0
d

2
+

H
−
d

2

+
1
8

g
2
1 + g
2
2

H
0
u

2
+

H
+
u

2
−

H
0
d

2
−

H
−
d

2
2
+
1
2
g
2
2

H
+
u H
0
d
⋆
+ H
0
uH
−
d
⋆

2
+ m2
S
|S|
2 +

λAλ

H
+
u H
−
d − H
0
uH
0
d

S +
1
3
κAκS
3 + h.c.

.
(3.8)
The neutral physical Higgs states are defined through the relations
H
0
u = vu +
1
√
2
(HuR + iHuI ), H0
d = vd +
1
√
2
(HdR + iHdI ),
S = s +
1
√
2
(SR + iSI ),
3.2.1 Higgs sector 47
where vu, vd and s are, respectively, the real vevs of Hu, Hd and S, which have to be
obtained from the minima of the scalar potential (3.8), after expanding the fields using
Eq. (3.9). We notice that when S acquires a vev, a term µeffHbu · Hbd appears in the
superpotential, with
µeff = λs, (3.10)
solving the µ-problem.
Therefore, the Higgs sector of the NMSSM is characterized by the seven parameters
λ, κ, m2
Hu
, m2
Hd
, m2
S
, Aλ and Aκ. One can express the three soft masses by the three
vevs using the minimization equations of the Higgs potential (3.8), which are given by
vu

m2
Hu + µ
2
eff + λ
2
v
2
d +
1
2
g
2

v
2
u − v
2
d

− vdµeff(Aλ + κs) = 0
vd

m2
Hd + µ
2
eff + λ
2
v
2
u +
1
2
g
2

v
2
d − v
2
u

− vuµeff(Aλ + κs) = 0
s

m2
S + κAκs + 2κ
2σ
2 + λ
2

v
2
u + v
2
d

− 2λκvuvd

− λAλvuvd = 0,
(3.11)
where we have defined
g
2 ≡
1
2

g
2
1 + g
2
2

. (3.12)
One can also define the β angle by
tan β =
vu
vd
. (3.13)
The Z boson mass is given by MZ = gv with v
2 = v
2
u + v
2
d ≃ (174 GeV)2
. Hence, with
MZ given, the set of parameters that describes the Higgs sector of the NMSSM can be
chosen to be the following
λ, κ, Aλ, Aκ, tan b and µeff. (3.14)
CP-even Higgs masses
One can obtain the Higgs mass matrices at tree level by expanding the Higgs potential
(3.8) around the vevs, using Eq. (3.9). We begin by writing3
the squared mass matrix
M2
S
of the scalar Higgses in the basis (HdR, HuR, SR):
M2
S =


g
2
v
2
d + µ tan βBeff (2λ
2 − g
2
) vuvd − µBeff 2λµvd − λ (Aλ + 2κs) vu
g
2
v
2
u +
µ
tan βBeff 2λµvu − λ (Aλ + 2κs) vd
λAλ
vuvd
s + κAκs + (2κs)
2

 ,
(3.15)
where we have defined Beff ≡ Aλ + κs (it plays the role of the B parameter of the
MSSM).
3For economy of space, we omit in this expression the subscript from µ
48 The Next-to-Minimal Supersymmetric Standard Model
Although an analytical diagonalization of the above 3 × 3 mass matrix is lengthy,
there is a crucial conclusion that comes from the approximate diagonalization of the
upper 2 × 2 submatrix. If it is rotated by an angle β, one of its diagonal elements
is M2
Z
(cos2 2β +
λ
2
g
2 sin2
2β) which is an upper bound for its lightest eigenvalue. The
first term is the same one as in the MSSM. The conclusion is that in the NMSSM
the lightest CP-even Higgs can be heavier than the corresponding of the MSSM, as
long as λ is large and tan β relatively small. Therefore, it is much easier to explain
the observed mass of the SM-like Higgs. However, λ is bounded from above in order
to avoid the appearance of the Landau pole below the GUT scale. Depending on the
other free parameters, λ should obey λ <∼ 0.7.
CP-odd Higgs masses
For the pseudoscalar case, the squared mass matrix in the basis (HdI , HuI , SI ) is
M2
P =


µeff (Aλ + κs) tan β µeff (Aλ + κs) λvu (Aλ − 2κs)
µeff
tan β
(Aλ + κs) λvd (Aλ − 2κs)
λ (Aλ + 4κs)
vuvd
s − 3κAκs

 . (3.16)
One eigenstate of this matrix corresponds to an unphysical massless Goldstone
boson G. In order to drop the Goldstone boson, we write the matrix in the basis
(A, G, SI ) by rotating the upper 2 × 2 submatrix by an angle β. After dropping the
massless mode, the 2 × 2 squared mass matrix turns out to be
M2
P =
2µeff
sin 2β
(Aλ + κs) λ (Aλ − 2κs) v
λ (Aλ + 4κs)
vuvd
s − 3Aκs
!
. (3.17)
Charged Higgs mass
The charged Higgs squared mass matrix is given, in the basis (H+
u
, H−
d
⋆
), by
M2
± =

µeff (Aλ + κs) + vuvd

1
2
g
2
2 − λ

cot β 1
1 tan β
!
, (3.18)
which contains one Goldstone boson and one physical mass eigenstate H± with eigenvalue
m2
± =
2µeff
sin 2β
(Aλ + κs) + v
2

1
2
g
2
2 − λ

. (3.19)
3.2.2 Sfermion sector
The mass matrix for the up-type squarks is given in the basis (ueR, ueL) by
Mu =

m2
u + h
2
u
v
2
u −
1
3
(v
2
u − v
2
d
) g
2
1 hu (Auvu − µeffvd)
hu (Auvu − µeffvd) m2
Q + h
2
u
v
2
u +
1
12 (v
2
u − v
2
d
) (g
2
1 − 3g
2
2
)
!
, (3.20)
3.2.3 Gaugino and higgsino sector 49
whereas for down-type squarks the mass matrix reads in the basis (deR, deL)
Md =

m2
d + h
2
d
v
2
d −
1
6
(v
2
u − v
2
d
) g
2
1 hd (Advd − µeffvu)
hd (Advd − µeffvu) m2
Q + h
2
d
v
2
d +
1
12 (v
2
u − v
2
d
) (g
2
1 − 3g
2
2
)
!
. (3.21)
The off-diagonal terms are proportional to the Yukawa coupling hu for the up-type
squarks and hd for the down-type ones. Therefore, the two lightest generations remain
approximately unmixed. For the third generation, the mass matrices are diagonalized
by a rotation by an angle θT and θB, respectively, for the stop and sbottom. The mass
eigenstates are, then, given by
et1 = cos θT
etL + sin θT
etR, et2 = cos θT
etL − sin θT
etR, (3.22)
eb1 = cos θB
ebL + sin θB
ebR, eb2 = cos θB
ebL − sin θB
ebR. (3.23)
In the slepton sector, for a similar reason, only the left- and right-handed staus are
mixed and their mass matrix
Mτ =

m2
E3 + h
2
τ
v
2
d −
1
2
(v
2
u − v
2
d
) g
2
1 hτ (Aτ vd − µeffvu)
hτ (Aτ vd − µeffvu) m2
L3 + h
2
τ
v
2
d −
1
4
(v
2
u − v
2
d
) (g
2
1 − g
2
2
)
!
(3.24)
is diagonalized after a rotation by an angle θτ . Their mass eigenstates are given by
τe1 = cos θτ τeL + sin θτ τeR, τe2 = cos θτ τeL − sin θτ τeR. (3.25)
Finally, the sneutrino masses are
mνe = m2
L −
1
4

v
2
u − v
2
d
g
2
1 + g
2
2

. (3.26)
3.2.3 Gaugino and higgsino sector
The gauginos λ1 and λ
3
2 mix with the neutral higgsinos ψ
0
d
, ψ
0
u
and ψS to form neutral
particles, the neutralinos. The 5 × 5 mass matrix of the neutralinos is written in the
basis
(−iλ1, −iλ3
2
, ψ0
d
, ψ0
u
, ψS) ≡ (B, e W , f He0
d
, He0
u
, Se) (3.27)
as
M0 =


M1 0 − √
1
2
g1vd √
1
2
g1vu 0
M2 √
1
2
g2vd − √
1
2
g2vu 0
0 −µeff −λvu
0 −λvd
2κs


. (3.28)
The mass matrix (3.28) is diagonalized by an orthogonal matrix Nij . The mass eigenstates of the neutralinos are usually denoted by χ
0
i
, with i = 1, . . . , 5, with increasing
masses (i = 1 corresponds to the lightest neutralino). These are given by
χ
0
i = Ni1Be + Ni2Wf + Ni3He0
d + Ni4He0
u + Ni5S. e (3.2
50 The Next-to-Minimal Supersymmetric Standard Model
We use the convention of a real matrix Nij , so that the physical masses mχ
0
i
are real,
but not necessarily positive.
In the charged sector, the SU(2)L charged gauginos λ
− = √
1
2
(λ
1
2 + iλ2
2
), λ
+ =
√
1
2
(λ
1
2 − iλ2
2
) mix with the charged higgsinos ψ
−
d
and ψ
+
u
, forming the charginos ψ
±:
ψ
± =

−iλ±
ψ
±
u
!
. (3.30)
The chargino mass matrix in the basis (ψ
−, ψ+) is
M± =

M2 g2vu
g2vd µeff !
. (3.31)
Since it is not symmetric, the diagonalization requires different rotations of ψ
− and
ψ
+. We denote these rotations by U and V , respectively, so that the mass eigenstates
are obtained by
χ
− = Uψ−, χ+ = V ψ+. (3.32)
3.3 DM Candidates in the NMSSM
Let us first review the characteristics that a DM candidate particle should have. First,
it should be massive in order to account for the missing mass in the galaxies. Second,
it must be electrically and color neutral. Otherwise, it would have condensed with
baryonic matter, forming anomalous heavy isotopes. Such isotopes are absent in nature. Finally, it should be stable and interact only weakly, in order to fit the observed
relic density.
In the NMSSM there are two possible candidates. Both can be stable particles if
they are the LSPs of the supersymmetric spectrum. The first one is the sneutrino (see
[148,149] for early discussions on MSSM sneutrino LSP). However, although sneutrinos
are WIMPs, their large coupling to the Z bosons result in a too large annihilation cross
section. Hence, if they were the DM particles, their relic density would have been very
small compared to the observed value. Exceptions are very massive sneutrinos, heavier
than about 5 TeV [150]. Furthermore, the same coupling result in a large scattering
cross section off the nuclei of the detectors. Therefore, sneutrinos are also excluded by
direct detection experiments.
The other possibility is the lightest neutralino. Neutralinos fulfill successfully, at
least in principle, all the requirements for a DM candidate. However, the resulting
relic density, although weakly interacting, may vary over many orders of magnitude as
a function of the free parameters of the theory. In the next sections we will investigate
further the properties of the lightest neutralino as the DM particle. We begin by
studying its annihilation that determines the DM relic density.
3.4 Neutralino relic density 51
3.4 Neutralino relic density
We remind that the neutralinos are mixed states composed of bino, wino, higgsinos
and the singlino. The exact content of the lightest neutralino determines its pair
annihilation channels and, therefore, its relic density (for detailed analyses, we refer
to [151–154]). Subsequently, we will briefly describe the neutralino pair annihilation
in various scenarios. We classify these scenarios with respect to the lightest neutralino
content.
Before we proceed, we should mention another mechanism that affects the neutralino LSP relic density. If there is a supersymmetric particle with mass close to the
LSP (but always larger), it would be abundant during the freeze-out and LSP coannihilations with this particle would contribute to the total annihilation cross section.
This particle, which is the Next-to-Lightest Supersymmetric Particle (NLSP), is most
commonly a stau or a stop. In the above sense, coannihilations refer not only to the
LSP–NLSP coannihilations, but also to the NLSP–NLSP annihilations, since the latter
reduce the number density of the NLSPs [155].
• Bino-like LSP
In principle, if the lightest neutralino is mostly bino-like, the total annihilation
cross section is expected to be small. Therefore, a bino-like neutralino LSP would
have been overabundant. The reason for this is that there is only one available
annihilation channel via t-channel sfermion exchange, since all couplings to gauge
bosons require a higgsino component. The cross section is even more reduced
when the sfermion mass is large.
However, there are still two ways to achieve the correct relic density. The first one
is using the coannihilation effect: if there is a sfermion with a mass slightly larger
(some GeV) than the LSP mass, their coannihilations can be proved to reduce
efficiently the relic density to the desired value. The second one concerns a binolike LSP, with a very small but non-negligible higgsino component. In this case,
if in addition the lightest CP-odd Higgs A1 is light enough, the annihilation to a
pair A1A1 (through an s-channel CP-even Higgs Hi exchange) can be enhanced
via Hi resonances. In this scenario a fine-tuning of the masses is necessary.
• Higgsino-like LSP
A mostly higgsino LSP is as well problematic. The strong couplings of the higgsinos to the gauge bosons lead to very large annihilation cross section. Therefore,
a possible higgsino LSP would have a very small relic density.
• Mixed bino–higgsino LSP
In this case, as it was probably expected, one can easily fit the relic density to
the observed value. This kind of LSP annihilates to W+W−, ZZ, W±H∓, ZHi
,
HiAj
, b
¯b and τ
+τ
− through s-channel Z or Higgs boson exchange or t-channel
neutralino or chargino exchange. The last two channels are the dominant ones
when the Higgs coupling to down-type fermions is enhanced, which occurs more
commonly in the regime of relatively large tan β. The annihilation channel to a
52 The Next-to-Minimal Supersymmetric Standard Model
pair of top quarks also contributes to the total cross section, if it is kinematically
allowed. However, in order to achieve the correct relic density, the higgsino
component cannot be very large.
• Singlino-like LSP
Since a mostly singlino LSP has small couplings to SM particles, the resulting relic
density is expected to be large. However, there are some annihilation channels
that can be enhanced in order to reduce the relic density. These include the
s-channel (scalar or pseudoscalar) Higgs exchange and the t-channel neutralino
exchange.
For the former, any Higgs with sufficient large singlet component gives an important contribution to the cross section, increasing with the parameter κ (since
the singlino-singlino-singlet coupling is proportional to κ). Concerning the latter
annihilation, in order to enhance it, one needs large values of the parameter λ.
In this case, the neutralino-neutralino-singlet coupling, which is proportional to
λ, is large and the annihilation proceeds giving a pair of scalar HsHs or a pair
of pseudoscalar AsAs singlet like Higgs.
As in the case of bino-like LSP, one can also use the effect of s-channel resonances
or coannihilations. In the latter case, an efficient NLSP can be the neutralino
χ
0
2
or the lightest stau τe1. In the case that the neutralino NLSP is higgsinolike, the LSP-NLSP coannihilation through a (doublet-like) Higgs exchange can
be proved very efficient. A stau NLSP reduces the relic density through NLSPNLSP annihilations, which is the only possibility in the case that both parameters
κ and λ are small. We refer to [156,157] for further discussion on this possibility.
Assuming universality conditions the wino mass M2 has to be larger than the bino
mass M1 (M2 ∼ 2M1). This is the reason that we have not considered a wino-like LSP.
3.5 Detection of neutralino DM
3.5.1 Direct detection
Since neutralinos are Majorana fermions, the effective Lagrangian describing their
elastic scattering with a quark in a nucleon can be written, in the Dirac fermion
notation, as [158]
Leff = a
SI
i χ¯
0
1χ
0
1
q¯iqi + a
SD
i χ¯
0
1γ5γµχ
0
1
q¯iγ5γ
µ
qi
, (3.33)
with i = u, d corresponding to up- and down-type quarks, respectively. The Lagrangian has to be understood as summing over the quark generations.
In this expression, we have omitted terms containing the operator ψγ¯
5ψ or a combination of ψγ¯
5γµψ and ψγ¯
µψ (with ψ = χ, q). This is a well qualified assumption:
Due to the small velocity of WIMPs, the momentum transfer ~q is very small compared
3.5.1 Direct detection 53
to the reduced mass of the WIMP-nucleus system. In the extreme limit of zero momentum transfer, the above operators vanish4
. Hence, we are left with the Lagrangian
(3.33) consisting of two terms, the first one corresponding to spin-independent (SI)
interactions and the second to spin-dependent (SD) ones. In the following, we will
focus again only to SI scattering, since the detector sensitivity to SD scattering is low,
as it has been already mentioned in Sec. 1.5.1.
The SI cross section for the neutralino-nucleus scattering can be written as [158]
(see, also, [159])
σ
SI
tot =
4m2
r
π
[Zfp + (A − Z)fn]
2
. (3.34)
mr is the neutralino-nucleus reduced mass mr =
mχmN
mχ+mN
, and Z, A are the atomic and
the nucleon number, respectively. It is more common, however, to use an expression
for the cross section normalized to the nucleon. In this case, on has for the neutralinoproton scattering
σ
SI
p =
4
π

mpmχ
0
1
mp + mχ
0
1
!2
f
2
p ≃
4m2
χ
0
1
π
f
2
p
, (3.35)
with a similar expression for the neutron.
The form factor fp is related to the couplings a to quarks through the expression
(omitting the “SI” superscripts)
fp
mp
=
X
q=u,d,s
f
p
T q
aq
mq
+
2
27
fT G X
q=c,b,t
aq
mq
. (3.36)
A similar expression may be obtained for the neutron form factor fn, by the replacement
p → n in the previous expression (henceforth, we focus to neutralino-proton scattering).
The parameters fT q are defined by the quark mass matrix elements
hp| mqqq¯ |pi = mpfT q, (3.37)
which corresponds to the contribution of the quark q to the proton mass and the
parameter fT G is related to them by
fT G = 1 −
X
q=u,d,s
fT q. (3.38)
The above parameters can be obtained by the following quantities
σπN =
1
2
(mu + md)(Bu + Bd) and σ0 =
1
2
(mu + md)(Bu + Bd − 2Bs,) (3.39)
with Bq = hN| qq¯ |Ni, which are obtained by chiral perturbation theory [160] or by
lattice simulations. Unfortunately, the uncertainties on the values of these quantities
are large (see [161], for more recent values and error bars).
4While there are possible circumstances in which the operators of (3.33) are also suppressed and,
therefore, comparable to the operators omitted, they are not phenomenologically interesting.
54 The Next-to-Minimal Supersymmetric Standard Model
χ
0
1
χ
0
1
χ
0
1 χ
0
1
qe
q q
q q
Hi
Figure 3.1: Feynman diagrams contributing to the elastic neutralino-quark scalar scattering amplitude at tree level.
The SI neutralino-nucleon interactions arise from t-channel Higgs exchange and
s-channel squark exchange at tree level (see Fig. 3.1), with one-loop contributions from
neutralino-gluon interactions. In practice, the s-channel Higgs exchange contribution
to the scattering amplitude dominates, especially due to the large masses of squarks.
In this case, the effective couplings a are given by
a
SI
d =
X
3
i=1
1
m2
Hi
C
1
i Cχ
0
1χ
0
1Hi
, aSI
u =
X
3
i=1
1
m2
Hi
C
2
i Cχ
0
1χ
0
1Hi
. (3.40)
C
1
i
and C
2
i
are the Higgs Hi couplings to down- and up-type quarks, respectively, given
by
C
1
i =
g2md
2MW cos β
Si1, C2
i =
g2mu
2MW sin β
Si2, (3.41)
with S the mixing matrix of the CP-even Higgs mass eigenstates and md, mu the
corresponding quark mass. We see from Eqs. (3.36) and (3.41) that the final cross
section (3.35) is independent of each quark mass. We write for completeness the
neutralino-neutralino-Higgs coupling Cχ
0
1χ
0
1Hi
:
Cχ
0
1χ
0
1Hi =
√
2λ (Si1N14N15 + Si2N13N15 + Si3N13N14) −
√
2κSi3N
2
15
+ g1 (Si1N11N13 − Si2N11N14) − g2 (Si1N12N13 − Si2N12N14), (3.42)
with N the neutralino mixing matrix given in (3.29).
The resulting cross section is proportional to m−4
Hi
. In the NMSSM, it is possible
for the lightest scalar Higgs eigenstate to be quite light, evading detection due to its
singlet nature. This scenario can give rise to large values of SI scattering cross section,
provided that the doublet components of th

[V]

@I.G.Y
Je me suis peut-être mal exprimé, mais comme dit plus-haut, un goût distinctif n’est pas un goût insincère. Évidemment que l’écoute de Bach leur procure des choses, ce n’est pas incompatible. Mais en aimant quelque chose, on y voit aussi son propre reflet. On s’aime en aimant. Et dans une modalité sociale, ça donne une logique de distinction par rapport au goût d’autrui. Le fait est qu’aujourd’hui (pour aller très vite) on se perçoit moins positivement en écoutant Bach qu’en écoutant du rap ou du reggaeton. Et de la même manière, l’écoute de ces musiques procure des sensations positives. Bref, le goût purement spontané n’existe pas – ce qui ne l’invalide pas, encore une fois.

[I.G.Y.]

« en aimant quelque chose, on y voit aussi son propre reflet ».

Oui tout à fait. Je pense juste que l’attitude que tu pointais dans la phrase que j’ai relevée permet bien de caractériser les logiques de distinction mais en toute généralité, pas seulement la distinction bourgeoise. La tendance à valoriser les mêmes choses que ceux qu’on valorise (puisqu’on se valorise inévitablement au travers du regard des autres, bref c’est ce que tu disais).

Tout à fait d’accord pour dire qu’on ne valorise jamais entièrement les choses pour ce qu’elles sont (même en étant très sincère/passionné).

[françois bégaudeau]

Je rappelle quand même que la discussion de départ reposait sur la notion de culture légitime
La distinction comme principe structurel, ce n’est pas périmé.
Mais cette autre fait structurel, la culture légitime, me semble en voie de péremption.

[maelstrom]

Mais ducoup c’est une bonne ou une mauvaise chose que la culture légitime devienne arriéré dans la bourgeoisie ?

[françois bégaudeau]

Avant de juger un phénomène, je propose qu’on tache de bien l’observer, nommer, préciser.

[Ludovic Bourgeois]

Tu évoques aussi ça dans ton livre sur l’art
Dont tu me parlais ?
J’ai horreur de la culture française promue
Aujourd’hui
A mes yeux
Soit elle exalte des affects femelles
Soit racailleuses débiles
Moi je le vois comme accompagnateur d’une idéologie politique
Hostile à l’homme blanc
Sommé de devenir soit fiotasse
Soit marron-mental

[maelstrom]

ludovic clique trois fois des yeux si tu est retenu en otage par jean monnaie

[maelstrom]

cligne*

[Ludovic Bourgeois]

Jean est plus « républicain » que moi et il se veut de droite absolument
Et il adore le « débat d’idées »
Trois trucs dont j’ai horreur
Mais sinon, oui j’aime bien

[maelstrom]

Qu’entend tu par marron-mental ?

[Ludovic Bourgeois]

Dans les pays de l’Est la virilité est
Associé à une froideur
Elle correspond à mes yeux à la psyché d’un homme blanc
En France, il y a la promotion de l’agressivité
Sanguine
Qui ne nous correspond pas

[maelstrom]

et le terme marron-mental correspond au quel donc ?

[Ludovic Bourgeois]

Joue pas au con
La jeunesse française est détruite par ces conneries

[maelstrom]

J’imagine que sa correspond a l’agressivité sanguine si je relis le message destiné a bégaudeau mais alors pourquoi donc utilisé le terme marron et pas rouge (pour le sang et la colère) ?

[Ludovic Bourgeois]

L’agressivité est une qualité en moyenne plus présente
Dans les populations nord africaines et africaines
Du fait d’un taux d’epitestosterone moyen ( présent chez les h mais aussi chez f) plus élevé

[maelstrom]

voila tu la dis, pourquoi autant tourner autour du pot quand c’est aussi bêtement simple

[Ludovic Bourgeois]

Ça me semblait évident dès le départ

[maelstrom]

je voulais que tu le dise clairement pour être sur mais tu est partit dans des explications alambiqué

[maelstrom]

ludovic:

[I.G.Y.]

@FB sur ce que tu pointes je suis assez d’accord, du moins j’ai le sentiment que tu touches du doigt quelque chose. N’ayant pas d’objection ni d’argument frappant, je me contente de vous lire. L’idée d’une culture, ou du moins de certaines formes de culture vues comme improductives donc délégitimées (et vice versa), c’est sans aucun doute une piste à creuser.

[V]

@i.g.y
Tu as raison, j’ai employé abusivement le verbe « caractériser ». On est donc d’accord.
Quant à ta remarque, @Francois Bégaudeau, le concept de culture légitime, entendu au sens de culture classique, devient suranné effectivement. Cela dit, le mécanisme de légitimité/illégitimité demeure. Et la bourgeoisie (du moins son pan majoritaire) légitime aujourd’hui un rapport davantage instrumental à la culture. Cela doit intriguer quand on sait que c’est précisement le « fonctionnalisme » des classes populaires que les bourgeois méprisaient.

[françois bégaudeau]

Je ne suis pas sûr qu’elle le « légitime » justement. Elle se contente de le pratiquer – sans vraiment être très consciente de ce qu’elle fait.
Je disais qu’elle délégitimait la culture. Ce n’est pas le mot. Elle la délaisse (elle commence de la délaisser). Elle n’envoie plus sa progéniture à normale sup, par exemple. Ou alors très à la marge.

[Charles]

Il est bien vrai que la culture légitime ne cesse de s’étendre et les illégitimes d’hier sont aujourd’hui portés aux nues ou tout du moins considérés : les jeux vidéos trouvent leur place dans les pages culture de Libé, le rap est devenu le genre musical central en France (la variété sans la déconsidération qui allait avec autrefois), le genre est célébré aussi bien dans la littérature (King traité comme un grand écrivain) que dans le cinéma (palme d’or et prix internationaux pour des films d’horreur) etc. Alors il y aura toujours des pratiques culturelles considérées comme légitimes que d’autres mais ce qui a changé c’est que maintenant il y a toujours un représentant du goût légitime pour prendre sa défense.

[maelstrom]

De qui tu parle précisement quand tu dis « mais ce qui a changé c’est que maintenant il y a toujours un représentant du goût légitime pour prendre sa défense. », a tu des exemples ?

[françois bégaudeau]

Il faudrait prendre des exemples d’artistes qui il y a quelques décennies auraient été évidemment délégitimés, et qui là ne le sont pas.
Tiens par exemple Jul. Un Jul, dans les années 90, à part ses fans tout le monde se serait foutu de sa gueule (moi le premier, dans une chanson par exemple). Là : respect partout (ou pour le moins non-moquerie, non-agression). Et la flamme des JO à porter.
Jul finira pas passer à Quotidien (arbitre parfait du bon gout), si ce n’est déjà fait. En tout cas zéro moquerie à son égard dans l’émission

[Charles]

Je parle des chroniqueurs culturels, télé ou presse, dans le genre d’Ariel Wizman et David Abiker sous un angle pop – je ne regarde presque plus la télé mais j’imagine que la chroniqueuse culture de Barthès fait la même chose. Libé aussi dans le genre.
Par exemple, la télé-réalité a aussi eu ses défenseurs – je ne reviens pas sur l’épisode Cahiers le connaissant mal et qui est sans doute plus compliqué que ce que je crois – mais je me souviens d’un Fred Bonnaud comparant les échanges entre candidats à du Rohmer.

[Demi Habile]

and also the definition of the unpolarized cross section to write
X
spins
Z
|M12→34|
2
(2π)
4
δ
4
(p1 + p2 − p3 − p4)
d
3p3
(2π)
32E3
d
3p4
(2π)
32E4
=
4F g1g2 σ12→34, (1.31)
where F ≡ [(p1 · p2)
2 − m2
1m2
2
]
1/2
and the spin factors g1, g2 come from the average
over initial spins. This way, the collision term (1.29) is written in a more compact form
g1
Z
C[f1]
d
3p1
(2π)
3
= −
Z
σvMøl (dn1dn2 − dn
eq
1 dn
eq
2
), (1.32)
where σ =
P
(all f)
σ12→f is the total annihilation cross section summed over all the
possible final states and vMøl ≡
F
E1E2
. The so called Møller velocity, vMøl, is defined in
such a way that the product vMøln1n2 is invariant under Lorentz transformations and,
in terms of particle velocities ~v1 and ~v2, it is given by the expression
vMøl =
h
~v2
1 − ~v2
2

2
− |~v1 × ~v2|
2
i1/2
. (1.33)
Due to symmetry considerations, the distributions in kinetic equilibrium are proportional to those in chemical equilibrium, with a proportionality factor independent of
the momentum. Therefore, the collision term (1.32), both before and after decoupling,
can be written in the form
g1
Z
C[f1]
d
3p1
(2π)
3
= −hσvMøli(n1n2 − n
eq
1 n
eq
2
), (1.34)
where the thermal averaged total annihilation cross section times the Møller velocity
has been defined by the expression
hσvMøli =
R
σvMøldn
eq
1 dn
eq
2
R
dn
eq
1 dn
eq
2
. (1.35)
We will come back to the thermal averaged cross section in the next subsection.
We are, now, able to write the full integrated Boltzmann equation, using the expressions (1.28), (1.34) that we have derived for the Liouville and the collision term,
respectively. In the simplified but interesting case of identical particles 1 and 2, the
Boltzmann equation is, finally, written as
n˙ + 3Hn = −hσvMøli(n
2 − n
2
eq). (1.36)
18 Dark Matter
However, instead of using n, it is more convenient to take the expansion of the universe
into account and calculate the number density per comoving volume Y , which can be
defined as the ratio of the number and entropy densities: Y ≡ n/s. The total entropy
density S = R3
s (R is the scale factor) remains constant, hence we can obtain a
differential equation for Y by dividing (1.36) by S. Before we write the final form
of the Boltzmann equation that it is used for the relic density calculations, we have
to change the variable that parametrizes the comoving density. In practice, the time
variable t is not convenient and the temperature of the Universe (actually the photon
temperature, since the photons were the last particles that went out of equilibrium) is
used instead. However, it proves even more useful to use as time variable the quantity
defined by x ≡ m/T with m the DM mass, so that Eq. (1.36) transforms into
dY
dx
=
1
3H
ds
dx
hσvMøli

Y
2 − Y
2
eq
. (1.37)
Last, using the Hubble parameter (1.2) for a radiation dominated Universe and the
expressions (1.20), (1.21) for the energy and entropy density, the Boltzmann equation
is written in its final form
dY
dx
= −
r
45GN
π
g
1/2
∗ m
x
2
hσvMøli

Y
2 − Y
2
eq
, (1.38)
where the effective degrees of freedom g
1/2
∗ have been defined by
g
1/2
∗ ≡
heff
g
1/2
eff

1 +
1
3
T
heff
dheff
dT

. (1.39)
The equilibrium density per comoving volume Yeq ≡ neq/s can be expressed as
Yeq(x) = 45g
4π
4
x
2K2(x)
heff(m/x)
, (1.40)
with K2 the modified Bessel function of second kind.
1.4.3 Thermal average of the annihilation cross section
We are going to derive a simple formula that one can use to calculate the thermal
average of the cross section times velocity, based again on the analysis of [38]. We will
use the assumption that equilibrium functions follow the Maxwell-Boltzmann distribution, instead of the actual Bose-Einstein or Fermi-Dirac. This is a well established
assumption if the freeze out occurs after T ≃ m/3 or for x >∼ 3, which is actually the
case for WIMPs. Under this assumption, the expression (1.35) gives, in the cosmic
comoving frame,
hσvMøli =
R
vMøle
−E1/T e
−E2/T d
3p1d
3p2
R
e
−E1/T e
−E2/T d
3p1d
3p2
. (1.4
1.4.3 Thermal average of the annihilation cross section 19
The volume element can be written as d3p1d
3p2 = 4πp1dE14πp2dE2
1
2
cos θ, with θ the
angle between ~p1 and ~p2. After changing the integration variables to E+, E−, s given
by
E+ = E1 + E2, E− = E1 − E2, s = 2m2 + 2E1E2 − 2p1p2 cos θ, (1.42)
(with s = −(p1 − p2)
2 one of the Mandelstam variables,) the volume element becomes
d
3p1d
3p2 = 2π
2E1E2dE+dE−ds and the initial integration region
{E1 > m, E2 > m, | cos θ| ≤ 1i
transforms into
|E−| ≤
1 −
4m2
s
1/2
(E
2
+ − s)
1/2
, E+ ≥
√
s, s ≥ 4m2
. (1.43)
After some algebraic calculations, it can be found that the quantity hσvMøliE1E2
depends only on s, specifically vMølE1E2 =
1
2
p
s(s − 4m2
). Hence, the numerator of the expression (1.41), which after changing the integration variables reads
2π
2
R
dE+
R
dE−
R
dsσvMølE1E2e
−E+/T , can be written, eventually, as
Z
vMøle
−E1/T e
−E2/T = 2π
2
Z ∞
4m2
dsσ(s − 4m2
)
Z
dE+e
−E+/T (E
2
+ − s)
1/2
. (1.44)
The integral over E+ can be written with the help of the modified Bessel function of
the first kind K1 as √
s T K1(
√
s/T). The denominator of (1.41) can be treated in a
similar way, so that the thermal average is, finally, given by the expression
hσvMøli =
1
8m4TK2
2
(x)
Z ∞
4m2
ds σ(s)(s − 4m2
)
√
s K1(
√
s/T). (1.45)
Eqs. (1.38)–(1.40) along with this last Eq. (1.45) are all we need in order to calculate
the relic density of a WIMP, if its total annihilation cross section in terms of the
Mandelstam variable s is known.
In many cases, in order to avoid the numerical integration in Eq. (1.45), an approximation for hσvMøli can be used. The thermal average is expanded in powers of x
−1
(or, equivalently, in powers of the squared WIMP velocity):
hσvMøli = a + bx−1 + . . . . (1.46)
(The coefficient a corresponds to the s-wave contribution to the cross section, the
coefficient b to the p-wave contribution, and so on.) This partial wave expansion gives
a quite good approximation, provided there are no s-channel resonances and thresholds
for the final states [39].
In [40], it was shown that, after expanding the integrands of Eq. (1.41) in powers
of x
−1
, all the integrations can be performed analytically. As we saw, the expression
20 Dark Matter
vMølE1E2 depends on momenta only through s. Therefore, one can form the Lorentz
invariant quantity
w(s) ≡ σ(s)vMølE1E2 =
1
2
σ(s)
p
s(s − 4m2
). (1.47)
The integration involves the Taylor expansion of this quantity w around s/4m2 = 1
and the general formula for the partial wave expansion of the thermal average is [40]
hσvMøli =
1
m2

w −
3
2
(2w − w
′
)x
−1 +
3
8
(16w − 8w
′ + 5w
′′)x
−2
−
5
16
(30w − 15w
′ + 3w
′′ − 7x
′′′)x
−3 + O(x
−4
)

s/4m2=1
, (1.48)
where primes denote derivatives with respect to s/4m2 and all quantities have to be
evaluated at s = 4m2
.
1.5 Direct Detection of DM
Since the beginning of 1980s, it has been realized that besides the numerous facts showing evidence for the existence of these new dark particles, it is also possible to detect
them directly. Already in 1985, two pioneering articles [41, 42] appeared, describing
the detection methods for WIMPs. Since WIMPs are expected to cluster gravitationally together with ordinary stars in the Milky Way halo, they would pass also through
Earth and, in principle, they can be detected through scattering with the nuclei in a
detector’s material. In practice, one has to measure the recoil energy deposited by this
scattering.
However, although one can deduce from rotation curves that DM dominates the
dark halo in the outer parts of our galaxy, it is not so obvious from direct measurements
whether there is any substantial amount of DM inside the solar radius R0 ≃ 8 kpc.
Using indirect methods (involving the determination of the gravitational potential,
through the measuring of the kinematics of stars, both near the mid-plane of the
galactic disk and at heights several times the disk thickness), it is almost certain
that the DM is also present in the solar system, with a local density ρ0 = (0.3 ±
0.1) GeV cm−3
[43].
This value for the local density implies that for a WIMP mass of order ∼ 100 GeV,
the local number density is n0 ∼ 10−3
cm−3
. It is also expected that the WIMPs
velocity is similar to the velocity with which the Sun orbits around the galactic center
(v0 ≃ 220 km s−1
), since they are both moving under the same gravitational potential.
These two quantities allow to estimate the order of magnitude of the incident flux
of WIMPs on the Earth: J0 = n0v0 ∼ 105
cm−2
s
−1
. This value is manifestly large,
but the very weak interactions of the DM particles with ordinary matter makes their
detection a difficult, although in principle feasible, task. In order to compensate for
the very low WIMP-nucleus scattering cross section, very large detectors are required.
1.5.1 Elastic scattering event rate 21
1.5.1 Elastic scattering event rate
In the following, we will confine ourselves to the elastic scattering with nuclei. Although
inelastic scattering of WIMPs off nuclei in a detector or off orbital electrons producing
an excited state is possible, the event rate of these processes is quite suppressed. In
contrast, during an elastic scattering the nucleus recoils as a whole.
The direct detection experiments measure the number of events per day and per
kilogram of the detector material, as a function of the amount of energy Q deposited
in the detector. This event rate would be given by R = nWIMP nnuclei σv in a simplified
model with WIMPs moving with a constant velocity v. The number density of WIMPs
is nWIMP = ρ0/mX and the number density of nuclei is just the ratio of the detector’s
mass over the nuclear mass mN .
For accurate calculations, one should take into account that the WIMPs move in the
halo not with a uniform velocity, but rather following a velocity distribution f(v). The
Earth’s motion in the solar system should be included into this distribution function.
The scattering cross section σ also depends on the velocity. Actually, the cross section
can be parametrized by a nuclear form factor F(Q) as
dσ =
σ
4m2
r
v
2
F
2
(Q)d|~q|
2
, (1.49)
where |~q|
2 = 2m2
r
v
2
(1 − cos θ) is the momentum transferred during the scattering,
mr =
mXmN
mX+mN
is the reduced mass of the WIMP – nucleus system and θ is the scattering
angle in the center of momentum frame. Therefore, one can write a general expression
for the differential event rate per unit detector mass as
dR =
ρ0
mX
1
mN
σF2
(Q)d|~q|
2
4m2
r
v
2
vf(v)dv. (1.50)
The energy deposited in the detector (transferred to the nucleus through one elastic
scattering) is
Q =
|~q|
2
2mN
=
m2
r
v
2
mN
(1 − cos θ). (1.51)
Therefore, the differential event rate over deposited energy can be written, using the
equations (1.50) and (1.51), as
dR
dQ
=
σρ0
√
πv0mXm2
r
F
2
(Q)T(Q), (1.52)
where, following [37], we have defined the dimensionless quantity T(Q) as
T(Q) ≡
√
π
2
v0
Z ∞
vmin
f(v)
v
dv, (1.53)
with the minimum velocity given by vmin =
qQmN
2m2
r
, obtained by Eq. (1.51). Finally,
the event rate R can be calculated by integrating (1.52) over the energy
R =
Z ∞
ET
dR
dQ
dQ. (1.54)
22 Dark Matter
The integration is performed for energies larger than the threshold energy ET of the
detector, below which it is insensitive to WIMP-nucleus recoils.
Using Eqs. (1.54) and (1.52), one can derive the scattering cross section from the
event rate. The experimental collaborations prefer to give their results already in terms
of the scattering cross section as a function of the WIMP mass. To be more precise,
the WIMP-nucleus total cross section consists of two parts: the spin-dependent (SD)
cross section and the spin-independent (SI) one. The former comes from axial current
couplings, whereas the latter comes from scalar-scalar and vector-vector couplings.
The SD cross section is much suppressed compared to the SI one in the case of heavy
nuclei targets and it vanishes if the nucleus contains an even number of nucleons, since
in this case the total nuclear spin is zero.
We see that two uncertainties enter the above calculation: the exact value of the
local density ρ0 and the exact form of the velocity distribution f(v). To these, one
has to include one more. The cross section σ that appears in the previous expressions
concerns the WIMP-nucleon cross section. The couplings of a WIMP with the various
quarks that constitute the nucleon are not the same and the WIMP-nucleon cross
section depends strongly on the exact quark content of the nucleon. To be more
precise, the largest uncertainty lies on the strange content of the nucleon, but we shall
return to this point when we will calculate the cross section in a specific particle theory,
the Next-to-Minimal Supersymmetric Standard Model, in Sec. 3.5.1.
1.5.2 Experimental status
The situation of the experimental results from direct DM searches is a bit confusing. The null observations in most of the experiments led them to set upper limits
on the WIMP-nucleon cross section. These bounds are quite stringent for the spinindependent cross section7
, especially in the regime of WIMP masses of the order of
100 GeV. However, some collaborations have already reported possible DM signals,
mainly in the low mass regime. The preferred regions of these experiments do not
coincide, while some of them have been already excluded by other experiments. The
present picture, for WIMP masses ranging from 5 to 1000 GeV, is summarized in Fig.
1.5, 1.6.
Fig. 1.5 mainly presents upper bounds coming from XENON100 [44]. XENON100
[46] is an experiment located at the Gran Sasso underground laboratory in Italy. It
contains in total 165 kg of liquid Xenon, with 65 kg acting as target mass and the
rest shielding the detector from background radiation. For these upper limits, 225
live days of data were used. The minimum value for the predicted upper bounds on
the cross section is 2 · 10−45 cm2
for WIMP mass ∼ 55 GeV (at 90% confidence level),
almost one order of magnitude lower than the previously released limits [47] by the
same collaboration, using 100 live days of data.
The stringent upper bounds up-to-date (at least for WIMP mass larger than about
7 GeV) come from the first results of the LUX experiment (see Fig. 1.6), after the first
7For the spin-dependent scattering, the exclusion limits are quite relaxed. Hence, we will focus on
the SI cross sections.
1.5.2 Experimental status 23
Figure 1.5: The XENON100 exclusion limit (thick blue line), along with the expected
sensitivity in green (1σ) and yellow (2σ) band. Other upper bounds are also shown as
well as detection claims. From [44].
85.3 live-days of its operation [45]. LUX [53] is a detector containing liquid Xenon, as
XENON100, but in larger quantity, with total mass 370 kg. Its operation started on
April 2013 with a goal to clearly detect or exclude WIMPs with a spin independent
cross section ∼ 2 · 10−46 cm2
.
In Fig. 1.5, except of the XENON100 bounds and other experimental limits on larger
WIMP-nucleon cross section, some detection claims also appear. These come from
DAMA [48,49], CoGeNT [50] and CRESST-II [51] experiments. The first positive result
came from DAMA [52], back in 2000. Since then, the experiment has accumulated 1.17
ton-yr of data over 13 years of operation. DAMA consists of 250 kg of radio pure NaI
scintillator and looks for the annual modulation of the WIMP flux in order to reduce
the influence of the background.
The annual modulation of the DM flux (see [54] for a recent review) is due to the
Earth’s orbital motion relative to the rotation of the galactic disk. The galactic disk
rotation through an essentially non-rotating DM halo, creates an effective DM wind in
the solar frame. During the earth’s heliocentric orbit, this wind reaches a maximum
when the Earth is moving fastest in the direction of the disk rotation (this happens
in the beginning of June) and a minimum when it is moving fastest in the opposite
direction (beginning of December).
DAMA claims an 8.9σ annual modulation with a minimum flux on May 26±7 days,
consistent with the expectation. Since the detector’s target consists of two different
nuclei and the experiment cannot distinguish between sodium and iodine recoils, there
24 Dark Matter
Figure 1.6: The LUX 90% confidence exclusion limit (blue line) with the 1σ range
(shaded area). The XENON100 upper bound is represented by the red line. The inset
shows also preferred regions by CoGeNT (shaded light red), CDMS II silicon detector
(shaded green), CRESST II (shaded yellow) and DAMA (shaded gray). From [45].
is no model independent way to determine the exact region in the cross section versus
WIMP mass plane to which the observed modulation corresponds. However, one can
assume two cases: one that the WIMP scattering off the sodium nucleus dominates the
recoil energy and the other with the iodine recoils dominating. The former corresponds
[55] to a light WIMP (∼ 10 GeV) and quite large scattering cross section and the latter
to a heavier WIMP (∼ 50 to 100 GeV) with smaller cross section (see Fig. 1.5).
The positive result of DAMA was followed many years later by the ones of CoGeNT
and CRESST-II, and more recently by the silicon detector of CDMS [56] (Fig. 1.7).
The discrepancy of the results raised a lot of debates among the experiments (for
example, [64–67]) and by some the positive results are regarded as controversial. On
the other hand, it also raised an effort to find a physical explanation behind this
inconsistency (see, for example, [68–71]).
1.6 Indirect Methods for DM Detection
The same annihilation processes that determined the DM relic abundance in the early
Universe also occur today in galactic regions where the DM concentration is higher.
This fact rises the possibility of detecting potential WIMP pair annihilations indirectly
through their imprints on the cosmic rays. Therefore, the indirect DM searches aim
at the detection of an excess over the known astrophysical background of charged
particles, photons or neutrinos.
Charged particles – electrons, protons and their antiparticles – may originate from
direct products (pair of SM particles) of WIMP annihilations, after their decay and
1.6 Indirect Methods for DM Detection 25
Figure 1.7: The blue contours represent preferred regions for a possible signal at 68%
and 90% C.L. using the silicon detector of CMDS [56]. The blue dotted line represents
the upper limit obtained by the same analysis and the blue solid line is the combined
limit with the silicon CDMS data set reported in [57]. Other limits also appear:
from the CMDS standard germanium detector (light and dark red dashed line, for
standard [58] and low threshold analysis [59], respectively), EDELWEISS [60] (dashed
orange), XENON10 [61] (dash-dotted green) and XENON100 [44] (long-dash-dotted
green). The filled regions identify possible signal regions associated with data from
CoGeNT [62] (dashed yellow, 90% C.L.), DAMA [49,55] (dotted tan, 99.7% C.L.) and
CRESST-II [51, 63] (dash-dotted pink, 95.45% C.L.) experiments. Taken from [56].
through the process of showering and hadronization. Although the exact shape of the
resulting spectrum would depend on the specific process, it is expected to show a steep
cutoff at the WIMP mass. Once produced in the DM halo, the charged particles have
to travel to the point of detection through the turbulent galactic field, which will cause
diffusion. Apart from that, a lot of processes disturb the propagation of the charged
particles, such as bremsstrahlung, inverse Compton scattering with CMB photons and
many others. Therefore, the uncertainties that enter the propagation of the charged
flux until it reaches the telescope are important (contrary to the case of photons and
neutrinos that propagate almost unperturbed through the galaxy).
As in the case of direct detection, the experimental status of charged particle detection concerning the DM is confusing. After some hints from HEAT [72] and AMS01 [73] (the former a far-infrared telescope in Antarctica, the latter a spectrometer,
prototype for AMS-02 mounted on the International Space Station [74]), the PAMELA
satellite observed [75, 76] a steep increase in the energy spectrum of positron fraction
e
+/(e
+ + e
−)
8
. Later FERMI satellite [77] and AMS-02 [78] confirmed the results up
8The searches for charged particles focus on the antiparticles in order to have a reduced background,
26 Dark Matter
Figure 1.8: A compilation of data of charged cosmic rays, together with plausible but
uncertain astrophysical backgrounds, taken from [79]. Left: Positron flux. Center:
Antiproton flux. Right: Sum of electrons and positrons.
to energies of ∼ 200 GeV. However, the excess of positrons is not followed by an excess
of antiprotons, whose flux seems to coincide with the predicted background [75]. In
Fig. 1.8, three plots summarizing the situation are shown [79].
The observed excess is very difficult to explain in terms of DM [79]. To begin with,
the annihilation cross section required to reproduce the excess is quite large, many
orders of magnitude larger than the thermal cross section. Moreover, an “ordinary”
WIMP with large annihilation cross section giving rise to charged leptons is expected
to give, additionally, a large number of antiprotons, a fact in contradiction with the
observations. Although a lot of work has been done to fit a DM particle to the observed
pattern, it is quite possible that the excesses come from a yet unknown astrophysical
source. We are not going to discuss further this matter, but we end with a comment.
If this excess is due to a source other than DM, then a possible DM positron excess
would be lost under this formidable background.
A last hint for DM came from the detection of highly energetic photons. However,
we will interrupt this discussion, since this signal and a possible explanation is the
subject of Ch. 4. There, we will also see the upper bounds on the annihilation cross
section being set due to the absence of excesses in diffuse γ radiation.
since they are much less abundant than the corresponding particles.
CHAPTER 2
PARTICLE PHYSICS
Since the DM comprises of particles, it should be explained by a general particle physics
theory. We start in the following section by describing the Standard Model (SM) of
particle physics. Although the SM describes so far the fundamental particles and their
interactions quite accurately, it cannot provide a DM candidate. Besides, the SM
suffers from some theoretical problems, which we discuss in Sec. 2.2. We will see that
these problems can be solved if one introduces a new symmetry, the supersymmetry,
which we describe in Sec. 2.3. We finish this chapter by briefly describing in Sec. 2.4 a
supersymmetric extension of the SM with the minimal additional particle content, the
Minimal Supersymmetric Standard Model (MSSM).
2.1 The Standard Model of Particle Physics
The Standard Model (SM) of particle physics1
consists of two well developed theories,
the quantum chromodynamics (QCD) and the electroweak (EW) theory. The former
describes the strong interactions among the quarks, whereas the latter describes the
electroweak interactions (the weak and electromagnetic interactions in a unified context) between fermions. The EW theory took its final form in the late 1960s by the
introduction by S. Weinberg [85] and A. Salam [86] of the Higgs mechanism that gives
masses to the SM particles, which followed the unification of electromagnetic and weak
interactions [87,88]. At the same time, the EW model preserves the gauge invariance,
making the theory renormalizable, as shown later by ’t Hooft [89]. On the other hand,
QCD obtained its final form some years later, after the confirmation of the existence
of quarks. Of course, the history of the SM is much longer and it can be traced back to
1920s with the formulation of a theoretical basis for a Quantum Field Theory (QFT).
Since then, the SM had many successes. The SM particle content was completed with
the discovery of the heaviest of the quarks, the top quark [90,91], in 1995 and, recently,
with the discovery of the Higgs boson [92, 93].
1There are many good textbooks on the SM and Quantum Field Theory, e.g. [80–84].
28 Particle Physics
The key concept within the SM, as in every QFT, is that of symmetries. Each
interaction respects a gauge symmetry, based on a Lie algebra. The strong interaction is
described by an SU(3)c symmetry, where the subscript c stands for color, the conserved
charge of strong interactions. The EW interactions, on the other hand, are based on
a SU(2)L × U(1)Y Lie algebra. Here, as we will subsequently see, L refers to the
left-handed fermions and Y is the hypercharge, the conserved charge under the U(1).
SU(2)L conserves a quantity known as weak isospin I. Therefore, the SM contains the
internal symmetries of the unitary product group
SU(2)L × U(1)Y × SU(3)c. (2.1)
2.1.1 The particle content of the SM
We mention for completeness that particles are divided into two main classes according
to the statistics they follow. The bosons are particles with integer spin and follow the
Bose-Einstein distribution, whereas fermions have half-integer spin and follow the
Dirac-Einstein statistics, obeying the Pauli exclusion principle. In the SM, all the
fermions have spin 1/2, whereas the bosons have spin 1 with only exception the Higgs
boson, which is a scalar (spin zero). We begin the description of the SM particles with
the fermions.
Each fermion is classified in irreducible representations of each individual Lie algebra, according to the conserved quantum numbers, i.e. the color C, the weak isospin
I and the hypercharge Y . A first classification of fermions can be done into leptons
and quarks, which transform differently under the SU(3)c. Leptons are singlets under
this transformation, while quarks act as triplets (the fundamental representation of
this group). The EW interactions violate maximally the parity symmetry and SU(2)L
acts only on states with negative chirality (left-handed). A Dirac spinor Ψ can be
decomposed into left and right chirality components using, respectively, the projection
operators PL =
1
2
(1 − γ5) and PR =
1
2
(1 + γ5):
ΨL = PLΨ and ΨR = PRΨ. (2.2)
Left-handed fermions have I = 1/2, with a third component of the isospin I3 = ±1/2.
Fermions with positive I3 are called up-type fermions and those with negative are
called down-type. These behave the same way under SU(2)L and form doublets with
one fermion of each type. On the other hand, right-handed fermions have I = 0 and
form singlets that do not undergo weak interactions. The hypercharge is written in
terms of the electric charge Q and the third component of the isospin I3 through the
Gell-Mann–Nishijima relation:
Q = I3 + Y/2. (2.3)
Therefore, left- and right-handed components transform differently under the U(1)Y ,
since they have different hypercharge.
The fermionic sector of the SM comprises three generations of fermions, transforming as spinors under Lorentz transformations. Each generation has the same structure.
For leptons, it is an SU(2)L doublet with components consisting of one left-handed
2.1.2 The SM Lagrangian 29
charged lepton and one neutrino (neutrinos are only left-handed in the SM), along
with a gauge singlet right-handed charged lepton. The quark doublet consists of an
up- (u) and a down-type (d) (left-handed) quark and the pattern is completed by the
two corresponding SU(2)L singlet right-handed quarks. We write these representations
as
Quarks: Q ≡

u
i
L
d
i
L
!
, ui
R, di
R Leptons: L ≡

ν
i
L
e
i
L
!
, ei
R, (2.4)
with i = 1, 2, 3 the generation index.
Having briefly described the fermionic sector, we turn to the bosonic sector of
the SM. It consists of the gauge bosons that mediate the interactions and the Higgs
boson that gives masses to the particles through a spontaneous symmetry breaking,
the electroweak symmetry breaking (EWSB) [94–98], which we shall describe in Sec.
2.1.3. Before the EWSB, these bosons are
• three Wa
µ
(a = 1, 2, 3) weak bosons, associated with the generators of SU(2)L,
• one neutral Bµ boson, associated with the generator of U(1)Y ,
• eight gluons Ga
µ
(a = 1, . . . , 8), associated with the generators of SU(3)c, and
• the complex scalar Higgs doublet Φ =
φ
+
φ
0
!
.
After the EWSB, the EW boson states mix and give the two W± bosons, the neutral
Z boson and the massless photon γ. From the symmetry breaking, one scalar degree of
freedom remains which is the famous (neutral) Higgs boson [97–99]. We will return to
the mixed physical states, after describing the Higgs mechanism for symmetry breaking.
A complete list of the SM particles (the physical states after EWSB) is shown in Table
2.1.
2.1.2 The SM Lagrangian
The gauge bosons are responsible for the mediation of the interactions and are associated with the generators of the corresponding symmetry. The EW gauge bosons Bµ
and Wa
µ
are associated, respectively, with the generator Y of the U(1)Y and the three
generators T
a
2
of the SU(2)L. The latter are defined as half of the Pauli matrices τ
a
(T
a
2 =
1
2
τ
a
) and they obey the algebra

T
a
2
, Tb
2

= iǫabcT
c
2
, (2.5)
where ǫ
abc is the fully antisymmetric Levi-Civita tensor. The eight gluons are associated
with an equal number of generators T
a
3
(Gell-Mann matrices) of SU(3)c and obey the
Lie algebra

T
a
3
, Tb
3

= if abcT
c
3
, with Tr
T
a
3 T
b
3

=
1
2
δ
ab
, (2.6)
30 Particle Physics
Name symbol mass charge (|e|) spin
Leptons
electron e 0.511 MeV −1 1/2
electron neutrino νe 0 (<2 eV) 0 1/2
muon µ 105.7 MeV −1 1/2
muon neutrino νµ 0 (<2 eV) 0 1/2
tau τ 1.777 GeV −1 1/2
tau neutrino ντ 0 (<2 eV) 0 1/2
Quarks
up u 2.7
+0.7
−0.5 MeV 2/3 1/2
down d 4.8
+0.7
−0.3 MeV −1/3 1/2
strange s (95 ± 5) MeV −1/3 1/2
charm c (1.275 ± 0.025) GeV 2/3 1/2
bottom b (4.18 ± 0.03) GeV −1/3 1/2
top t (173.5 ± 0.6 ± 0.8) GeV 2/3 1/2
Bosons
photon γ 0 (<10−18 eV) 0 (<10−35) 1
W boson W± (80.385 ± 0.015) GeV ±1 1
Z boson Z (91.1876 ± 0.0021) GeV 0 1
gluon g 0 (.O(1) MeV) 0 1
Higgs H
(125.3 ± 0.4 ± 0.5) GeV
0 0
(126.0 ± 0.4 ± 0.4) GeV
Table 2.1: The particle content of the SM. All values are those given in [100], except of
the Higgs mass that is taken from [92, 93] (up and down row, respectively), assuming
that the observed excess corresponds to the SM Higgs. The u, d and s quark masses
are estimates of so-called “current-quark masses” in a mass-independent subtraction
scheme as MS at a scale ∼ 2 GeV. The c and b quark masses are the running masses
in the MS scheme. The values in the parenthesis are the current experimental limits.
with f
abc the structure constants of the group.
Using the structure constants of the corresponding groups, we define the field
strengths for the gauge bosons as
Bµν ≡ ∂µBν − ∂νBµ, (2.7a)
Wµν ≡ ∂µWa
ν − ∂νWa
µ + g2ǫ
abcWb
µWc
ν
(2.7b)
and
G
a
µν ≡ ∂µG
a
ν − ∂νG
a
µ + g3f
abcG
b
µG
c
ν
. (2.7c)
2.1.2 The SM Lagrangian 31
We use the notation g1, g2 and g3 for the coupling constants of U(1)Y , SU(2)L and
SU(3)c, respectively. As in any Yang-Mills theory, the non-abelian gauge groups lead
to self-interactions, which is not the case for the abelian U(1)Y group.
Before we finally write the full Lagrangian, we have to introduce the covariant
derivative for fermions, which in a general form can be written as
DµΨ =
∂µ − ig1
1
2
Y Bµ − ig2T
a
2 Wa
µ − ig3T
a
3 G
a
µ

Ψ. (2.8)
This form has to be understood as that, depending on Ψ, only the relevant terms
apply, hence for SU(2)L singlet leptons only the two first terms inside the parenthesis
are relevant, for doublet leptons the three first terms and for the corresponding quark
singlets and doublets the last term also participates. We also have to notice that in
order to retain the gauge symmetry, mass terms are forbidden in the Lagrangian. For
example, the mass term mψψ¯ = m

ψ¯
LψR + ψ¯
RψL

(with ψ¯ ≡ ψ
†γ
0
) is not invariant
under SU(2)L. This paradox is solved by the introduction of the Higgs scalar field
(see next subsection). The SM Lagrangian can be now written2
, split for simplicity in
three parts, each describing the gauge bosons, the fermions and the scalar sector,
LSM = Lgauge + Lfermion + Lscalar, (2.9)
with
Lgauge = −
1
4
G
a
µνG
µν
a −
1
4
Wa
µνWµν
a −
1
4
BµνB
µν
, (2.10a)
Lfermion = iL¯Dµγ
µL + ie¯RDµγµeR
+ iQ¯Dµγ
µQ + iu¯RDµγ
µuR + i
¯dRDµγ
µ
dR
−

heL¯ΦeR + hdQ¯ΦdR + huQ¯ΦeuR + h.c.

(2.10b)
and
Lscalar = (DµΦ)†
(DµΦ) − V (Φ†Φ), (2.10c)
where
V (Φ†Φ) = µ
2Φ
†Φ + λ

Φ
†Φ
2
(2.11)
is the scalar Higgs potential. Φ is the conjugate of Φ, related to the charge conjugate e
by Φ =e iτ2Φ
⋆
, with τi the Pauli matrices. The covariant derivative acting on the Higgs
scalar field gives
DµΦ =
∂µ − ig1
1
2
Y Bµ − ig2T
a
2 Wa
µ

Φ. (2.12)
Before we proceed to the description of the Higgs mechanism, a last comment concerning the SM Lagrangian is in order. If we restore the generation indices, we see that
2For simplicity, from now on we are going to omit the generations indice
32 Particle Physics
the Yukawa couplings h are 3 × 3, in general complex, matrices. As any complex matrix, they can be diagonalized with the help of two unitary matrices VL and VR, which
are related by VR = U
†VL with U again a unitary matrix. The diagonalization in the
quark sector to the mass eigenstates induces a mixing among the flavors (generations),
described by the Cabibbo–Kobayashi–Maskawa (CKM) matrix [101, 102]. The CKM
matrix is defined by
VCKM ≡ V
u
L
†
V
d
L
†
, (2.13)
where V
u
L
, V
d
L
are the unitary matrices that diagonalize the Yukawa couplings Hu
, Hd
,
respectively. This product of the two matrices appears in the charged current when it
is expressed in terms of the observable mass eigenstates.
2.1.3 Mass generation through the Higgs mechanism
We will start by examining the scalar potential (2.11). The vacuum expectation value
(vev) of the Higgs field hΦi ≡ h0|Φ|0i is given by the minimum of the potential. For
µ
2 > 0, the potential is always non-negative and Φ has a zero vev. The hypothesis of
the Higgs mechanism is that µ
2 < 0. In this case, the field Φ will acquire a vev
hΦi =
1
2

0
v
!
with v =
r
−
µ2
λ
. (2.14)
Since the charged component of Φ still has a zero vev, the U(1)Q symmetry of quantum
electrodynamics (QED) remains unbroken.
We expand the field Φ around the minima v in terms of real fields, and at leading
order we have
Φ(x) =
θ2(x) + iθ1(x)
√
1
2
(v + H(x)) − iθ3(x)
!
=
1
√
2
e
iθa(x)τ
a

0
v + H(x)
!
. (2.15)
We can eliminate the unphysical degrees of freedom θa, using the fact that the theory
remains gauge invariant. Therefore, we perform the following SU(2)L gauge transformation on Φ (unitary gauge)
Φ(x) → e
−iθa(x)τ
a
Φ(x), (2.16)
so that
Φ(x) = 1
√
2

0
v + H(x)
!
. (2.17)
We are going to use the following definitions for the gauge fields
W±
µ ≡
1
2

W1
µ ∓ iW2
µ

, (2.18a)
Zµ ≡
1
p
g
2
1 + g
2
2

g2W3
µ − g1Bµ

, (2.18b)
Aµ ≡
1
p
g
2
1 + g
2
2

g1W3
µ + g2Bµ

, (2.1
2.2 Limits of the SM and the emergence of supersymmetry 33
Then, the kinetic term for Φ (see Eq. (2.10c)) can be written in the unitary gauge as
(DµΦ)†
(D
µΦ) = 1
2
(∂µH)
2 + M2
W W+
µ W−µ +
1
2
M2
ZZµZ
µ
, (2.19)
with
MW ≡
1
2
g2v and MZ ≡
1
2
q
g
2
1 + g
2
2
v. (2.20)
We see that the definitions (2.18) correspond to the physical states of the gauge bosons
that have acquired masses due to the non-zero Higgs vev, given by (2.20). The photon
has remained massless, which reflects the fact that after the spontaneous breakdown of
SU(2)L × U(1)Y the U(1)Q remained unbroken. Among the initial degrees of freedom
of the complex scalar field Φ, three were absorbed by W± and Z and one remained as
the neutral Higgs particle with squared mass
m2
H = 2λv2
. (2.21)
We note that λ should be positive so that the scalar potential (2.11) is bounded from
below.
Fermions also acquire masses due to the Higgs mechanism. The Yukawa terms in
the fermionic part (2.10b) of the SM Lagrangian are written, after expanding around
the vev in the unitary gauge,
LY = −
1
√
2
hee¯L(v + H)eR −
1
√
2
hd
¯dL(v + H)dR −
1
√
2
huu¯L(v + H)uR + h.c. . (2.22)
Therefore, we can identify the masses of the fermions as
me
i =
h
i
e
v
√
2
, md
i =
h
i
d
v
√
2
, mui =
h
i
u
v
√
2
, (2.23)
where we have written explicitly the generation indices.
2.2 Limits of the SM and the emergence of supersymmetry
2.2.1 General discussion of the SM problems
The SM has been proven extremely successful and has been tested in high precision
in many different experiments. It has predicted many new particles before their final
discovery and also explained how the particles gain their masses. Its last triumph was
of course the discovery of a boson that seems to be very similar to the Higgs boson of
the SM. However, it is generally accepted that the SM cannot be the ultimate theory. It
is not only observed phenomena that the SM does not explain; SM also faces important
theoretical issues.
The most prominent among the inconsistencies of the SM with observations is the
oscillations among neutrinos of different generations. In order for the oscillations to
34 Particle Physics
φ φ
k
Figure 2.1: The scalar one-loop diagram giving rise to quadratic divergences.
occur, neutrinos should have non-zero masses. However, minimal modifications of the
SM are able to fit with the data of neutrino physics. Another issue that a more complete theory has to face is the matter asymmetry, the observed dominance of matter
over antimatter in the Universe. In addition, in order to comply with the standard
cosmological model, it has to provide the appropriate particle(s) that drove the inflation. Last, but not least, we saw that in order to explain the DM that dominates the
Universe, a massive, stable weakly interacting particle must exist. Such a particle is
not present in the SM.
On the other hand, the SM also suffers from a theoretical perspective. For example,
the SM counts 19 free parameters; one expects that a fundamental theory would have
a much smaller number of free parameters. Simple modifications of the SM have been
proposed relating some of these parameters. Grand unified theories (GUTs) unify
the gauge couplings at a high scale ∼ 1016 GeV. However, this unification is only
approximate unless the GUT is embedded in a supersymmetric framework. Another
serious problem of the SM is that of naturalness. This will be the topic of the following
subsection.
2.2.2 The naturalness problem of the SM
The presence of fundamental scalar fields, like the Higgs, gives rise to quadratic divergences. The diagram of Fig. 2.1 contributes to the squared mass of the scalar
δm2 = λ
Z Λ
d
4k
(2π)
4
k
−2
. (2.24)
This contribution is approximated by δm2 ∼ λΛ
2/(16π
2
), quadratic in a cut-off Λ,
which should be finite. For the case of the Higgs scalar field, one has to include its
couplings to the gauge fields and the top quark3
. Therefore,
δm2
H =
3Λ2
8π
2v
2

4m2
t − 2M2
W − M2
Z − m2
H

+ O(ln Λ
µ
)

, (2.25)
where we have used Eq. (2.21) and m2
H ≡ m2
0 + δm2
H.
3Since the contribution to the squared mass correction are quadratic in the Yukawa couplings (or
quark masses), the lighter quarks can be neglected
2.2.3 A way out 35
Taking Λ as a fundamental scale Λ ∼ MP l ∼ 1019 GeV we have
m2
0 = m2
H −
3Λ2
8π
2v
2

4m2
t − 2M2
W − M2
Z − m2
H

(2.26)
and we can see that m2
0 has to be adjusted to a precision of about 30 orders of magnitude
in order to achieve an EW scale Higgs mass. This is considered as an intolerable finetuning, which is against the general belief that the observable properties of a theory
have to be stable under small variations of the fundamental (bare) parameters. It is
exactly the above behavior that is considered as unnatural. Although the SM could
be self-consistent without imposing a large scale, grand unification of the parameters
introduce a hierarchy problem between the different scales.
A more strict definition of naturalness comes from ’t Hooft [103], which we rewrite
here:
At an energy scale µ, a physical parameter or set of physical parameters
αi(µ) is allowed to be very small only if the replacement αi(µ) = 0 would
increase the symmetry of the system.
Clearly, this is not the case here. Although mH is small compared to the fundamental
scale Λ, it is not protected by any symmetry and a fine-tuning is necessary.
2.2.3 A way out
The naturalness in the ’t Hooft sense is inspired by quantum electrodynamics, which is
the archetype for a natural theory. For example, the corrections to the electron mass
me are themselves proportional to me, with a dimensionless proportionality factor that
behaves like ∼ ln Λ. In general, fermion masses are protected by the chiral symmetry; small values (compared to the fundamental scale) of these masses enhances the
symmetry.
If a new symmetry exists in nature, relating fermion fields to scalar fields, then each
scalar mass would be related somehow to the corresponding fermion mass. Therefore,
the scalar mass itself can be naturally small compared to Λ, since this would mean
that the fermion mass is small, which enhances the chiral symmetry. Such a symmetry,
relating bosons to fermions and vice versa, is known as supersymmetry [104, 105].
Actually, as we will see later, if this new symmetry remains unbroken, the masses of
the conjugate bosons and fermions would have to be equal.
In order to make the above statement more concrete, we consider a toy model with
two additional complex scalar fields feL and feR. We will discuss only the quadratic
divergences that come from corrections to the Higgs mass due to a fermion. The
generalization for the contributions from the gauge bosons or the self-interaction is
straightforward. The interactions in this toy model of the new scalar fields with the
Higgs are described by the Lagrangian
Lfefφe = λfe|φ|
2

|feL|
2 + |feR|
2

. (2.27
36 Particle Physics
It can be easily checked that the quadratic divergence coming from a fermion at one
loop is exactly canceled, as long as the new quartic coupling λfe obeys the relation
λfe = −λ
2
f
(λf is the Yukawa coupling for the fermion f).
2.3 A brief summary of Supersymmetry
Supersymmetry (SUSY) is a symmetry relating fermions and bosons. The supersymmetry transformation should turn a boson state into a fermion state and vice versa. If
Q is the operator that generates such transformations, then
Q |bosoni = |fermioni Q |fermioni = |bosoni. (2.28)
Due to commutation and anticommutation rules of bosons and fermions, Q has to
be an anticommuting spinor operator, carrying spin angular momentum 1/2. Since
spinors are complex objects, the hermitian conjugate Q†
is also a symmetry operator4
.
There is a no-go theorem, the Coleman-Mandula theorem [106], that restricts the
conserved charges which transform as tensors under the Lorentz group to the generators
of translations Pµ and the generators of Lorentz transformations Mµν. Although this
theorem can be evaded in the case of supersymmetry due to the anticommutation
properties of Q, Q†
[107], it restricts the underlying algebra of supersymmetry [108].
Therefore, the basic supersymmetric algebra can be written as5
{Q, Q†
} = P
µ
, (2.29a)
{Q, Q} = {Q
†
, Q†
} = 0, (2.29b)
[P
µ
, Q] = [P
µ
, Q] = 0. (2.29c)
In the following, we summarize the basic conclusions derived from this algebra.
• The single-particle states of a supersymmetric theory fall into irreducible representations of the SUSY algebra, called supermultiplets. A supermultiplet contains
both fermion and boson states, called superpartners.
• Superpartners must have equal masses: Consider |Ωi and |Ω
′
i as the superpartners, |Ω
′
i should be proportional to some combination of the Q and Q† operators
acting on |Ωi, up to a space-time translation or rotation. Since −P
2
commutes
with Q, Q† and all space-time translation and rotation operators, |Ωi, |Ω
′
i will
have equal eigenvalues of −P
2 and thus equal masses.
• Superpartners must be in the same representation of gauge groups, since Q, Q†
commute with the generators of gauge transformations. This means that they
have equal charges, weak isospin and color degrees of freedom.
4We will confine ourselves to the phenomenologically more interesting case of N = 1 supersymmetry, with N referring to the number of distinct copies of Q, Q†
.
5We present a simplified version, omitting spinor indices in Q and Q†
.
2.3 A brief summary of Supersymmetry 37
• Each supermultiplet contains an equal number of fermion and boson degrees of
freedom (nF and nB, respectively): Consider the operator (−1)2s
, with s the spin
angular momentum, and the states |ii that have the same eigenvalue p
µ of P
µ
.
Then, using the SUSY algebra (2.29) and the completeness relation P
i
|ii hi| =
1, we have P
i
hi|(−1)2sP
µ
|ii = 0. On the other hand, P
i
hi|(−1)2sP
µ
|ii =
p
µTr [(−1)2s
] ∝ nB − nF . Therefore, nF = nB.
As addendum to the last point, we see that two kind of supermultiplets are possible
(neglecting gravity):
• A chiral (or matter or scalar ) supermultiplet, which consists of a single Weyl
fermion (with two spin helicity states, nF = 2) and two real scalars (each with
nB = 1), which can be replaced by a single complex scalar field.
• A gauge (or vector ) supermultiplet, which consists of a massless spin 1 boson
(two helicity states, nB = 2) and a massless spin 1/2 fermion (nF = 2).
Other combinations either are reduced to combinations of the above supermultiplets
or lead to non-renormalizable interactions.
It is possible to study supersymmetry in a geometric approach, using a space-time
manifold extended by four fermionic (Grassmann) coordinates. This manifold is called
superspace. The fields, in turn, expressed in terms of the extended set of coordinates
are called superfields. We are not going to discuss the technical details of this topic
(the interested reader may refer to the rich bibliography, for example [109–111]).
However, it is important to mention a very useful function of the superfields, the
superpotential. A generic form of a (renormalizable) superpotential in terms of the
superfields Φ is the following b
W =
1
2
MijΦbiΦbj +
1
6
y
ijkΦbiΦbjΦbk. (2.30)
The Lagrangian density can always be written according to the superpotential. The
superpotential has also to fulfill some requirements. In order for the Lagrangian to
be supersymmetric invariant, W has to be holomorphic in the complex scalar fields
(it does not involve hermitian conjugates Φb† of the superfields). Conventionally, W
involves only left chiral superfields. Instead of the SU(2)L singlet right chiral fermion
fields, one can use their left chiral charge conjugates.
As we mentioned before, the members of a supermultiplet have equal masses. This
contradicts our experience, since the partners of the light SM particles would have been
detected long time ago. Hence, the supersymmetry should be broken at a large energy
scale. The common approach is that SUSY is broken in a hidden sector, very weakly
coupled to the visible sector. Then, one has to explain how the SUSY breaking mediated to the visible sector. The two most popular scenarios are the gravity mediation
scenario [112–114] and the Gauge-Mediated SUSY Breaking (GSMB) [113, 115–117],
where the mediation occurs through gauge interactions.
There are two approaches with which one can address the SUSY breaking. In the
first approach, one refers to a GUT unification and determines the supersymmetric
38 Particle Physics
breaking parameters at low energies through the renormalization group equations.
This approach results in a small number of free parameters. In the second approach,
the starting point is the low energy scale. In this case, the SUSY breaking has to be
parametrized by the addition of breaking terms to the low energy Lagrangian. This
results in a larger set of free parameters. These terms should not reintroduce quadratic
divergences to the scalar masses, since the cancellation of these divergences was the
main motivation for SUSY. Then, one talks about soft breaking terms.
2.4 The Minimal Supersymmetric Standard Model
One can construct a supersymmetric version of the standard model with a minimal
content of particles. This model is known as the Minimal Supersymmetric Standard
Model (MSSM). In a SUSY extension of the SM, each of the SM particles is either in a
chiral or in a gauge supermultiplet, and should have a superpartner with spin differing
by 1/2.
The spin-0 partners of quarks and leptons are called squarks and sleptons, respectively (or collectively sfermions), and they have to reside in chiral supermultiplets.
The left- and right-handed components of fermions are distinct 2-component Weyl
fermions with different gauge transformations in the SM, so that each must have its
own complex scalar superpartner. The gauge bosons of the SM reside in gauge supermultiplets, along with their spin-1/2 superpartners, which are called gauginos. Every
gaugino field, like its gauge boson partner, transforms as the adjoint representation of
the corresponding gauge group. They have left- and right-handed components which
are charge conjugates of each other: (λeL)
c = λeR.
The Higgs boson, since it is a spin-0 particle, should reside in a chiral supermultiplet. However, we saw (in the fermionic part of the SM Lagrangian, Eq. (2.10b))
that the Y = 1/2 Higgs in the SM can give mass to both up- and down-type quarks,
only if the conjugate Higgs field with Y = −1/2 is involved. Since in the superpotential there are no conjugate fields, two Higgs doublets have to be introduced. Each
Higgs supermultiplet would have hypercharge Y = +1/2 or Y = −1/2. The Higgs
with the negative hypercharge gives mass to the down-type fermions and it is called
down-type Higgs (Hd, or H1 in the SLHA convention [118]) and the other one gives
mass to up-type fermions and it is called up-type Higgs (Hu, or H2).
The MSSM respects a discrete Z2 symmetry, the R-parity. If one writes the most
general terms in the supersymmetric Lagrangian (still gauge-invariant and holomorphic), some of them would lead to non-observed processes. The most obvious constraint
comes from the non-observed proton decay, which arises from a term that violates both
lepton and baryon numbers (L and B, respectively) by one unit. In order to avoid these
terms, R-parity, a multiplicative conserved quantum number, is introduced, defined as
PR = (−1)3(B−L)+2s
, (2.31)
with s the spin of the particle.
The R even particles are the SM particles, whereas the R odd are the new particles
introduced by the MSSM and are called supersymmetric particles. Due to R-parity,
2.4 The Minimal Supersymmetric Standard Model 39
if it is exactly conserved, there can be no mixing among odd and even particles and,
additionally, each interaction vertex in the theory can only involve an even number of
supersymmetric particles. The phenomenological consequences are quite important.
First, the lightest among the odd-parity particles is stable. This particle is known
as the lightest supersymmetric particle (LSP). Second, in collider experiments, supersymmetric particles can only be produced in pairs. The first of these consequences
was a breakthrough for the incorporation of DM into a general theory. If the LSP is
electrically neutral, it interacts only weakly and it consists an attractive candidate for
DM.
We are not going to enter further into the details of the MSSM6
. Although MSSM
offers a possible DM candidate, there is a strong theoretical reason to move from the
minimal model. This reason is the so-called µ-problem of the MSSM, with which we
begin the discussion of the next chapter, where we shall describe more thoroughly the
Next-to-Minimal Supersymmetric Standard Model.
6We refer to [110] for an excellent and detailed description of MSSM.
40 Particle Physics
Part II
Dark Matter in the
Next-to-Minimal Supersymmetric
Standard Model

CHAPTER 3
THE NEXT-TO-MINIMAL
SUPERSYMMETRIC STANDARD
MODEL
The Next-to-Minimal Supersymmetric Standard Model (NMSSM) is an extension of
the MSSM by a chiral, SU(2)L singlet superfield Sb (see [119, 120] for reviews). The
introduction of this field solves the µ-problem1
from which the MSSM suffers, but
also leads to a different phenomenology from that of the minimal model. The scalar
component of the additional field mixes with the scalar Higgs doublets, leading to three
CP-even mass eigenstates and two CP-odd eigenstates (as in the MSSM a doublet-like
pair of charged Higgs also exists). On the other hand, the fermionic component of the
singlet (singlino) mixes with gauginos and higgsinos, forming five neutral states, the
neutralinos.
Concerning the CP-even sector, a new possibility opens. The lightest Higgs mass
eigenstate may have evaded the detection due to a sizeable singlet component. Besides,
the SM-like Higgs is naturally heavier than in the MSSM [123–126]. Therefore, a SMlike Higgs mass ∼ 125 GeV is much easier to explain [127–141]. The singlet component
of the CP-odd Higgs also allows for a potentially very light pseudoscalar with suppressed couplings to SM particles, with various consequences, especially on low energy
observables (for example, [142–145]). The singlino component of the neutralino may
also play an important role for both collider phenomenology and DM. This is the case
when the neutralino is the LSP and the lightest neutralino has a significant singlino
component.
We start the discussion about the NMSSM by describing the µ-problem and how
this is solved in the context of the NMSSM. In Sec. 3.2 we introduce the NMSSM
Lagrangian and we write the mass matrices of the Higgs sector particles and the su1However, historically, the introduction of a singlet field preceded the µ-problem, e.g. [104, 105,
121, 122].
44 The Next-to-Minimal Supersymmetric Standard Model
persymmetric particles, at tree level. We continue by examining, in Sec. 3.3, the DM
candidates in the NMSSM and particularly the neutralino. The processes which determine the neutralino relic density are described in Sec. 3.4. The detection possibilities
of a potential NMSSM neutralino as DM are discussed in (Sec. 3.5). We close this
chapter (Sec. 3.6) by examining possible ways to include non-zero neutrino masses and
the additional DM candidates that are introduced.
3.1 Motivation – The µ-problem of the MSSM
As we saw, the minimal extension of the SM, the MSSM, contains two Higgs SU(2)L
doublets Hu and Hd. The Lagrangian of the MSSM should contain a supersymmetric
mass term, µHuHd, for these two doublets. There are several reasons, which we will
subsequently review, that require the existence of such a term. On the other hand,
the fact that |µ| cannot be very large, actually it should be of the order of the EW
scale, brings back the problem of naturalness. A parameter of the model should be
much smaller than the “natural” scale (the GUT or the Planck scale) before the EW
symmetry breaking. This leads to the so-called µ-problem of the MSSM [146].
The reasons that such a term should exist in the Lagrangian of the MSSM are
mainly phenomenological. The doublets Hu and Hd are components of chiral superfields that also contain fermionic SU(2)L doublets. Their electrically charged components mix with the superpartners of the W± bosons, forming two charged Dirac
fermions, the charginos. The unsuccessful searches for charginos in LEP have excluded
charginos with masses almost up to its kinetic limit (∼ 104 GeV) [147]. Since the µ term
determines the mass of the charginos, µ cannot be zero and actually |µ| >∼ 100 GeV,
independently of the other free parameters of the model. Moreover, µ = 0 would result
in a Peccei-Quinn symmetry of the Higgs sector and an undesirable massless axion.
Finally, there is one more reason for µ 6= 0 related to the mass generation by the Higgs
mechanism. The term µHuHd will be accompanied by a soft SUSY breaking term
BµHuHd. This term is necessary so that both neutral components of Hu and Hd are
non-vanishing at the minimum of the potential.
The Higgs mechanism also requires that µ is not too large. In order to generate
the EW symmetry breaking, the Higgs potential has to be unstable at its origin Hu =
Hd = 0. Soft SUSY breaking terms for Hu and Hd of the order of the SUSY breaking
scale generate such an instability. However, the µ induced squared masses for Hu,
Hd are always positive and would destroy the instability in case they dominate the
negative soft mass terms.
The NMSSM is able to solve the µ-problem by dynamically generating the mass
µ. This is achieved by the introduction of an SU(2)L singlet scalar field S. When S
acquires a vev, a mass term for the Hu and Hd emerges with an effective mass µeff of
the correct order, as long as the vev is of the order of the SUSY breaking scale. This
can be obtained in a more “natural” way through the soft SUSY breaking terms.
3.2 The NMSSM Lagrangian 45
3.2 The NMSSM Lagrangian
All the necessary information for the Lagrangian of the NMSSM can be extracted from
the superpotential and the soft SUSY breaking Lagrangian, containing the soft gaugino and scalar masses, and the trilinear couplings. We begin with the superpotential,
writing all the interactions of the NMSSM superfields, which include the MSSM superfields and the additional gauge singlet chiral superfield2 Sb. Hence, the superpotential
reads
W = λSbHbu · Hbd +
1
3
κSb3
+ huQb · HbuUbc
R + hdHbd · QbDbc
R + heHbd · LbEbc
R.
(3.1)
The couplings to quarks and leptons have to be understood as 3 × 3 matrices and the
quark and lepton fields as vectors in the flavor space. The SU(2)L doublet superfields
are given (as in the MSSM) by
Qb =

UbL
DbL
!
, Lb =

νb
EbL
!
, Hbu =

Hb +
u
Hb0
u
!
, Hbd =

Hb0
d
Hb −
d
!
(3.2)
and the product of two doublets is given by, for example, Qb · Hbu = UbLHb0
u − Hb +
u DbL.
An important fact to note is that the superpotential given by (3.1) does not include all possible renormalizable couplings (which respect R-parity). The most general
superpotential would also include the terms
W ⊃ µHbu · Hbd +
1
2
µ
′Sb2 + ξF s, b (3.3)
with the first two terms corresponding to supersymmetric masses and the third one,
with ξF of dimension mass2
, to a tadpole term. However, the above dimensionful
parameters µ, µ
′ and ξF should be of the order of the SUSY breaking scale, a fact
that contradicts the motivation behind the NMSSM. Here, we omit these terms and
we will work with the scale invariant superpotential (3.1). The Lagrangian of a scale
invariant superpotential possesses an accidental Z3 symmetry, which corresponds to a
multiplication of all the components of all chiral fields by a phase ei2π/3
.
The corresponding soft SUSY breaking masses and couplings are
−Lsof t = m2
Hu
|Hu|
2 + m2
Hd
|Hd|
2 + m2
S
|S|
2
+ m2
Q|Q|
2 + m2
D|DR|
2 + m2
U
|UR|
2 + m2
L
|L|
2 + m2
E|ER|
2
+

huAuQ · HuU
c
R − hdAdQ · HdD
c
R − heAeL · HdE
c
R
+λAλHu · HdS +
1
3
κAκS
3 + h.c.

+
1
2
M1λ1λ1 +
1
2
M2λ
i
2λ
i
2 +
1
2
M3λ
a
3λ
a
3
,
(3.4)
2Here, the hatted capital letters denote chiral superfields, whereas the corresponding unhatted
ones indicate their complex scalar components.
46 The Next-to-Minimal Supersymmetric Standard Model
where we have also included the soft breaking masses for the gauginos. λ1 is the U(1)Y
gaugino (bino), λ
i
2 with i = 1, 2, 3 is the SU(2)L gaugino (winos) and, finally, the λ
a
3
with a = 1, . . . , 8 denotes the SU(3)c gaugino (gluinos).
The scalar potential, expressed by the so-called D and F terms, can be written
explicitly using the general formula
V =
1
2

D
aD
a + D
′2

+ F
⋆
i Fi
, (3.5)
where
D
a = g2Φ
∗
i T
a
ijΦj (3.6a)
D
′ =
1
2
g1YiΦ
∗
i Φi (3.6b)
Fi =
∂W
∂Φi
. (3.6c)
We remind that T
a are the SU(2)L generators and Yi the hypercharge of the scalar
field Φi
. The Yukawa interactions and fermion mass terms are given by the general
Lagrangian
LY ukawa = −
1
2
∂
2W
∂Φi∂Φj
ψiψj + h.c.
, (3.7)
using the superpotential (3.1). The two-component spinor ψi
is the superpartner of
the scalar Φi
.
3.2.1 Higgs sector
Using the general form of the scalar potential, the following Higgs potential is derived
VHiggs =

λ

H
+
u H
−
d − H
0
uH
0
d

+ κS2

2
+

m2
Hu + |λS|
2

H
0
u

2
+

H
+
u

2

+

m2
Hd + |λS|
2

H
0
d

2
+

H
−
d

2

+
1
8

g
2
1 + g
2
2

H
0
u

2
+

H
+
u

2
−

H
0
d

2
−

H
−
d

2
2
+
1
2
g
2
2

H
+
u H
0
d
⋆
+ H
0
uH
−
d
⋆

2
+ m2
S
|S|
2 +

λAλ

H
+
u H
−
d − H
0
uH
0
d

S +
1
3
κAκS
3 + h.c.

.
(3.8)
The neutral physical Higgs states are defined through the relations
H
0
u = vu +
1
√
2
(HuR + iHuI ), H0
d = vd +
1
√
2
(HdR + iHdI ),
S = s +
1
√
2
(SR + iSI ),
3.2.1 Higgs sector 47
where vu, vd and s are, respectively, the real vevs of Hu, Hd and S, which have to be
obtained from the minima of the scalar potential (3.8), after expanding the fields using
Eq. (3.9). We notice that when S acquires a vev, a term µeffHbu · Hbd appears in the
superpotential, with
µeff = λs, (3.10)
solving the µ-problem.
Therefore, the Higgs sector of the NMSSM is characterized by the seven parameters
λ, κ, m2
Hu
, m2
Hd
, m2
S
, Aλ and Aκ. One can express the three soft masses by the three
vevs using the minimization equations of the Higgs potential (3.8), which are given by
vu

m2
Hu + µ
2
eff + λ
2
v
2
d +
1
2
g
2

v
2
u − v
2
d

− vdµeff(Aλ + κs) = 0
vd

m2
Hd + µ
2
eff + λ
2
v
2
u +
1
2
g
2

v
2
d − v
2
u

− vuµeff(Aλ + κs) = 0
s

m2
S + κAκs + 2κ
2σ
2 + λ
2

v
2
u + v
2
d

− 2λκvuvd

− λAλvuvd = 0,
(3.11)
where we have defined
g
2 ≡
1
2

g
2
1 + g
2
2

. (3.12)
One can also define the β angle by
tan β =
vu
vd
. (3.13)
The Z boson mass is given by MZ = gv with v
2 = v
2
u + v
2
d ≃ (174 GeV)2
. Hence, with
MZ given, the set of parameters that describes the Higgs sector of the NMSSM can be
chosen to be the following
λ, κ, Aλ, Aκ, tan b and µeff. (3.14)
CP-even Higgs masses
One can obtain the Higgs mass matrices at tree level by expanding the Higgs potential
(3.8) around the vevs, using Eq. (3.9). We begin by writing3
the squared mass matrix
M2
S
of the scalar Higgses in the basis (HdR, HuR, SR):
M2
S =


g
2
v
2
d + µ tan βBeff (2λ
2 − g
2
) vuvd − µBeff 2λµvd − λ (Aλ + 2κs) vu
g
2
v
2
u +
µ
tan βBeff 2λµvu − λ (Aλ + 2κs) vd
λAλ
vuvd
s + κAκs + (2κs)
2

 ,
(3.15)
where we have defined Beff ≡ Aλ + κs (it plays the role of the B parameter of the
MSSM).
3For economy of space, we omit in this expression the subscript from µ
48 The Next-to-Minimal Supersymmetric Standard Model
Although an analytical diagonalization of the above 3 × 3 mass matrix is lengthy,
there is a crucial conclusion that comes from the approximate diagonalization of the
upper 2 × 2 submatrix. If it is rotated by an angle β, one of its diagonal elements
is M2
Z
(cos2 2β +
λ
2
g
2 sin2
2β) which is an upper bound for its lightest eigenvalue. The
first term is the same one as in the MSSM. The conclusion is that in the NMSSM
the lightest CP-even Higgs can be heavier than the corresponding of the MSSM, as
long as λ is large and tan β relatively small. Therefore, it is much easier to explain
the observed mass of the SM-like Higgs. However, λ is bounded from above in order
to avoid the appearance of the Landau pole below the GUT scale. Depending on the
other free parameters, λ should obey λ <∼ 0.7.
CP-odd Higgs masses
For the pseudoscalar case, the squared mass matrix in the basis (HdI , HuI , SI ) is
M2
P =


µeff (Aλ + κs) tan β µeff (Aλ + κs) λvu (Aλ − 2κs)
µeff
tan β
(Aλ + κs) λvd (Aλ − 2κs)
λ (Aλ + 4κs)
vuvd
s − 3κAκs

 . (3.16)
One eigenstate of this matrix corresponds to an unphysical massless Goldstone
boson G. In order to drop the Goldstone boson, we write the matrix in the basis
(A, G, SI ) by rotating the upper 2 × 2 submatrix by an angle β. After dropping the
massless mode, the 2 × 2 squared mass matrix turns out to be
M2
P =
2µeff
sin 2β
(Aλ + κs) λ (Aλ − 2κs) v
λ (Aλ + 4κs)
vuvd
s − 3Aκs
!
. (3.17)
Charged Higgs mass
The charged Higgs squared mass matrix is given, in the basis (H+
u
, H−
d
⋆
), by
M2
± =

µeff (Aλ + κs) + vuvd

1
2
g
2
2 − λ

cot β 1
1 tan β
!
, (3.18)
which contains one Goldstone boson and one physical mass eigenstate H± with eigenvalue
m2
± =
2µeff
sin 2β
(Aλ + κs) + v
2

1
2
g
2
2 − λ

. (3.19)
3.2.2 Sfermion sector
The mass matrix for the up-type squarks is given in the basis (ueR, ueL) by
Mu =

m2
u + h
2
u
v
2
u −
1
3
(v
2
u − v
2
d
) g
2
1 hu (Auvu − µeffvd)
hu (Auvu − µeffvd) m2
Q + h
2
u
v
2
u +
1
12 (v
2
u − v
2
d
) (g
2
1 − 3g
2
2
)
!
, (3.20)
3.2.3 Gaugino and higgsino sector 49
whereas for down-type squarks the mass matrix reads in the basis (deR, deL)
Md =

m2
d + h
2
d
v
2
d −
1
6
(v
2
u − v
2
d
) g
2
1 hd (Advd − µeffvu)
hd (Advd − µeffvu) m2
Q + h
2
d
v
2
d +
1
12 (v
2
u − v
2
d
) (g
2
1 − 3g
2
2
)
!
. (3.21)
The off-diagonal terms are proportional to the Yukawa coupling hu for the up-type
squarks and hd for the down-type ones. Therefore, the two lightest generations remain
approximately unmixed. For the third generation, the mass matrices are diagonalized
by a rotation by an angle θT and θB, respectively, for the stop and sbottom. The mass
eigenstates are, then, given by
et1 = cos θT
etL + sin θT
etR, et2 = cos θT
etL − sin θT
etR, (3.22)
eb1 = cos θB
ebL + sin θB
ebR, eb2 = cos θB
ebL − sin θB
ebR. (3.23)
In the slepton sector, for a similar reason, only the left- and right-handed staus are
mixed and their mass matrix
Mτ =

m2
E3 + h
2
τ
v
2
d −
1
2
(v
2
u − v
2
d
) g
2
1 hτ (Aτ vd − µeffvu)
hτ (Aτ vd − µeffvu) m2
L3 + h
2
τ
v
2
d −
1
4
(v
2
u − v
2
d
) (g
2
1 − g
2
2
)
!
(3.24)
is diagonalized after a rotation by an angle θτ . Their mass eigenstates are given by
τe1 = cos θτ τeL + sin θτ τeR, τe2 = cos θτ τeL − sin θτ τeR. (3.25)
Finally, the sneutrino masses are
mνe = m2
L −
1
4

v
2
u − v
2
d
g
2
1 + g
2
2

. (3.26)
3.2.3 Gaugino and higgsino sector
The gauginos λ1 and λ
3
2 mix with the neutral higgsinos ψ
0
d
, ψ
0
u
and ψS to form neutral
particles, the neutralinos. The 5 × 5 mass matrix of the neutralinos is written in the
basis
(−iλ1, −iλ3
2
, ψ0
d
, ψ0
u
, ψS) ≡ (B, e W , f He0
d
, He0
u
, Se) (3.27)
as
M0 =


M1 0 − √
1
2
g1vd √
1
2
g1vu 0
M2 √
1
2
g2vd − √
1
2
g2vu 0
0 −µeff −λvu
0 −λvd
2κs


. (3.28)
The mass matrix (3.28) is diagonalized by an orthogonal matrix Nij . The mass eigenstates of the neutralinos are usually denoted by χ
0
i
, with i = 1, . . . , 5, with increasing
masses (i = 1 corresponds to the lightest neutralino). These are given by
χ
0
i = Ni1Be + Ni2Wf + Ni3He0
d + Ni4He0
u + Ni5S. e (3.2
50 The Next-to-Minimal Supersymmetric Standard Model
We use the convention of a real matrix Nij , so that the physical masses mχ
0
i
are real,
but not necessarily positive.
In the charged sector, the SU(2)L charged gauginos λ
− = √
1
2
(λ
1
2 + iλ2
2
), λ
+ =
√
1
2
(λ
1
2 − iλ2
2
) mix with the charged higgsinos ψ
−
d
and ψ
+
u
, forming the charginos ψ
±:
ψ
± =

−iλ±
ψ
±
u
!
. (3.30)
The chargino mass matrix in the basis (ψ
−, ψ+) is
M± =

M2 g2vu
g2vd µeff !
. (3.31)
Since it is not symmetric, the diagonalization requires different rotations of ψ
− and
ψ
+. We denote these rotations by U and V , respectively, so that the mass eigenstates
are obtained by
χ
− = Uψ−, χ+ = V ψ+. (3.32)
3.3 DM Candidates in the NMSSM
Let us first review the characteristics that a DM candidate particle should have. First,
it should be massive in order to account for the missing mass in the galaxies. Second,
it must be electrically and color neutral. Otherwise, it would have condensed with
baryonic matter, forming anomalous heavy isotopes. Such isotopes are absent in nature. Finally, it should be stable and interact only weakly, in order to fit the observed
relic density.
In the NMSSM there are two possible candidates. Both can be stable particles if
they are the LSPs of the supersymmetric spectrum. The first one is the sneutrino (see
[148,149] for early discussions on MSSM sneutrino LSP). However, although sneutrinos
are WIMPs, their large coupling to the Z bosons result in a too large annihilation cross
section. Hence, if they were the DM particles, their relic density would have been very
small compared to the observed value. Exceptions are very massive sneutrinos, heavier
than about 5 TeV [150]. Furthermore, the same coupling result in a large scattering
cross section off the nuclei of the detectors. Therefore, sneutrinos are also excluded by
direct detection experiments.
The other possibility is the lightest neutralino. Neutralinos fulfill successfully, at
least in principle, all the requirements for a DM candidate. However, the resulting
relic density, although weakly interacting, may vary over many orders of magnitude as
a function of the free parameters of the theory. In the next sections we will investigate
further the properties of the lightest neutralino as the DM particle. We begin by
studying its annihilation that determines the DM relic density.
3.4 Neutralino relic density 51
3.4 Neutralino relic density
We remind that the neutralinos are mixed states composed of bino, wino, higgsinos
and the singlino. The exact content of the lightest neutralino determines its pair
annihilation channels and, therefore, its relic density (for detailed analyses, we refer
to [151–154]). Subsequently, we will briefly describe the neutralino pair annihilation
in various scenarios. We classify these scenarios with respect to the lightest neutralino
content.
Before we proceed, we should mention another mechanism that affects the neutralino LSP relic density. If there is a supersymmetric particle with mass close to the
LSP (but always larger), it would be abundant during the freeze-out and LSP coannihilations with this particle would contribute to the total annihilation cross section.
This particle, which is the Next-to-Lightest Supersymmetric Particle (NLSP), is most
commonly a stau or a stop. In the above sense, coannihilations refer not only to the
LSP–NLSP coannihilations, but also to the NLSP–NLSP annihilations, since the latter
reduce the number density of the NLSPs [155].
• Bino-like LSP
In principle, if the lightest neutralino is mostly bino-like, the total annihilation
cross section is expected to be small. Therefore, a bino-like neutralino LSP would
have been overabundant. The reason for this is that there is only one available
annihilation channel via t-channel sfermion exchange, since all couplings to gauge
bosons require a higgsino component. The cross section is even more reduced
when the sfermion mass is large.
However, there are still two ways to achieve the correct relic density. The first one
is using the coannihilation effect: if there is a sfermion with a mass slightly larger
(some GeV) than the LSP mass, their coannihilations can be proved to reduce
efficiently the relic density to the desired value. The second one concerns a binolike LSP, with a very small but non-negligible higgsino component. In this case,
if in addition the lightest CP-odd Higgs A1 is light enough, the annihilation to a
pair A1A1 (through an s-channel CP-even Higgs Hi exchange) can be enhanced
via Hi resonances. In this scenario a fine-tuning of the masses is necessary.
• Higgsino-like LSP
A mostly higgsino LSP is as well problematic. The strong couplings of the higgsinos to the gauge bosons lead to very large annihilation cross section. Therefore,
a possible higgsino LSP would have a very small relic density.
• Mixed bino–higgsino LSP
In this case, as it was probably expected, one can easily fit the relic density to
the observed value. This kind of LSP annihilates to W+W−, ZZ, W±H∓, ZHi
,
HiAj
, b
¯b and τ
+τ
− through s-channel Z or Higgs boson exchange or t-channel
neutralino or chargino exchange. The last two channels are the dominant ones
when the Higgs coupling to down-type fermions is enhanced, which occurs more
commonly in the regime of relatively large tan β. The annihilation channel to a
52 The Next-to-Minimal Supersymmetric Standard Model
pair of top quarks also contributes to the total cross section, if it is kinematically
allowed. However, in order to achieve the correct relic density, the higgsino
component cannot be very large.
• Singlino-like LSP
Since a mostly singlino LSP has small couplings to SM particles, the resulting relic
density is expected to be large. However, there are some annihilation channels
that can be enhanced in order to reduce the relic density. These include the
s-channel (scalar or pseudoscalar) Higgs exchange and the t-channel neutralino
exchange.
For the former, any Higgs with sufficient large singlet component gives an important contribution to the cross section, increasing with the parameter κ (since
the singlino-singlino-singlet coupling is proportional to κ). Concerning the latter
annihilation, in order to enhance it, one needs large values of the parameter λ.
In this case, the neutralino-neutralino-singlet coupling, which is proportional to
λ, is large and the annihilation proceeds giving a pair of scalar HsHs or a pair
of pseudoscalar AsAs singlet like Higgs.
As in the case of bino-like LSP, one can also use the effect of s-channel resonances
or coannihilations. In the latter case, an efficient NLSP can be the neutralino
χ
0
2
or the lightest stau τe1. In the case that the neutralino NLSP is higgsinolike, the LSP-NLSP coannihilation through a (doublet-like) Higgs exchange can
be proved very efficient. A stau NLSP reduces the relic density through NLSPNLSP annihilations, which is the only possibility in the case that both parameters
κ and λ are small. We refer to [156,157] for further discussion on this possibility.
Assuming universality conditions the wino mass M2 has to be larger than the bino
mass M1 (M2 ∼ 2M1). This is the reason that we have not considered a wino-like LSP.
3.5 Detection of neutralino DM
3.5.1 Direct detection
Since neutralinos are Majorana fermions, the effective Lagrangian describing their
elastic scattering with a quark in a nucleon can be written, in the Dirac fermion
notation, as [158]
Leff = a
SI
i χ¯
0
1χ
0
1
q¯iqi + a
SD
i χ¯
0
1γ5γµχ
0
1
q¯iγ5γ
µ
qi
, (3.33)
with i = u, d corresponding to up- and down-type quarks, respectively. The Lagrangian has to be understood as summing over the quark generations.
In this expression, we have omitted terms containing the operator ψγ¯
5ψ or a combination of ψγ¯
5γµψ and ψγ¯
µψ (with ψ = χ, q). This is a well qualified assumption:
Due to the small velocity of WIMPs, the momentum transfer ~q is very small compared
3.5.1 Direct detection 53
to the reduced mass of the WIMP-nucleus system. In the extreme limit of zero momentum transfer, the above operators vanish4
. Hence, we are left with the Lagrangian
(3.33) consisting of two terms, the first one corresponding to spin-independent (SI)
interactions and the second to spin-dependent (SD) ones. In the following, we will
focus again only to SI scattering, since the detector sensitivity to SD scattering is low,
as it has been already mentioned in Sec. 1.5.1.
The SI cross section for the neutralino-nucleus scattering can be written as [158]
(see, also, [159])
σ
SI
tot =
4m2
r
π
[Zfp + (A − Z)fn]
2
. (3.34)
mr is the neutralino-nucleus reduced mass mr =
mχmN
mχ+mN
, and Z, A are the atomic and
the nucleon number, respectively. It is more common, however, to use an expression
for the cross section normalized to the nucleon. In this case, on has for the neutralinoproton scattering
σ
SI
p =
4
π

mpmχ
0
1
mp + mχ
0
1
!2
f
2
p ≃
4m2
χ
0
1
π
f
2
p
, (3.35)
with a similar expression for the neutron.
The form factor fp is related to the couplings a to quarks through the expression
(omitting the “SI” superscripts)
fp
mp
=
X
q=u,d,s
f
p
T q
aq
mq
+
2
27
fT G X
q=c,b,t
aq
mq
. (3.36)
A similar expression may be obtained for the neutron form factor fn, by the replacement
p → n in the previous expression (henceforth, we focus to neutralino-proton scattering).
The parameters fT q are defined by the quark mass matrix elements
hp| mqqq¯ |pi = mpfT q, (3.37)
which corresponds to the contribution of the quark q to the proton mass and the
parameter fT G is related to them by
fT G = 1 −
X
q=u,d,s
fT q. (3.38)
The above parameters can be obtained by the following quantities
σπN =
1
2
(mu + md)(Bu + Bd) and σ0 =
1
2
(mu + md)(Bu + Bd − 2Bs,) (3.39)
with Bq = hN| qq¯ |Ni, which are obtained by chiral perturbation theory [160] or by
lattice simulations. Unfortunately, the uncertainties on the values of these quantities
are large (see [161], for more recent values and error bars).
4While there are possible circumstances in which the operators of (3.33) are also suppressed and,
therefore, comparable to the operators omitted, they are not phenomenologically interesting.
54 The Next-to-Minimal Supersymmetric Standard Model
χ
0
1
χ
0
1
χ
0
1 χ
0
1
qe
q q
q q
Hi
Figure 3.1: Feynman diagrams contributing to the elastic neutralino-quark scalar scattering amplitude at tree level.
The SI neutralino-nucleon interactions arise from t-channel Higgs exchange and
s-channel squark exchange at tree level (see Fig. 3.1), with one-loop contributions from
neutralino-gluon interactions. In practice, the s-channel Higgs exchange contribution
to the scattering amplitude dominates, especially due to the large masses of squarks.
In this case, the effective couplings a are given by
a
SI
d =
X
3
i=1
1
m2
Hi
C
1
i Cχ
0
1χ
0
1Hi
, aSI
u =
X
3
i=1
1
m2
Hi
C
2
i Cχ
0
1χ
0
1Hi
. (3.40)
C
1
i
and C
2
i
are the Higgs Hi couplings to down- and up-type quarks, respectively, given
by
C
1
i =
g2md
2MW cos β
Si1, C2
i =
g2mu
2MW sin β
Si2, (3.41)
with S the mixing matrix of the CP-even Higgs mass eigenstates and md, mu the
corresponding quark mass. We see from Eqs. (3.36) and (3.41) that the final cross
section (3.35) is independent of each quark mass. We write for completeness the
neutralino-neutralino-Higgs coupling Cχ
0
1χ
0
1Hi
:
Cχ
0
1χ
0
1Hi =
√
2λ (Si1N14N15 + Si2N13N15 + Si3N13N14) −
√
2κSi3N
2
15
+ g1 (Si1N11N13 − Si2N11N14) − g2 (Si1N12N13 − Si2N12N14),
with N the neutralino mixing matrix given in (3.29).
The resulting cross section is proportional to m−4
Hi
. In the NMSSM, it is possible
for the lightest scalar Higgs eigenstate to be quite light, evading detection due to its
singlet nature. This scenario can give rise to large values of SI scattering cross section,
provided that the doublet components of th

[Jeanmonnaie]

SHB vient de se faire écraser comme une mouche. 🪰
Aura-t-il la sagesse de tourner sept fois sa langue avant de poster, ou persistera-t-il par orgueil à défendre sa position bancale ?
Vous le saurez au prochain épisode.

[JeanMonnaie]

Essayons de déminer ensemble ce sujet. Nous savons que même avant la Révolution française, la porosité entre culture bourgeoise et populaire était mince. La Flûte enchantée de Mozart était un opéra populaire. Dans Les Aventures de Huckleberry Finn, Twain inclut une scène célèbre où deux personnages, le roi et le duc, escrocs itinérants, prétendent être des acteurs et montent une représentation de plusieurs scènes de Shakespeare. Ils récitent des passages de pièces telles que Hamlet et Roméo et Juliette. Pour parodier Shakespeare dans un livre pour enfants, cela veut dire que tout le monde à l’époque connaissait les livres originaux.

Les appréciations d’un livre changent selon les époques. Ce qui est populaire peut être perçu comme élitisme l’autre jour. Exemple avec Don Quichotte. Chaque époque a porté un point de vue différent sur le roman. À l’époque de sa première publication, il était généralement considéré comme un roman comique. Après la Révolution française, il fut populaire en partie à cause de son éthique : les individus peuvent avoir raison contre une société tout entière. Au XIXe siècle, il était considéré comme un commentaire social. Au XXe siècle, il fut rangé dans la catégorie des classiques littéraires. Nous voyons que, loin d’un complot des élites pour s’approprier un savoir bourgeois, la culture s’est toujours diffusée dans toutes les strates de la société.

Aujourd’hui, nous sommes dans un apparent paradoxe : jamais la culture et l’école n’ont été aussi démocratisées dans le monde et nous assistons à une idiocratie généralisée. Là encore, je ferais une lecture anti-marxiste car la baisse du niveau à l’école touche aussi bien les classes populaires que les élites bourgeoises. Ce qui nous prouve qu’au niveau culturel, le niveau scolaire suit le même mouvement que le reste de la société. Une baisse d’exigence chez les uns conduit irrémédiablement à une baisse de niveau chez les autres. Les enfants de Zemmour écoutent du rap, cette musique était vantée par Roselyne Bachelot à l’époque. Donc, même si elle le faisait pour des raisons électoralistes, ses enfants, eux, écoutent cette musique. Je le redis encore une fois, les valeurs culturelles sont plus une question d’époque que de classe.

Seule l’aristocratie pouvait avoir ses précarrés culturels, mais la bourgeoisie a promu l’éducation de masse, rendant la culture plus accessible à un plus grand nombre de personnes. Avec les avancées technologiques et la globalisation, une culture de masse a émergé, brouillant les distinctions entre élitisme culturel et culture populaire. Les médias de masse (cinéma, télévision, internet) ont joué un rôle clé dans cette transformation. Tout le monde peut avoir accès à tout. Certes, la classe populaire n’aura ni le temps ni l’envie de regarder un film exigeant, mais je ne suis pas sûr que les bourgeois le fassent non plus. Ceux qui pourront le faire, ce sont les étudiants, les fonctionnaires comme les professeurs qui se voient comme une petite noblesse intellectuelle, etc., qui sont en réalité pour la plupart des classes moyennes.

Là où il y a un terrain à défricher et où je veux retomber sur mes pattes de droite, c’est que cette noblesse culturelle de classe moyenne, qui n’a plus l’argent pour se démarquer mais qui possède la culture, veut mimer l’aristocratie de l’Ancien Régime. Il y a nous et les autres. La gauche, dominante dans la culture, influence également la gauche radicale. Comme je l’ai démontré, il y a toujours une porosité culturelle. C’est ce qui explique en partie le mépris de la gauche pour la culture populaire ou cette jouissance à utiliser des tournures de phrases compliquées, voire à faire un minimum de démonstration pour être sûr de ne pas être compris.

J’avais expliqué que ce qui plaisait chez Lordon n’était pas ce qu’il disait en lui-même, mais le plaisir de décrypter ses tournures alambiquées, comme Champollion et ses pierres de Rosette. Je vais vous mettre un passage de Thierry dans le topic de Francis Cousin :

Thierry :

« Une petite dizaine d’années de ça, j’ai bien failli me faire cousiner le cerveau. Je ne sais pas si c’est toujours le cas, mais à l’époque, il proposait des séances de type psychanalyse à 100.- l’heure via Skype. Il prétendait que la dépression ambiante et les troubles mentaux légers étaient la conséquence directe du malaise de la vie dans un système capitaliste assommant et mortifère (en substance). Séduisant pour n’importe quel jeune traversant une période compliquée. Bref, j’ai finalement passé mon chemin pour plusieurs raisons. Comme souligné par certains, il y a un côté très sectaire dans la proposition de Cousin. Déjà de par l’enrobage pompeux du verbe qui t’endort et te donne le sentiment d’accéder à une vérité une fois décryptée. (Pas si compliqué que ça après quelques efforts), mais également par le côté spirituel et ésotérique étrange qui accompagne son travail discrètement. Symboliques païennes sur ses bouquins… »

Tout est dit, et c’est ce chantier que la gauche doit ouvrir. Que signifie ce besoin de morale et de culture dans la gauche radicale quand on prétend être le mouvement des ouvriers ? Que signifie ce besoin d’élitisme dans ce mouvement ? Quelles en sont les conséquences ? Les 8 % d’ouvriers qui votent à gauche ont-ils un rapport avec cela ? Il y a là d’innombrables portes à ouvrir que vous percevez sans jamais vouloir les ouvrir complètement, de peur de ce que vous pourriez découvrir.

[maelstrom]

Ta du cherché longtemps pour retrouver le topic sur francis cousin derrière tout les topics deleatur 34

[Jeanmonnaie]

5mn

[Demi Habile]

and also the definition of the unpolarized cross section to write
X
spins
Z
|M12→34|
2
(2π)
4
δ
4
(p1 + p2 − p3 − p4)
d
3p3
(2π)
32E3
d
3p4
(2π)
32E4
=
4F g1g2 σ12→34, (1.31)
where F ≡ [(p1 · p2)
2 − m2
1m2
2
]
1/2
and the spin factors g1, g2 come from the average
over initial spins. This way, the collision term (1.29) is written in a more compact form
g1
Z
C[f1]
d
3p1
(2π)
3
= −
Z
σvMøl (dn1dn2 − dn
eq
1 dn
eq
2
), (1.32)
where σ =
P
(all f)
σ12→f is the total annihilation cross section summed over all the
possible final states and vMøl ≡
F
E1E2
. The so called Møller velocity, vMøl, is defined in
such a way that the product vMøln1n2 is invariant under Lorentz transformations and,
in terms of particle velocities ~v1 and ~v2, it is given by the expression
vMøl =
h
~v2
1 − ~v2
2

2
− |~v1 × ~v2|
2
i1/2
. (1.33)
Due to symmetry considerations, the distributions in kinetic equilibrium are proportional to those in chemical equilibrium, with a proportionality factor independent of
the momentum. Therefore, the collision term (1.32), both before and after decoupling,
can be written in the form
g1
Z
C[f1]
d
3p1
(2π)
3
= −hσvMøli(n1n2 − n
eq
1 n
eq
2
), (1.34)
where the thermal averaged total annihilation cross section times the Møller velocity
has been defined by the expression
hσvMøli =
R
σvMøldn
eq
1 dn
eq
2
R
dn
eq
1 dn
eq
2
. (1.35)
We will come back to the thermal averaged cross section in the next subsection.
We are, now, able to write the full integrated Boltzmann equation, using the expressions (1.28), (1.34) that we have derived for the Liouville and the collision term,
respectively. In the simplified but interesting case of identical particles 1 and 2, the
Boltzmann equation is, finally, written as
n˙ + 3Hn = −hσvMøli(n
2 − n
2
eq). (1.36)
18 Dark Matter
However, instead of using n, it is more convenient to take the expansion of the universe
into account and calculate the number density per comoving volume Y , which can be
defined as the ratio of the number and entropy densities: Y ≡ n/s. The total entropy
density S = R3
s (R is the scale factor) remains constant, hence we can obtain a
differential equation for Y by dividing (1.36) by S. Before we write the final form
of the Boltzmann equation that it is used for the relic density calculations, we have
to change the variable that parametrizes the comoving density. In practice, the time
variable t is not convenient and the temperature of the Universe (actually the photon
temperature, since the photons were the last particles that went out of equilibrium) is
used instead. However, it proves even more useful to use as time variable the quantity
defined by x ≡ m/T with m the DM mass, so that Eq. (1.36) transforms into
dY
dx
=
1
3H
ds
dx
hσvMøli

Y
2 − Y
2
eq
. (1.37)
Last, using the Hubble parameter (1.2) for a radiation dominated Universe and the
expressions (1.20), (1.21) for the energy and entropy density, the Boltzmann equation
is written in its final form
dY
dx
= −
r
45GN
π
g
1/2
∗ m
x
2
hσvMøli

Y
2 − Y
2
eq
, (1.38)
where the effective degrees of freedom g
1/2
∗ have been defined by
g
1/2
∗ ≡
heff
g
1/2
eff

1 +
1
3
T
heff
dheff
dT

. (1.39)
The equilibrium density per comoving volume Yeq ≡ neq/s can be expressed as
Yeq(x) = 45g
4π
4
x
2K2(x)
heff(m/x)
, (1.40)
with K2 the modified Bessel function of second kind.
1.4.3 Thermal average of the annihilation cross section
We are going to derive a simple formula that one can use to calculate the thermal
average of the cross section times velocity, based again on the analysis of [38]. We will
use the assumption that equilibrium functions follow the Maxwell-Boltzmann distribution, instead of the actual Bose-Einstein or Fermi-Dirac. This is a well established
assumption if the freeze out occurs after T ≃ m/3 or for x >∼ 3, which is actually the
case for WIMPs. Under this assumption, the expression (1.35) gives, in the cosmic
comoving frame,
hσvMøli =
R
vMøle
−E1/T e
−E2/T d
3p1d
3p2
R
e
−E1/T e
−E2/T d
3p1d
3p2
. (1.4
1.4.3 Thermal average of the annihilation cross section 19
The volume element can be written as d3p1d
3p2 = 4πp1dE14πp2dE2
1
2
cos θ, with θ the
angle between ~p1 and ~p2. After changing the integration variables to E+, E−, s given
by
E+ = E1 + E2, E− = E1 − E2, s = 2m2 + 2E1E2 − 2p1p2 cos θ, (1.42)
(with s = −(p1 − p2)
2 one of the Mandelstam variables,) the volume element becomes
d
3p1d
3p2 = 2π
2E1E2dE+dE−ds and the initial integration region
{E1 > m, E2 > m, | cos θ| ≤ 1i
transforms into
|E−| ≤
1 −
4m2
s
1/2
(E
2
+ − s)
1/2
, E+ ≥
√
s, s ≥ 4m2
. (1.43)
After some algebraic calculations, it can be found that the quantity hσvMøliE1E2
depends only on s, specifically vMølE1E2 =
1
2
p
s(s − 4m2
). Hence, the numerator of the expression (1.41), which after changing the integration variables reads
2π
2
R
dE+
R
dE−
R
dsσvMølE1E2e
−E+/T , can be written, eventually, as
Z
vMøle
−E1/T e
−E2/T = 2π
2
Z ∞
4m2
dsσ(s − 4m2
)
Z
dE+e
−E+/T (E
2
+ − s)
1/2
. (1.44)
The integral over E+ can be written with the help of the modified Bessel function of
the first kind K1 as √
s T K1(
√
s/T). The denominator of (1.41) can be treated in a
similar way, so that the thermal average is, finally, given by the expression
hσvMøli =
1
8m4TK2
2
(x)
Z ∞
4m2
ds σ(s)(s − 4m2
)
√
s K1(
√
s/T). (1.45)
Eqs. (1.38)–(1.40) along with this last Eq. (1.45) are all we need in order to calculate
the relic density of a WIMP, if its total annihilation cross section in terms of the
Mandelstam variable s is known.
In many cases, in order to avoid the numerical integration in Eq. (1.45), an approximation for hσvMøli can be used. The thermal average is expanded in powers of x
−1
(or, equivalently, in powers of the squared WIMP velocity):
hσvMøli = a + bx−1 + . . . . (1.46)
(The coefficient a corresponds to the s-wave contribution to the cross section, the
coefficient b to the p-wave contribution, and so on.) This partial wave expansion gives
a quite good approximation, provided there are no s-channel resonances and thresholds
for the final states [39].
In [40], it was shown that, after expanding the integrands of Eq. (1.41) in powers
of x
−1
, all the integrations can be performed analytically. As we saw, the expression
20 Dark Matter
vMølE1E2 depends on momenta only through s. Therefore, one can form the Lorentz
invariant quantity
w(s) ≡ σ(s)vMølE1E2 =
1
2
σ(s)
p
s(s − 4m2
). (1.47)
The integration involves the Taylor expansion of this quantity w around s/4m2 = 1
and the general formula for the partial wave expansion of the thermal average is [40]
hσvMøli =
1
m2

w −
3
2
(2w − w
′
)x
−1 +
3
8
(16w − 8w
′ + 5w
′′)x
−2
−
5
16
(30w − 15w
′ + 3w
′′ − 7x
′′′)x
−3 + O(x
−4
)

s/4m2=1
, (1.48)
where primes denote derivatives with respect to s/4m2 and all quantities have to be
evaluated at s = 4m2
.
1.5 Direct Detection of DM
Since the beginning of 1980s, it has been realized that besides the numerous facts showing evidence for the existence of these new dark particles, it is also possible to detect
them directly. Already in 1985, two pioneering articles [41, 42] appeared, describing
the detection methods for WIMPs. Since WIMPs are expected to cluster gravitationally together with ordinary stars in the Milky Way halo, they would pass also through
Earth and, in principle, they can be detected through scattering with the nuclei in a
detector’s material. In practice, one has to measure the recoil energy deposited by this
scattering.
However, although one can deduce from rotation curves that DM dominates the
dark halo in the outer parts of our galaxy, it is not so obvious from direct measurements
whether there is any substantial amount of DM inside the solar radius R0 ≃ 8 kpc.
Using indirect methods (involving the determination of the gravitational potential,
through the measuring of the kinematics of stars, both near the mid-plane of the
galactic disk and at heights several times the disk thickness), it is almost certain
that the DM is also present in the solar system, with a local density ρ0 = (0.3 ±
0.1) GeV cm−3
[43].
This value for the local density implies that for a WIMP mass of order ∼ 100 GeV,
the local number density is n0 ∼ 10−3
cm−3
. It is also expected that the WIMPs
velocity is similar to the velocity with which the Sun orbits around the galactic center
(v0 ≃ 220 km s−1
), since they are both moving under the same gravitational potential.
These two quantities allow to estimate the order of magnitude of the incident flux
of WIMPs on the Earth: J0 = n0v0 ∼ 105
cm−2
s
−1
. This value is manifestly large,
but the very weak interactions of the DM particles with ordinary matter makes their
detection a difficult, although in principle feasible, task. In order to compensate for
the very low WIMP-nucleus scattering cross section, very large detectors are required.
1.5.1 Elastic scattering event rate 21
1.5.1 Elastic scattering event rate
In the following, we will confine ourselves to the elastic scattering with nuclei. Although
inelastic scattering of WIMPs off nuclei in a detector or off orbital electrons producing
an excited state is possible, the event rate of these processes is quite suppressed. In
contrast, during an elastic scattering the nucleus recoils as a whole.
The direct detection experiments measure the number of events per day and per
kilogram of the detector material, as a function of the amount of energy Q deposited
in the detector. This event rate would be given by R = nWIMP nnuclei σv in a simplified
model with WIMPs moving with a constant velocity v. The number density of WIMPs
is nWIMP = ρ0/mX and the number density of nuclei is just the ratio of the detector’s
mass over the nuclear mass mN .
For accurate calculations, one should take into account that the WIMPs move in the
halo not with a uniform velocity, but rather following a velocity distribution f(v). The
Earth’s motion in the solar system should be included into this distribution function.
The scattering cross section σ also depends on the velocity. Actually, the cross section
can be parametrized by a nuclear form factor F(Q) as
dσ =
σ
4m2
r
v
2
F
2
(Q)d|~q|
2
, (1.49)
where |~q|
2 = 2m2
r
v
2
(1 − cos θ) is the momentum transferred during the scattering,
mr =
mXmN
mX+mN
is the reduced mass of the WIMP – nucleus system and θ is the scattering
angle in the center of momentum frame. Therefore, one can write a general expression
for the differential event rate per unit detector mass as
dR =
ρ0
mX
1
mN
σF2
(Q)d|~q|
2
4m2
r
v
2
vf(v)dv. (1.50)
The energy deposited in the detector (transferred to the nucleus through one elastic
scattering) is
Q =
|~q|
2
2mN
=
m2
r
v
2
mN
(1 − cos θ). (1.51)
Therefore, the differential event rate over deposited energy can be written, using the
equations (1.50) and (1.51), as
dR
dQ
=
σρ0
√
πv0mXm2
r
F
2
(Q)T(Q), (1.52)
where, following [37], we have defined the dimensionless quantity T(Q) as
T(Q) ≡
√
π
2
v0
Z ∞
vmin
f(v)
v
dv, (1.53)
with the minimum velocity given by vmin =
qQmN
2m2
r
, obtained by Eq. (1.51). Finally,
the event rate R can be calculated by integrating (1.52) over the energy
R =
Z ∞
ET
dR
dQ
dQ. (1.54)
22 Dark Matter
The integration is performed for energies larger than the threshold energy ET of the
detector, below which it is insensitive to WIMP-nucleus recoils.
Using Eqs. (1.54) and (1.52), one can derive the scattering cross section from the
event rate. The experimental collaborations prefer to give their results already in terms
of the scattering cross section as a function of the WIMP mass. To be more precise,
the WIMP-nucleus total cross section consists of two parts: the spin-dependent (SD)
cross section and the spin-independent (SI) one. The former comes from axial current
couplings, whereas the latter comes from scalar-scalar and vector-vector couplings.
The SD cross section is much suppressed compared to the SI one in the case of heavy
nuclei targets and it vanishes if the nucleus contains an even number of nucleons, since
in this case the total nuclear spin is zero.
We see that two uncertainties enter the above calculation: the exact value of the
local density ρ0 and the exact form of the velocity distribution f(v). To these, one
has to include one more. The cross section σ that appears in the previous expressions
concerns the WIMP-nucleon cross section. The couplings of a WIMP with the various
quarks that constitute the nucleon are not the same and the WIMP-nucleon cross
section depends strongly on the exact quark content of the nucleon. To be more
precise, the largest uncertainty lies on the strange content of the nucleon, but we shall
return to this point when we will calculate the cross section in a specific particle theory,
the Next-to-Minimal Supersymmetric Standard Model, in Sec. 3.5.1.
1.5.2 Experimental status
The situation of the experimental results from direct DM searches is a bit confusing. The null observations in most of the experiments led them to set upper limits
on the WIMP-nucleon cross section. These bounds are quite stringent for the spinindependent cross section7
, especially in the regime of WIMP masses of the order of
100 GeV. However, some collaborations have already reported possible DM signals,
mainly in the low mass regime. The preferred regions of these experiments do not
coincide, while some of them have been already excluded by other experiments. The
present picture, for WIMP masses ranging from 5 to 1000 GeV, is summarized in Fig.
1.5, 1.6.
Fig. 1.5 mainly presents upper bounds coming from XENON100 [44]. XENON100
[46] is an experiment located at the Gran Sasso underground laboratory in Italy. It
contains in total 165 kg of liquid Xenon, with 65 kg acting as target mass and the
rest shielding the detector from background radiation. For these upper limits, 225
live days of data were used. The minimum value for the predicted upper bounds on
the cross section is 2 · 10−45 cm2
for WIMP mass ∼ 55 GeV (at 90% confidence level),
almost one order of magnitude lower than the previously released limits [47] by the
same collaboration, using 100 live days of data.
The stringent upper bounds up-to-date (at least for WIMP mass larger than about
7 GeV) come from the first results of the LUX experiment (see Fig. 1.6), after the first
7For the spin-dependent scattering, the exclusion limits are quite relaxed. Hence, we will focus on
the SI cross sections.
1.5.2 Experimental status 23
Figure 1.5: The XENON100 exclusion limit (thick blue line), along with the expected
sensitivity in green (1σ) and yellow (2σ) band. Other upper bounds are also shown as
well as detection claims. From [44].
85.3 live-days of its operation [45]. LUX [53] is a detector containing liquid Xenon, as
XENON100, but in larger quantity, with total mass 370 kg. Its operation started on
April 2013 with a goal to clearly detect or exclude WIMPs with a spin independent
cross section ∼ 2 · 10−46 cm2
.
In Fig. 1.5, except of the XENON100 bounds and other experimental limits on larger
WIMP-nucleon cross section, some detection claims also appear. These come from
DAMA [48,49], CoGeNT [50] and CRESST-II [51] experiments. The first positive result
came from DAMA [52], back in 2000. Since then, the experiment has accumulated 1.17
ton-yr of data over 13 years of operation. DAMA consists of 250 kg of radio pure NaI
scintillator and looks for the annual modulation of the WIMP flux in order to reduce
the influence of the background.
The annual modulation of the DM flux (see [54] for a recent review) is due to the
Earth’s orbital motion relative to the rotation of the galactic disk. The galactic disk
rotation through an essentially non-rotating DM halo, creates an effective DM wind in
the solar frame. During the earth’s heliocentric orbit, this wind reaches a maximum
when the Earth is moving fastest in the direction of the disk rotation (this happens
in the beginning of June) and a minimum when it is moving fastest in the opposite
direction (beginning of December).
DAMA claims an 8.9σ annual modulation with a minimum flux on May 26±7 days,
consistent with the expectation. Since the detector’s target consists of two different
nuclei and the experiment cannot distinguish between sodium and iodine recoils, there
24 Dark Matter
Figure 1.6: The LUX 90% confidence exclusion limit (blue line) with the 1σ range
(shaded area). The XENON100 upper bound is represented by the red line. The inset
shows also preferred regions by CoGeNT (shaded light red), CDMS II silicon detector
(shaded green), CRESST II (shaded yellow) and DAMA (shaded gray). From [45].
is no model independent way to determine the exact region in the cross section versus
WIMP mass plane to which the observed modulation corresponds. However, one can
assume two cases: one that the WIMP scattering off the sodium nucleus dominates the
recoil energy and the other with the iodine recoils dominating. The former corresponds
[55] to a light WIMP (∼ 10 GeV) and quite large scattering cross section and the latter
to a heavier WIMP (∼ 50 to 100 GeV) with smaller cross section (see Fig. 1.5).
The positive result of DAMA was followed many years later by the ones of CoGeNT
and CRESST-II, and more recently by the silicon detector of CDMS [56] (Fig. 1.7).
The discrepancy of the results raised a lot of debates among the experiments (for
example, [64–67]) and by some the positive results are regarded as controversial. On
the other hand, it also raised an effort to find a physical explanation behind this
inconsistency (see, for example, [68–71]).
1.6 Indirect Methods for DM Detection
The same annihilation processes that determined the DM relic abundance in the early
Universe also occur today in galactic regions where the DM concentration is higher.
This fact rises the possibility of detecting potential WIMP pair annihilations indirectly
through their imprints on the cosmic rays. Therefore, the indirect DM searches aim
at the detection of an excess over the known astrophysical background of charged
particles, photons or neutrinos.
Charged particles – electrons, protons and their antiparticles – may originate from
direct products (pair of SM particles) of WIMP annihilations, after their decay and
1.6 Indirect Methods for DM Detection 25
Figure 1.7: The blue contours represent preferred regions for a possible signal at 68%
and 90% C.L. using the silicon detector of CMDS [56]. The blue dotted line represents
the upper limit obtained by the same analysis and the blue solid line is the combined
limit with the silicon CDMS data set reported in [57]. Other limits also appear:
from the CMDS standard germanium detector (light and dark red dashed line, for
standard [58] and low threshold analysis [59], respectively), EDELWEISS [60] (dashed
orange), XENON10 [61] (dash-dotted green) and XENON100 [44] (long-dash-dotted
green). The filled regions identify possible signal regions associated with data from
CoGeNT [62] (dashed yellow, 90% C.L.), DAMA [49,55] (dotted tan, 99.7% C.L.) and
CRESST-II [51, 63] (dash-dotted pink, 95.45% C.L.) experiments. Taken from [56].
through the process of showering and hadronization. Although the exact shape of the
resulting spectrum would depend on the specific process, it is expected to show a steep
cutoff at the WIMP mass. Once produced in the DM halo, the charged particles have
to travel to the point of detection through the turbulent galactic field, which will cause
diffusion. Apart from that, a lot of processes disturb the propagation of the charged
particles, such as bremsstrahlung, inverse Compton scattering with CMB photons and
many others. Therefore, the uncertainties that enter the propagation of the charged
flux until it reaches the telescope are important (contrary to the case of photons and
neutrinos that propagate almost unperturbed through the galaxy).
As in the case of direct detection, the experimental status of charged particle detection concerning the DM is confusing. After some hints from HEAT [72] and AMS01 [73] (the former a far-infrared telescope in Antarctica, the latter a spectrometer,
prototype for AMS-02 mounted on the International Space Station [74]), the PAMELA
satellite observed [75, 76] a steep increase in the energy spectrum of positron fraction
e
+/(e
+ + e
−)
8
. Later FERMI satellite [77] and AMS-02 [78] confirmed the results up
8The searches for charged particles focus on the antiparticles in order to have a reduced background,
26 Dark Matter
Figure 1.8: A compilation of data of charged cosmic rays, together with plausible but
uncertain astrophysical backgrounds, taken from [79]. Left: Positron flux. Center:
Antiproton flux. Right: Sum of electrons and positrons.
to energies of ∼ 200 GeV. However, the excess of positrons is not followed by an excess
of antiprotons, whose flux seems to coincide with the predicted background [75]. In
Fig. 1.8, three plots summarizing the situation are shown [79].
The observed excess is very difficult to explain in terms of DM [79]. To begin with,
the annihilation cross section required to reproduce the excess is quite large, many
orders of magnitude larger than the thermal cross section. Moreover, an “ordinary”
WIMP with large annihilation cross section giving rise to charged leptons is expected
to give, additionally, a large number of antiprotons, a fact in contradiction with the
observations. Although a lot of work has been done to fit a DM particle to the observed
pattern, it is quite possible that the excesses come from a yet unknown astrophysical
source. We are not going to discuss further this matter, but we end with a comment.
If this excess is due to a source other than DM, then a possible DM positron excess
would be lost under this formidable background.
A last hint for DM came from the detection of highly energetic photons. However,
we will interrupt this discussion, since this signal and a possible explanation is the
subject of Ch. 4. There, we will also see the upper bounds on the annihilation cross
section being set due to the absence of excesses in diffuse γ radiation.
since they are much less abundant than the corresponding particles.
CHAPTER 2
PARTICLE PHYSICS
Since the DM comprises of particles, it should be explained by a general particle physics
theory. We start in the following section by describing the Standard Model (SM) of
particle physics. Although the SM describes so far the fundamental particles and their
interactions quite accurately, it cannot provide a DM candidate. Besides, the SM
suffers from some theoretical problems, which we discuss in Sec. 2.2. We will see that
these problems can be solved if one introduces a new symmetry, the supersymmetry,
which we describe in Sec. 2.3. We finish this chapter by briefly describing in Sec. 2.4 a
supersymmetric extension of the SM with the minimal additional particle content, the
Minimal Supersymmetric Standard Model (MSSM).
2.1 The Standard Model of Particle Physics
The Standard Model (SM) of particle physics1
consists of two well developed theories,
the quantum chromodynamics (QCD) and the electroweak (EW) theory. The former
describes the strong interactions among the quarks, whereas the latter describes the
electroweak interactions (the weak and electromagnetic interactions in a unified context) between fermions. The EW theory took its final form in the late 1960s by the
introduction by S. Weinberg [85] and A. Salam [86] of the Higgs mechanism that gives
masses to the SM particles, which followed the unification of electromagnetic and weak
interactions [87,88]. At the same time, the EW model preserves the gauge invariance,
making the theory renormalizable, as shown later by ’t Hooft [89]. On the other hand,
QCD obtained its final form some years later, after the confirmation of the existence
of quarks. Of course, the history of the SM is much longer and it can be traced back to
1920s with the formulation of a theoretical basis for a Quantum Field Theory (QFT).
Since then, the SM had many successes. The SM particle content was completed with
the discovery of the heaviest of the quarks, the top quark [90,91], in 1995 and, recently,
with the discovery of the Higgs boson [92, 93].
1There are many good textbooks on the SM and Quantum Field Theory, e.g. [80–84].
28 Particle Physics
The key concept within the SM, as in every QFT, is that of symmetries. Each
interaction respects a gauge symmetry, based on a Lie algebra. The strong interaction is
described by an SU(3)c symmetry, where the subscript c stands for color, the conserved
charge of strong interactions. The EW interactions, on the other hand, are based on
a SU(2)L × U(1)Y Lie algebra. Here, as we will subsequently see, L refers to the
left-handed fermions and Y is the hypercharge, the conserved charge under the U(1).
SU(2)L conserves a quantity known as weak isospin I. Therefore, the SM contains the
internal symmetries of the unitary product group
SU(2)L × U(1)Y × SU(3)c. (2.1)
2.1.1 The particle content of the SM
We mention for completeness that particles are divided into two main classes according
to the statistics they follow. The bosons are particles with integer spin and follow the
Bose-Einstein distribution, whereas fermions have half-integer spin and follow the
Dirac-Einstein statistics, obeying the Pauli exclusion principle. In the SM, all the
fermions have spin 1/2, whereas the bosons have spin 1 with only exception the Higgs
boson, which is a scalar (spin zero). We begin the description of the SM particles with
the fermions.
Each fermion is classified in irreducible representations of each individual Lie algebra, according to the conserved quantum numbers, i.e. the color C, the weak isospin
I and the hypercharge Y . A first classification of fermions can be done into leptons
and quarks, which transform differently under the SU(3)c. Leptons are singlets under
this transformation, while quarks act as triplets (the fundamental representation of
this group). The EW interactions violate maximally the parity symmetry and SU(2)L
acts only on states with negative chirality (left-handed). A Dirac spinor Ψ can be
decomposed into left and right chirality components using, respectively, the projection
operators PL =
1
2
(1 − γ5) and PR =
1
2
(1 + γ5):
ΨL = PLΨ and ΨR = PRΨ. (2.2)
Left-handed fermions have I = 1/2, with a third component of the isospin I3 = ±1/2.
Fermions with positive I3 are called up-type fermions and those with negative are
called down-type. These behave the same way under SU(2)L and form doublets with
one fermion of each type. On the other hand, right-handed fermions have I = 0 and
form singlets that do not undergo weak interactions. The hypercharge is written in
terms of the electric charge Q and the third component of the isospin I3 through the
Gell-Mann–Nishijima relation:
Q = I3 + Y/2. (2.3)
Therefore, left- and right-handed components transform differently under the U(1)Y ,
since they have different hypercharge.
The fermionic sector of the SM comprises three generations of fermions, transforming as spinors under Lorentz transformations. Each generation has the same structure.
For leptons, it is an SU(2)L doublet with components consisting of one left-handed
2.1.2 The SM Lagrangian 29
charged lepton and one neutrino (neutrinos are only left-handed in the SM), along
with a gauge singlet right-handed charged lepton. The quark doublet consists of an
up- (u) and a down-type (d) (left-handed) quark and the pattern is completed by the
two corresponding SU(2)L singlet right-handed quarks. We write these representations
as
Quarks: Q ≡

u
i
L
d
i
L
!
, ui
R, di
R Leptons: L ≡

ν
i
L
e
i
L
!
, ei
R, (2.4)
with i = 1, 2, 3 the generation index.
Having briefly described the fermionic sector, we turn to the bosonic sector of
the SM. It consists of the gauge bosons that mediate the interactions and the Higgs
boson that gives masses to the particles through a spontaneous symmetry breaking,
the electroweak symmetry breaking (EWSB) [94–98], which we shall describe in Sec.
2.1.3. Before the EWSB, these bosons are
• three Wa
µ
(a = 1, 2, 3) weak bosons, associated with the generators of SU(2)L,
• one neutral Bµ boson, associated with the generator of U(1)Y ,
• eight gluons Ga
µ
(a = 1, . . . , 8), associated with the generators of SU(3)c, and
• the complex scalar Higgs doublet Φ =
φ
+
φ
0
!
.
After the EWSB, the EW boson states mix and give the two W± bosons, the neutral
Z boson and the massless photon γ. From the symmetry breaking, one scalar degree of
freedom remains which is the famous (neutral) Higgs boson [97–99]. We will return to
the mixed physical states, after describing the Higgs mechanism for symmetry breaking.
A complete list of the SM particles (the physical states after EWSB) is shown in Table
2.1.
2.1.2 The SM Lagrangian
The gauge bosons are responsible for the mediation of the interactions and are associated with the generators of the corresponding symmetry. The EW gauge bosons Bµ
and Wa
µ
are associated, respectively, with the generator Y of the U(1)Y and the three
generators T
a
2
of the SU(2)L. The latter are defined as half of the Pauli matrices τ
a
(T
a
2 =
1
2
τ
a
) and they obey the algebra

T
a
2
, Tb
2

= iǫabcT
c
2
, (2.5)
where ǫ
abc is the fully antisymmetric Levi-Civita tensor. The eight gluons are associated
with an equal number of generators T
a
3
(Gell-Mann matrices) of SU(3)c and obey the
Lie algebra

T
a
3
, Tb
3

= if abcT
c
3
, with Tr
T
a
3 T
b
3

=
1
2
δ
ab
, (2.6)
30 Particle Physics
Name symbol mass charge (|e|) spin
Leptons
electron e 0.511 MeV −1 1/2
electron neutrino νe 0 (<2 eV) 0 1/2
muon µ 105.7 MeV −1 1/2
muon neutrino νµ 0 (<2 eV) 0 1/2
tau τ 1.777 GeV −1 1/2
tau neutrino ντ 0 (<2 eV) 0 1/2
Quarks
up u 2.7
+0.7
−0.5 MeV 2/3 1/2
down d 4.8
+0.7
−0.3 MeV −1/3 1/2
strange s (95 ± 5) MeV −1/3 1/2
charm c (1.275 ± 0.025) GeV 2/3 1/2
bottom b (4.18 ± 0.03) GeV −1/3 1/2
top t (173.5 ± 0.6 ± 0.8) GeV 2/3 1/2
Bosons
photon γ 0 (<10−18 eV) 0 (<10−35) 1
W boson W± (80.385 ± 0.015) GeV ±1 1
Z boson Z (91.1876 ± 0.0021) GeV 0 1
gluon g 0 (.O(1) MeV) 0 1
Higgs H
(125.3 ± 0.4 ± 0.5) GeV
0 0
(126.0 ± 0.4 ± 0.4) GeV
Table 2.1: The particle content of the SM. All values are those given in [100], except of
the Higgs mass that is taken from [92, 93] (up and down row, respectively), assuming
that the observed excess corresponds to the SM Higgs. The u, d and s quark masses
are estimates of so-called “current-quark masses” in a mass-independent subtraction
scheme as MS at a scale ∼ 2 GeV. The c and b quark masses are the running masses
in the MS scheme. The values in the parenthesis are the current experimental limits.
with f
abc the structure constants of the group.
Using the structure constants of the corresponding groups, we define the field
strengths for the gauge bosons as
Bµν ≡ ∂µBν − ∂νBµ, (2.7a)
Wµν ≡ ∂µWa
ν − ∂νWa
µ + g2ǫ
abcWb
µWc
ν
(2.7b)
and
G
a
µν ≡ ∂µG
a
ν − ∂νG
a
µ + g3f
abcG
b
µG
c
ν
. (2.7c)
2.1.2 The SM Lagrangian 31
We use the notation g1, g2 and g3 for the coupling constants of U(1)Y , SU(2)L and
SU(3)c, respectively. As in any Yang-Mills theory, the non-abelian gauge groups lead
to self-interactions, which is not the case for the abelian U(1)Y group.
Before we finally write the full Lagrangian, we have to introduce the covariant
derivative for fermions, which in a general form can be written as
DµΨ =
∂µ − ig1
1
2
Y Bµ − ig2T
a
2 Wa
µ − ig3T
a
3 G
a
µ

Ψ. (2.8)
This form has to be understood as that, depending on Ψ, only the relevant terms
apply, hence for SU(2)L singlet leptons only the two first terms inside the parenthesis
are relevant, for doublet leptons the three first terms and for the corresponding quark
singlets and doublets the last term also participates. We also have to notice that in
order to retain the gauge symmetry, mass terms are forbidden in the Lagrangian. For
example, the mass term mψψ¯ = m

ψ¯
LψR + ψ¯
RψL

(with ψ¯ ≡ ψ
†γ
0
) is not invariant
under SU(2)L. This paradox is solved by the introduction of the Higgs scalar field
(see next subsection). The SM Lagrangian can be now written2
, split for simplicity in
three parts, each describing the gauge bosons, the fermions and the scalar sector,
LSM = Lgauge + Lfermion + Lscalar, (2.9)
with
Lgauge = −
1
4
G
a
µνG
µν
a −
1
4
Wa
µνWµν
a −
1
4
BµνB
µν
, (2.10a)
Lfermion = iL¯Dµγ
µL + ie¯RDµγµeR
+ iQ¯Dµγ
µQ + iu¯RDµγ
µuR + i
¯dRDµγ
µ
dR
−

heL¯ΦeR + hdQ¯ΦdR + huQ¯ΦeuR + h.c.

(2.10b)
and
Lscalar = (DµΦ)†
(DµΦ) − V (Φ†Φ), (2.10c)
where
V (Φ†Φ) = µ
2Φ
†Φ + λ

Φ
†Φ
2
(2.11)
is the scalar Higgs potential. Φ is the conjugate of Φ, related to the charge conjugate e
by Φ =e iτ2Φ
⋆
, with τi the Pauli matrices. The covariant derivative acting on the Higgs
scalar field gives
DµΦ =
∂µ − ig1
1
2
Y Bµ − ig2T
a
2 Wa
µ

Φ. (2.12)
Before we proceed to the description of the Higgs mechanism, a last comment concerning the SM Lagrangian is in order. If we restore the generation indices, we see that
2For simplicity, from now on we are going to omit the generations indice
32 Particle Physics
the Yukawa couplings h are 3 × 3, in general complex, matrices. As any complex matrix, they can be diagonalized with the help of two unitary matrices VL and VR, which
are related by VR = U
†VL with U again a unitary matrix. The diagonalization in the
quark sector to the mass eigenstates induces a mixing among the flavors (generations),
described by the Cabibbo–Kobayashi–Maskawa (CKM) matrix [101, 102]. The CKM
matrix is defined by
VCKM ≡ V
u
L
†
V
d
L
†
, (2.13)
where V
u
L
, V
d
L
are the unitary matrices that diagonalize the Yukawa couplings Hu
, Hd
,
respectively. This product of the two matrices appears in the charged current when it
is expressed in terms of the observable mass eigenstates.
2.1.3 Mass generation through the Higgs mechanism
We will start by examining the scalar potential (2.11). The vacuum expectation value
(vev) of the Higgs field hΦi ≡ h0|Φ|0i is given by the minimum of the potential. For
µ
2 > 0, the potential is always non-negative and Φ has a zero vev. The hypothesis of
the Higgs mechanism is that µ
2 < 0. In this case, the field Φ will acquire a vev
hΦi =
1
2

0
v
!
with v =
r
−
µ2
λ
. (2.14)
Since the charged component of Φ still has a zero vev, the U(1)Q symmetry of quantum
electrodynamics (QED) remains unbroken.
We expand the field Φ around the minima v in terms of real fields, and at leading
order we have
Φ(x) =
θ2(x) + iθ1(x)
√
1
2
(v + H(x)) − iθ3(x)
!
=
1
√
2
e
iθa(x)τ
a

0
v + H(x)
!
. (2.15)
We can eliminate the unphysical degrees of freedom θa, using the fact that the theory
remains gauge invariant. Therefore, we perform the following SU(2)L gauge transformation on Φ (unitary gauge)
Φ(x) → e
−iθa(x)τ
a
Φ(x), (2.16)
so that
Φ(x) = 1
√
2

0
v + H(x)
!
. (2.17)
We are going to use the following definitions for the gauge fields
W±
µ ≡
1
2

W1
µ ∓ iW2
µ

, (2.18a)
Zµ ≡
1
p
g
2
1 + g
2
2

g2W3
µ − g1Bµ

, (2.18b)
Aµ ≡
1
p
g
2
1 + g
2
2

g1W3
µ + g2Bµ

, (2.1
2.2 Limits of the SM and the emergence of supersymmetry 33
Then, the kinetic term for Φ (see Eq. (2.10c)) can be written in the unitary gauge as
(DµΦ)†
(D
µΦ) = 1
2
(∂µH)
2 + M2
W W+
µ W−µ +
1
2
M2
ZZµZ
µ
, (2.19)
with
MW ≡
1
2
g2v and MZ ≡
1
2
q
g
2
1 + g
2
2
v. (2.20)
We see that the definitions (2.18) correspond to the physical states of the gauge bosons
that have acquired masses due to the non-zero Higgs vev, given by (2.20). The photon
has remained massless, which reflects the fact that after the spontaneous breakdown of
SU(2)L × U(1)Y the U(1)Q remained unbroken. Among the initial degrees of freedom
of the complex scalar field Φ, three were absorbed by W± and Z and one remained as
the neutral Higgs particle with squared mass
m2
H = 2λv2
. (2.21)
We note that λ should be positive so that the scalar potential (2.11) is bounded from
below.
Fermions also acquire masses due to the Higgs mechanism. The Yukawa terms in
the fermionic part (2.10b) of the SM Lagrangian are written, after expanding around
the vev in the unitary gauge,
LY = −
1
√
2
hee¯L(v + H)eR −
1
√
2
hd
¯dL(v + H)dR −
1
√
2
huu¯L(v + H)uR + h.c. . (2.22)
Therefore, we can identify the masses of the fermions as
me
i =
h
i
e
v
√
2
, md
i =
h
i
d
v
√
2
, mui =
h
i
u
v
√
2
, (2.23)
where we have written explicitly the generation indices.
2.2 Limits of the SM and the emergence of supersymmetry
2.2.1 General discussion of the SM problems
The SM has been proven extremely successful and has been tested in high precision
in many different experiments. It has predicted many new particles before their final
discovery and also explained how the particles gain their masses. Its last triumph was
of course the discovery of a boson that seems to be very similar to the Higgs boson of
the SM. However, it is generally accepted that the SM cannot be the ultimate theory. It
is not only observed phenomena that the SM does not explain; SM also faces important
theoretical issues.
The most prominent among the inconsistencies of the SM with observations is the
oscillations among neutrinos of different generations. In order for the oscillations to
34 Particle Physics
φ φ
k
Figure 2.1: The scalar one-loop diagram giving rise to quadratic divergences.
occur, neutrinos should have non-zero masses. However, minimal modifications of the
SM are able to fit with the data of neutrino physics. Another issue that a more complete theory has to face is the matter asymmetry, the observed dominance of matter
over antimatter in the Universe. In addition, in order to comply with the standard
cosmological model, it has to provide the appropriate particle(s) that drove the inflation. Last, but not least, we saw that in order to explain the DM that dominates the
Universe, a massive, stable weakly interacting particle must exist. Such a particle is
not present in the SM.
On the other hand, the SM also suffers from a theoretical perspective. For example,
the SM counts 19 free parameters; one expects that a fundamental theory would have
a much smaller number of free parameters. Simple modifications of the SM have been
proposed relating some of these parameters. Grand unified theories (GUTs) unify
the gauge couplings at a high scale ∼ 1016 GeV. However, this unification is only
approximate unless the GUT is embedded in a supersymmetric framework. Another
serious problem of the SM is that of naturalness. This will be the topic of the following
subsection.
2.2.2 The naturalness problem of the SM
The presence of fundamental scalar fields, like the Higgs, gives rise to quadratic divergences. The diagram of Fig. 2.1 contributes to the squared mass of the scalar
δm2 = λ
Z Λ
d
4k
(2π)
4
k
−2
. (2.24)
This contribution is approximated by δm2 ∼ λΛ
2/(16π
2
), quadratic in a cut-off Λ,
which should be finite. For the case of the Higgs scalar field, one has to include its
couplings to the gauge fields and the top quark3
. Therefore,
δm2
H =
3Λ2
8π
2v
2

4m2
t − 2M2
W − M2
Z − m2
H

+ O(ln Λ
µ
)

, (2.25)
where we have used Eq. (2.21) and m2
H ≡ m2
0 + δm2
H.
3Since the contribution to the squared mass correction are quadratic in the Yukawa couplings (or
quark masses), the lighter quarks can be neglected
2.2.3 A way out 35
Taking Λ as a fundamental scale Λ ∼ MP l ∼ 1019 GeV we have
m2
0 = m2
H −
3Λ2
8π
2v
2

4m2
t − 2M2
W − M2
Z − m2
H

(2.26)
and we can see that m2
0 has to be adjusted to a precision of about 30 orders of magnitude
in order to achieve an EW scale Higgs mass. This is considered as an intolerable finetuning, which is against the general belief that the observable properties of a theory
have to be stable under small variations of the fundamental (bare) parameters. It is
exactly the above behavior that is considered as unnatural. Although the SM could
be self-consistent without imposing a large scale, grand unification of the parameters
introduce a hierarchy problem between the different scales.
A more strict definition of naturalness comes from ’t Hooft [103], which we rewrite
here:
At an energy scale µ, a physical parameter or set of physical parameters
αi(µ) is allowed to be very small only if the replacement αi(µ) = 0 would
increase the symmetry of the system.
Clearly, this is not the case here. Although mH is small compared to the fundamental
scale Λ, it is not protected by any symmetry and a fine-tuning is necessary.
2.2.3 A way out
The naturalness in the ’t Hooft sense is inspired by quantum electrodynamics, which is
the archetype for a natural theory. For example, the corrections to the electron mass
me are themselves proportional to me, with a dimensionless proportionality factor that
behaves like ∼ ln Λ. In general, fermion masses are protected by the chiral symmetry; small values (compared to the fundamental scale) of these masses enhances the
symmetry.
If a new symmetry exists in nature, relating fermion fields to scalar fields, then each
scalar mass would be related somehow to the corresponding fermion mass. Therefore,
the scalar mass itself can be naturally small compared to Λ, since this would mean
that the fermion mass is small, which enhances the chiral symmetry. Such a symmetry,
relating bosons to fermions and vice versa, is known as supersymmetry [104, 105].
Actually, as we will see later, if this new symmetry remains unbroken, the masses of
the conjugate bosons and fermions would have to be equal.
In order to make the above statement more concrete, we consider a toy model with
two additional complex scalar fields feL and feR. We will discuss only the quadratic
divergences that come from corrections to the Higgs mass due to a fermion. The
generalization for the contributions from the gauge bosons or the self-interaction is
straightforward. The interactions in this toy model of the new scalar fields with the
Higgs are described by the Lagrangian
Lfefφe = λfe|φ|
2

|feL|
2 + |feR|
2

. (2.27
36 Particle Physics
It can be easily checked that the quadratic divergence coming from a fermion at one
loop is exactly canceled, as long as the new quartic coupling λfe obeys the relation
λfe = −λ
2
f
(λf is the Yukawa coupling for the fermion f).
2.3 A brief summary of Supersymmetry
Supersymmetry (SUSY) is a symmetry relating fermions and bosons. The supersymmetry transformation should turn a boson state into a fermion state and vice versa. If
Q is the operator that generates such transformations, then
Q |bosoni = |fermioni Q |fermioni = |bosoni. (2.28)
Due to commutation and anticommutation rules of bosons and fermions, Q has to
be an anticommuting spinor operator, carrying spin angular momentum 1/2. Since
spinors are complex objects, the hermitian conjugate Q†
is also a symmetry operator4
.
There is a no-go theorem, the Coleman-Mandula theorem [106], that restricts the
conserved charges which transform as tensors under the Lorentz group to the generators
of translations Pµ and the generators of Lorentz transformations Mµν. Although this
theorem can be evaded in the case of supersymmetry due to the anticommutation
properties of Q, Q†
[107], it restricts the underlying algebra of supersymmetry [108].
Therefore, the basic supersymmetric algebra can be written as5
{Q, Q†
} = P
µ
, (2.29a)
{Q, Q} = {Q
†
, Q†
} = 0, (2.29b)
[P
µ
, Q] = [P
µ
, Q] = 0. (2.29c)
In the following, we summarize the basic conclusions derived from this algebra.
• The single-particle states of a supersymmetric theory fall into irreducible representations of the SUSY algebra, called supermultiplets. A supermultiplet contains
both fermion and boson states, called superpartners.
• Superpartners must have equal masses: Consider |Ωi and |Ω
′
i as the superpartners, |Ω
′
i should be proportional to some combination of the Q and Q† operators
acting on |Ωi, up to a space-time translation or rotation. Since −P
2
commutes
with Q, Q† and all space-time translation and rotation operators, |Ωi, |Ω
′
i will
have equal eigenvalues of −P
2 and thus equal masses.
• Superpartners must be in the same representation of gauge groups, since Q, Q†
commute with the generators of gauge transformations. This means that they
have equal charges, weak isospin and color degrees of freedom.
4We will confine ourselves to the phenomenologically more interesting case of N = 1 supersymmetry, with N referring to the number of distinct copies of Q, Q†
.
5We present a simplified version, omitting spinor indices in Q and Q†
.
2.3 A brief summary of Supersymmetry 37
• Each supermultiplet contains an equal number of fermion and boson degrees of
freedom (nF and nB, respectively): Consider the operator (−1)2s
, with s the spin
angular momentum, and the states |ii that have the same eigenvalue p
µ of P
µ
.
Then, using the SUSY algebra (2.29) and the completeness relation P
i
|ii hi| =
1, we have P
i
hi|(−1)2sP
µ
|ii = 0. On the other hand, P
i
hi|(−1)2sP
µ
|ii =
p
µTr [(−1)2s
] ∝ nB − nF . Therefore, nF = nB.
As addendum to the last point, we see that two kind of supermultiplets are possible
(neglecting gravity):
• A chiral (or matter or scalar ) supermultiplet, which consists of a single Weyl
fermion (with two spin helicity states, nF = 2) and two real scalars (each with
nB = 1), which can be replaced by a single complex scalar field.
• A gauge (or vector ) supermultiplet, which consists of a massless spin 1 boson
(two helicity states, nB = 2) and a massless spin 1/2 fermion (nF = 2).
Other combinations either are reduced to combinations of the above supermultiplets
or lead to non-renormalizable interactions.
It is possible to study supersymmetry in a geometric approach, using a space-time
manifold extended by four fermionic (Grassmann) coordinates. This manifold is called
superspace. The fields, in turn, expressed in terms of the extended set of coordinates
are called superfields. We are not going to discuss the technical details of this topic
(the interested reader may refer to the rich bibliography, for example [109–111]).
However, it is important to mention a very useful function of the superfields, the
superpotential. A generic form of a (renormalizable) superpotential in terms of the
superfields Φ is the following b
W =
1
2
MijΦbiΦbj +
1
6
y
ijkΦbiΦbjΦbk. (2.30)
The Lagrangian density can always be written according to the superpotential. The
superpotential has also to fulfill some requirements. In order for the Lagrangian to
be supersymmetric invariant, W has to be holomorphic in the complex scalar fields
(it does not involve hermitian conjugates Φb† of the superfields). Conventionally, W
involves only left chiral superfields. Instead of the SU(2)L singlet right chiral fermion
fields, one can use their left chiral charge conjugates.
As we mentioned before, the members of a supermultiplet have equal masses. This
contradicts our experience, since the partners of the light SM particles would have been
detected long time ago. Hence, the supersymmetry should be broken at a large energy
scale. The common approach is that SUSY is broken in a hidden sector, very weakly
coupled to the visible sector. Then, one has to explain how the SUSY breaking mediated to the visible sector. The two most popular scenarios are the gravity mediation
scenario [112–114] and the Gauge-Mediated SUSY Breaking (GSMB) [113, 115–117],
where the mediation occurs through gauge interactions.
There are two approaches with which one can address the SUSY breaking. In the
first approach, one refers to a GUT unification and determines the supersymmetric
38 Particle Physics
breaking parameters at low energies through the renormalization group equations.
This approach results in a small number of free parameters. In the second approach,
the starting point is the low energy scale. In this case, the SUSY breaking has to be
parametrized by the addition of breaking terms to the low energy Lagrangian. This
results in a larger set of free parameters. These terms should not reintroduce quadratic
divergences to the scalar masses, since the cancellation of these divergences was the
main motivation for SUSY. Then, one talks about soft breaking terms.
2.4 The Minimal Supersymmetric Standard Model
One can construct a supersymmetric version of the standard model with a minimal
content of particles. This model is known as the Minimal Supersymmetric Standard
Model (MSSM). In a SUSY extension of the SM, each of the SM particles is either in a
chiral or in a gauge supermultiplet, and should have a superpartner with spin differing
by 1/2.
The spin-0 partners of quarks and leptons are called squarks and sleptons, respectively (or collectively sfermions), and they have to reside in chiral supermultiplets.
The left- and right-handed components of fermions are distinct 2-component Weyl
fermions with different gauge transformations in the SM, so that each must have its
own complex scalar superpartner. The gauge bosons of the SM reside in gauge supermultiplets, along with their spin-1/2 superpartners, which are called gauginos. Every
gaugino field, like its gauge boson partner, transforms as the adjoint representation of
the corresponding gauge group. They have left- and right-handed components which
are charge conjugates of each other: (λeL)
c = λeR.
The Higgs boson, since it is a spin-0 particle, should reside in a chiral supermultiplet. However, we saw (in the fermionic part of the SM Lagrangian, Eq. (2.10b))
that the Y = 1/2 Higgs in the SM can give mass to both up- and down-type quarks,
only if the conjugate Higgs field with Y = −1/2 is involved. Since in the superpotential there are no conjugate fields, two Higgs doublets have to be introduced. Each
Higgs supermultiplet would have hypercharge Y = +1/2 or Y = −1/2. The Higgs
with the negative hypercharge gives mass to the down-type fermions and it is called
down-type Higgs (Hd, or H1 in the SLHA convention [118]) and the other one gives
mass to up-type fermions and it is called up-type Higgs (Hu, or H2).
The MSSM respects a discrete Z2 symmetry, the R-parity. If one writes the most
general terms in the supersymmetric Lagrangian (still gauge-invariant and holomorphic), some of them would lead to non-observed processes. The most obvious constraint
comes from the non-observed proton decay, which arises from a term that violates both
lepton and baryon numbers (L and B, respectively) by one unit. In order to avoid these
terms, R-parity, a multiplicative conserved quantum number, is introduced, defined as
PR = (−1)3(B−L)+2s
, (2.31)
with s the spin of the particle.
The R even particles are the SM particles, whereas the R odd are the new particles
introduced by the MSSM and are called supersymmetric particles. Due to R-parity,
2.4 The Minimal Supersymmetric Standard Model 39
if it is exactly conserved, there can be no mixing among odd and even particles and,
additionally, each interaction vertex in the theory can only involve an even number of
supersymmetric particles. The phenomenological consequences are quite important.
First, the lightest among the odd-parity particles is stable. This particle is known
as the lightest supersymmetric particle (LSP). Second, in collider experiments, supersymmetric particles can only be produced in pairs. The first of these consequences
was a breakthrough for the incorporation of DM into a general theory. If the LSP is
electrically neutral, it interacts only weakly and it consists an attractive candidate for
DM.
We are not going to enter further into the details of the MSSM6
. Although MSSM
offers a possible DM candidate, there is a strong theoretical reason to move from the
minimal model. This reason is the so-called µ-problem of the MSSM, with which we
begin the discussion of the next chapter, where we shall describe more thoroughly the
Next-to-Minimal Supersymmetric Standard Model.
6We refer to [110] for an excellent and detailed description of MSSM.
40 Particle Physics
Part II
Dark Matter in the
Next-to-Minimal Supersymmetric
Standard Model

CHAPTER 3
THE NEXT-TO-MINIMAL
SUPERSYMMETRIC STANDARD
MODEL
The Next-to-Minimal Supersymmetric Standard Model (NMSSM) is an extension of
the MSSM by a chiral, SU(2)L singlet superfield Sb (see [119, 120] for reviews). The
introduction of this field solves the µ-problem1
from which the MSSM suffers, but
also leads to a different phenomenology from that of the minimal model. The scalar
component of the additional field mixes with the scalar Higgs doublets, leading to three
CP-even mass eigenstates and two CP-odd eigenstates (as in the MSSM a doublet-like
pair of charged Higgs also exists). On the other hand, the fermionic component of the
singlet (singlino) mixes with gauginos and higgsinos, forming five neutral states, the
neutralinos.
Concerning the CP-even sector, a new possibility opens. The lightest Higgs mass
eigenstate may have evaded the detection due to a sizeable singlet component. Besides,
the SM-like Higgs is naturally heavier than in the MSSM [123–126]. Therefore, a SMlike Higgs mass ∼ 125 GeV is much easier to explain [127–141]. The singlet component
of the CP-odd Higgs also allows for a potentially very light pseudoscalar with suppressed couplings to SM particles, with various consequences, especially on low energy
observables (for example, [142–145]). The singlino component of the neutralino may
also play an important role for both collider phenomenology and DM. This is the case
when the neutralino is the LSP and the lightest neutralino has a significant singlino
component.
We start the discussion about the NMSSM by describing the µ-problem and how
this is solved in the context of the NMSSM. In Sec. 3.2 we introduce the NMSSM
Lagrangian and we write the mass matrices of the Higgs sector particles and the su1However, historically, the introduction of a singlet field preceded the µ-problem, e.g. [104, 105,
121, 122].
44 The Next-to-Minimal Supersymmetric Standard Model
persymmetric particles, at tree level. We continue by examining, in Sec. 3.3, the DM
candidates in the NMSSM and particularly the neutralino. The processes which determine the neutralino relic density are described in Sec. 3.4. The detection possibilities
of a potential NMSSM neutralino as DM are discussed in (Sec. 3.5). We close this
chapter (Sec. 3.6) by examining possible ways to include non-zero neutrino masses and
the additional DM candidates that are introduced.
3.1 Motivation – The µ-problem of the MSSM
As we saw, the minimal extension of the SM, the MSSM, contains two Higgs SU(2)L
doublets Hu and Hd. The Lagrangian of the MSSM should contain a supersymmetric
mass term, µHuHd, for these two doublets. There are several reasons, which we will
subsequently review, that require the existence of such a term. On the other hand,
the fact that |µ| cannot be very large, actually it should be of the order of the EW
scale, brings back the problem of naturalness. A parameter of the model should be
much smaller than the “natural” scale (the GUT or the Planck scale) before the EW
symmetry breaking. This leads to the so-called µ-problem of the MSSM [146].
The reasons that such a term should exist in the Lagrangian of the MSSM are
mainly phenomenological. The doublets Hu and Hd are components of chiral superfields that also contain fermionic SU(2)L doublets. Their electrically charged components mix with the superpartners of the W± bosons, forming two charged Dirac
fermions, the charginos. The unsuccessful searches for charginos in LEP have excluded
charginos with masses almost up to its kinetic limit (∼ 104 GeV) [147]. Since the µ term
determines the mass of the charginos, µ cannot be zero and actually |µ| >∼ 100 GeV,
independently of the other free parameters of the model. Moreover, µ = 0 would result
in a Peccei-Quinn symmetry of the Higgs sector and an undesirable massless axion.
Finally, there is one more reason for µ 6= 0 related to the mass generation by the Higgs
mechanism. The term µHuHd will be accompanied by a soft SUSY breaking term
BµHuHd. This term is necessary so that both neutral components of Hu and Hd are
non-vanishing at the minimum of the potential.
The Higgs mechanism also requires that µ is not too large. In order to generate
the EW symmetry breaking, the Higgs potential has to be unstable at its origin Hu =
Hd = 0. Soft SUSY breaking terms for Hu and Hd of the order of the SUSY breaking
scale generate such an instability. However, the µ induced squared masses for Hu,
Hd are always positive and would destroy the instability in case they dominate the
negative soft mass terms.
The NMSSM is able to solve the µ-problem by dynamically generating the mass
µ. This is achieved by the introduction of an SU(2)L singlet scalar field S. When S
acquires a vev, a mass term for the Hu and Hd emerges with an effective mass µeff of
the correct order, as long as the vev is of the order of the SUSY breaking scale. This
can be obtained in a more “natural” way through the soft SUSY breaking terms.
3.2 The NMSSM Lagrangian 45
3.2 The NMSSM Lagrangian
All the necessary information for the Lagrangian of the NMSSM can be extracted from
the superpotential and the soft SUSY breaking Lagrangian, containing the soft gaugino and scalar masses, and the trilinear couplings. We begin with the superpotential,
writing all the interactions of the NMSSM superfields, which include the MSSM superfields and the additional gauge singlet chiral superfield2 Sb. Hence, the superpotential
reads
W = λSbHbu · Hbd +
1
3
κSb3
+ huQb · HbuUbc
R + hdHbd · QbDbc
R + heHbd · LbEbc
R.
(3.1)
The couplings to quarks and leptons have to be understood as 3 × 3 matrices and the
quark and lepton fields as vectors in the flavor space. The SU(2)L doublet superfields
are given (as in the MSSM) by
Qb =

UbL
DbL
!
, Lb =

νb
EbL
!
, Hbu =

Hb +
u
Hb0
u
!
, Hbd =

Hb0
d
Hb −
d
!
(3.2)
and the product of two doublets is given by, for example, Qb · Hbu = UbLHb0
u − Hb +
u DbL.
An important fact to note is that the superpotential given by (3.1) does not include all possible renormalizable couplings (which respect R-parity). The most general
superpotential would also include the terms
W ⊃ µHbu · Hbd +
1
2
µ
′Sb2 + ξF s, b (3.3)
with the first two terms corresponding to supersymmetric masses and the third one,
with ξF of dimension mass2
, to a tadpole term. However, the above dimensionful
parameters µ, µ
′ and ξF should be of the order of the SUSY breaking scale, a fact
that contradicts the motivation behind the NMSSM. Here, we omit these terms and
we will work with the scale invariant superpotential (3.1). The Lagrangian of a scale
invariant superpotential possesses an accidental Z3 symmetry, which corresponds to a
multiplication of all the components of all chiral fields by a phase ei2π/3
.
The corresponding soft SUSY breaking masses and couplings are
−Lsof t = m2
Hu
|Hu|
2 + m2
Hd
|Hd|
2 + m2
S
|S|
2
+ m2
Q|Q|
2 + m2
D|DR|
2 + m2
U
|UR|
2 + m2
L
|L|
2 + m2
E|ER|
2
+

huAuQ · HuU
c
R − hdAdQ · HdD
c
R − heAeL · HdE
c
R
+λAλHu · HdS +
1
3
κAκS
3 + h.c.

+
1
2
M1λ1λ1 +
1
2
M2λ
i
2λ
i
2 +
1
2
M3λ
a
3λ
a
3
,
(3.4)
2Here, the hatted capital letters denote chiral superfields, whereas the corresponding unhatted
ones indicate their complex scalar components.
46 The Next-to-Minimal Supersymmetric Standard Model
where we have also included the soft breaking masses for the gauginos. λ1 is the U(1)Y
gaugino (bino), λ
i
2 with i = 1, 2, 3 is the SU(2)L gaugino (winos) and, finally, the λ
a
3
with a = 1, . . . , 8 denotes the SU(3)c gaugino (gluinos).
The scalar potential, expressed by the so-called D and F terms, can be written
explicitly using the general formula
V =
1
2

D
aD
a + D
′2

+ F
⋆
i Fi
, (3.5)
where
D
a = g2Φ
∗
i T
a
ijΦj (3.6a)
D
′ =
1
2
g1YiΦ
∗
i Φi (3.6b)
Fi =
∂W
∂Φi
. (3.6c)
We remind that T
a are the SU(2)L generators and Yi the hypercharge of the scalar
field Φi
. The Yukawa interactions and fermion mass terms are given by the general
Lagrangian
LY ukawa = −
1
2
∂
2W
∂Φi∂Φj
ψiψj + h.c.
, (3.7)
using the superpotential (3.1). The two-component spinor ψi
is the superpartner of
the scalar Φi
.
3.2.1 Higgs sector
Using the general form of the scalar potential, the following Higgs potential is derived
VHiggs =

λ

H
+
u H
−
d − H
0
uH
0
d

+ κS2

2
+

m2
Hu + |λS|
2

H
0
u

2
+

H
+
u

2

+

m2
Hd + |λS|
2

H
0
d

2
+

H
−
d

2

+
1
8

g
2
1 + g
2
2

H
0
u

2
+

H
+
u

2
−

H
0
d

2
−

H
−
d

2
2
+
1
2
g
2
2

H
+
u H
0
d
⋆
+ H
0
uH
−
d
⋆

2
+ m2
S
|S|
2 +

λAλ

H
+
u H
−
d − H
0
uH
0
d

S +
1
3
κAκS
3 + h.c.

.
(3.8)
The neutral physical Higgs states are defined through the relations
H
0
u = vu +
1
√
2
(HuR + iHuI ), H0
d = vd +
1
√
2
(HdR + iHdI ),
S = s +
1
√
2
(SR + iSI ),
3.2.1 Higgs sector 47
where vu, vd and s are, respectively, the real vevs of Hu, Hd and S, which have to be
obtained from the minima of the scalar potential (3.8), after expanding the fields using
Eq. (3.9). We notice that when S acquires a vev, a term µeffHbu · Hbd appears in the
superpotential, with
µeff = λs, (3.10)
solving the µ-problem.
Therefore, the Higgs sector of the NMSSM is characterized by the seven parameters
λ, κ, m2
Hu
, m2
Hd
, m2
S
, Aλ and Aκ. One can express the three soft masses by the three
vevs using the minimization equations of the Higgs potential (3.8), which are given by
vu

m2
Hu + µ
2
eff + λ
2
v
2
d +
1
2
g
2

v
2
u − v
2
d

− vdµeff(Aλ + κs) = 0
vd

m2
Hd + µ
2
eff + λ
2
v
2
u +
1
2
g
2

v
2
d − v
2
u

− vuµeff(Aλ + κs) = 0
s

m2
S + κAκs + 2κ
2σ
2 + λ
2

v
2
u + v
2
d

− 2λκvuvd

− λAλvuvd = 0,
(3.11)
where we have defined
g
2 ≡
1
2

g
2
1 + g
2
2

. (3.12)
One can also define the β angle by
tan β =
vu
vd
. (3.13)
The Z boson mass is given by MZ = gv with v
2 = v
2
u + v
2
d ≃ (174 GeV)2
. Hence, with
MZ given, the set of parameters that describes the Higgs sector of the NMSSM can be
chosen to be the following
λ, κ, Aλ, Aκ, tan b and µeff. (3.14)
CP-even Higgs masses
One can obtain the Higgs mass matrices at tree level by expanding the Higgs potential
(3.8) around the vevs, using Eq. (3.9). We begin by writing3
the squared mass matrix
M2
S
of the scalar Higgses in the basis (HdR, HuR, SR):
M2
S =


g
2
v
2
d + µ tan βBeff (2λ
2 − g
2
) vuvd − µBeff 2λµvd − λ (Aλ + 2κs) vu
g
2
v
2
u +
µ
tan βBeff 2λµvu − λ (Aλ + 2κs) vd
λAλ
vuvd
s + κAκs + (2κs)
2

 ,
(3.15)
where we have defined Beff ≡ Aλ + κs (it plays the role of the B parameter of the
MSSM).
3For economy of space, we omit in this expression the subscript from µ
48 The Next-to-Minimal Supersymmetric Standard Model
Although an analytical diagonalization of the above 3 × 3 mass matrix is lengthy,
there is a crucial conclusion that comes from the approximate diagonalization of the
upper 2 × 2 submatrix. If it is rotated by an angle β, one of its diagonal elements
is M2
Z
(cos2 2β +
λ
2
g
2 sin2
2β) which is an upper bound for its lightest eigenvalue. The
first term is the same one as in the MSSM. The conclusion is that in the NMSSM
the lightest CP-even Higgs can be heavier than the corresponding of the MSSM, as
long as λ is large and tan β relatively small. Therefore, it is much easier to explain
the observed mass of the SM-like Higgs. However, λ is bounded from above in order
to avoid the appearance of the Landau pole below the GUT scale. Depending on the
other free parameters, λ should obey λ <∼ 0.7.
CP-odd Higgs masses
For the pseudoscalar case, the squared mass matrix in the basis (HdI , HuI , SI ) is
M2
P =


µeff (Aλ + κs) tan β µeff (Aλ + κs) λvu (Aλ − 2κs)
µeff
tan β
(Aλ + κs) λvd (Aλ − 2κs)
λ (Aλ + 4κs)
vuvd
s − 3κAκs

 . (3.16)
One eigenstate of this matrix corresponds to an unphysical massless Goldstone
boson G. In order to drop the Goldstone boson, we write the matrix in the basis
(A, G, SI ) by rotating the upper 2 × 2 submatrix by an angle β. After dropping the
massless mode, the 2 × 2 squared mass matrix turns out to be
M2
P =
2µeff
sin 2β
(Aλ + κs) λ (Aλ − 2κs) v
λ (Aλ + 4κs)
vuvd
s − 3Aκs
!
. (3.17)
Charged Higgs mass
The charged Higgs squared mass matrix is given, in the basis (H+
u
, H−
d
⋆
), by
M2
± =

µeff (Aλ + κs) + vuvd

1
2
g
2
2 − λ

cot β 1
1 tan β
!
, (3.18)
which contains one Goldstone boson and one physical mass eigenstate H± with eigenvalue
m2
± =
2µeff
sin 2β
(Aλ + κs) + v
2

1
2
g
2
2 − λ

. (3.19)
3.2.2 Sfermion sector
The mass matrix for the up-type squarks is given in the basis (ueR, ueL) by
Mu =

m2
u + h
2
u
v
2
u −
1
3
(v
2
u − v
2
d
) g
2
1 hu (Auvu − µeffvd)
hu (Auvu − µeffvd) m2
Q + h
2
u
v
2
u +
1
12 (v
2
u − v
2
d
) (g
2
1 − 3g
2
2
)
!
, (3.20)
3.2.3 Gaugino and higgsino sector 49
whereas for down-type squarks the mass matrix reads in the basis (deR, deL)
Md =

m2
d + h
2
d
v
2
d −
1
6
(v
2
u − v
2
d
) g
2
1 hd (Advd − µeffvu)
hd (Advd − µeffvu) m2
Q + h
2
d
v
2
d +
1
12 (v
2
u − v
2
d
) (g
2
1 − 3g
2
2
)
!
. (3.21)
The off-diagonal terms are proportional to the Yukawa coupling hu for the up-type
squarks and hd for the down-type ones. Therefore, the two lightest generations remain
approximately unmixed. For the third generation, the mass matrices are diagonalized
by a rotation by an angle θT and θB, respectively, for the stop and sbottom. The mass
eigenstates are, then, given by
et1 = cos θT
etL + sin θT
etR, et2 = cos θT
etL − sin θT
etR, (3.22)
eb1 = cos θB
ebL + sin θB
ebR, eb2 = cos θB
ebL − sin θB
ebR. (3.23)
In the slepton sector, for a similar reason, only the left- and right-handed staus are
mixed and their mass matrix
Mτ =

m2
E3 + h
2
τ
v
2
d −
1
2
(v
2
u − v
2
d
) g
2
1 hτ (Aτ vd − µeffvu)
hτ (Aτ vd − µeffvu) m2
L3 + h
2
τ
v
2
d −
1
4
(v
2
u − v
2
d
) (g
2
1 − g
2
2
)
!
(3.24)
is diagonalized after a rotation by an angle θτ . Their mass eigenstates are given by
τe1 = cos θτ τeL + sin θτ τeR, τe2 = cos θτ τeL − sin θτ τeR. (3.25)
Finally, the sneutrino masses are
mνe = m2
L −
1
4

v
2
u − v
2
d
g
2
1 + g
2
2

. (3.26)
3.2.3 Gaugino and higgsino sector
The gauginos λ1 and λ
3
2 mix with the neutral higgsinos ψ
0
d
, ψ
0
u
and ψS to form neutral
particles, the neutralinos. The 5 × 5 mass matrix of the neutralinos is written in the
basis
(−iλ1, −iλ3
2
, ψ0
d
, ψ0
u
, ψS) ≡ (B, e W , f He0
d
, He0
u
, Se) (3.27)
as
M0 =


M1 0 − √
1
2
g1vd √
1
2
g1vu 0
M2 √
1
2
g2vd − √
1
2
g2vu 0
0 −µeff −λvu
0 −λvd
2κs


. (3.28)
The mass matrix (3.28) is diagonalized by an orthogonal matrix Nij . The mass eigenstates of the neutralinos are usually denoted by χ
0
i
, with i = 1, . . . , 5, with increasing
masses (i = 1 corresponds to the lightest neutralino). These are given by
χ
0
i = Ni1Be + Ni2Wf + Ni3He0
d + Ni4He0
u + Ni5S. e (3.2
50 The Next-to-Minimal Supersymmetric Standard Model
We use the convention of a real matrix Nij , so that the physical masses mχ
0
i
are real,
but not necessarily positive.
In the charged sector, the SU(2)L charged gauginos λ
− = √
1
2
(λ
1
2 + iλ2
2
), λ
+ =
√
1
2
(λ
1
2 − iλ2
2
) mix with the charged higgsinos ψ
−
d
and ψ
+
u
, forming the charginos ψ
±:
ψ
± =

−iλ±
ψ
±
u
!
. (3.30)
The chargino mass matrix in the basis (ψ
−, ψ+) is
M± =

M2 g2vu
g2vd µeff !
. (3.31)
Since it is not symmetric, the diagonalization requires different rotations of ψ
− and
ψ
+. We denote these rotations by U and V , respectively, so that the mass eigenstates
are obtained by
χ
− = Uψ−, χ+ = V ψ+. (3.32)
3.3 DM Candidates in the NMSSM
Let us first review the characteristics that a DM candidate particle should have. First,
it should be massive in order to account for the missing mass in the galaxies. Second,
it must be electrically and color neutral. Otherwise, it would have condensed with
baryonic matter, forming anomalous heavy isotopes. Such isotopes are absent in nature. Finally, it should be stable and interact only weakly, in order to fit the observed
relic density.
In the NMSSM there are two possible candidates. Both can be stable particles if
they are the LSPs of the supersymmetric spectrum. The first one is the sneutrino (see
[148,149] for early discussions on MSSM sneutrino LSP). However, although sneutrinos
are WIMPs, their large coupling to the Z bosons result in a too large annihilation cross
section. Hence, if they were the DM particles, their relic density would have been very
small compared to the observed value. Exceptions are very massive sneutrinos, heavier
than about 5 TeV [150]. Furthermore, the same coupling result in a large scattering
cross section off the nuclei of the detectors. Therefore, sneutrinos are also excluded by
direct detection experiments.
The other possibility is the lightest neutralino. Neutralinos fulfill successfully, at
least in principle, all the requirements for a DM candidate. However, the resulting
relic density, although weakly interacting, may vary over many orders of magnitude as
a function of the free parameters of the theory. In the next sections we will investigate
further the properties of the lightest neutralino as the DM particle. We begin by
studying its annihilation that determines the DM relic density.
3.4 Neutralino relic density 51
3.4 Neutralino relic density
We remind that the neutralinos are mixed states composed of bino, wino, higgsinos
and the singlino. The exact content of the lightest neutralino determines its pair
annihilation channels and, therefore, its relic density (for detailed analyses, we refer
to [151–154]). Subsequently, we will briefly describe the neutralino pair annihilation
in various scenarios. We classify these scenarios with respect to the lightest neutralino
content.
Before we proceed, we should mention another mechanism that affects the neutralino LSP relic density. If there is a supersymmetric particle with mass close to the
LSP (but always larger), it would be abundant during the freeze-out and LSP coannihilations with this particle would contribute to the total annihilation cross section.
This particle, which is the Next-to-Lightest Supersymmetric Particle (NLSP), is most
commonly a stau or a stop. In the above sense, coannihilations refer not only to the
LSP–NLSP coannihilations, but also to the NLSP–NLSP annihilations, since the latter
reduce the number density of the NLSPs [155].
• Bino-like LSP
In principle, if the lightest neutralino is mostly bino-like, the total annihilation
cross section is expected to be small. Therefore, a bino-like neutralino LSP would
have been overabundant. The reason for this is that there is only one available
annihilation channel via t-channel sfermion exchange, since all couplings to gauge
bosons require a higgsino component. The cross section is even more reduced
when the sfermion mass is large.
However, there are still two ways to achieve the correct relic density. The first one
is using the coannihilation effect: if there is a sfermion with a mass slightly larger
(some GeV) than the LSP mass, their coannihilations can be proved to reduce
efficiently the relic density to the desired value. The second one concerns a binolike LSP, with a very small but non-negligible higgsino component. In this case,
if in addition the lightest CP-odd Higgs A1 is light enough, the annihilation to a
pair A1A1 (through an s-channel CP-even Higgs Hi exchange) can be enhanced
via Hi resonances. In this scenario a fine-tuning of the masses is necessary.
• Higgsino-like LSP
A mostly higgsino LSP is as well problematic. The strong couplings of the higgsinos to the gauge bosons lead to very large annihilation cross section. Therefore,
a possible higgsino LSP would have a very small relic density.
• Mixed bino–higgsino LSP
In this case, as it was probably expected, one can easily fit the relic density to
the observed value. This kind of LSP annihilates to W+W−, ZZ, W±H∓, ZHi
,
HiAj
, b
¯b and τ
+τ
− through s-channel Z or Higgs boson exchange or t-channel
neutralino or chargino exchange. The last two channels are the dominant ones
when the Higgs coupling to down-type fermions is enhanced, which occurs more
commonly in the regime of relatively large tan β. The annihilation channel to a
52 The Next-to-Minimal Supersymmetric Standard Model
pair of top quarks also contributes to the total cross section, if it is kinematically
allowed. However, in order to achieve the correct relic density, the higgsino
component cannot be very large.
• Singlino-like LSP
Since a mostly singlino LSP has small couplings to SM particles, the resulting relic
density is expected to be large. However, there are some annihilation channels
that can be enhanced in order to reduce the relic density. These include the
s-channel (scalar or pseudoscalar) Higgs exchange and the t-channel neutralino
exchange.
For the former, any Higgs with sufficient large singlet component gives an important contribution to the cross section, increasing with the parameter κ (since
the singlino-singlino-singlet coupling is proportional to κ). Concerning the latter
annihilation, in order to enhance it, one needs large values of the parameter λ.
In this case, the neutralino-neutralino-singlet coupling, which is proportional to
λ, is large and the annihilation proceeds giving a pair of scalar HsHs or a pair
of pseudoscalar AsAs singlet like Higgs.
As in the case of bino-like LSP, one can also use the effect of s-channel resonances
or coannihilations. In the latter case, an efficient NLSP can be the neutralino
χ
0
2
or the lightest stau τe1. In the case that the neutralino NLSP is higgsinolike, the LSP-NLSP coannihilation through a (doublet-like) Higgs exchange can
be proved very efficient. A stau NLSP reduces the relic density through NLSPNLSP annihilations, which is the only possibility in the case that both parameters
κ and λ are small. We refer to [156,157] for further discussion on this possibility.
Assuming universality conditions the wino mass M2 has to be larger than the bino
mass M1 (M2 ∼ 2M1). This is the reason that we have not considered a wino-like LSP.
3.5 Detection of neutralino DM
3.5.1 Direct detection
Since neutralinos are Majorana fermions, the effective Lagrangian describing their
elastic scattering with a quark in a nucleon can be written, in the Dirac fermion
notation, as [158]
Leff = a
SI
i χ¯
0
1χ
0
1
q¯iqi + a
SD
i χ¯
0
1γ5γµχ
0
1
q¯iγ5γ
µ
qi
, (3.33)
with i = u, d corresponding to up- and down-type quarks, respectively. The Lagrangian has to be understood as summing over the quark generations.
In this expression, we have omitted terms containing the operator ψγ¯
5ψ or a combination of ψγ¯
5γµψ and ψγ¯
µψ (with ψ = χ, q). This is a well qualified assumption:
Due to the small velocity of WIMPs, the momentum transfer ~q is very small compared
3.5.1 Direct detection 53
to the reduced mass of the WIMP-nucleus system. In the extreme limit of zero momentum transfer, the above operators vanish4
. Hence, we are left with the Lagrangian
(3.33) consisting of two terms, the first one corresponding to spin-independent (SI)
interactions and the second to spin-dependent (SD) ones. In the following, we will
focus again only to SI scattering, since the detector sensitivity to SD scattering is low,
as it has been already mentioned in Sec. 1.5.1.
The SI cross section for the neutralino-nucleus scattering can be written as [158]
(see, also, [159])
σ
SI
tot =
4m2
r
π
[Zfp + (A − Z)fn]
2
. (3.34)
mr is the neutralino-nucleus reduced mass mr =
mχmN
mχ+mN
, and Z, A are the atomic and
the nucleon number, respectively. It is more common, however, to use an expression
for the cross section normalized to the nucleon. In this case, on has for the neutralinoproton scattering
σ
SI
p =
4
π

mpmχ
0
1
mp + mχ
0
1
!2
f
2
p ≃
4m2
χ
0
1
π
f
2
p
, (3.35)
with a similar expression for the neutron.
The form factor fp is related to the couplings a to quarks through the expression
(omitting the “SI” superscripts)
fp
mp
=
X
q=u,d,s
f
p
T q
aq
mq
+
2
27
fT G X
q=c,b,t
aq
mq
. (3.36)
A similar expression may be obtained for the neutron form factor fn, by the replacement
p → n in the previous expression (henceforth, we focus to neutralino-proton scattering).
The parameters fT q are defined by the quark mass matrix elements
hp| mqqq¯ |pi = mpfT q, (3.37)
which corresponds to the contribution of the quark q to the proton mass and the
parameter fT G is related to them by
fT G = 1 −
X
q=u,d,s
fT q. (3.38)
The above parameters can be obtained by the following quantities
σπN =
1
2
(mu + md)(Bu + Bd) and σ0 =
1
2
(mu + md)(Bu + Bd − 2Bs,) (3.39)
with Bq = hN| qq¯ |Ni, which are obtained by chiral perturbation theory [160] or by
lattice simulations. Unfortunately, the uncertainties on the values of these quantities
are large (see [161], for more recent values and error bars).
4While there are possible circumstances in which the operators of (3.33) are also suppressed and,
therefore, comparable to the operators omitted, they are not phenomenologically interesting.
54 The Next-to-Minimal Supersymmetric Standard Model
χ
0
1
χ
0
1
χ
0
1 χ
0
1
qe
q q
q q
Hi
Figure 3.1: Feynman diagrams contributing to the elastic neutralino-quark scalar scattering amplitude at tree level.
The SI neutralino-nucleon interactions arise from t-channel Higgs exchange and
s-channel squark exchange at tree level (see Fig. 3.1), with one-loop contributions from
neutralino-gluon interactions. In practice, the s-channel Higgs exchange contribution
to the scattering amplitude dominates, especially due to the large masses of squarks.
In this case, the effective couplings a are given by
a
SI
d =
X
3
i=1
1
m2
Hi
C
1
i Cχ
0
1χ
0
1Hi
, aSI
u =
X
3
i=1
1
m2
Hi
C
2
i Cχ
0
1χ
0
1Hi
. (3.40)
C
1
i
and C
2
i
are the Higgs Hi couplings to down- and up-type quarks, respectively, given
by
C
1
i =
g2md
2MW cos β
Si1, C2
i =
g2mu
2MW sin β
Si2, (3.41)
with S the mixing matrix of the CP-even Higgs mass eigenstates and md, mu the
corresponding quark mass. We see from Eqs. (3.36) and (3.41) that the final cross
section (3.35) is independent of each quark mass. We write for completeness the
neutralino-neutralino-Higgs coupling Cχ
0
1χ
0
1Hi
:
Cχ
0
1χ
0
1Hi =
√
2λ (Si1N14N15 + Si2N13N15 + Si3N13N14) −
√
2κSi3N
2
15
+ g1 (Si1N11N13 − Si2N11N14) − g2 (Si1N12N13 − Si2N12N14), (3.42)
with N the neutralino mixing matrix given in (3.29).
The resulting cross section is proportional to m−4
Hi
In the NMSSM, it is possible
for the lightest scalar Higgs eigenstate to be quite light, evading detection due to its
singlet nature. This scenario can give rise to large values of SI scattering cross section,
provided that the doublet components of th

[pinakia.colada]

Voici un travail de Philippe Coulangeon, qui propose – concernant les préférences musicales selon la classe sociale – une grille univore/omnivore, et que j’avais trouvé plutôt convaincant.
.
Son article, ici.
.
Au regard de cette analyse, l’écoute du rap dans la bourgeoisie n’a rien d’étonnant.

[françois bégaudeau]

On est bien au coeur du sujet

La distinction continue à exister. Ainsi que la distribution sociale des gouts. Et le mépris qui entoure certains gouts.
Evidemment
Ce qui est sérieusement étiolé, c’est précisément la notion de « gout légitime »
D’ailleurs je vois que ce qui est en cours au sein de la bourgeoisie, et qui n’en est qu’au début, c’est la déligitimation de la culture elle-même – en tant qu’improductive, féminine, etc. S’accrocher au rocher de la légitimité c’est rater ce phénomène là.

[Demi Habile]

Dark matter in the Next-to-Minimal Supersymmetric
Standard Model
Pantelis Mitropoulos
To cite this version:
Pantelis Mitropoulos. Dark matter in the Next-to-Minimal Supersymmetric Standard Model. Other
[cond-mat.other]. Université Paris Sud – Paris XI, 2013. English. ffNNT : 2013PA112341ff. fftel00952344ff
Universit´e Paris-Sud
ECOLE DOCTORALE: ´ Particules, Noyaux et Cosmos (517)
Laboratoire de Physique Th´eorique d’Orsay
DISCIPLINE Physique Th´eorique
THESE DE DOCTORAT `
soutenue le 10/12/2013
par
Pantelis MITROPOULOS
Dark Matter in the Next-to-Minimal
Supersymmetric Standard Model
Directeur de th`ese: Ulrich ELLWANGER Enseignant-chercheur (LPT)
Composition du jury:
Pr´esidente du jury: Asmˆaa ABADA Enseignant-chercheur (LPT)
Rapporteurs: Genevi`eve BELANGER Chercheur (LAPTH) ´
Michel TYTGAT Enseignant-chercheur (Service de Physique Th´eorique, ULB)
Examinateur: Aldo DEANDREA Enseignant-chercheur (IPNL)

ACKNOWLEDGMENTS
I am very grateful to my advisor Ulrich Ellwanger for his priceless support and his
patience during all these years. I feel exceptionally lucky having had the opportunity
to do research under his guidance.
I would also like to thank all the members of our group for the warm working
environment they provided me, but I am especially grateful to Yann Mambrini and
Adam Falkowski, the organizers of the journal club of our group, for the inspiration
they provided. Of course, I cannot forget to thank my previous colleague but still
friend Debottam Das for his warm welcome when I first came to the lab and his help
during my work.
Last but not least, I would like to thank Asmaa Abada, Genevieve Belanger, Aldo
Deandrea and Michel Tytgat who did me the honor to participate in my jury.
I acknowledge financial support from the Greek State Scholarship Foundation (I.K.Y.).
iv
CONTENTS
Introduction ix
I Particle Dark Matter 1
1 Dark Matter 3
1.1 The Standard Big Bang Cosmological Model . . . . . . . . . . . . . . . 4
1.2 Evidence of DM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.1 Galactic rotation velocities . . . . . . . . . . . . . . . . . . . . . 6
1.2.2 Gravitational lensing . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.3 CMB radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.4 Other evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3 Particle DM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4 The Standard Thermal Mechanism . . . . . . . . . . . . . . . . . . . . 13
1.4.1 Relic Abundance, thermal cross section and WIMPs . . . . . . . 13
1.4.2 The Boltzmann equation . . . . . . . . . . . . . . . . . . . . . . 15
1.4.3 Thermal average of the annihilation cross section . . . . . . . . 18
1.5 Direct Detection of DM . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.5.1 Elastic scattering event rate . . . . . . . . . . . . . . . . . . . . 21
1.5.2 Experimental status . . . . . . . . . . . . . . . . . . . . . . . . 22
1.6 Indirect Methods for DM Detection . . . . . . . . . . . . . . . . . . . . 24
2 Particle Physics 27
2.1 The Standard Model of Particle Physics . . . . . . . . . . . . . . . . . 27
2.1.1 The particle content of the SM . . . . . . . . . . . . . . . . . . 28
2.1.2 The SM Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.1.3 Mass generation through the Higgs mechanism . . . . . . . . . . 32
2.2 Limits of the SM and the emergence of supersymmetry . . . . . . . . . 33
2.2.1 General discussion of the SM problems . . . . . . . . . . . . . . 33
vi CONTENTS
2.2.2 The naturalness problem of the SM . . . . . . . . . . . . . . . . 34
2.2.3 A way out . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3 A brief summary of Supersymmetry . . . . . . . . . . . . . . . . . . . . 36
2.4 The Minimal Supersymmetric Standard Model . . . . . . . . . . . . . . 38
II Dark Matter in the Next-to-Minimal Supersymmetric
Standard Model 41
3 The Next-to-Minimal Supersymmetric Standard Model 43
3.1 Motivation – The µ-problem of the MSSM . . . . . . . . . . . . . . . . 44
3.2 The NMSSM Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2.1 Higgs sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.2.2 Sfermion sector . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.2.3 Gaugino and higgsino sector . . . . . . . . . . . . . . . . . . . . 49
3.3 DM Candidates in the NMSSM . . . . . . . . . . . . . . . . . . . . . . 50
3.4 Neutralino relic density . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.5 Detection of neutralino DM . . . . . . . . . . . . . . . . . . . . . . . . 52
3.5.1 Direct detection . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.5.2 Indirect detection . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.6 Neutrino masses and more DM candidates . . . . . . . . . . . . . . . . 55
4 A possible indirect indication for Dark Matter 59
4.1 Photon Radiation and Detection . . . . . . . . . . . . . . . . . . . . . . 60
4.2 Photon Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.3 The 130 GeV Fermi line . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.4 Upper bounds from diffuse γ-rays . . . . . . . . . . . . . . . . . . . . . 68
4.5 A 130 GeV photon line in the NMSSM . . . . . . . . . . . . . . . . . . 70
4.5.1 General aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.5.2 Implementation for the Fermi Line . . . . . . . . . . . . . . . . 71
4.5.3 Constraints from direct DM searches . . . . . . . . . . . . . . . 72
4.5.4 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.5.5 Update for the latest direct detection constraints . . . . . . . . 77
4.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
III Asymmetric Dark Matter 83
5 Asymmetric DM and upper bounds on its self-annihilation 85
5.1 Chemical potential and number densities . . . . . . . . . . . . . . . . . 86
5.2 Asymmetric DM self-annihilation . . . . . . . . . . . . . . . . . . . . . 88
5.3 Boltzmann equations for asymmetric DM . . . . . . . . . . . . . . . . . 89
5.3.1 Qualitative analysis . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.3.2 Numerical solution . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.4 Implications for specific models . . . . . . . . . . . . . . . . . . . . . . 93
CONTENTS vii
5.4.1 Sneutrino ADM . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.4.2 Higgsino ADM . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.4.3 The ∆W ∼ XXHL/Λ model . . . . . . . . . . . . . . . . . . . 95
5.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
6 A specific model for asymmetric DM 101
6.1 Sneutrinos as asymmetric DM . . . . . . . . . . . . . . . . . . . . . . . 101
6.2 Big Bang Nucleosynthesis and neutrinos . . . . . . . . . . . . . . . . . 103
6.3 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6.3.1 Constraints from lepton flavour violation and BBN . . . . . . . 108
6.4 Right-handed sneutrinos as ADM . . . . . . . . . . . . . . . . . . . . . 109
6.4.1 Asymmetry from sphaleron processes and the ADM mass . . . . 109
6.4.2 Constraints from oscillations, self and pair annihilation . . . . . 112
6.4.3 ADM Detection: prospects and constraints . . . . . . . . . . . . 114
6.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
Conclusion 117
Appendices 119
A Relativistic degrees of freedom 121
A.1 Energy Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
A.2 Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
A.3 Entropy density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
A.4 Calculation of the effective degrees of freedom . . . . . . . . . . . . . . 123
B Cross section for the neutralino annihilation to photons 127
B.1 χ
0
1χ
0
1 → γγ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
B.2 χ
0
1χ
0
1 → Zγ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
Bibliography 131
viii CONTENTS
INTRODUCTION
One of the current major puzzles of theoretical physics is the explanation of a nonluminous and yet unknown form of matter present throughout the Universe, called
Dark Matter (DM). Although the evidence for its existence, originating from various
gravitational effects, are so far only implicit observations, they are strong enough to
consider with great certainty that more than about 80% of the total matter in the
Universe is dark. Moreover, this evidence suggests that DM consists of non-baryonic,
massive long-lived particles which interact only through gravity and weak interactions.
None of the particles described by the Standard Model (SM) of particle physics do
meet the required specifications to account for dark matter. Many models that extend
the standard theory have been proposed, in an effort to incorporate particles with the
desired characteristics.
Among the numerous possibilities, supersymmetry seems to be quite appealing.
Supersymmetry is a symmetry between bosons and fermions, introduced to solve theoretical problems of the Standard Model. In most cases, supersymmetric extensions
of the Standard Model also conserve a discrete symmetry, the R-parity, in order to
conform with particle physics phenomenology, especially the non-observation of the
proton decay. A new possibility appeared in this class of models: one of the new particles is stable and neutral and, in principle, it is possible to be a viable DM candidate
with the observed abundance.
Many experiments are running around the world, aiming either at the direct detection of DM particles or at the detection of indirect signals coming from them. The
latter originate from dark matter annihilation in regions of the Universe that it is expected to be more condensed. The results of these experiments constitute a test of the
various theoretical models proposed to explain the DM problem.
Another puzzling fact is the agreement of the values, at the level of order of magnitude, of the DM abundance and the abundance of baryonic matter. If this is not just
a coincidence, these two forms of matter should have something in common. In order
to explain this coincidence, the possibility that the DM particles carry a conserved
quantum number related to baryon number has recently attracted a lot of attention.
Then, it is in principle possible that the DM current abundance is the asymmetry
x CONTENTS
between DM particles and antiparticles, as it is in the case of baryons. The difference
for the two densities comes simply from the difference in their masses.
In the first part of the current dissertation we deal in general with particle Dark
Matter. In Chapter 1 we review the DM physics. We give the evidence for the DM
existence and explain why particle DM is more favorable among other possibilities. We
also describe the common mechanism that determines the DM relic abundance and,
finally, we examine the DM detection methods and present the current experimental
status. In order to explain the DM, foremost, one needs a theory that describes
successfully the known fundamental particles. In Chapter 2 we describe the theory
that has been established during the last decades as the Standard Model of particle
physics. In the same chapter, we also discuss the theoretical problems from which this
model suffers and motivate the supersymmetric extensions of the SM.
In the second part we examine the DM in the context of a specific supersymmetric
model, the Next-to-Minimal Supersymmetric Standard Model (NMSSM). There are
good theoretical and phenomenological reasons to move from the minimal supersymmetric model to the NMSSM. These are described in Chapter 3. We also describe the
Lagrangian of the model and the possible DM candidates this model provides, exploring the general DM characteristics and detection. In the subsequent Chapter 4, we
attempt to explain in the NMSSM a possible indirect DM signal, a monochromatic
photon excess, that may originate from DM annihilation.
The last, third, part of the thesis is devoted to asymmetric DM. In Chapter 5
we introduce the asymmetric DM and explore the conditions under which the DM
current density is determined indeed by its asymmetry. We derive quite severe upper
bounds on the DM particle-particle or antiparticle-antiparticle self-annihilation, which
constrain the asymmetric DM models. Subsequently, we propose in Chapter 6 a specific
asymmetric DM model, obtained by an extension of the NMSSM, which respects the
self-annihilation bounds. We investigate, in the same chapter, the properties of the
proposed DM and discuss possible bounds coming from collider physics, cosmology
and DM detection experiments.
Note: The original work of this thesis is included in the last three chapters (Ch. 4,
5 and 6), which are based on the publications [1–3].
INTRODUCTION
La Matiere Noire (MN) est une forme inconnue de matiere non-lumineuse et r´epandue
dans toute Univers. L’explication de la MN figure parmi les d´efis principales de la
physique th´eorique. Bien que les ´evidences de son existence sont jusqu’`a maintenant
que des observations implicites, d’origine d’une vari´et´e des effets gravitationnelles, ils
sont assez robustes pour consid´erer avec grande certitude que la MN constitue plus
que le 80% de la matiere totale de l’Univers. En plus, les ´evidences suggerent que la
MN est constitu´ee par particules massifs, non-baryoniques, a vie longue, lesquels interagissent seulementa travers la gravit´e et des interactions faibles. Aucun des particules
d´ecrites par le Mod`ele Standard (MS) de la physique des particules ne correspond pas
aux sp´ecificit´es de la MN. Plusieurs mod`eles ont ´et´e propos´e en s’´etendant la th´eorie
standard et ayant comme but d’inclure les particules pr´esentant les caract´eristiques
d´esir´es.
Parmi les plusieurs possibilit´es, la th´eorie de supersym´etrie semble ˆetre la plus attirante. La supersym´etrie est une sym´etrie entre les bosons et les fermions, introduite
pour r´esoudre les problemes th´eoriques du MS. Dans la majorit´e de cas, les extensions supersym´etriques du MS conservent une sym´etrie discrete, la R-parit´e, afin de
se conformer avec la ph´enom´enologie de la physique des particules, sp´ecialement en
ce qui concerne l’absence d’observation de la d´esint´egration du proton. Une nouvelle
possibilit´e a ´et´e apparue dans cette classe des mod`eles: un de nouveaux particules est
stable et neutre et, en principe, il est possible d’ˆetre un candidat pour expliquer la
MN, viable avec l’abondance observ´ee.
Plusieurs exp´eriences sont effectu´ees au monde, ayant comme but soit la d´etection
directe des particules de MN, soit la d´etection de signaux indirects d’origine des particules de MN. Les r´esultats de ces exp´eriences constituent un test des diff´erents mod`eles
th´eoriques qui proposent et qui expliquent le probl`eme de la MN.
Un autre probl`eme est l’accord au niveau de l’ordre de magnitude entre les valeurs
d’abondance de MN et de l’abondance de mati`ere baryonique. S’il s’agit pas d’une
co¨ıncidence, ces deux formes de mati`eres devraient avoir quelque chose en commun.
En cons´equence, afin d’expliquer cette co¨ıncidence, la possibilit´e que les particules de
MN portent un nombre quantique qui est conserv´e en relation avec le nombre bary-
xii CONTENTS
onique a r´ecemment attir´e beaucoup d’attention. Il est donc possible que l’abondance
actuelle de MN est expliqu´ee par l’asym´etrie entre les particules de MN et les antiparticules, comme c’est le cas pour les baryons. La diff´erence entre les deux densit´es vient
simplement par la diff´erence entre les masses.
Dans la premiere partie de cette these nous traitons de mani`ere g´en´erale les particules de la MN. Dans Chapitre 1 nous effectuons une r´evision de la physique de MN.
Nous fournissons les preuves pour l’existence de la MN et nous expliquons pourquoi
les particules de MN sont plus favorable parmi les autres possibilit´es. Nous d´ecrivons
aussi le m´ecanisme commun qui d´etermine l’abondance de la MN et, filialement nous
examinons les m´ethodes de d´etection de MN et pr´esentons l’´etat de l’art actuel des
exp´eriences. Afin d’expliquer la MN, il faut utiliser la th´eorie qui d´ecrit en succ`es les
particules fondamentales d´ej`a connus. Dans le Chapitre 2 nous d´ecrivons la th´eorie
qui a ´et´e ´etablie dans les derniers d´ecennies selon le MS de la physique des particules.
Dans le mˆeme chapitre, nous discutons aussi les probl`emes th´eoriques du MS et ceux
qui motivent les extensions supersym´etriques du MS.
Dans la deuxieme partie, nous examinons la MN, dans le contexte d’un modele sp´ecifique de supersym´etre, le Next-to-Minimal Supersymmetric Standard Model (NMSSM).
Il y a des bonnes raisons th´eoriques ainsi que ph´enom´enologiques pour passer du mod`ele
supersym´etrique minimal au NMSSM. Ces raisons sont d´ecrites dans le Chapitre 3.
Nous d´ecrivons aussi le Lagrangien du modele ainsi que les particules candidat possible de la MN que ce modele nous offre, en explorant les caract´eristiques g´en´erales de la
MN. Au prochain Chapitre 4, nous tentons `a expliquer dans NMSSM un signal de MN
indirect, un exc`es de photons monochromatique, qui peuvent provenir de l’annihilation
de la MN.
La derniere partie de cette these est d´evou´ee aux asym´etries de la MN. Dans le
Chapitre 5 nous introduisons la MN asym´etrique et nous explorons les conditions
sous lesquels la densit´e actuelle de la MN est en effet d´etermin´ee par son asym´etrie.
Nous trouvons des limites sup´erieures assez s´ev`eres sur l’auto-annihilation de particuleparticule ou de antiparticule-antiparticule. En plus, nous proposons dans Chapitre 6 un
mod`ele d’asym´etrie sp´ecifique de MN, obtenu par l’extension de NMSSM, qui respect
l’auto annihilation des limites. Nous investiguons, dans le mˆeme chapitre, les propri´et´es
de la MN telles que propos´ees et nous discutons les limites possibles de l’origine de
physique des collisionneurs, de la cosmologie et des exp´eriences de d´etection de la MN.
Note: Le travail original de cette th`ese est inclue dans les derniers trois chapitres
(Ch. 4, 5 et 6), lesquels sont bas´es sur des publications [1–3].
Part I
Particle Dark Matter

CHAPTER 1
DARK MATTER
The latest results from the Linear Hadron Collider (LHC) and the Planck satellite offered an amazing verification of the Standard Models of both Particle Physics (henceforth, denoted just as SM) and Cosmology. The discovery of the Higgs boson completed
the detection of all particles predicted by the SM and put an end to any potential
doubts about it. On the other hand, the Cosmic Microwave Background radiation
observed by Planck is consistent in high precision with the standard cosmology. But
at the same time, Planck confirmed once more the fact that the total matter of the
Universe is dominated by one yet unknown form of matter, the so-called Dark Matter
(DM). The nature of DM constitutes one of the major puzzles of the theoretical physics
today.
The story of DM is not new. In 1970s, it was realized that the measured rotational
velocity of isolated stars or gas clouds in the outer parts of galaxies was not as one
should expect from the gravitational attraction of the known matter. This fact brought
back to light an old idea about a non-luminous form of matter and forced to take it
seriously. It was back in 1933 that Zwicky [4,5] observed that the mass of the luminous
matter (stars, gas etc.) in the Coma system, a cluster of about one thousand galaxies,
was not adequate to explain the motion of cluster member galaxies. The idea, however,
of a non-luminous form of matter preexisted [6] and it was actually used one year earlier
by Oort [7] to explain his observations, which nevertheless proved erroneous. However,
today, the existence of this non-luminous, dark matter is considered unquestionable
due to various kinds of evidence, many of them independent of the others. It is almost
certain nowadays that DM does not only cluster with stellar matter forming the galactic
halos, but it also exists as a background throughout the entire Universe.
The evidence for the DM will be the subject of the next but one section (Sec.
1.2). Meanwhile, we have to give a brief review of the standard cosmological model.
In Sec. 1.3 we discuss the possible DM candidates and the reason that a particle DM
is most favorable. Subsequently, in Sec. 1.4 we review the standard mechanism that
determines the density of the DM particles, a quantity that has been calculated quite
accurately by astrophysical observations. We finish this chapter by describing, in the
4 Dark Matter
last two sections 1.5 and 1.6, the detection methods of particle DM and the current
experimental status.
1.1 The Standard Big Bang Cosmological Model
In this section we are going to review the standard cosmological model based on the Big
Bang theory and on general relativity. However, it is not going to be an introduction to
the general theory of relativity, but rather a very brief review of notions and formulas
that we need for the description of DM.
A basic characteristic of the standard cosmological model is the evidence that the
universe is expanding. The expansion was discovered at the late 1920’s [8] by observing
the spectra of distant galaxies. A local observer that detects light from a distant object
sees a redshift z in the frequency, which corresponds to the motion of the object away
from the observer due to the Doppler effect. All of the observed galaxy spectra up
to the present time (except of few coming from very nearby galaxies) are red-shifted,
a fact stressing the universality of the expansion. The redshift z can be written in
power series in terms of the luminosity distance dL ≡

L
4πF
1/2
(where L is the object’s
luminosity and F the measured flux) as
z = H0dL +
1
2
(q0 − 1) (H0dL)
2
, (1.1)
where H0 is the present expansion rate of the Universe, known as the Hubble constant
and q0 is a parameter that represents the deviation from the linear Hubble law and
measures the deceleration of the Universe. Usually, the Hubble parameter is taken to
be
H0 = 100h km s−1 Mpc−1
, (1.2)
with the numerical uncertainties moved to the dimensionless parameter h, which takes
the value h = 0.673 ± 0.012 [9].
The expansion of the Universe may originate naturally from an isotropic and homogeneous cosmological model based on general relativity. Although Einstein imposed
these two assumptions without any observational evidence, today they are general
thought as undeniable. The best evidence for isotropy comes from the observation of
the Cosmic Microwave Background (CMB) radiation, which exhibits a temperature
uniformity. Testing the homogeneity of the Universe is not so straightforward, but
sky surveys have confirmed it with large accuracy [10]. The validity of these assumptions form the modern cosmological principle, which reflects the fact that all spatial
positions in the Universe are essentially equivalent.
Isotropy and homogeneity are playing an essential role, since they allow the description of the space-time of the Universe in terms of only two parameters denoted
by R(t) and k, accounting, respectively, for its overall expansion (or contraction) and
its spatial curvature. The most general expression for a space-time metric, known as
Friedmann-Robertson-Walker or FRW metric, can be written as (see, for example, [11])
ds
2 = dt
2 − R(t)

dr
2
1 − kr2
+ r
2

dθ
2 + sin2
θdφ
2

, (1.
1.1 The Standard Big Bang Cosmological Model 5
where as usual r, θ, φ and t are the spherical and time coordinates, respectively. The
curvature constant k takes only the discrete values +1, 0, −1, corresponding to closed,
(spatially) flat and open geometries. R(t) is the cosmological scale factor and determines proper distances in terms of the comoving coordinates. Usually, it is convenient
to define a dimensionless scale factor a(t) ≡
R(t)
R0
, where R0 is the present-day value of
R. The Hubble parameter can be defined through the scale factor as
H(t) ≡
R˙(t)
R(t)
=
a˙(t)
a(t)
. (1.4)
We can use the metric (1.3) in order to show that the cosmological redshift is a
direct consequence of the Hubble expansion. The redshift is defined as
z =
f1 − f2
f2
, (1.5)
with f1 the frequency of the emitted light and f2 the frequency of the observed light.
For scales smaller than cosmological, so that the expansion velocity v12 (corresponding
to the velocity with which the distant object moves away from the observer) is not
relativistic, the redshift is approximated as z ≃
v12
c
. Using the metric (1.3) for a light
signal (ds
2 = 0), we eventually arrive at the simple relation 1 + z =
R2
R1
between the
redshift z and the scale factor R.
The evolution of the Universe can be described by two rather simple equations,
known as Friedmann–Lemaˆıtre equations. Assuming the matter content of the Universe
as a perfect fluid, the energy–momentum tensor is written as
Tµν = −pgµν + (p + ρ)uµuν, (1.6)
where gµν is the metric tensor related to the metric (1.3), p the isotropic pressure, ρ
the energy density and u = (1, 0, 0, 0) the velocity vector for the isotropic fluid in
comoving coordinates. The Einstein’s equations lead to the following expressions:
H
2 =
8π
3
GN ρ −
k
R2
+
Λ
3
(1.7)
and
R¨
R
= −
4π
3
GN (ρ + 3p) + Λ
3
, (1.8)
where GN is the gravitational constant and Λ the cosmological constant, which can be
interpreted to correspond to the energy of the vacuum. (The first of these equations
is often called the Friedmann equation.) The energy–momentum conservation leads to
a third equation:
ρ˙ = −3H(p + ρ). (1.9)
Examining (1.7), we see that in the absence of a cosmological constant (Λ = 0), the
expansion or contraction of the Universe is determined solely by the value of k. For
k = +1 it will recollapse, while it is going to expand indefinitely if k = 0 or k = −1.
6 Dark Matter
This way, one can define the following expression that gives the critical density, such
that k = 0 (when Λ = 0)
ρC ≡
3H2
8πGN
. (1.10)
Finally, the cosmological density parameter Ωtot is defined as the energy density relative
to its critical value
Ωtot ≡
ρ
ρC
. (1.11)
The Friedmann equation can be rewritten in terms of the density parameter as k
R2 =
H2
(Ωtot − 1). It is often useful to distinguish the origin of the contribution to the total
density. In this sense,
Ωtot = Ωmat + Ωrad + ΩΛ, (1.12)
where Ωmat is the contribution from pressureless matter, Ωrad comes from relativistic
particles (radiation) and ΩΛ is due to the cosmological constant. The matter density is
further divided to the contribution from baryonic matter (Ωb) and from (non-baryonic)
DM (ΩDM).
It is important to note that much of the history of the Universe can be described
by assuming that either matter or radiation dominates the total energy density. By
defining the parameter w =
p
ρ
, Eq. (1.9) is written in terms of w as ˙ρ = −3(1 + w)ρ
R˙
R
.
After integration, it gives
ρ ∝ R
−3(1+w)
. (1.13)
In the radiation dominated era of the Universe w = 1/3, while during matter domination w = 0, so that ρ ∝ R−4
(radiation dominated) and ρ ∝ R−3
(matter dominated),
respectively.
1.2 Evidence of DM
1.2.1 Galactic rotation velocities
As it was mentioned before, the first strong evidence for the existence of DM were
the galactic rotation velocities [12]. The mass distribution of a spiral galaxy can be
approximated as spherical or ellipsoidal. Applying the Newton’s law, which is sufficient
for such large distances, we can see that at a distance r from the galactic center the
rotation velocity obeys the equation v
2
r =
GNM(r)
r
2
, where M(r) is the mass distribution
in the galaxy. Taking r much larger than the radius of the luminous mass, so that
M(r) corresponds to the total galactic mass, Newton’s law implies that v ∝ 1/
√
r.
However, galaxy observations based on the Doppler effect show that the velocity rises
with r towards a constant value vconst ≃ 100 − 200 km s−1
. The first galaxy in which
this behavior observed was Messier 33, a spiral galaxy about 3 million light years (ly)
away. Its rotation curve can be seen in Fig. 1.1 (left). Along with the observed curve,
the expected rotation velocity due to the luminous mass has also been plotted. The
same phenomenon has already been observed for a plethora of galaxies, including our
galaxy [13] (see Fig. 1.1 – right).
1.2.2 Gravitational lensing 7
Figure 1.1: Left: The rotation curve for the M33 dwarf galaxy, superimposed on its
optical image, as observed by starlight and 21 cm hydrogen spectrum lines, and the
expected rotation curve due to the luminous amount of mass. From [14,15]. Right: The
rotation velocities for the Milky Way, the NGC 4258 and M31 galaxies as a function
of the distance from the galactic center. From [13].
Returning to the Newton’s law, we can easily check that the aforementioned disagreement would have been resolved, if the mass distribution was growing linearly with
r, M(r) ∝ r. Actually, a self-gravitating ball of an ideal gas at a uniform temperature
kT =
1
2mXvconst, with mX the mass of the particles that constitute the gas and vconst
the asymptotic value of the rotation velocity, would have exactly this mass profile [16].
Therefore, a simple solution to the missing mass problem is the assumption that the
disk galaxies are immersed in extended DM halos. Current analyses of rotation curves
imply that Ωmat ≃ 0.1 (see [17] for a review), while observations of the luminous matter
constrain its density to only Ωlum <∼ 0.01. Hence, about 90% of the total mass of the
galaxies is dark.
1.2.2 Gravitational lensing
Since DM interacts gravitationally, its mass warps the space-time causing the distortion
of a passing beam of light. Hence, although dark, the presence of DM should be visible
through the “bending” of the light coming from behind sources. This fact is used in the
so-called gravitational lensing: large clusters of galaxies can be used as astrophysical
lenses that bend and magnify the light coming from galaxies far behind them. The
distorted picture can give an estimate for the mass distribution of the lens. Since
lensing does not rely on the dynamics of the observed systems, it is a completely
independent method of predicting the DM density.
In contrast to optical lenses, a gravitational lens has no single focal point, but
instead a focal line. The maximum bending occurs closest to the center of the lens,
and the minimum furthest from it. In the ideal case that the light source (a distant
galaxy), the lens (the cluster of galaxies) and the telescope lie in a straight line, the
source galaxy would appear as a ring around the lensing object. In fact, partially
because of a misalignment of the three objects, but also due to the complex mass
8 Dark Matter
Figure 1.2: Left: Abell 1689 acting as gravitational lens that bends and magnifies the
light of the galaxies located far behind it. Some of the faintest objects in the picture
are probably over 13 billion light-years away (redshift value 6). This color image is a
composite of visible-light and near-infrared exposures taken by the Hubble telescope in
June 2002. According to NASA, it reveals 10 times more arcs than would be seen by
a ground-based telescope. Courtesy of the Space Telescope Science Institute (STScI).
Right: A masked region of Abell 1689. Cluster members were selected using color
information and then masked over, so that these regions do not affect the surface
density estimate of background sources. The background galaxies are also shown as
open circles. Superimposed are the concentric bins used to calculate the radial profile,
centered on the peak in the light distribution. From [18].
distribution of the lensing cluster, the source resembles partial arcs scattered around
the lens. Fig. 1.2 is an example of the arcs formed as the light of distant galaxies passed
through the cluster Abell 1689, one of the most massive known galaxy clusters, acting
as a 2-million-light-year-wide lens in space.
In many cases, the distortion of the light of background sources is too weak to
form arcs and can be detected only by analyzing a large number of sources and using
statistical methods. This kind of lensing is known as weak lensing. The lensing shows
up statistically as a preferred stretching of the distant objects perpendicular to the
direction towards the center of the lens. By measuring the shapes and orientations of
large numbers of distant galaxies, their orientations can be averaged to measure the
shear of the lensing field in any region. For a population of unlensed galaxies, the shear
pattern should be, on average, randomly distributed. In the presence of lensing, the
shear field is polarized and, since it is related non-locally to the surface mass density,
it can be used to estimate the mass distribution.
Perhaps the most compelling evidence for DM came applying these weak lensing
techniques on the colliding system of Bullet cluster [19,20]. The Bullet cluster consists
of two primary galaxy concentrations, a less massive subcluster that is currently moving
away from a more massive main cluster. The X-ray image reveals the relative motion
1.2.3 CMB radiation 9
Figure 1.3: The left panel is a color image from the Magellan images of the merging
Bullet cluster, with the white bar indicating 200 kpc at the distance of the cluster. The
right panel is an X-ray Chandra image of the same cluster. The contours represent
the weak lensing mass reconstruction. The separation between the location of the
luminous interacting X-ray halo and the location of gravitating matter can be clearly
seen. From [20].
of the two systems. Comparing with the line-of-sight velocity differences of the two
components, it can be deduced that the two cores passed through each other about
100 million years ago and that the merger is occurring in the plane of the sky.
The cluster observation reveals that its mass is partially made of baryons observable
in optical and infrared data, but it is dominated by baryons observable in X-rays.
During the merger, the galaxies, which correspond to the small amount of optical
baryons, remain collisionless, while the X-ray halo is affected by ram pressure. The
mass distribution of the system was reconstructed by means of weak lensing. In the
absence of DM, one should expect that the reconstructed mass distribution would
exhibit a primary peak coincident with the dominant X-ray gas, which is spatially
offset from the galaxy distribution (right panel of Fig. 1.3). However, as it can be seen
in the left panel of Fig. 1.3, the mass maps created from weak lensing have the primary
mass peaks in good spatial agreement with the galaxies.
The analysis performed in [20] is in agreement with the other astrophysical observations: only 12% of the total mass of the cluster is due to baryons (from which
1% is visible in optical spectrum and the rest is the X-ray halo) and 88% is the DM.
Combining all the astrophysical bounds, one can make a rough estimation for the DM
density, which lies on the range
0.1 <∼ Ω
astr
DM h
2 <∼ 0.3. (1.14)
1.2.3 CMB radiation
The most precise prediction of the DM density is coming, however, from analyses of
the Cosmic Microwave Background (CMB) spectrum. The most recent observation
of CMB by the Planck satellite (which improved previous results [21, 22] by WMAP)
10 Dark Matter
constrained the DM density in the interval [9]
ΩDMh
2 = 0.1199 ± 0.0027. (1.15)
This result plays a key role for testing possible DM candidates and we are going to use
it many times throughout this work. In the following, we will describe how DM affects
the CMB spectrum. Once again, the detailed analysis leading to the above calculation
is complicated and goes well beyond the scope of this thesis. We will rather try to
give a qualitative picture of the relation among DM and the shape of the observed
spectrum.
The CMB that we observe today consists of photons that have started a free travel
through space since their last scattering with matter, early in the history of the Universe (see, for example, [23,24]). Even earlier, while the Universe was made up from a
very hot interacting plasma of photons, electrons and baryons, the large temperature
of photons was preventing the electrons to combine with protons to form hydrogen
atoms. As the Universe was expanding, the photon temperature was decreasing, and
at some point the formation of atoms was possible. This corresponds to the recombination epoch of the Universe. After then, the photons no longer interacted with the
neutral plasma and their free propagation started, with a temperature that is redshifting following the expansion of the Universe. The value of this temperature today is
∼ 2.73 K [25].
Although the CMB radiation is highly isotropic1
, small anisotropies appear if one
concentrates on smaller scales, which correspond to smaller angles in the sky, later led
to structure formation in the Universe. In order to study these anisotropies (see for
example [26,27]), the temperature, which is a function of the polar coordinates defining
the direction on the sky, is expanded in spherical harmonics:
T(θ, φ) = X
l,|m|≤l
almYlm(θ, φ). (1.16)
The coefficients alm describe temperature variations on angular scales l ∼ π/∆θ.
The l = 0 term is the isotropic temperature, while l = 1 is the dipole anisotropy
corresponding to the motion of the solar system. The variance of the temperature
h∆T
2
i ≡ h(T − hTi)
2
i is written, using the orthogonality of the spherical harmonics,
as
h∆T
2
i =
1
4π
X
l>1
(2l + 1)Cl
, (1.17)
where we Cl ≡ h|alm|
2
im is the average of the coefficients alm over m. The quantity
D
2
l ≡
l(l + 1)
2π
Cl (1.18)
gives the contribution to the temperature fluctuations per interval of ln l. The CMB
power spectrum – the plot of Dl versus l – as observed by the Planck satellite is shown
in Fig. 1.4.
1About 1 part in 100, 000, after subtracting the uninteresting dipole anisotropy, which is due to
the Doppler effect caused by the solar system’s motion.
1.2.3 CMB radiation 11
2 50 500 1000 1500 2000 2500
ℓ
102
103
104
Dℓ
[µ
K
2
]
lensed CMB
30 to 353
70
100
143
143×217
217
353×143
Figure 1.4: The Planck power spectra. The dashed line indicates the best-fit Planck
spectrum. From [28].
We are ready to reach the main point of this section, to wit, how these anisotropies
were generated and, eventually, why the existence of DM is necessary to explain the
observed spectrum. To do so, we have to go back once again to the study of the
early Universe. Before recombination, the CMB photons and the baryons acted as
a nearly perfect fluid. Gravitational potential wells, caused by random fluctuations,
had been stretched to cosmic scales during inflation. The photon-baryon fluid was
under the influence of this potential. While gravity was compressing the fluid, its
radiation pressure was resisting, resulting in acoustic oscillations. The sound waves
were changing the photon temperature; it was rising during compression and it was
falling during rarefaction. The oscillations stopped at recombination as the photons
were released from the fluid, and what we observe today is actually a frozen picture
of this procedure. The peaks are caused by modes that have reached extrema of
compression and rarefaction at the time of last scattering. The first peak corresponds
to modes that have had enough time to oscillate through exactly one half of a period
before last scattering, the second peak is caused by modes oscillated through a full
period (half the wavelength of the first mode), and so on.
Much information can be deduced from the CMB power spectrum. For example,
without entering into the details, the position of the first peak is related to the spatially
geometry of the Universe, whereas the relative height of the second peak, compared to
the first one, is related to the baryonic density [29]. Here, we will focus on the effect
of DM on the power spectrum.
We start without assuming a priori the existence of DM. When radiation dominated
over matter, the density fluctuation stabilizes as the radiation pressure prevents further
compression, causing the decay of the gravitational potential. Since the potential well
lowers after the compression, the amplitude of the rarefaction will be larger. We note
that modes with smaller wavelength (higher multipoles) started oscillating first, so that
12 Dark Matter
it is expected that each even peak would be higher than the successive odd peak. In the
presence of a collisionless cold (non-relativistic) fluid, the density fluctuation remains
after the compression and the gravitational potential does not decay. Therefore, in the
presence of (cold) DM, the third peak is expected to be comparable or higher than the
second one2
. Indeed, this is the case of the observed CMB power spectrum (Fig. 1.4).
In practice, the effect of the various phenomena determining the shape of the power
spectrum is more complicated than the above simplified qualitative analysis. One has
to apply statistical methods in order to fit a cosmological model to the observed CMB
spectrum. The best fit to the power spectrum as observed by Planck [9] is a flat
ΛCDM model3
, with baryonic density Ωbh
2 = 0.02205 ± 0.00028, dark matter density
ΩDMh
2 = 0.1199±0.0027 and energy density of the cosmological constant (dark energy
density) ΩΛ = 0.685+0.018
−0.016.
1.2.4 Other evidence
The clues for the existence of DM are not limited to the three aforesaid phenomena. For
example, sky surveys of Baryon Acoustic Oscillations (BAO) – periodic fluctuations
of the baryonic density caused by acoustic oscillations in the early Universe – are
consistent with the results extracted by the CMB spectrum. The velocity dispersion of
galaxies in galactic clusters indicate a large mass-to-light ratio, giving another evidence
for DM. Furthermore, numerical simulations require a significant amount of cold DM
in order to reproduce the large scale structure of the Universe.
1.3 Particle DM
Before we proceed to possible DM candidates, we have to refer to an attempt for
an alternative explanation of the above phenomena, without the introduction of DM.
Mainly in order to explain the anomalous galactic rotation curves, Milgrom proposed
in 1983 [31] a modified version of Newton’s law in galactic scales. This theory is known
as Modified Newtonian Dynamics (MoND) and it has gained a lot of attention since
then (see, for example, [32] for a review). However, MoND seem insufficient to account
for the necessity of DM at scales larger than the galactic ones [17,33,34]. Furthermore,
weak lensing of the Bullet cluster disfavors these theories [19], since in the case of
MoND the X-ray gas would be the dominant component of the total mass and the
separation indicated in Fig. 1.3 (right panel) would not have been observed.
One of the first possibilities examined for DM candidates were astrophysical objects
that may count for DM. These were collectively called MAssive Compact Halo Objects
(MACHOs) and such examples are brown or white dwarfs, neutron stars and stellar
black hole remnants. These objects contribute to the density of baryonic DM. However,
Big Bang nucleosynthesis and the CMB have set a limit on this density, which is
also confirmed by the observation of MACHOs in the Milky Way halo through their
2The higher multipoles are affected by a damping effect [30].
3The standard cosmological model with a cosmological constant Λ and Cold Dark Matter.
1.4 The Standard Thermal Mechanism 13
gravitational lensing effect. This limit is far below the required value in order to fit
the DM observations. As a consequence, non-baryonic DM is a necessary ingredient of
the Universe.
Since the astrophysical objects are not adequate to count for the main component
of DM, the attention has focused on possible particles that can play the role of this
non-luminous matter. The only known particle that fits the criteria for DM is the
neutrino. Although neutrinos are massless in the SM of particle physics, oscillations
among their various flavors suggest a small but non-zero mass. However, a universe
dominated by particles with such small mass would form large structures first, with
the small structures forming later by fragmentation of the larger objects. This time
scale, in which the galaxies form last and quite recently, seems incompatible with our
current view of galactic evolution.
Nevertheless, extensions of the SM, essential to solve some of its theoretical drawbacks, provide particles that can, in principle, successfully solve the DM problem. In
the next section, we will see that favorable candidates are Weakly Interacting Massive
Particles (WIMPs). Supersymmetric theories that respect a discrete symmetry, the Rparity, provide a very promising WIMP, the neutralino. We will not extend here, since
we are going to discuss neutralinos in more detail in the following chapters. However,
WIMPs are also predicted by other, non-supersymmetric theories, such as models with
TeV scale extra dimensions.
For completeness, we will finish this section by just mentioning the axions, although
we will not deal with them in the rest of this thesis. Axions are neutral scalar hypothetical particles associated with the spontaneous breaking of the global U(1) Peccei-Quinn
symmetry [35, 36], introduced to dynamically solve the strong CP problem. Their
very small coupling to ordinary matter gives a large lifetime to axions, larger than
the age of the Universe. Axions were never in thermal equilibrium and were always
non-relativistic. These characteristics allow them to be possible DM candidates.
1.4 The Standard Thermal Mechanism
1.4.1 Relic Abundance, thermal cross section and WIMPs
We shall discuss subsequently the mechanism that is widely considered responsible for
the current DM density, in case of particle DM, as well as the requirements in order
to fit this density to the observed value. We will also see why WIMPs are favorable
DM candidates. This subsection will remain descriptive; a more detailed analysis will
follow.
We assume a particle X with mass mX that is neutral and stable. X would be
the DM particle for this analysis. Early in the history of the Universe, when its
temperature was much larger than the particle’s mass (T ≫ mX), Xs were abundant
with a density comparable with the photon’s density. Due to pair annihilations with
their antiparticles, they were rapidly converting to lighter particles and vice versa.
The annihilations were in equilibrium, without affecting the density of the X particles.
Shortly after T drops below the mass mX, the number density of X started to drop
14 Dark Matter
very fast, since lighter particles do not have enough energy anymore to produce X
particles and pair annihilation continued to destroy them. The equilibrium particle
density is given by
n
eq
X =
g
(2π
3
)
Z
f(~p) d3
~p, (1.19)
where g is the number of internal (spin) degrees of freedom of the particle and f(~p) is
the Bose-Einstein or the Fermi-Dirac distribution function in terms of the momentum
~p. We will see4
that Eq. (1.19) gives (after integration) n
eq
X ∝ T
3
, for T ≫ mX,
whereas for T ≪ mX the particle density is Boltzmann (exponentially) suppressed
with n
eq
X ∝ e
−mX/T
.
As the Universe is expanding and the X particle density decreases, the pair annihilations of X particles become more rare, until they eventually stop when their rate
Γ drops below the expansion rate, Γ <∼ H. The rate of a pair annihilation Γ is proportional to the density of the annihilating particles, more precisely Γ = nhσvi, where hσvi
is the thermal average of the annihilation cross section σ times the particles relative
velocity v (we will return to this in more detail in the following subsection). At the
point where the Xs cease to annihilate, they fall out of equilibrium with the thermal
plasma and what remains is their relic cosmological abundance, almost constant since
then. It is customary to say that the DM density froze-out and call the temperature
where this occurred the freeze-out temperature, henceforth Tf o.
We can use the freeze-out condition Γ ≃ H to approximate the DM relic density in
terms of the thermal averaged annihilation cross section (we reproduce the calculation
performed originally in [37]). For this purpose, we will need the expressions for the
energy and entropy density, which are defined in the App. A and which we rewrite here
ρ(T) = π
2
30
geff(T) T
4
(1.20)
and
s(T) = 2π
2
45
heff(T) T
3
. (1.21)
We recall (see App. A, for more details) that geff and heff are effective relativistic degrees
of freedom. Assuming that there is no significant entropy production since the freezeout, the entropy per comoving volume remains constant, so that the ratio nX/s remains
also constant (since the freeze-out). Hence, the present-day DM particle density is given
by nX0 = s0

nX
s

f o, with s0 ≃ 4 · 103
cm−3
the current entropy density. Therefore, we
have to compute the ratio nX/s during freeze-out.
The early Universe is radiation dominated, hence Eq. (1.2) reads, after replacing
the energy density by Eq. (1.20), as H =
2π
3
pπ
5GN g
1/2
eff T
2
. The freeze-out condition
4Number densities will be discussed again much later in this thesis, in Sec. 5.1, in the presence of
chemical potentials
1.4.2 The Boltzmann equation 15
gives, then,
nX
s

f o =
45
3π
pπ
5GN
g
1/2
eff
heff
(Tf ohσvi)
−1
, which evaluates5
to
nX
s

f o
≃ 7 · 10−9 GeV
mX
10−27 cm3
s
−1
hσvi
. (1.22)
We remind that ΩX ≡
ρX
ρc
=
m
ρc

nX
s

f o s0, where the critical density today is ρc =
10−5h
2 GeV cm−3
, so that, finally, the relic density is
ΩXh
2 ≃
3 · 10−27 cm3
s
−1
hσvi
, (1.23)
independently of the DM mass mX.
In order to reproduce the observed relic density (1.15), the annihilation cross section
during the freeze-out has to be
hσvith ≃ 3 · 10−26 cm3
s
−1
. (1.24)
This quantity is known as thermal cross section. The scale of this value is remarkably
close to the cross section of weakly interacting particles, which can be estimated to be
hσweakvi ∼ α
2
m2
W
∼ 10−25 cm3
s
−1
, with α a generic weak coupling. This fact established
the WIMPs as the most favorable DM candidates.
1.4.2 The Boltzmann equation6
Although a weakly interacting particle has, in principle, the correct order of magnitude
of the annihilation cross section for the correct order of relic density, in practice, the
final result may vary over many orders of magnitude. This is the reason that a more
detailed analysis is required in order to be able to calculate the precise value of the
DM relic density.
The density of a species is governed by the Boltzmann equation, which can be
written in compact operator form as
L[f] = C[f], (1.25)
with L and C the Liouville and collision operators, respectively. f = f(~p, ~x) is the
phase-space density, which is, in general, a function of the momentum and space-time
coordinates and it is defined as
f =
(2π)
3
g
dN
d
3p d
3x
, (1.26)
with N the number of particles. It is normalized in such a way that f = 1 corresponds
to the maximum phase-space density allowed by the Pauli principle for a fermion. In
5
In this evaluation, we have used the expected relation between the freeze-out temperature and
the mass mX of the particle, Tfo ∼
mX
20 . However, we notice that the exact value of the denominator
depends on the annihilation cross section.
6
In this part, we follow part of the analysis performed in [38
16 Dark Matter
the special case of the spatially homogeneous and isotropic FRW cosmology, the phasespace density has the same symmetries and depends only on the particle energy E and
the time t, i.e. f = f(E, t).
The Liouville operator gives the net rate of change in time of f and the collision
operator describes the number of particles per phase-space volume that are lost or
gained per unit time due to collisions with other particles. The particle number density
n =
R
dN
d3x
is given through (1.26) by the integral (1.19) of f(E, t) over all momenta and
sum over all spin degrees of freedom. We will perform the same integral and sum in
the Boltzmann equation (1.25), in order to write it in a more convenient form involving
the particle densities.
First, the Liouville term for f = f(E, T) is written as
L[f] = ∂f
∂t − H
|p|
2
E
∂f
∂E . (1.27)
Integrating it and summing over all the spin degrees of freedom, it becomes
g1
Z
L[f1]
d
3p1
(2π
3
)
=
∂
∂t Z
f1
g1d
3p1
(2π)
3
− Hg1
Z
|p1|
2
E1
4π|p1|
2 dp1
(2π
3
)
= ˙n −
Hg1
(2π)
3
4π
Z
|p1|
3
∂f1
∂E1
dE1
= ˙n + 3Hn,
(1.28)
where we have used Eq. (1.27) and (1.19), pdp = EdE and in the last step we have
performed a partial integration.
Now we turn to the collision term, which in integrated form and summed over spins
is written, in the case of annihilation of two particles 1 and 2 to two others, 3 and 4,
as
g1
Z
C[f1]
d
3p1
(2π)
3
=
−
X
spins

f1f2(1 ± f3)(1 ± f4)|M12→34|
2 − f3f4(1 ± f1)(1 ± f2)|M34→12|
2

· (2π)
4
δ
4
(p1 + p2 − p3 − p4)
d
3p1
(2π)
32E1
d
3p2
(2π)
32E2
d
3p3
(2π)
32E3
d
3p4
(2π)
32E4
, (1.29)
where the “+” sign applies for bosons and “−” for fermions. We assume that the
annihilation products 3 and 4 go quickly into equilibrium with the thermal plasma, such
that the density functions f3 and f4 in Eq. (1.29) can be replaced by the equilibrium
densities f
eq
3
and f
eq
4
, respectively. Furthermore, the δ-function in the integral enforces
E1 + E2 = E3 + E4 and, since f
eq
3
f
eq
4 ∝ exp
−
E3+E4
T

, the product f
eq
3
f
eq
4
is replaced
by the corresponding product of the annihilating particle densities f
eq
1
f
eq
2
(principle of
detailed balance). In order to simplify the expression (1.29), we will apply the unitarit
1.4.2 The Boltzmann equation 17
condition which yields
X
spins
Z
|M34→12|
2
(2π)
4
δ
4
(p1 + p2 − p3 − p4)
d
3p3
(2π)
32E3
d
3p4
(2π)
32E4
=
X
spins
Z
|M12→34|
2
(2π)
4
δ
4
(p1 + p2 − p3 − p4)
d
3p3
(2π)
32E3
d
3p4
(2π)
32E4
(1.30)
and also the definition of the unpolarized cross section to write
X
spins
Z
|M12→34|
2
(2π)
4
δ
4
(p1 + p2 − p3 − p4)
d
3p3
(2π)
32E3
d
3p4
(2π)
32E4
=
4F g1g2 σ12→34, (1.31)
where F ≡ [(p1 · p2)
2 − m2
1m2
2
]
1/2
and the spin factors g1, g2 come from the average
over initial spins. This way, the collision term (1.29) is written in a more compact form
g1
Z
C[f1]
d
3p1
(2π)
3
= −
Z
σvMøl (dn1dn2 − dn
eq
1 dn
eq
2
), (1.32)
where σ =
P
(all f)
σ12→f is the total annihilation cross section summed over all the
possible final states and vMøl ≡
F
E1E2
. The so called Møller velocity, vMøl, is defined in
such a way that the product vMøln1n2 is invariant under Lorentz transformations and,
in terms of particle velocities ~v1 and ~v2, it is given by the expression
vMøl =
h
~v2
1 − ~v2
2

2
− |~v1 × ~v2|
2
i1/2
. (1.33)
Due to symmetry considerations, the distributions in kinetic equilibrium are proportional to those in chemical equilibrium, with a proportionality factor independent of
the momentum. Therefore, the collision term (1.32), both before and after decoupling,
can be written in the form
g1
Z
C[f1]
d
3p1
(2π)
3
= −hσvMøli(n1n2 − n
eq
1 n
eq
2
), (1.34)
where the thermal averaged total annihilation cross section times the Møller velocity
has been defined by the expression
hσvMøli =
R
σvMøldn
eq
1 dn
eq
2
R
dn
eq
1 dn
eq
2
. (1.35)
We will come back to the thermal averaged cross section in the next subsection.
We are, now, able to write the full integrated Boltzmann equation, using the expressions (1.28), (1.34) that we have derived for the Liouville and the collision term,
respectively. In the simplified but interesting case of identical particles 1 and 2, the
Boltzmann equation is, finally, written as
n˙ + 3Hn = −hσvMøli(n
2 − n
2
eq). (1.36)
18 Dark Matter
However, instead of using n, it is more convenient to take the expansion of the universe
into account and calculate the number density per comoving volume Y , which can be
defined as the ratio of the number and entropy densities: Y ≡ n/s. The total entropy
density S = R3
s (R is the scale factor) remains constant, hence we can obtain a
differential equation for Y by dividing (1.36) by S. Before we write the final form
of the Boltzmann equation that it is used for the relic density calculations, we have
to change the variable that parametrizes the comoving density. In practice, the time
variable t is not convenient and the temperature of the Universe (actually the photon
temperature, since the photons were the last particles that went out of equilibrium) is
used instead. However, it proves even more useful to use as time variable the quantity
defined by x ≡ m/T with m the DM mass, so that Eq. (1.36) transforms into
dY
dx
=
1
3H
ds
dx
hσvMøli

Y
2 − Y
2
eq
. (1.37)
Last, using the Hubble parameter (1.2) for a radiation dominated Universe and the
expressions (1.20), (1.21) for the energy and entropy density, the Boltzmann equation
is written in its final form
dY
dx
= −
r
45GN
π
g
1/2
∗ m
x
2
hσvMøli

Y
2 − Y
2
eq
, (1.38)
where the effective degrees of freedom g
1/2
∗ have been defined by
g
1/2
∗ ≡
heff
g
1/2
eff

1 +
1
3
T
heff
dheff
dT

. (1.39)
The equilibrium density per comoving volume Yeq ≡ neq/s can be expressed as
Yeq(x) = 45g
4π
4
x
2K2(x)
heff(m/x)
, (1.40)
with K2 the modified Bessel function of second kind.
1.4.3 Thermal average of the annihilation cross section
We are going to derive a simple formula that one can use to calculate the thermal
average of the cross section times velocity, based again on the analysis of [38]. We will
use the assumption that equilibrium functions follow the Maxwell-Boltzmann distribution, instead of the actual Bose-Einstein or Fermi-Dirac. This is a well established
assumption if the freeze out occurs after T ≃ m/3 or for x >∼ 3, which is actually the
case for WIMPs. Under this assumption, the expression (1.35) gives, in the cosmic
comoving frame,
hσvMøli =
R
vMøle
−E1/T e
−E2/T d
3p1d
3p2
R
e
−E1/T e
−E2/T d
3p1d
3p2
. (1.4
1.4.3 Thermal average of the annihilation cross section 19
The volume element can be written as d3p1d
3p2 = 4πp1dE14πp2dE2
1
2
cos θ, with θ the
angle between ~p1 and ~p2. After changing the integration variables to E+, E−, s given
by
E+ = E1 + E2, E− = E1 − E2, s = 2m2 + 2E1E2 − 2p1p2 cos θ, (1.42)
(with s = −(p1 − p2)
2 one of the Mandelstam variables,) the volume element becomes
d
3p1d
3p2 = 2π
2E1E2dE+dE−ds and the initial integration region
{E1 > m, E2 > m, | cos θ| ≤ 1i
transforms into
|E−| ≤
1 −
4m2
s
1/2
(E
2
+ − s)
1/2
, E+ ≥
√
s, s ≥ 4m2
. (1.43)
After some algebraic calculations, it can be found that the quantity hσvMøliE1E2
depends only on s, specifically vMølE1E2 =
1
2
p
s(s − 4m2
). Hence, the numerator of the expression (1.41), which after changing the integration variables reads
2π
2
R
dE+
R
dE−
R
dsσvMølE1E2e
−E+/T , can be written, eventually, as
Z
vMøle
−E1/T e
−E2/T = 2π
2
Z ∞
4m2
dsσ(s − 4m2
)
Z
dE+e
−E+/T (E
2
+ − s)
1/2
. (1.44)
The integral over E+ can be written with the help of the modified Bessel function of
the first kind K1 as √
s T K1(
√
s/T). The denominator of (1.41) can be treated in a
similar way, so that the thermal average is, finally, given by the expression
hσvMøli =
1
8m4TK2
2
(x)
Z ∞
4m2
ds σ(s)(s − 4m2
)
√
s K1(
√
s/T). (1.45)
Eqs. (1.38)–(1.40) along with this last Eq. (1.45) are all we need in order to calculate
the relic density of a WIMP, if its total annihilation cross section in terms of the
Mandelstam variable s is known.
In many cases, in order to avoid the numerical integration in Eq. (1.45), an approximation for hσvMøli can be used. The thermal average is expanded in powers of x
−1
(or, equivalently, in powers of the squared WIMP velocity):
hσvMøli = a + bx−1 + . . . . (1.46)
(The coefficient a corresponds to the s-wave contribution to the cross section, the
coefficient b to the p-wave contribution, and so on.) This partial wave expansion gives
a quite good approximation, provided there are no s-channel resonances and thresholds
for the final states [39].
In [40], it was shown that, after expanding the integrands of Eq. (1.41) in powers
of x
−1
, all the integrations can be performed analytically. As we saw, the expression
20 Dark Matter
vMølE1E2 depends on momenta only through s. Therefore, one can form the Lorentz
invariant quantity
w(s) ≡ σ(s)vMølE1E2 =
1
2
σ(s)
p
s(s − 4m2
). (1.47)
The integration involves the Taylor expansion of this quantity w around s/4m2 = 1
and the general formula for the partial wave expansion of the thermal average is [40]
hσvMøli =
1
m2

w −
3
2
(2w − w
′
)x
−1 +
3
8
(16w − 8w
′ + 5w
′′)x
−2
−
5
16
(30w − 15w
′ + 3w
′′ − 7x
′′′)x
−3 + O(x
−4
)

s/4m2=1
, (1.48)
where primes denote derivatives with respect to s/4m2 and all quantities have to be
evaluated at s = 4m2
.
1.5 Direct Detection of DM
Since the beginning of 1980s, it has been realized that besides the numerous facts showing evidence for the existence of these new dark particles, it is also possible to detect
them directly. Already in 1985, two pioneering articles [41, 42] appeared, describing
the detection methods for WIMPs. Since WIMPs are expected to cluster gravitationally together with ordinary stars in the Milky Way halo, they would pass also through
Earth and, in principle, they can be detected through scattering with the nuclei in a
detector’s material. In practice, one has to measure the recoil energy deposited by this
scattering.
However, although one can deduce from rotation curves that DM dominates the
dark halo in the outer parts of our galaxy, it is not so obvious from direct measurements
whether there is any substantial amount of DM inside the solar radius R0 ≃ 8 kpc.
Using indirect methods (involving the determination of the gravitational potential,
through the measuring of the kinematics of stars, both near the mid-plane of the
galactic disk and at heights several times the disk thickness), it is almost certain
that the DM is also present in the solar system, with a local density ρ0 = (0.3 ±
0.1) GeV cm−3
[43].
This value for the local density implies that for a WIMP mass of order ∼ 100 GeV,
the local number density is n0 ∼ 10−3
cm−3
. It is also expected that the WIMPs
velocity is similar to the velocity with which the Sun orbits around the galactic center
(v0 ≃ 220 km s−1
), since they are both moving under the same gravitational potential.
These two quantities allow to estimate the order of magnitude of the incident flux
of WIMPs on the Earth: J0 = n0v0 ∼ 105
cm−2
s
−1
. This value is manifestly large,
but the very weak interactions of the DM particles with ordinary matter makes their
detection a difficult, although in principle feasible, task. In order to compensate for
the very low WIMP-nucleus scattering cross section, very large detectors are required.
1.5.1 Elastic scattering event rate 21
1.5.1 Elastic scattering event rate
In the following, we will confine ourselves to the elastic scattering with nuclei. Although
inelastic scattering of WIMPs off nuclei in a detector or off orbital electrons producing
an excited state is possible, the event rate of these processes is quite suppressed. In
contrast, during an elastic scattering the nucleus recoils as a whole.
The direct detection experiments measure the number of events per day and per
kilogram of the detector material, as a function of the amount of energy Q deposited
in the detector. This event rate would be given by R = nWIMP nnuclei σv in a simplified
model with WIMPs moving with a constant velocity v. The number density of WIMPs
is nWIMP = ρ0/mX and the number density of nuclei is just the ratio of the detector’s
mass over the nuclear mass mN .
For accurate calculations, one should take into account that the WIMPs move in the
halo not with a uniform velocity, but rather following a velocity distribution f(v). The
Earth’s motion in the solar system should be included into this distribution function.
The scattering cross section σ also depends on the velocity. Actually, the cross section
can be parametrized by a nuclear form factor F(Q) as
dσ =
σ
4m2
r
v
2
F
2
(Q)d|~q|
2
, (1.49)
where |~q|
2 = 2m2
r
v
2
(1 − cos θ) is the momentum transferred during the scattering,
mr =
mXmN
mX+mN
is the reduced mass of the WIMP – nucleus system and θ is the scattering
angle in the center of momentum frame. Therefore, one can write a general expression
for the differential event rate per unit detector mass as
dR =
ρ0
mX
1
mN
σF2
(Q)d|~q|
2
4m2
r
v
2
vf(v)dv. (1.50)
The energy deposited in the detector (transferred to the nucleus through one elastic
scattering) is
Q =
|~q|
2
2mN
=
m2
r
v
2
mN
(1 − cos θ). (1.51)
Therefore, the differential event rate over deposited energy can be written, using the
equations (1.50) and (1.51), as
dR
dQ
=
σρ0
√
πv0mXm2
r
F
2
(Q)T(Q), (1.52)
where, following [37], we have defined the dimensionless quantity T(Q) as
T(Q) ≡
√
π
2
v0
Z ∞
vmin
f(v)
v
dv, (1.53)
with the minimum velocity given by vmin =
qQmN
2m2
r
, obtained by Eq. (1.51). Finally,
the event rate R can be calculated by integrating (1.52) over the energy
R =
Z ∞
ET
dR
dQ
dQ. (1.54)
22 Dark Matter
The integration is performed for energies larger than the threshold energy ET of the
detector, below which it is insensitive to WIMP-nucleus recoils.
Using Eqs. (1.54) and (1.52), one can derive the scattering cross section from the
event rate. The experimental collaborations prefer to give their results already in terms
of the scattering cross section as a function of the WIMP mass. To be more precise,
the WIMP-nucleus total cross section consists of two parts: the spin-dependent (SD)
cross section and the spin-independent (SI) one. The former comes from axial current
couplings, whereas the latter comes from scalar-scalar and vector-vector couplings.
The SD cross section is much suppressed compared to the SI one in the case of heavy
nuclei targets and it vanishes if the nucleus contains an even number of nucleons, since
in this case the total nuclear spin is zero.
We see that two uncertainties enter the above calculation: the exact value of the
local density ρ0 and the exact form of the velocity distribution f(v). To these, one
has to include one more. The cross section σ that appears in the previous expressions
concerns the WIMP-nucleon cross section. The couplings of a WIMP with the various
quarks that constitute the nucleon are not the same and the WIMP-nucleon cross
section depends strongly on the exact quark content of the nucleon. To be more
precise, the largest uncertainty lies on the strange content of the nucleon, but we shall
return to this point when we will calculate the cross section in a specific particle theory,
the Next-to-Minimal Supersymmetric Standard Model, in Sec. 3.5.1.
1.5.2 Experimental status
The situation of the experimental results from direct DM searches is a bit confusing. The null observations in most of the experiments led them to set upper limits
on the WIMP-nucleon cross section. These bounds are quite stringent for the spinindependent cross section7
, especially in the regime of WIMP masses of the order of
100 GeV. However, some collaborations have already reported possible DM signals,
mainly in the low mass regime. The preferred regions of these experiments do not
coincide, while some of them have been already excluded by other experiments. The
present picture, for WIMP masses ranging from 5 to 1000 GeV, is summarized in Fig.
1.5, 1.6.
Fig. 1.5 mainly presents upper bounds coming from XENON100 [44]. XENON100
[46] is an experiment located at the Gran Sasso underground laboratory in Italy. It
contains in total 165 kg of liquid Xenon, with 65 kg acting as target mass and the
rest shielding the detector from background radiation. For these upper limits, 225
live days of data were used. The minimum value for the predicted upper bounds on
the cross section is 2 · 10−45 cm2
for WIMP mass ∼ 55 GeV (at 90% confidence level),
almost one order of magnitude lower than the previously released limits [47] by the
same collaboration, using 100 live days of data.
The stringent upper bounds up-to-date (at least for WIMP mass larger than about
7 GeV) come from the first results of the LUX experiment (see Fig. 1.6), after the first
7For the spin-dependent scattering, the exclusion limits are quite relaxed. Hence, we will focus on
the SI cross sections.
1.5.2 Experimental status 23
Figure 1.5: The XENON100 exclusion limit (thick blue line), along with the expected
sensitivity in green (1σ) and yellow (2σ) band. Other upper bounds are also shown as
well as detection claims. From [44].
85.3 live-days of its operation [45]. LUX [53] is a detector containing liquid Xenon, as
XENON100, but in larger quantity, with total mass 370 kg. Its operation started on
April 2013 with a goal to clearly detect or exclude WIMPs with a spin independent
cross section ∼ 2 · 10−46 cm2
.
In Fig. 1.5, except of the XENON100 bounds and other experimental limits on larger
WIMP-nucleon cross section, some detection claims also appear. These come from
DAMA [48,49], CoGeNT [50] and CRESST-II [51] experiments. The first positive result
came from DAMA [52], back in 2000. Since then, the experiment has accumulated 1.17
ton-yr of data over 13 years of operation. DAMA consists of 250 kg of radio pure NaI
scintillator and looks for the annual modulation of the WIMP flux in order to reduce
the influence of the background.
The annual modulation of the DM flux (see [54] for a recent review) is due to the
Earth’s orbital motion relative to the rotation of the galactic disk. The galactic disk
rotation through an essentially non-rotating DM halo, creates an effective DM wind in
the solar frame. During the earth’s heliocentric orbit, this wind reaches a maximum
when the Earth is moving fastest in the direction of the disk rotation (this happens
in the beginning of June) and a minimum when it is moving fastest in the opposite
direction (beginning of December).
DAMA claims an 8.9σ annual modulation with a minimum flux on May 26±7 days,
consistent with the expectation. Since the detector’s target consists of two different
nuclei and the experiment cannot distinguish between sodium and iodine recoils, there
24 Dark Matter
Figure 1.6: The LUX 90% confidence exclusion limit (blue line) with the 1σ range
(shaded area). The XENON100 upper bound is represented by the red line. The inset
shows also preferred regions by CoGeNT (shaded light red), CDMS II silicon detector
(shaded green), CRESST II (shaded yellow) and DAMA (shaded gray). From [45].
is no model independent way to determine the exact region in the cross section versus
WIMP mass plane to which the observed modulation corresponds. However, one can
assume two cases: one that the WIMP scattering off the sodium nucleus dominates the
recoil energy and the other with the iodine recoils dominating. The former corresponds
[55] to a light WIMP (∼ 10 GeV) and quite large scattering cross section and the latter
to a heavier WIMP (∼ 50 to 100 GeV) with smaller cross section (see Fig. 1.5).
The positive result of DAMA was followed many years later by the ones of CoGeNT
and CRESST-II, and more recently by the silicon detector of CDMS [56] (Fig. 1.7).
The discrepancy of the results raised a lot of debates among the experiments (for
example, [64–67]) and by some the positive results are regarded as controversial. On
the other hand, it also raised an effort to find a physical explanation behind this
inconsistency (see, for example, [68–71]).
1.6 Indirect Methods for DM Detection
The same annihilation processes that determined the DM relic abundance in the early
Universe also occur today in galactic regions where the DM concentration is higher.
This fact rises the possibility of detecting potential WIMP pair annihilations indirectly
through their imprints on the cosmic rays. Therefore, the indirect DM searches aim
at the detection of an excess over the known astrophysical background of charged
particles, photons or neutrinos.
Charged particles – electrons, protons and their antiparticles – may originate from
direct products (pair of SM particles) of WIMP annihilations, after their decay and
1.6 Indirect Methods for DM Detection 25
Figure 1.7: The blue contours represent preferred regions for a possible signal at 68%
and 90% C.L. using the silicon detector of CMDS [56]. The blue dotted line represents
the upper limit obtained by the same analysis and the blue solid line is the combined
limit with the silicon CDMS data set reported in [57]. Other limits also appear:
from the CMDS standard germanium detector (light and dark red dashed line, for
standard [58] and low threshold analysis [59], respectively), EDELWEISS [60] (dashed
orange), XENON10 [61] (dash-dotted green) and XENON100 [44] (long-dash-dotted
green). The filled regions identify possible signal regions associated with data from
CoGeNT [62] (dashed yellow, 90% C.L.), DAMA [49,55] (dotted tan, 99.7% C.L.) and
CRESST-II [51, 63] (dash-dotted pink, 95.45% C.L.) experiments. Taken from [56].
through the process of showering and hadronization. Although the exact shape of the
resulting spectrum would depend on the specific process, it is expected to show a steep
cutoff at the WIMP mass. Once produced in the DM halo, the charged particles have
to travel to the point of detection through the turbulent galactic field, which will cause
diffusion. Apart from that, a lot of processes disturb the propagation of the charged
particles, such as bremsstrahlung, inverse Compton scattering with CMB photons and
many others. Therefore, the uncertainties that enter the propagation of the charged
flux until it reaches the telescope are important (contrary to the case of photons and
neutrinos that propagate almost unperturbed through the galaxy).
As in the case of direct detection, the experimental status of charged particle detection concerning the DM is confusing. After some hints from HEAT [72] and AMS01 [73] (the former a far-infrared telescope in Antarctica, the latter a spectrometer,
prototype for AMS-02 mounted on the International Space Station [74]), the PAMELA
satellite observed [75, 76] a steep increase in the energy spectrum of positron fraction
e
+/(e
+ + e
−)
8
. Later FERMI satellite [77] and AMS-02 [78] confirmed the results up
8The searches for charged particles focus on the antiparticles in order to have a reduced background,
26 Dark Matter
Figure 1.8: A compilation of data of charged cosmic rays, together with plausible but
uncertain astrophysical backgrounds, taken from [79]. Left: Positron flux. Center:
Antiproton flux. Right: Sum of electrons and positrons.
to energies of ∼ 200 GeV. However, the excess of positrons is not followed by an excess
of antiprotons, whose flux seems to coincide with the predicted background [75]. In
Fig. 1.8, three plots summarizing the situation are shown [79].
The observed excess is very difficult to explain in terms of DM [79]. To begin with,
the annihilation cross section required to reproduce the excess is quite large, many
orders of magnitude larger than the thermal cross section. Moreover, an “ordinary”
WIMP with large annihilation cross section giving rise to charged leptons is expected
to give, additionally, a large number of antiprotons, a fact in contradiction with the
observations. Although a lot of work has been done to fit a DM particle to the observed
pattern, it is quite possible that the excesses come from a yet unknown astrophysical
source. We are not going to discuss further this matter, but we end with a comment.
If this excess is due to a source other than DM, then a possible DM positron excess
would be lost under this formidable background.
A last hint for DM came from the detection of highly energetic photons. However,
we will interrupt this discussion, since this signal and a possible explanation is the
subject of Ch. 4. There, we will also see the upper bounds on the annihilation cross
section being set due to the absence of excesses in diffuse γ radiation.
since they are much less abundant than the corresponding particles.
CHAPTER 2
PARTICLE PHYSICS
Since the DM comprises of particles, it should be explained by a general particle physics
theory. We start in the following section by describing the Standard Model (SM) of
particle physics. Although the SM describes so far the fundamental particles and their
interactions quite accurately, it cannot provide a DM candidate. Besides, the SM
suffers from some theoretical problems, which we discuss in Sec. 2.2. We will see that
these problems can be solved if one introduces a new symmetry, the supersymmetry,
which we describe in Sec. 2.3. We finish this chapter by briefly describing in Sec. 2.4 a
supersymmetric extension of the SM with the minimal additional particle content, the
Minimal Supersymmetric Standard Model (MSSM).
2.1 The Standard Model of Particle Physics
The Standard Model (SM) of particle physics1
consists of two well developed theories,
the quantum chromodynamics (QCD) and the electroweak (EW) theory. The former
describes the strong interactions among the quarks, whereas the latter describes the
electroweak interactions (the weak and electromagnetic interactions in a unified context) between fermions. The EW theory took its final form in the late 1960s by the
introduction by S. Weinberg [85] and A. Salam [86] of the Higgs mechanism that gives
masses to the SM particles, which followed the unification of electromagnetic and weak
interactions [87,88]. At the same time, the EW model preserves the gauge invariance,
making the theory renormalizable, as shown later by ’t Hooft [89]. On the other hand,
QCD obtained its final form some years later, after the confirmation of the existence
of quarks. Of course, the history of the SM is much longer and it can be traced back to
1920s with the formulation of a theoretical basis for a Quantum Field Theory (QFT).
Since then, the SM had many successes. The SM particle content was completed with
the discovery of the heaviest of the quarks, the top quark [90,91], in 1995 and, recently,
with the discovery of the Higgs boson [92, 93].
1There are many good textbooks on the SM and Quantum Field Theory, e.g. [80–84].
28 Particle Physics
The key concept within the SM, as in every QFT, is that of symmetries. Each
interaction respects a gauge symmetry, based on a Lie algebra. The strong interaction is
described by an SU(3)c symmetry, where the subscript c stands for color, the conserved
charge of strong interactions. The EW interactions, on the other hand, are based on
a SU(2)L × U(1)Y Lie algebra. Here, as we will subsequently see, L refers to the
left-handed fermions and Y is the hypercharge, the conserved charge under the U(1).
SU(2)L conserves a quantity known as weak isospin I. Therefore, the SM contains the
internal symmetries of the unitary product group
SU(2)L × U(1)Y × SU(3)c. (2.1)
2.1.1 The particle content of the SM
We mention for completeness that particles are divided into two main classes according
to the statistics they follow. The bosons are particles with integer spin and follow the
Bose-Einstein distribution, whereas fermions have half-integer spin and follow the
Dirac-Einstein statistics, obeying the Pauli exclusion principle. In the SM, all the
fermions have spin 1/2, whereas the bosons have spin 1 with only exception the Higgs
boson, which is a scalar (spin zero). We begin the description of the SM particles with
the fermions.
Each fermion is classified in irreducible representations of each individual Lie algebra, according to the conserved quantum numbers, i.e. the color C, the weak isospin
I and the hypercharge Y . A first classification of fermions can be done into leptons
and quarks, which transform differently under the SU(3)c. Leptons are singlets under
this transformation, while quarks act as triplets (the fundamental representation of
this group). The EW interactions violate maximally the parity symmetry and SU(2)L
acts only on states with negative chirality (left-handed). A Dirac spinor Ψ can be
decomposed into left and right chirality components using, respectively, the projection
operators PL =
1
2
(1 − γ5) and PR =
1
2
(1 + γ5):
ΨL = PLΨ and ΨR = PRΨ. (2.2)
Left-handed fermions have I = 1/2, with a third component of the isospin I3 = ±1/2.
Fermions with positive I3 are called up-type fermions and those with negative are
called down-type. These behave the same way under SU(2)L and form doublets with
one fermion of each type. On the other hand, right-handed fermions have I = 0 and
form singlets that do not undergo weak interactions. The hypercharge is written in
terms of the electric charge Q and the third component of the isospin I3 through the
Gell-Mann–Nishijima relation:
Q = I3 + Y/2. (2.3)
Therefore, left- and right-handed components transform differently under the U(1)Y ,
since they have different hypercharge.
The fermionic sector of the SM comprises three generations of fermions, transforming as spinors under Lorentz transformations. Each generation has the same structure.
For leptons, it is an SU(2)L doublet with components consisting of one left-handed
2.1.2 The SM Lagrangian 29
charged lepton and one neutrino (neutrinos are only left-handed in the SM), along
with a gauge singlet right-handed charged lepton. The quark doublet consists of an
up- (u) and a down-type (d) (left-handed) quark and the pattern is completed by the
two corresponding SU(2)L singlet right-handed quarks. We write these representations
as
Quarks: Q ≡

u
i
L
d
i
L
!
, ui
R, di
R Leptons: L ≡

ν
i
L
e
i
L
!
, ei
R, (2.4)
with i = 1, 2, 3 the generation index.
Having briefly described the fermionic sector, we turn to the bosonic sector of
the SM. It consists of the gauge bosons that mediate the interactions and the Higgs
boson that gives masses to the particles through a spontaneous symmetry breaking,
the electroweak symmetry breaking (EWSB) [94–98], which we shall describe in Sec.
2.1.3. Before the EWSB, these bosons are
• three Wa
µ
(a = 1, 2, 3) weak bosons, associated with the generators of SU(2)L,
• one neutral Bµ boson, associated with the generator of U(1)Y ,
• eight gluons Ga
µ
(a = 1, . . . , 8), associated with the generators of SU(3)c, and
• the complex scalar Higgs doublet Φ =
φ
+
φ
0
!
.
After the EWSB, the EW boson states mix and give the two W± bosons, the neutral
Z boson and the massless photon γ. From the symmetry breaking, one scalar degree of
freedom remains which is the famous (neutral) Higgs boson [97–99]. We will return to
the mixed physical states, after describing the Higgs mechanism for symmetry breaking.
A complete list of the SM particles (the physical states after EWSB) is shown in Table
2.1.
2.1.2 The SM Lagrangian
The gauge bosons are responsible for the mediation of the interactions and are associated with the generators of the corresponding symmetry. The EW gauge bosons Bµ
and Wa
µ
are associated, respectively, with the generator Y of the U(1)Y and the three
generators T
a
2
of the SU(2)L. The latter are defined as half of the Pauli matrices τ
a
(T
a
2 =
1
2
τ
a
) and they obey the algebra

T
a
2
, Tb
2

= iǫabcT
c
2
, (2.5)
where ǫ
abc is the fully antisymmetric Levi-Civita tensor. The eight gluons are associated
with an equal number of generators T
a
3
(Gell-Mann matrices) of SU(3)c and obey the
Lie algebra

T
a
3
, Tb
3

= if abcT
c
3
, with Tr
T
a
3 T
b
3

=
1
2
δ
ab
, (2.6)
30 Particle Physics
Name symbol mass charge (|e|) spin
Leptons
electron e 0.511 MeV −1 1/2
electron neutrino νe 0 (<2 eV) 0 1/2
muon µ 105.7 MeV −1 1/2
muon neutrino νµ 0 (<2 eV) 0 1/2
tau τ 1.777 GeV −1 1/2
tau neutrino ντ 0 (<2 eV) 0 1/2
Quarks
up u 2.7
+0.7
−0.5 MeV 2/3 1/2
down d 4.8
+0.7
−0.3 MeV −1/3 1/2
strange s (95 ± 5) MeV −1/3 1/2
charm c (1.275 ± 0.025) GeV 2/3 1/2
bottom b (4.18 ± 0.03) GeV −1/3 1/2
top t (173.5 ± 0.6 ± 0.8) GeV 2/3 1/2
Bosons
photon γ 0 (<10−18 eV) 0 (<10−35) 1
W boson W± (80.385 ± 0.015) GeV ±1 1
Z boson Z (91.1876 ± 0.0021) GeV 0 1
gluon g 0 (.O(1) MeV) 0 1
Higgs H
(125.3 ± 0.4 ± 0.5) GeV
0 0
(126.0 ± 0.4 ± 0.4) GeV
Table 2.1: The particle content of the SM. All values are those given in [100], except of
the Higgs mass that is taken from [92, 93] (up and down row, respectively), assuming
that the observed excess corresponds to the SM Higgs. The u, d and s quark masses
are estimates of so-called “current-quark masses” in a mass-independent subtraction
scheme as MS at a scale ∼ 2 GeV. The c and b quark masses are the running masses
in the MS scheme. The values in the parenthesis are the current experimental limits.
with f
abc the structure constants of the group.
Using the structure constants of the corresponding groups, we define the field
strengths for the gauge bosons as
Bµν ≡ ∂µBν − ∂νBµ, (2.7a)
Wµν ≡ ∂µWa
ν − ∂νWa
µ + g2ǫ
abcWb
µWc
ν
(2.7b)
and
G
a
µν ≡ ∂µG
a
ν − ∂νG
a
µ + g3f
abcG
b
µG
c
ν
. (2.7c)
2.1.2 The SM Lagrangian 31
We use the notation g1, g2 and g3 for the coupling constants of U(1)Y , SU(2)L and
SU(3)c, respectively. As in any Yang-Mills theory, the non-abelian gauge groups lead
to self-interactions, which is not the case for the abelian U(1)Y group.
Before we finally write the full Lagrangian, we have to introduce the covariant
derivative for fermions, which in a general form can be written as
DµΨ =
∂µ − ig1
1
2
Y Bµ − ig2T
a
2 Wa
µ − ig3T
a
3 G
a
µ

Ψ. (2.8)
This form has to be understood as that, depending on Ψ, only the relevant terms
apply, hence for SU(2)L singlet leptons only the two first terms inside the parenthesis
are relevant, for doublet leptons the three first terms and for the corresponding quark
singlets and doublets the last term also participates. We also have to notice that in
order to retain the gauge symmetry, mass terms are forbidden in the Lagrangian. For
example, the mass term mψψ¯ = m

ψ¯
LψR + ψ¯
RψL

(with ψ¯ ≡ ψ
†γ
0
) is not invariant
under SU(2)L. This paradox is solved by the introduction of the Higgs scalar field
(see next subsection). The SM Lagrangian can be now written2
, split for simplicity in
three parts, each describing the gauge bosons, the fermions and the scalar sector,
LSM = Lgauge + Lfermion + Lscalar, (2.9)
with
Lgauge = −
1
4
G
a
µνG
µν
a −
1
4
Wa
µνWµν
a −
1
4
BµνB
µν
, (2.10a)
Lfermion = iL¯Dµγ
µL + ie¯RDµγµeR
+ iQ¯Dµγ
µQ + iu¯RDµγ
µuR + i
¯dRDµγ
µ
dR
−

heL¯ΦeR + hdQ¯ΦdR + huQ¯ΦeuR + h.c.

(2.10b)
and
Lscalar = (DµΦ)†
(DµΦ) − V (Φ†Φ), (2.10c)
where
V (Φ†Φ) = µ
2Φ
†Φ + λ

Φ
†Φ
2
(2.11)
is the scalar Higgs potential. Φ is the conjugate of Φ, related to the charge conjugate e
by Φ =e iτ2Φ
⋆
, with τi the Pauli matrices. The covariant derivative acting on the Higgs
scalar field gives
DµΦ =
∂µ − ig1
1
2
Y Bµ − ig2T
a
2 Wa
µ

Φ. (2.12)
Before we proceed to the description of the Higgs mechanism, a last comment concerning the SM Lagrangian is in order. If we restore the generation indices, we see that
2For simplicity, from now on we are going to omit the generations indice
32 Particle Physics
the Yukawa couplings h are 3 × 3, in general complex, matrices. As any complex matrix, they can be diagonalized with the help of two unitary matrices VL and VR, which
are related by VR = U
†VL with U again a unitary matrix. The diagonalization in the
quark sector to the mass eigenstates induces a mixing among the flavors (generations),
described by the Cabibbo–Kobayashi–Maskawa (CKM) matrix [101, 102]. The CKM
matrix is defined by
VCKM ≡ V
u
L
†
V
d
L
†
, (2.13)
where V
u
L
, V
d
L
are the unitary matrices that diagonalize the Yukawa couplings Hu
, Hd
,
respectively. This product of the two matrices appears in the charged current when it
is expressed in terms of the observable mass eigenstates.
2.1.3 Mass generation through the Higgs mechanism
We will start by examining the scalar potential (2.11). The vacuum expectation value
(vev) of the Higgs field hΦi ≡ h0|Φ|0i is given by the minimum of the potential. For
µ
2 > 0, the potential is always non-negative and Φ has a zero vev. The hypothesis of
the Higgs mechanism is that µ
2 < 0. In this case, the field Φ will acquire a vev
hΦi =
1
2

0
v
!
with v =
r
−
µ2
λ
. (2.14)
Since the charged component of Φ still has a zero vev, the U(1)Q symmetry of quantum
electrodynamics (QED) remains unbroken.
We expand the field Φ around the minima v in terms of real fields, and at leading
order we have
Φ(x) =
θ2(x) + iθ1(x)
√
1
2
(v + H(x)) − iθ3(x)
!
=
1
√
2
e
iθa(x)τ
a

0
v + H(x)
!
. (2.15)
We can eliminate the unphysical degrees of freedom θa, using the fact that the theory
remains gauge invariant. Therefore, we perform the following SU(2)L gauge transformation on Φ (unitary gauge)
Φ(x) → e
−iθa(x)τ
a
Φ(x), (2.16)
so that
Φ(x) = 1
√
2

0
v + H(x)
!
. (2.17)
We are going to use the following definitions for the gauge fields
W±
µ ≡
1
2

W1
µ ∓ iW2
µ

, (2.18a)
Zµ ≡
1
p
g
2
1 + g
2
2

g2W3
µ − g1Bµ

, (2.18b)
Aµ ≡
1
p
g
2
1 + g
2
2

g1W3
µ + g2Bµ

, (2.1
2.2 Limits of the SM and the emergence of supersymmetry 33
Then, the kinetic term for Φ (see Eq. (2.10c)) can be written in the unitary gauge as
(DµΦ)†
(D
µΦ) = 1
2
(∂µH)
2 + M2
W W+
µ W−µ +
1
2
M2
ZZµZ
µ
, (2.19)
with
MW ≡
1
2
g2v and MZ ≡
1
2
q
g
2
1 + g
2
2
v. (2.20)
We see that the definitions (2.18) correspond to the physical states of the gauge bosons
that have acquired masses due to the non-zero Higgs vev, given by (2.20). The photon
has remained massless, which reflects the fact that after the spontaneous breakdown of
SU(2)L × U(1)Y the U(1)Q remained unbroken. Among the initial degrees of freedom
of the complex scalar field Φ, three were absorbed by W± and Z and one remained as
the neutral Higgs particle with squared mass
m2
H = 2λv2
. (2.21)
We note that λ should be positive so that the scalar potential (2.11) is bounded from
below.
Fermions also acquire masses due to the Higgs mechanism. The Yukawa terms in
the fermionic part (2.10b) of the SM Lagrangian are written, after expanding around
the vev in the unitary gauge,
LY = −
1
√
2
hee¯L(v + H)eR −
1
√
2
hd
¯dL(v + H)dR −
1
√
2
huu¯L(v + H)uR + h.c. . (2.22)
Therefore, we can identify the masses of the fermions as
me
i =
h
i
e
v
√
2
, md
i =
h
i
d
v
√
2
, mui =
h
i
u
v
√
2
, (2.23)
where we have written explicitly the generation indices.
2.2 Limits of the SM and the emergence of supersymmetry
2.2.1 General discussion of the SM problems
The SM has been proven extremely successful and has been tested in high precision
in many different experiments. It has predicted many new particles before their final
discovery and also explained how the particles gain their masses. Its last triumph was
of course the discovery of a boson that seems to be very similar to the Higgs boson of
the SM. However, it is generally accepted that the SM cannot be the ultimate theory. It
is not only observed phenomena that the SM does not explain; SM also faces important
theoretical issues.
The most prominent among the inconsistencies of the SM with observations is the
oscillations among neutrinos of different generations. In order for the oscillations to
34 Particle Physics
φ φ
k
Figure 2.1: The scalar one-loop diagram giving rise to quadratic divergences.
occur, neutrinos should have non-zero masses. However, minimal modifications of the
SM are able to fit with the data of neutrino physics. Another issue that a more complete theory has to face is the matter asymmetry, the observed dominance of matter
over antimatter in the Universe. In addition, in order to comply with the standard
cosmological model, it has to provide the appropriate particle(s) that drove the inflation. Last, but not least, we saw that in order to explain the DM that dominates the
Universe, a massive, stable weakly interacting particle must exist. Such a particle is
not present in the SM.
On the other hand, the SM also suffers from a theoretical perspective. For example,
the SM counts 19 free parameters; one expects that a fundamental theory would have
a much smaller number of free parameters. Simple modifications of the SM have been
proposed relating some of these parameters. Grand unified theories (GUTs) unify
the gauge couplings at a high scale ∼ 1016 GeV. However, this unification is only
approximate unless the GUT is embedded in a supersymmetric framework. Another
serious problem of the SM is that of naturalness. This will be the topic of the following
subsection.
2.2.2 The naturalness problem of the SM
The presence of fundamental scalar fields, like the Higgs, gives rise to quadratic divergences. The diagram of Fig. 2.1 contributes to the squared mass of the scalar
δm2 = λ
Z Λ
d
4k
(2π)
4
k
−2
. (2.24)
This contribution is approximated by δm2 ∼ λΛ
2/(16π
2
), quadratic in a cut-off Λ,
which should be finite. For the case of the Higgs scalar field, one has to include its
couplings to the gauge fields and the top quark3
. Therefore,
δm2
H =
3Λ2
8π
2v
2

4m2
t − 2M2
W − M2
Z − m2
H

+ O(ln Λ
µ
)

, (2.25)
where we have used Eq. (2.21) and m2
H ≡ m2
0 + δm2
H.
3Since the contribution to the squared mass correction are quadratic in the Yukawa couplings (or
quark masses), the lighter quarks can be neglected
2.2.3 A way out 35
Taking Λ as a fundamental scale Λ ∼ MP l ∼ 1019 GeV we have
m2
0 = m2
H −
3Λ2
8π
2v
2

4m2
t − 2M2
W − M2
Z − m2
H

(2.26)
and we can see that m2
0 has to be adjusted to a precision of about 30 orders of magnitude
in order to achieve an EW scale Higgs mass. This is considered as an intolerable finetuning, which is against the general belief that the observable properties of a theory
have to be stable under small variations of the fundamental (bare) parameters. It is
exactly the above behavior that is considered as unnatural. Although the SM could
be self-consistent without imposing a large scale, grand unification of the parameters
introduce a hierarchy problem between the different scales.
A more strict definition of naturalness comes from ’t Hooft [103], which we rewrite
here:
At an energy scale µ, a physical parameter or set of physical parameters
αi(µ) is allowed to be very small only if the replacement αi(µ) = 0 would
increase the symmetry of the system.
Clearly, this is not the case here. Although mH is small compared to the fundamental
scale Λ, it is not protected by any symmetry and a fine-tuning is necessary.
2.2.3 A way out
The naturalness in the ’t Hooft sense is inspired by quantum electrodynamics, which is
the archetype for a natural theory. For example, the corrections to the electron mass
me are themselves proportional to me, with a dimensionless proportionality factor that
behaves like ∼ ln Λ. In general, fermion masses are protected by the chiral symmetry; small values (compared to the fundamental scale) of these masses enhances the
symmetry.
If a new symmetry exists in nature, relating fermion fields to scalar fields, then each
scalar mass would be related somehow to the corresponding fermion mass. Therefore,
the scalar mass itself can be naturally small compared to Λ, since this would mean
that the fermion mass is small, which enhances the chiral symmetry. Such a symmetry,
relating bosons to fermions and vice versa, is known as supersymmetry [104, 105].
Actually, as we will see later, if this new symmetry remains unbroken, the masses of
the conjugate bosons and fermions would have to be equal.
In order to make the above statement more concrete, we consider a toy model with
two additional complex scalar fields feL and feR. We will discuss only the quadratic
divergences that come from corrections to the Higgs mass due to a fermion. The
generalization for the contributions from the gauge bosons or the self-interaction is
straightforward. The interactions in this toy model of the new scalar fields with the
Higgs are described by the Lagrangian
Lfefφe = λfe|φ|
2

|feL|
2 + |feR|
2

. (2.27
36 Particle Physics
It can be easily checked that the quadratic divergence coming from a fermion at one
loop is exactly canceled, as long as the new quartic coupling λfe obeys the relation
λfe = −λ
2
f
(λf is the Yukawa coupling for the fermion f).
2.3 A brief summary of Supersymmetry
Supersymmetry (SUSY) is a symmetry relating fermions and bosons. The supersymmetry transformation should turn a boson state into a fermion state and vice versa. If
Q is the operator that generates such transformations, then
Q |bosoni = |fermioni Q |fermioni = |bosoni. (2.28)
Due to commutation and anticommutation rules of bosons and fermions, Q has to
be an anticommuting spinor operator, carrying spin angular momentum 1/2. Since
spinors are complex objects, the hermitian conjugate Q†
is also a symmetry operator4
.
There is a no-go theorem, the Coleman-Mandula theorem [106], that restricts the
conserved charges which transform as tensors under the Lorentz group to the generators
of translations Pµ and the generators of Lorentz transformations Mµν. Although this
theorem can be evaded in the case of supersymmetry due to the anticommutation
properties of Q, Q†
[107], it restricts the underlying algebra of supersymmetry [108].
Therefore, the basic supersymmetric algebra can be written as5
{Q, Q†
} = P
µ
, (2.29a)
{Q, Q} = {Q
†
, Q†
} = 0, (2.29b)
[P
µ
, Q] = [P
µ
, Q] = 0. (2.29c)
In the following, we summarize the basic conclusions derived from this algebra.
• The single-particle states of a supersymmetric theory fall into irreducible representations of the SUSY algebra, called supermultiplets. A supermultiplet contains
both fermion and boson states, called superpartners.
• Superpartners must have equal masses: Consider |Ωi and |Ω
′
i as the superpartners, |Ω
′
i should be proportional to some combination of the Q and Q† operators
acting on |Ωi, up to a space-time translation or rotation. Since −P
2
commutes
with Q, Q† and all space-time translation and rotation operators, |Ωi, |Ω
′
i will
have equal eigenvalues of −P
2 and thus equal masses.
• Superpartners must be in the same representation of gauge groups, since Q, Q†
commute with the generators of gauge transformations. This means that they
have equal charges, weak isospin and color degrees of freedom.
4We will confine ourselves to the phenomenologically more interesting case of N = 1 supersymmetry, with N referring to the number of distinct copies of Q, Q†
.
5We present a simplified version, omitting spinor indices in Q and Q†
.
2.3 A brief summary of Supersymmetry 37
• Each supermultiplet contains an equal number of fermion and boson degrees of
freedom (nF and nB, respectively): Consider the operator (−1)2s
, with s the spin
angular momentum, and the states |ii that have the same eigenvalue p
µ of P
µ
.
Then, using the SUSY algebra (2.29) and the completeness relation P
i
|ii hi| =
1, we have P
i
hi|(−1)2sP
µ
|ii = 0. On the other hand, P
i
hi|(−1)2sP
µ
|ii =
p
µTr [(−1)2s
] ∝ nB − nF . Therefore, nF = nB.
As addendum to the last point, we see that two kind of supermultiplets are possible
(neglecting gravity):
• A chiral (or matter or scalar ) supermultiplet, which consists of a single Weyl
fermion (with two spin helicity states, nF = 2) and two real scalars (each with
nB = 1), which can be replaced by a single complex scalar field.
• A gauge (or vector ) supermultiplet, which consists of a massless spin 1 boson
(two helicity states, nB = 2) and a massless spin 1/2 fermion (nF = 2).
Other combinations either are reduced to combinations of the above supermultiplets
or lead to non-renormalizable interactions.
It is possible to study supersymmetry in a geometric approach, using a space-time
manifold extended by four fermionic (Grassmann) coordinates. This manifold is called
superspace. The fields, in turn, expressed in terms of the extended set of coordinates
are called superfields. We are not going to discuss the technical details of this topic
(the interested reader may refer to the rich bibliography, for example [109–111]).
However, it is important to mention a very useful function of the superfields, the
superpotential. A generic form of a (renormalizable) superpotential in terms of the
superfields Φ is the following b
W =
1
2
MijΦbiΦbj +
1
6
y
ijkΦbiΦbjΦbk. (2.30)
The Lagrangian density can always be written according to the superpotential. The
superpotential has also to fulfill some requirements. In order for the Lagrangian to
be supersymmetric invariant, W has to be holomorphic in the complex scalar fields
(it does not involve hermitian conjugates Φb† of the superfields). Conventionally, W
involves only left chiral superfields. Instead of the SU(2)L singlet right chiral fermion
fields, one can use their left chiral charge conjugates.
As we mentioned before, the members of a supermultiplet have equal masses. This
contradicts our experience, since the partners of the light SM particles would have been
detected long time ago. Hence, the supersymmetry should be broken at a large energy
scale. The common approach is that SUSY is broken in a hidden sector, very weakly
coupled to the visible sector. Then, one has to explain how the SUSY breaking mediated to the visible sector. The two most popular scenarios are the gravity mediation
scenario [112–114] and the Gauge-Mediated SUSY Breaking (GSMB) [113, 115–117],
where the mediation occurs through gauge interactions.
There are two approaches with which one can address the SUSY breaking. In the
first approach, one refers to a GUT unification and determines the supersymmetric
38 Particle Physics
breaking parameters at low energies through the renormalization group equations.
This approach results in a small number of free parameters. In the second approach,
the starting point is the low energy scale. In this case, the SUSY breaking has to be
parametrized by the addition of breaking terms to the low energy Lagrangian. This
results in a larger set of free parameters. These terms should not reintroduce quadratic
divergences to the scalar masses, since the cancellation of these divergences was the
main motivation for SUSY. Then, one talks about soft breaking terms.
2.4 The Minimal Supersymmetric Standard Model
One can construct a supersymmetric version of the standard model with a minimal
content of particles. This model is known as the Minimal Supersymmetric Standard
Model (MSSM). In a SUSY extension of the SM, each of the SM particles is either in a
chiral or in a gauge supermultiplet, and should have a superpartner with spin differing
by 1/2.
The spin-0 partners of quarks and leptons are called squarks and sleptons, respectively (or collectively sfermions), and they have to reside in chiral supermultiplets.
The left- and right-handed components of fermions are distinct 2-component Weyl
fermions with different gauge transformations in the SM, so that each must have its
own complex scalar superpartner. The gauge bosons of the SM reside in gauge supermultiplets, along with their spin-1/2 superpartners, which are called gauginos. Every
gaugino field, like its gauge boson partner, transforms as the adjoint representation of
the corresponding gauge group. They have left- and right-handed components which
are charge conjugates of each other: (λeL)
c = λeR.
The Higgs boson, since it is a spin-0 particle, should reside in a chiral supermultiplet. However, we saw (in the fermionic part of the SM Lagrangian, Eq. (2.10b))
that the Y = 1/2 Higgs in the SM can give mass to both up- and down-type quarks,
only if the conjugate Higgs field with Y = −1/2 is involved. Since in the superpotential there are no conjugate fields, two Higgs doublets have to be introduced. Each
Higgs supermultiplet would have hypercharge Y = +1/2 or Y = −1/2. The Higgs
with the negative hypercharge gives mass to the down-type fermions and it is called
down-type Higgs (Hd, or H1 in the SLHA convention [118]) and the other one gives
mass to up-type fermions and it is called up-type Higgs (Hu, or H2).
The MSSM respects a discrete Z2 symmetry, the R-parity. If one writes the most
general terms in the supersymmetric Lagrangian (still gauge-invariant and holomorphic), some of them would lead to non-observed processes. The most obvious constraint
comes from the non-observed proton decay, which arises from a term that violates both
lepton and baryon numbers (L and B, respectively) by one unit. In order to avoid these
terms, R-parity, a multiplicative conserved quantum number, is introduced, defined as
PR = (−1)3(B−L)+2s
, (2.31)
with s the spin of the particle.
The R even particles are the SM particles, whereas the R odd are the new particles
introduced by the MSSM and are called supersymmetric particles. Due to R-parity,
2.4 The Minimal Supersymmetric Standard Model 39
if it is exactly conserved, there can be no mixing among odd and even particles and,
additionally, each interaction vertex in the theory can only involve an even number of
supersymmetric particles. The phenomenological consequences are quite important.
First, the lightest among the odd-parity particles is stable. This particle is known
as the lightest supersymmetric particle (LSP). Second, in collider experiments, supersymmetric particles can only be produced in pairs. The first of these consequences
was a breakthrough for the incorporation of DM into a general theory. If the LSP is
electrically neutral, it interacts only weakly and it consists an attractive candidate for
DM.
We are not going to enter further into the details of the MSSM6
. Although MSSM
offers a possible DM candidate, there is a strong theoretical reason to move from the
minimal model. This reason is the so-called µ-problem of the MSSM, with which we
begin the discussion of the next chapter, where we shall describe more thoroughly the
Next-to-Minimal Supersymmetric Standard Model.

[maelstrom]

Mais d’ou demi habile sors t’il tout ses copier coller ?

[Demi Habile]

« thèse doctorat physique théorique »
.
C’est ce que je rentre dans Google.

[maelstrom]

Bégaudeau:Je comprend comment la culture est déligitimer depuis quelque années parce que improductive mais je vois pas dans la bourgeoisie qui la déligitime parce que féminine ? peut tu expliquer plus précisement ta pensée

[françois bégaudeau]

Improductive et féminine doivent se lire ensemble, comme si l’un faisait résonner l’autre.
Féminine étant ici à comprendre intuitivement.
Le fait objectif étant que le champ culturel, comme déjà dit, s’est considérablement féminisée depuis 50 ans. Symptome et cause de moindre valeur sur le marché – symptome et cause, analogiquement, d’un désintéret croissant de la bourgeoisie pour la culture (désintéret prenant ici tout son sens).
Répondit Bégaudeau.

[JeanMonnaie]

Outre le fait que cet article, indigeste pour un non-chercheur, a été lu en diagonale par tout le monde ici et date de 20 ans, ce qui mérite une profonde mise à jour de ses analyses, je remarque qu’il n’utilise pas le terme bourgeois mais élite et dominant. Autrement dit, ce ne sont pas ceux qui détiennent les moyens de production, donc les bourgeois, qui souhaitent garder une distance particulière avec les prolétaires, mais les prolétaires eux-mêmes, avec les autres prolétaires, du professeur au médecin, pour simplifier. Les questions soulevées précédemment restent toujours d’actualité.

[maelstrom]

toi même tu partage des articles libé sur lutte ouvrière qui date des année 90

[Demi Habile]

and also the definition of the unpolarized cross section to write
X
spin
Z
|M12→34|
2
(2π)
4
δ
4
(p1 + p2 − p3 − p4)
d
3p3
(2π)
32E3
d
3p4
(2π)
32E4
=
4F g1g2 σ12→34, (1.31)
where F ≡ [(p1 · p2)
2 − m2
1m2
2
]
1/2
and the spin factors g1, g2 come from the average
over initial spins. This way, the collision term (1.29) is written in a more compact form
g1
Z
C[f1]
d
3p1
(2π)
3
= −
Z
σvMøl (dn1dn2 − dn
eq
1 dn
eq
2
), (1.32)
where σ =
P
(all f)
σ12→f is the total annihilation cross section summed over all the
possible final states and vMøl ≡
F
E1E2
. The so called Møller velocity, vMøl, is defined in
such a way that the product vMøln1n2 is invariant under Lorentz transformations and,
in terms of particle velocities ~v1 and ~v2, it is given by the expression
vMøl =
h
~v2
1 − ~v2
2

2
− |~v1 × ~v2|
2
i1/2
. (1.33)
Due to symmetry considerations, the distributions in kinetic equilibrium are proportional to those in chemical equilibrium, with a proportionality factor independent of
the momentum. Therefore, the collision term (1.32), both before and after decoupling,
can be written in the form
g1
Z
C[f1]
d
3p1
(2π)
3
= −hσvMøli(n1n2 − n
eq
1 n
eq
2
), (1.34)
where the thermal averaged total annihilation cross section times the Møller velocity
has been defined by the expression
hσvMøli =
R
σvMøldn
eq
1 dn
eq
2
R
dn
eq
1 dn
eq
2
. (1.35)
We will come back to the thermal averaged cross section in the next subsection.
We are, now, able to write the full integrated Boltzmann equation, using the expressions (1.28), (1.34) that we have derived for the Liouville and the collision term,
respectively. In the simplified but interesting case of identical particles 1 and 2, the
Boltzmann equation is, finally, written as
n˙ + 3Hn = −hσvMøli(n
2 − n
2
eq). (1.36)
18 Dark Matter
However, instead of using n, it is more convenient to take the expansion of the universe
into account and calculate the number density per comoving volume Y , which can be
defined as the ratio of the number and entropy densities: Y ≡ n/s. The total entropy
density S = R3
s (R is the scale factor) remains constant, hence we can obtain a
differential equation for Y by dividing (1.36) by S. Before we write the final form
of the Boltzmann equation that it is used for the relic density calculations, we have
to change the variable that parametrizes the comoving density. In practice, the time
variable t is not convenient and the temperature of the Universe (actually the photon
temperature, since the photons were the last particles that went out of equilibrium) is
used instead. However, it proves even more useful to use as time variable the quantity
defined by x ≡ m/T with m the DM mass, so that Eq. (1.36) transforms into
dY
dx
=
1
3H
ds
dx
hσvMøli

Y
2 − Y
2
eq
. (1.37)
Last, using the Hubble parameter (1.2) for a radiation dominated Universe and the
expressions (1.20), (1.21) for the energy and entropy density, the Boltzmann equation
is written in its final form
dY
dx
= −
r
45GN
π
g
1/2
∗ m
x
2
hσvMøli

Y
2 − Y
2
eq
, (1.38)
where the effective degrees of freedom g
1/2
∗ have been defined by
g
1/2
∗ ≡
heff
g
1/2
eff

1 +
1
3
T
heff
dheff
dT

. (1.39)
The equilibrium density per comoving volume Yeq ≡ neq/s can be expressed as
Yeq(x) = 45g
4π
4
x
2K2(x)
heff(m/x)
, (1.40)
with K2 the modified Bessel function of second kind.
1.4.3 Thermal average of the annihilation cross section
We are going to derive a simple formula that one can use to calculate the thermal
average of the cross section times velocity, based again on the analysis of [38]. We will
use the assumption that equilibrium functions follow the Maxwell-Boltzmann distribution, instead of the actual Bose-Einstein or Fermi-Dirac. This is a well established
assumption if the freeze out occurs after T ≃ m/3 or for x >∼ 3, which is actually the
case for WIMPs. Under this assumption, the expression (1.35) gives, in the cosmic
comoving frame,
hσvMøli =
R
vMøle
−E1/T e
−E2/T d
3p1d
3p2
R
e
−E1/T e
−E2/T d
3p1d
3p2
. (1.4
1.4.3 Thermal average of the annihilation cross section 19
The volume element can be written as d3p1d
3p2 = 4πp1dE14πp2dE2
1
2
cos θ, with θ the
angle between ~p1 and ~p2. After changing the integration variables to E+, E−, s given
by
E+ = E1 + E2, E− = E1 − E2, s = 2m2 + 2E1E2 − 2p1p2 cos θ, (1.42)
(with s = −(p1 − p2)
2 one of the Mandelstam variables,) the volume element becomes
d
3p1d
3p2 = 2π
2E1E2dE+dE−ds and the initial integration region
{E1 > m, E2 > m, | cos θ| ≤ 1i
transforms into
|E−| ≤
1 −
4m2
s
1/2
(E
2
+ − s)
1/2
, E+ ≥
√
s, s ≥ 4m2
. (1.43)
After some algebraic calculations, it can be found that the quantity hσvMøliE1E2
depends only on s, specifically vMølE1E2 =
1
2
p
s(s − 4m2
). Hence, the numerator of the expression (1.41), which after changing the integration variables reads
2π
2
R
dE+
R
dE−
R
dsσvMølE1E2e
−E+/T , can be written, eventually, as
Z
vMøle
−E1/T e
−E2/T = 2π
2
Z ∞
4m2
dsσ(s − 4m2
)
Z
dE+e
−E+/T (E
2
+ − s)
1/2
. (1.44)
The integral over E+ can be written with the help of the modified Bessel function of
the first kind K1 as √
s T K1(
√
s/T). The denominator of (1.41) can be treated in a
similar way, so that the thermal average is, finally, given by the expression
hσvMøli =
1
8m4TK2
2
(x)
Z ∞
4m2
ds σ(s)(s − 4m2
)
√
s K1(
√
s/T). (1.45)
Eqs. (1.38)–(1.40) along with this last Eq. (1.45) are all we need in order to calculate
the relic density of a WIMP, if its total annihilation cross section in terms of the
Mandelstam variable s is known.
In many cases, in order to avoid the numerical integration in Eq. (1.45), an approximation for hσvMøli can be used. The thermal average is expanded in powers of x
−1
(or, equivalently, in powers of the squared WIMP velocity):
hσvMøli = a + bx−1 + . . . . (1.46)
(The coefficient a corresponds to the s-wave contribution to the cross section, the
coefficient b to the p-wave contribution, and so on.) This partial wave expansion gives
a quite good approximation, provided there are no s-channel resonances and thresholds
for the final states [39].
In [40], it was shown that, after expanding the integrands of Eq. (1.41) in powers
of x
−1
, all the integrations can be performed analytically. As we saw, the expression
20 Dark Matter
vMølE1E2 depends on momenta only through s. Therefore, one can form the Lorentz
invariant quantity
w(s) ≡ σ(s)vMølE1E2 =
1
2
σ(s)
p
s(s − 4m2
). (1.47)
The integration involves the Taylor expansion of this quantity w around s/4m2 = 1
and the general formula for the partial wave expansion of the thermal average is [40]
hσvMøli =
1
m2

w −
3
2
(2w − w
′
)x
−1 +
3
8
(16w − 8w
′ + 5w
′′)x
−2
−
5
16
(30w − 15w
′ + 3w
′′ − 7x
′′′)x
−3 + O(x
−4
)

s/4m2=1
, (1.48)
where primes denote derivatives with respect to s/4m2 and all quantities have to be
evaluated at s = 4m2
.
1.5 Direct Detection of DM
Since the beginning of 1980s, it has been realized that besides the numerous facts showing evidence for the existence of these new dark particles, it is also possible to detect
them directly. Already in 1985, two pioneering articles [41, 42] appeared, describing
the detection methods for WIMPs. Since WIMPs are expected to cluster gravitationally together with ordinary stars in the Milky Way halo, they would pass also through
Earth and, in principle, they can be detected through scattering with the nuclei in a
detector’s material. In practice, one has to measure the recoil energy deposited by this
scattering.
However, although one can deduce from rotation curves that DM dominates the
dark halo in the outer parts of our galaxy, it is not so obvious from direct measurements
whether there is any substantial amount of DM inside the solar radius R0 ≃ 8 kpc.
Using indirect methods (involving the determination of the gravitational potential,
through the measuring of the kinematics of stars, both near the mid-plane of the
galactic disk and at heights several times the disk thickness), it is almost certain
that the DM is also present in the solar system, with a local density ρ0 = (0.3 ±
0.1) GeV cm−3
[43].
This value for the local density implies that for a WIMP mass of order ∼ 100 GeV,
the local number density is n0 ∼ 10−3
cm−3
. It is also expected that the WIMPs
velocity is similar to the velocity with which the Sun orbits around the galactic center
(v0 ≃ 220 km s−1
), since they are both moving under the same gravitational potential.
These two quantities allow to estimate the order of magnitude of the incident flux
of WIMPs on the Earth: J0 = n0v0 ∼ 105
cm−2
s
−1
. This value is manifestly large,
but the very weak interactions of the DM particles with ordinary matter makes their
detection a difficult, although in principle feasible, task. In order to compensate for
the very low WIMP-nucleus scattering cross section, very large detectors are required.
1.5.1 Elastic scattering event rate 21
1.5.1 Elastic scattering event rate
In the following, we will confine ourselves to the elastic scattering with nuclei. Although
inelastic scattering of WIMPs off nuclei in a detector or off orbital electrons producing
an excited state is possible, the event rate of these processes is quite suppressed. In
contrast, during an elastic scattering the nucleus recoils as a whole.
The direct detection experiments measure the number of events per day and per
kilogram of the detector material, as a function of the amount of energy Q deposited
in the detector. This event rate would be given by R = nWIMP nnuclei σv in a simplified
model with WIMPs moving with a constant velocity v. The number density of WIMPs
is nWIMP = ρ0/mX and the number density of nuclei is just the ratio of the detector’s
mass over the nuclear mass mN .
For accurate calculations, one should take into account that the WIMPs move in the
halo not with a uniform velocity, but rather following a velocity distribution f(v). The
Earth’s motion in the solar system should be included into this distribution function.
The scattering cross section σ also depends on the velocity. Actually, the cross section
can be parametrized by a nuclear form factor F(Q) as
dσ =
σ
4m2
r
v
2
F
2
(Q)d|~q|
2
, (1.49)
where |~q|
2 = 2m2
r
v
2
(1 − cos θ) is the momentum transferred during the scattering,
mr =
mXmN
mX+mN
is the reduced mass of the WIMP – nucleus system and θ is the scattering
angle in the center of momentum frame. Therefore, one can write a general expression
for the differential event rate per unit detector mass as
dR =
ρ0
mX
1
mN
σF2
(Q)d|~q|
2
4m2
r
v
2
vf(v)dv. (1.50)
The energy deposited in the detector (transferred to the nucleus through one elastic
scattering) is
Q =
|~q|
2
2mN
=
m2
r
v
2
mN
(1 − cos θ). (1.51)
Therefore, the differential event rate over deposited energy can be written, using the
equations (1.50) and (1.51), as
dR
dQ
=
σρ0
√
πv0mXm2
r
F
2
(Q)T(Q), (1.52)
where, following [37], we have defined the dimensionless quantity T(Q) as
T(Q) ≡
√
π
2
v0
Z ∞
vmin
f(v)
v
dv, (1.53)
with the minimum velocity given by vmin =
qQmN
2m2
r
, obtained by Eq. (1.51). Finally,
the event rate R can be calculated by integrating (1.52) over the energy
R =
Z ∞
ET
dR
dQ
dQ. (1.54)
22 Dark Matter
The integration is performed for energies larger than the threshold energy ET of the
detector, below which it is insensitive to WIMP-nucleus recoils.
Using Eqs. (1.54) and (1.52), one can derive the scattering cross section from the
event rate. The experimental collaborations prefer to give their results already in terms
of the scattering cross section as a function of the WIMP mass. To be more precise,
the WIMP-nucleus total cross section consists of two parts: the spin-dependent (SD)
cross section and the spin-independent (SI) one. The former comes from axial current
couplings, whereas the latter comes from scalar-scalar and vector-vector couplings.
The SD cross section is much suppressed compared to the SI one in the case of heavy
nuclei targets and it vanishes if the nucleus contains an even number of nucleons, since
in this case the total nuclear spin is zero.
We see that two uncertainties enter the above calculation: the exact value of the
local density ρ0 and the exact form of the velocity distribution f(v). To these, one
has to include one more. The cross section σ that appears in the previous expressions
concerns the WIMP-nucleon cross section. The couplings of a WIMP with the various
quarks that constitute the nucleon are not the same and the WIMP-nucleon cross
section depends strongly on the exact quark content of the nucleon. To be more
precise, the largest uncertainty lies on the strange content of the nucleon, but we shall
return to this point when we will calculate the cross section in a specific particle theory,
the Next-to-Minimal Supersymmetric Standard Model, in Sec. 3.5.1.
1.5.2 Experimental status
The situation of the experimental results from direct DM searches is a bit confusing. The null observations in most of the experiments led them to set upper limits
on the WIMP-nucleon cross section. These bounds are quite stringent for the spinindependent cross section7
, especially in the regime of WIMP masses of the order of
100 GeV. However, some collaborations have already reported possible DM signals,
mainly in the low mass regime. The preferred regions of these experiments do not
coincide, while some of them have been already excluded by other experiments. The
present picture, for WIMP masses ranging from 5 to 1000 GeV, is summarized in Fig.
1.5, 1.6.
Fig. 1.5 mainly presents upper bounds coming from XENON100 [44]. XENON100
[46] is an experiment located at the Gran Sasso underground laboratory in Italy. It
contains in total 165 kg of liquid Xenon, with 65 kg acting as target mass and the
rest shielding the detector from background radiation. For these upper limits, 225
live days of data were used. The minimum value for the predicted upper bounds on
the cross section is 2 · 10−45 cm2
for WIMP mass ∼ 55 GeV (at 90% confidence level),
almost one order of magnitude lower than the previously released limits [47] by the
same collaboration, using 100 live days of data.
The stringent upper bounds up-to-date (at least for WIMP mass larger than about
7 GeV) come from the first results of the LUX experiment (see Fig. 1.6), after the first
7For the spin-dependent scattering, the exclusion limits are quite relaxed. Hence, we will focus on
the SI cross sections.
1.5.2 Experimental status 23
Figure 1.5: The XENON100 exclusion limit (thick blue line), along with the expected
sensitivity in green (1σ) and yellow (2σ) band. Other upper bounds are also shown as
well as detection claims. From [44].
85.3 live-days of its operation [45]. LUX [53] is a detector containing liquid Xenon, as
XENON100, but in larger quantity, with total mass 370 kg. Its operation started on
April 2013 with a goal to clearly detect or exclude WIMPs with a spin independent
cross section ∼ 2 · 10−46 cm2
.
In Fig. 1.5, except of the XENON100 bounds and other experimental limits on larger
WIMP-nucleon cross section, some detection claims also appear. These come from
DAMA [48,49], CoGeNT [50] and CRESST-II [51] experiments. The first positive result
came from DAMA [52], back in 2000. Since then, the experiment has accumulated 1.17
ton-yr of data over 13 years of operation. DAMA consists of 250 kg of radio pure NaI
scintillator and looks for the annual modulation of the WIMP flux in order to reduce
the influence of the background.
The annual modulation of the DM flux (see [54] for a recent review) is due to the
Earth’s orbital motion relative to the rotation of the galactic disk. The galactic disk
rotation through an essentially non-rotating DM halo, creates an effective DM wind in
the solar frame. During the earth’s heliocentric orbit, this wind reaches a maximum
when the Earth is moving fastest in the direction of the disk rotation (this happens
in the beginning of June) and a minimum when it is moving fastest in the opposite
direction (beginning of December).
DAMA claims an 8.9σ annual modulation with a minimum flux on May 26±7 days,
consistent with the expectation. Since the detector’s target consists of two different
nuclei and the experiment cannot distinguish between sodium and iodine recoils, there
24 Dark Matter
Figure 1.6: The LUX 90% confidence exclusion limit (blue line) with the 1σ range
(shaded area). The XENON100 upper bound is represented by the red line. The inset
shows also preferred regions by CoGeNT (shaded light red), CDMS II silicon detector
(shaded green), CRESST II (shaded yellow) and DAMA (shaded gray). From [45].
is no model independent way to determine the exact region in the cross section versus
WIMP mass plane to which the observed modulation corresponds. However, one can
assume two cases: one that the WIMP scattering off the sodium nucleus dominates the
recoil energy and the other with the iodine recoils dominating. The former corresponds
[55] to a light WIMP (∼ 10 GeV) and quite large scattering cross section and the latter
to a heavier WIMP (∼ 50 to 100 GeV) with smaller cross section (see Fig. 1.5).
The positive result of DAMA was followed many years later by the ones of CoGeNT
and CRESST-II, and more recently by the silicon detector of CDMS [56] (Fig. 1.7).
The discrepancy of the results raised a lot of debates among the experiments (for
example, [64–67]) and by some the positive results are regarded as controversial. On
the other hand, it also raised an effort to find a physical explanation behind this
inconsistency (see, for example, [68–71]).
1.6 Indirect Methods for DM Detection
The same annihilation processes that determined the DM relic abundance in the early
Universe also occur today in galactic regions where the DM concentration is higher.
This fact rises the possibility of detecting potential WIMP pair annihilations indirectly
through their imprints on the cosmic rays. Therefore, the indirect DM searches aim
at the detection of an excess over the known astrophysical background of charged
particles, photons or neutrinos.
Charged particles – electrons, protons and their antiparticles – may originate from
direct products (pair of SM particles) of WIMP annihilations, after their decay and
1.6 Indirect Methods for DM Detection 25
Figure 1.7: The blue contours represent preferred regions for a possible signal at 68%
and 90% C.L. using the silicon detector of CMDS [56]. The blue dotted line represents
the upper limit obtained by the same analysis and the blue solid line is the combined
limit with the silicon CDMS data set reported in [57]. Other limits also appear:
from the CMDS standard germanium detector (light and dark red dashed line, for
standard [58] and low threshold analysis [59], respectively), EDELWEISS [60] (dashed
orange), XENON10 [61] (dash-dotted green) and XENON100 [44] (long-dash-dotted
green). The filled regions identify possible signal regions associated with data from
CoGeNT [62] (dashed yellow, 90% C.L.), DAMA [49,55] (dotted tan, 99.7% C.L.) and
CRESST-II [51, 63] (dash-dotted pink, 95.45% C.L.) experiments. Taken from [56].
through the process of showering and hadronization. Although the exact shape of the
resulting spectrum would depend on the specific process, it is expected to show a steep
cutoff at the WIMP mass. Once produced in the DM halo, the charged particles have
to travel to the point of detection through the turbulent galactic field, which will cause
diffusion. Apart from that, a lot of processes disturb the propagation of the charged
particles, such as bremsstrahlung, inverse Compton scattering with CMB photons and
many others. Therefore, the uncertainties that enter the propagation of the charged
flux until it reaches the telescope are important (contrary to the case of photons and
neutrinos that propagate almost unperturbed through the galaxy).
As in the case of direct detection, the experimental status of charged particle detection concerning the DM is confusing. After some hints from HEAT [72] and AMS01 [73] (the former a far-infrared telescope in Antarctica, the latter a spectrometer,
prototype for AMS-02 mounted on the International Space Station [74]), the PAMELA
satellite observed [75, 76] a steep increase in the energy spectrum of positron fraction
e
+/(e
+ + e
−)
8
. Later FERMI satellite [77] and AMS-02 [78] confirmed the results up
8The searches for charged particles focus on the antiparticles in order to have a reduced background,
26 Dark Matter
Figure 1.8: A compilation of data of charged cosmic rays, together with plausible but
uncertain astrophysical backgrounds, taken from [79]. Left: Positron flux. Center:
Antiproton flux. Right: Sum of electrons and positrons.
to energies of ∼ 200 GeV. However, the excess of positrons is not followed by an excess
of antiprotons, whose flux seems to coincide with the predicted background [75]. In
Fig. 1.8, three plots summarizing the situation are shown [79].
The observed excess is very difficult to explain in terms of DM [79]. To begin with,
the annihilation cross section required to reproduce the excess is quite large, many
orders of magnitude larger than the thermal cross section. Moreover, an “ordinary”
WIMP with large annihilation cross section giving rise to charged leptons is expected
to give, additionally, a large number of antiprotons, a fact in contradiction with the
observations. Although a lot of work has been done to fit a DM particle to the observed
pattern, it is quite possible that the excesses come from a yet unknown astrophysical
source. We are not going to discuss further this matter, but we end with a comment.
If this excess is due to a source other than DM, then a possible DM positron excess
would be lost under this formidable background.
A last hint for DM came from the detection of highly energetic photons. However,
we will interrupt this discussion, since this signal and a possible explanation is the
subject of Ch. 4. There, we will also see the upper bounds on the annihilation cross
section being set due to the absence of excesses in diffuse γ radiation.
since they are much less abundant than the corresponding particles.
CHAPTER 2
PARTICLE PHYSICS
Since the DM comprises of particles, it should be explained by a general particle physics
theory. We start in the following section by describing the Standard Model (SM) of
particle physics. Although the SM describes so far the fundamental particles and their
interactions quite accurately, it cannot provide a DM candidate. Besides, the SM
suffers from some theoretical problems, which we discuss in Sec. 2.2. We will see that
these problems can be solved if one introduces a new symmetry, the supersymmetry,
which we describe in Sec. 2.3. We finish this chapter by briefly describing in Sec. 2.4 a
supersymmetric extension of the SM with the minimal additional particle content, the
Minimal Supersymmetric Standard Model (MSSM).
2.1 The Standard Model of Particle Physics
The Standard Model (SM) of particle physics1
consists of two well developed theories,
the quantum chromodynamics (QCD) and the electroweak (EW) theory. The former
describes the strong interactions among the quarks, whereas the latter describes the
electroweak interactions (the weak and electromagnetic interactions in a unified context) between fermions. The EW theory took its final form in the late 1960s by the
introduction by S. Weinberg [85] and A. Salam [86] of the Higgs mechanism that gives
masses to the SM particles, which followed the unification of electromagnetic and weak
interactions [87,88]. At the same time, the EW model preserves the gauge invariance,
making the theory renormalizable, as shown later by ’t Hooft [89]. On the other hand,
QCD obtained its final form some years later, after the confirmation of the existence
of quarks. Of course, the history of the SM is much longer and it can be traced back to
1920s with the formulation of a theoretical basis for a Quantum Field Theory (QFT).
Since then, the SM had many successes. The SM particle content was completed with
the discovery of the heaviest of the quarks, the top quark [90,91], in 1995 and, recently,
with the discovery of the Higgs boson [92, 93].
1There are many good textbooks on the SM and Quantum Field Theory, e.g. [80–84].
28 Particle Physics
The key concept within the SM, as in every QFT, is that of symmetries. Each
interaction respects a gauge symmetry, based on a Lie algebra. The strong interaction is
described by an SU(3)c symmetry, where the subscript c stands for color, the conserved
charge of strong interactions. The EW interactions, on the other hand, are based on
a SU(2)L × U(1)Y Lie algebra. Here, as we will subsequently see, L refers to the
left-handed fermions and Y is the hypercharge, the conserved charge under the U(1).
SU(2)L conserves a quantity known as weak isospin I. Therefore, the SM contains the
internal symmetries of the unitary product group
SU(2)L × U(1)Y × SU(3)c. (2.1)
2.1.1 The particle content of the SM
We mention for completeness that particles are divided into two main classes according
to the statistics they follow. The bosons are particles with integer spin and follow the
Bose-Einstein distribution, whereas fermions have half-integer spin and follow the
Dirac-Einstein statistics, obeying the Pauli exclusion principle. In the SM, all the
fermions have spin 1/2, whereas the bosons have spin 1 with only exception the Higgs
boson, which is a scalar (spin zero). We begin the description of the SM particles with
the fermions.
Each fermion is classified in irreducible representations of each individual Lie algebra, according to the conserved quantum numbers, i.e. the color C, the weak isospin
I and the hypercharge Y . A first classification of fermions can be done into leptons
and quarks, which transform differently under the SU(3)c. Leptons are singlets under
this transformation, while quarks act as triplets (the fundamental representation of
this group). The EW interactions violate maximally the parity symmetry and SU(2)L
acts only on states with negative chirality (left-handed). A Dirac spinor Ψ can be
decomposed into left and right chirality components using, respectively, the projection
operators PL =
1
2
(1 − γ5) and PR =
1
2
(1 + γ5):
ΨL = PLΨ and ΨR = PRΨ. (2.2)
Left-handed fermions have I = 1/2, with a third component of the isospin I3 = ±1/2.
Fermions with positive I3 are called up-type fermions and those with negative are
called down-type. These behave the same way under SU(2)L and form doublets with
one fermion of each type. On the other hand, right-handed fermions have I = 0 and
form singlets that do not undergo weak interactions. The hypercharge is written in
terms of the electric charge Q and the third component of the isospin I3 through the
Gell-Mann–Nishijima relation:
Q = I3 + Y/2. (2.3)
Therefore, left- and right-handed components transform differently under the U(1)Y ,
since they have different hypercharge.
The fermionic sector of the SM comprises three generations of fermions, transforming as spinors under Lorentz transformations. Each generation has the same structure.
For leptons, it is an SU(2)L doublet with components consisting of one left-handed
2.1.2 The SM Lagrangian 29
charged lepton and one neutrino (neutrinos are only left-handed in the SM), along
with a gauge singlet right-handed charged lepton. The quark doublet consists of an
up- (u) and a down-type (d) (left-handed) quark and the pattern is completed by the
two corresponding SU(2)L singlet right-handed quarks. We write these representations
as
Quarks: Q ≡

u
i
L
d
i
L
!
, ui
R, di
R Leptons: L ≡

ν
i
L
e
i
L
!
, ei
R, (2.4)
with i = 1, 2, 3 the generation index.
Having briefly described the fermionic sector, we turn to the bosonic sector of
the SM. It consists of the gauge bosons that mediate the interactions and the Higgs
boson that gives masses to the particles through a spontaneous symmetry breaking,
the electroweak symmetry breaking (EWSB) [94–98], which we shall describe in Sec.
2.1.3. Before the EWSB, these bosons are
• three Wa
µ
(a = 1, 2, 3) weak bosons, associated with the generators of SU(2)L,
• one neutral Bµ boson, associated with the generator of U(1)Y ,
• eight gluons Ga
µ
(a = 1, . . . , 8), associated with the generators of SU(3)c, and
• the complex scalar Higgs doublet Φ =
φ
+
φ
0
!
.
After the EWSB, the EW boson states mix and give the two W± bosons, the neutral
Z boson and the massless photon γ. From the symmetry breaking, one scalar degree of
freedom remains which is the famous (neutral) Higgs boson [97–99]. We will return to
the mixed physical states, after describing the Higgs mechanism for symmetry breaking.
A complete list of the SM particles (the physical states after EWSB) is shown in Table
2.1.
2.1.2 The SM Lagrangian
The gauge bosons are responsible for the mediation of the interactions and are associated with the generators of the corresponding symmetry. The EW gauge bosons Bµ
and Wa
µ
are associated, respectively, with the generator Y of the U(1)Y and the three
generators T
a
2
of the SU(2)L. The latter are defined as half of the Pauli matrices τ
a
(T
a
2 =
1
2
τ
a
) and they obey the algebra

T
a
2
, Tb
2

= iǫabcT
c
2
, (2.5)
where ǫ
abc is the fully antisymmetric Levi-Civita tensor. The eight gluons are associated
with an equal number of generators T
a
3
(Gell-Mann matrices) of SU(3)c and obey the
Lie algebra

T
a
3
, Tb
3

= if abcT
c
3
, with Tr
T
a
3 T
b
3

=
1
2
δ
ab
, (2.6)
30 Particle Physics
Name symbol mass charge (|e|) spin
Leptons
electron e 0.511 MeV −1 1/2
electron neutrino νe 0 (<2 eV) 0 1/2
muon µ 105.7 MeV −1 1/2
muon neutrino νµ 0 (<2 eV) 0 1/2
tau τ 1.777 GeV −1 1/2
tau neutrino ντ 0 (<2 eV) 0 1/2
Quarks
up u 2.7
+0.7
−0.5 MeV 2/3 1/2
down d 4.8
+0.7
−0.3 MeV −1/3 1/2
strange s (95 ± 5) MeV −1/3 1/2
charm c (1.275 ± 0.025) GeV 2/3 1/2
bottom b (4.18 ± 0.03) GeV −1/3 1/2
top t (173.5 ± 0.6 ± 0.8) GeV 2/3 1/2
Bosons
photon γ 0 (<10−18 eV) 0 (<10−35) 1
W boson W± (80.385 ± 0.015) GeV ±1 1
Z boson Z (91.1876 ± 0.0021) GeV 0 1
gluon g 0 (.O(1) MeV) 0 1
Higgs H
(125.3 ± 0.4 ± 0.5) GeV
0 0
(126.0 ± 0.4 ± 0.4) GeV
Table 2.1: The particle content of the SM. All values are those given in [100], except of
the Higgs mass that is taken from [92, 93] (up and down row, respectively), assuming
that the observed excess corresponds to the SM Higgs. The u, d and s quark masses
are estimates of so-called “current-quark masses” in a mass-independent subtraction
scheme as MS at a scale ∼ 2 GeV. The c and b quark masses are the running masses
in the MS scheme. The values in the parenthesis are the current experimental limits.
with f
abc the structure constants of the group.
Using the structure constants of the corresponding groups, we define the field
strengths for the gauge bosons as
Bµν ≡ ∂µBν − ∂νBµ, (2.7a)
Wµν ≡ ∂µWa
ν − ∂νWa
µ + g2ǫ
abcWb
µWc
ν
(2.7b)
and
G
a
µν ≡ ∂µG
a
ν − ∂νG
a
µ + g3f
abcG
b
µG
c
ν
. (2.7c)
2.1.2 The SM Lagrangian 31
We use the notation g1, g2 and g3 for the coupling constants of U(1)Y , SU(2)L and
SU(3)c, respectively. As in any Yang-Mills theory, the non-abelian gauge groups lead
to self-interactions, which is not the case for the abelian U(1)Y group.
Before we finally write the full Lagrangian, we have to introduce the covariant
derivative for fermions, which in a general form can be written as
DµΨ =
∂µ − ig1
1
2
Y Bµ − ig2T
a
2 Wa
µ − ig3T
a
3 G
a
µ

Ψ. (2.8)
This form has to be understood as that, depending on Ψ, only the relevant terms
apply, hence for SU(2)L singlet leptons only the two first terms inside the parenthesis
are relevant, for doublet leptons the three first terms and for the corresponding quark
singlets and doublets the last term also participates. We also have to notice that in
order to retain the gauge symmetry, mass terms are forbidden in the Lagrangian. For
example, the mass term mψψ¯ = m

ψ¯
LψR + ψ¯
RψL

(with ψ¯ ≡ ψ
†γ
0
) is not invariant
under SU(2)L. This paradox is solved by the introduction of the Higgs scalar field
(see next subsection). The SM Lagrangian can be now written2
, split for simplicity in
three parts, each describing the gauge bosons, the fermions and the scalar sector,
LSM = Lgauge + Lfermion + Lscalar, (2.9)
with
Lgauge = −
1
4
G
a
µνG
µν
a −
1
4
Wa
µνWµν
a −
1
4
BµνB
µν
, (2.10a)
Lfermion = iL¯Dµγ
µL + ie¯RDµγµeR
+ iQ¯Dµγ
µQ + iu¯RDµγ
µuR + i
¯dRDµγ
µ
dR
−

heL¯ΦeR + hdQ¯ΦdR + huQ¯ΦeuR + h.c.

(2.10b)
and
Lscalar = (DµΦ)†
(DµΦ) − V (Φ†Φ), (2.10c)
where
V (Φ†Φ) = µ
2Φ
†Φ + λ

Φ
†Φ
2
(2.11)
is the scalar Higgs potential. Φ is the conjugate of Φ, related to the charge conjugate e
by Φ =e iτ2Φ
⋆
, with τi the Pauli matrices. The covariant derivative acting on the Higgs
scalar field gives
DµΦ =
∂µ − ig1
1
2
Y Bµ − ig2T
a
2 Wa
µ

Φ. (2.12)
Before we proceed to the description of the Higgs mechanism, a last comment concerning the SM Lagrangian is in order. If we restore the generation indices, we see that
2For simplicity, from now on we are going to omit the generations indice
32 Particle Physics
the Yukawa couplings h are 3 × 3, in general complex, matrices. As any complex matrix, they can be diagonalized with the help of two unitary matrices VL and VR, which
are related by VR = U
†VL with U again a unitary matrix. The diagonalization in the
quark sector to the mass eigenstates induces a mixing among the flavors (generations),
described by the Cabibbo–Kobayashi–Maskawa (CKM) matrix [101, 102]. The CKM
matrix is defined by
VCKM ≡ V
u
L
†
V
d
L
†
, (2.13)
where V
u
L
, V
d
L
are the unitary matrices that diagonalize the Yukawa couplings Hu
, Hd
,
respectively. This product of the two matrices appears in the charged current when it
is expressed in terms of the observable mass eigenstates.
2.1.3 Mass generation through the Higgs mechanism
We will start by examining the scalar potential (2.11). The vacuum expectation value
(vev) of the Higgs field hΦi ≡ h0|Φ|0i is given by the minimum of the potential. For
µ
2 > 0, the potential is always non-negative and Φ has a zero vev. The hypothesis of
the Higgs mechanism is that µ
2 < 0. In this case, the field Φ will acquire a vev
hΦi =
1
2

0
v
!
with v =
r
−
µ2
λ
. (2.14)
Since the charged component of Φ still has a zero vev, the U(1)Q symmetry of quantum
electrodynamics (QED) remains unbroken.
We expand the field Φ around the minima v in terms of real fields, and at leading
order we have
Φ(x) =
θ2(x) + iθ1(x)
√
1
2
(v + H(x)) − iθ3(x)
!
=
1
√
2
e
iθa(x)τ
a

0
v + H(x)
!
. (2.15)
We can eliminate the unphysical degrees of freedom θa, using the fact that the theory
remains gauge invariant. Therefore, we perform the following SU(2)L gauge transformation on Φ (unitary gauge)
Φ(x) → e
−iθa(x)τ
a
Φ(x), (2.16)
so that
Φ(x) = 1
√
2

0
v + H(x)
!
. (2.17)
We are going to use the following definitions for the gauge fields
W±
µ ≡
1
2

W1
µ ∓ iW2
µ

, (2.18a)
Zµ ≡
1
p
g
2
1 + g
2
2

g2W3
µ − g1Bµ

, (2.18b)
Aµ ≡
1
p
g
2
1 + g
2
2

g1W3
µ + g2Bµ

, (2.1
2.2 Limits of the SM and the emergence of supersymmetry 33
Then, the kinetic term for Φ (see Eq. (2.10c)) can be written in the unitary gauge as
(DµΦ)†
(D
µΦ) = 1
2
(∂µH)
2 + M2
W W+
µ W−µ +
1
2
M2
ZZµZ
µ
, (2.19)
with
MW ≡
1
2
g2v and MZ ≡
1
2
q
g
2
1 + g
2
2
v. (2.20)
We see that the definitions (2.18) correspond to the physical states of the gauge bosons
that have acquired masses due to the non-zero Higgs vev, given by (2.20). The photon
has remained massless, which reflects the fact that after the spontaneous breakdown of
SU(2)L × U(1)Y the U(1)Q remained unbroken. Among the initial degrees of freedom
of the complex scalar field Φ, three were absorbed by W± and Z and one remained as
the neutral Higgs particle with squared mass
m2
H = 2λv2
. (2.21)
We note that λ should be positive so that the scalar potential (2.11) is bounded from
below.
Fermions also acquire masses due to the Higgs mechanism. The Yukawa terms in
the fermionic part (2.10b) of the SM Lagrangian are written, after expanding around
the vev in the unitary gauge,
LY = −
1
√
2
hee¯L(v + H)eR −
1
√
2
hd
¯dL(v + H)dR −
1
√
2
huu¯L(v + H)uR + h.c. . (2.22)
Therefore, we can identify the masses of the fermions as
me
i =
h
i
e
v
√
2
, md
i =
h
i
d
v
√
2
, mui =
h
i
u
v
√
2
, (2.23)
where we have written explicitly the generation indices.
2.2 Limits of the SM and the emergence of supersymmetry
2.2.1 General discussion of the SM problems
The SM has been proven extremely successful and has been tested in high precision
in many different experiments. It has predicted many new particles before their final
discovery and also explained how the particles gain their masses. Its last triumph was
of course the discovery of a boson that seems to be very similar to the Higgs boson of
the SM. However, it is generally accepted that the SM cannot be the ultimate theory. It
is not only observed phenomena that the SM does not explain; SM also faces important
theoretical issues.
The most prominent among the inconsistencies of the SM with observations is the
oscillations among neutrinos of different generations. In order for the oscillations to
34 Particle Physics
φ φ
k
Figure 2.1: The scalar one-loop diagram giving rise to quadratic divergences.
occur, neutrinos should have non-zero masses. However, minimal modifications of the
SM are able to fit with the data of neutrino physics. Another issue that a more complete theory has to face is the matter asymmetry, the observed dominance of matter
over antimatter in the Universe. In addition, in order to comply with the standard
cosmological model, it has to provide the appropriate particle(s) that drove the inflation. Last, but not least, we saw that in order to explain the DM that dominates the
Universe, a massive, stable weakly interacting particle must exist. Such a particle is
not present in the SM.
On the other hand, the SM also suffers from a theoretical perspective. For example,
the SM counts 19 free parameters; one expects that a fundamental theory would have
a much smaller number of free parameters. Simple modifications of the SM have been
proposed relating some of these parameters. Grand unified theories (GUTs) unify
the gauge couplings at a high scale ∼ 1016 GeV. However, this unification is only
approximate unless the GUT is embedded in a supersymmetric framework. Another
serious problem of the SM is that of naturalness. This will be the topic of the following
subsection.
2.2.2 The naturalness problem of the SM
The presence of fundamental scalar fields, like the Higgs, gives rise to quadratic divergences. The diagram of Fig. 2.1 contributes to the squared mass of the scalar
δm2 = λ
Z Λ
d
4k
(2π)
4
k
−2
. (2.24)
This contribution is approximated by δm2 ∼ λΛ
2/(16π
2
), quadratic in a cut-off Λ,
which should be finite. For the case of the Higgs scalar field, one has to include its
couplings to the gauge fields and the top quark3
. Therefore,
δm2
H =
3Λ2
8π
2v
2

4m2
t − 2M2
W − M2
Z − m2
H

+ O(ln Λ
µ
)

, (2.25)
where we have used Eq. (2.21) and m2
H ≡ m2
0 + δm2
H.
3Since the contribution to the squared mass correction are quadratic in the Yukawa couplings (or
quark masses), the lighter quarks can be neglected
2.2.3 A way out 35
Taking Λ as a fundamental scale Λ ∼ MP l ∼ 1019 GeV we have
m2
0 = m2
H −
3Λ2
8π
2v
2

4m2
t − 2M2
W − M2
Z − m2
H

(2.26)
and we can see that m2
0 has to be adjusted to a precision of about 30 orders of magnitude
in order to achieve an EW scale Higgs mass. This is considered as an intolerable finetuning, which is against the general belief that the observable properties of a theory
have to be stable under small variations of the fundamental (bare) parameters. It is
exactly the above behavior that is considered as unnatural. Although the SM could
be self-consistent without imposing a large scale, grand unification of the parameters
introduce a hierarchy problem between the different scales.
A more strict definition of naturalness comes from ’t Hooft [103], which we rewrite
here:
At an energy scale µ, a physical parameter or set of physical parameters
αi(µ) is allowed to be very small only if the replacement αi(µ) = 0 would
increase the symmetry of the system.
Clearly, this is not the case here. Although mH is small compared to the fundamental
scale Λ, it is not protected by any symmetry and a fine-tuning is necessary.
2.2.3 A way out
The naturalness in the ’t Hooft sense is inspired by quantum electrodynamics, which is
the archetype for a natural theory. For example, the corrections to the electron mass
me are themselves proportional to me, with a dimensionless proportionality factor that
behaves like ∼ ln Λ. In general, fermion masses are protected by the chiral symmetry; small values (compared to the fundamental scale) of these masses enhances the
symmetry.
If a new symmetry exists in nature, relating fermion fields to scalar fields, then each
scalar mass would be related somehow to the corresponding fermion mass. Therefore,
the scalar mass itself can be naturally small compared to Λ, since this would mean
that the fermion mass is small, which enhances the chiral symmetry. Such a symmetry,
relating bosons to fermions and vice versa, is known as supersymmetry [104, 105].
Actually, as we will see later, if this new symmetry remains unbroken, the masses of
the conjugate bosons and fermions would have to be equal.
In order to make the above statement more concrete, we consider a toy model with
two additional complex scalar fields feL and feR. We will discuss only the quadratic
divergences that come from corrections to the Higgs mass due to a fermion. The
generalization for the contributions from the gauge bosons or the self-interaction is
straightforward. The interactions in this toy model of the new scalar fields with the
Higgs are described by the Lagrangian
Lfefφe = λfe|φ|
2

|feL|
2 + |feR|
2

. (2.27
36 Particle Physics
It can be easily checked that the quadratic divergence coming from a fermion at one
loop is exactly canceled, as long as the new quartic coupling λfe obeys the relation
λfe = −λ
2
f
(λf is the Yukawa coupling for the fermion f).
2.3 A brief summary of Supersymmetry
Supersymmetry (SUSY) is a symmetry relating fermions and bosons. The supersymmetry transformation should turn a boson state into a fermion state and vice versa. If
Q is the operator that generates such transformations, then
Q |bosoni = |fermioni Q |fermioni = |bosoni. (2.28)
Due to commutation and anticommutation rules of bosons and fermions, Q has to
be an anticommuting spinor operator, carrying spin angular momentum 1/2. Since
spinors are complex objects, the hermitian conjugate Q†
is also a symmetry operator4
.
There is a no-go theorem, the Coleman-Mandula theorem [106], that restricts the
conserved charges which transform as tensors under the Lorentz group to the generators
of translations Pµ and the generators of Lorentz transformations Mµν. Although this
theorem can be evaded in the case of supersymmetry due to the anticommutation
properties of Q, Q†
[107], it restricts the underlying algebra of supersymmetry [108].
Therefore, the basic supersymmetric algebra can be written as5
{Q, Q†
} = P
µ
, (2.29a)
{Q, Q} = {Q
†
, Q†
} = 0, (2.29b)
[P
µ
, Q] = [P
µ
, Q] = 0. (2.29c)
In the following, we summarize the basic conclusions derived from this algebra.
• The single-particle states of a supersymmetric theory fall into irreducible representations of the SUSY algebra, called supermultiplets. A supermultiplet contains
both fermion and boson states, called superpartners.
• Superpartners must have equal masses: Consider |Ωi and |Ω
′
i as the superpartners, |Ω
′
i should be proportional to some combination of the Q and Q† operators
acting on |Ωi, up to a space-time translation or rotation. Since −P
2
commutes
with Q, Q† and all space-time translation and rotation operators, |Ωi, |Ω
′
i will
have equal eigenvalues of −P
2 and thus equal masses.
• Superpartners must be in the same representation of gauge groups, since Q, Q†
commute with the generators of gauge transformations. This means that they
have equal charges, weak isospin and color degrees of freedom.
4We will confine ourselves to the phenomenologically more interesting case of N = 1 supersymmetry, with N referring to the number of distinct copies of Q, Q†
.
5We present a simplified version, omitting spinor indices in Q and Q†
.
2.3 A brief summary of Supersymmetry 37
• Each supermultiplet contains an equal number of fermion and boson degrees of
freedom (nF and nB, respectively): Consider the operator (−1)2s
, with s the spin
angular momentum, and the states |ii that have the same eigenvalue p
µ of P
µ
.
Then, using the SUSY algebra (2.29) and the completeness relation P
i
|ii hi| =
1, we have P
i
hi|(−1)2sP
µ
|ii = 0. On the other hand, P
i
hi|(−1)2sP
µ
|ii =
p
µTr [(−1)2s
] ∝ nB − nF . Therefore, nF = nB.
As addendum to the last point, we see that two kind of supermultiplets are possible
(neglecting gravity):
• A chiral (or matter or scalar ) supermultiplet, which consists of a single Weyl
fermion (with two spin helicity states, nF = 2) and two real scalars (each with
nB = 1), which can be replaced by a single complex scalar field.
• A gauge (or vector ) supermultiplet, which consists of a massless spin 1 boson
(two helicity states, nB = 2) and a massless spin 1/2 fermion (nF = 2).
Other combinations either are reduced to combinations of the above supermultiplets
or lead to non-renormalizable interactions.
It is possible to study supersymmetry in a geometric approach, using a space-time
manifold extended by four fermionic (Grassmann) coordinates. This manifold is called
superspace. The fields, in turn, expressed in terms of the extended set of coordinates
are called superfields. We are not going to discuss the technical details of this topic
(the interested reader may refer to the rich bibliography, for example [109–111]).
However, it is important to mention a very useful function of the superfields, the
superpotential. A generic form of a (renormalizable) superpotential in terms of the
superfields Φ is the following b
W =
1
2
MijΦbiΦbj +
1
6
y
ijkΦbiΦbjΦbk. (2.30)
The Lagrangian density can always be written according to the superpotential. The
superpotential has also to fulfill some requirements. In order for the Lagrangian to
be supersymmetric invariant, W has to be holomorphic in the complex scalar fields
(it does not involve hermitian conjugates Φb† of the superfields). Conventionally, W
involves only left chiral superfields. Instead of the SU(2)L singlet right chiral fermion
fields, one can use their left chiral charge conjugates.
As we mentioned before, the members of a supermultiplet have equal masses. This
contradicts our experience, since the partners of the light SM particles would have been
detected long time ago. Hence, the supersymmetry should be broken at a large energy
scale. The common approach is that SUSY is broken in a hidden sector, very weakly
coupled to the visible sector. Then, one has to explain how the SUSY breaking mediated to the visible sector. The two most popular scenarios are the gravity mediation
scenario [112–114] and the Gauge-Mediated SUSY Breaking (GSMB) [113, 115–117],
where the mediation occurs through gauge interactions.
There are two approaches with which one can address the SUSY breaking. In the
first approach, one refers to a GUT unification and determines the supersymmetric
38 Particle Physics
breaking parameters at low energies through the renormalization group equations.
This approach results in a small number of free parameters. In the second approach,
the starting point is the low energy scale. In this case, the SUSY breaking has to be
parametrized by the addition of breaking terms to the low energy Lagrangian. This
results in a larger set of free parameters. These terms should not reintroduce quadratic
divergences to the scalar masses, since the cancellation of these divergences was the
main motivation for SUSY. Then, one talks about soft breaking terms.
2.4 The Minimal Supersymmetric Standard Model
One can construct a supersymmetric version of the standard model with a minimal
content of particles. This model is known as the Minimal Supersymmetric Standard
Model (MSSM). In a SUSY extension of the SM, each of the SM particles is either in a
chiral or in a gauge supermultiplet, and should have a superpartner with spin differing
by 1/2.
The spin-0 partners of quarks and leptons are called squarks and sleptons, respectively (or collectively sfermions), and they have to reside in chiral supermultiplets.
The left- and right-handed components of fermions are distinct 2-component Weyl
fermions with different gauge transformations in the SM, so that each must have its
own complex scalar superpartner. The gauge bosons of the SM reside in gauge supermultiplets, along with their spin-1/2 superpartners, which are called gauginos. Every
gaugino field, like its gauge boson partner, transforms as the adjoint representation of
the corresponding gauge group. They have left- and right-handed components which
are charge conjugates of each other: (λeL)
c = λeR.
The Higgs boson, since it is a spin-0 particle, should reside in a chiral supermultiplet. However, we saw (in the fermionic part of the SM Lagrangian, Eq. (2.10b))
that the Y = 1/2 Higgs in the SM can give mass to both up- and down-type quarks,
only if the conjugate Higgs field with Y = −1/2 is involved. Since in the superpotential there are no conjugate fields, two Higgs doublets have to be introduced. Each
Higgs supermultiplet would have hypercharge Y = +1/2 or Y = −1/2. The Higgs
with the negative hypercharge gives mass to the down-type fermions and it is called
down-type Higgs (Hd, or H1 in the SLHA convention [118]) and the other one gives
mass to up-type fermions and it is called up-type Higgs (Hu, or H2).
The MSSM respects a discrete Z2 symmetry, the R-parity. If one writes the most
general terms in the supersymmetric Lagrangian (still gauge-invariant and holomorphic), some of them would lead to non-observed processes. The most obvious constraint
comes from the non-observed proton decay, which arises from a term that violates both
lepton and baryon numbers (L and B, respectively) by one unit. In order to avoid these
terms, R-parity, a multiplicative conserved quantum number, is introduced, defined as
PR = (−1)3(B−L)+2s
, (2.31)
with s the spin of the particle.
The R even particles are the SM particles, whereas the R odd are the new particles
introduced by the MSSM and are called supersymmetric particles. Due to R-parity,
2.4 The Minimal Supersymmetric Standard Model 39
if it is exactly conserved, there can be no mixing among odd and even particles and,
additionally, each interaction vertex in the theory can only involve an even number of
supersymmetric particles. The phenomenological consequences are quite important.
First, the lightest among the odd-parity particles is stable. This particle is known
as the lightest supersymmetric particle (LSP). Second, in collider experiments, supersymmetric particles can only be produced in pairs. The first of these consequences
was a breakthrough for the incorporation of DM into a general theory. If the LSP is
electrically neutral, it interacts only weakly and it consists an attractive candidate for
DM.
We are not going to enter further into the details of the MSSM6
. Although MSSM
offers a possible DM candidate, there is a strong theoretical reason to move from the
minimal model. This reason is the so-called µ-problem of the MSSM, with which we
begin the discussion of the next chapter, where we shall describe more thoroughly the
Next-to-Minimal Supersymmetric Standard Model.
6We refer to [110] for an excellent and detailed description of MSSM.
40 Particle Physics
Part II
Dark Matter in the
Next-to-Minimal Supersymmetric
Standard Model

CHAPTER 3
THE NEXT-TO-MINIMAL
SUPERSYMMETRIC STANDARD
MODEL
The Next-to-Minimal Supersymmetric Standard Model (NMSSM) is an extension of
the MSSM by a chiral, SU(2)L singlet superfield Sb (see [119, 120] for reviews). The
introduction of this field solves the µ-problem1
from which the MSSM suffers, but
also leads to a different phenomenology from that of the minimal model. The scalar
component of the additional field mixes with the scalar Higgs doublets, leading to three
CP-even mass eigenstates and two CP-odd eigenstates (as in the MSSM a doublet-like
pair of charged Higgs also exists). On the other hand, the fermionic component of the
singlet (singlino) mixes with gauginos and higgsinos, forming five neutral states, the
neutralinos.
Concerning the CP-even sector, a new possibility opens. The lightest Higgs mass
eigenstate may have evaded the detection due to a sizeable singlet component. Besides,
the SM-like Higgs is naturally heavier than in the MSSM [123–126]. Therefore, a SMlike Higgs mass ∼ 125 GeV is much easier to explain [127–141]. The singlet component
of the CP-odd Higgs also allows for a potentially very light pseudoscalar with suppressed couplings to SM particles, with various consequences, especially on low energy
observables (for example, [142–145]). The singlino component of the neutralino may
also play an important role for both collider phenomenology and DM. This is the case
when the neutralino is the LSP and the lightest neutralino has a significant singlino
component.
We start the discussion about the NMSSM by describing the µ-problem and how
this is solved in the context of the NMSSM. In Sec. 3.2 we introduce the NMSSM
Lagrangian and we write the mass matrices of the Higgs sector particles and the su1However, historically, the introduction of a singlet field preceded the µ-problem, e.g. [104, 105,
121, 122].
44 The Next-to-Minimal Supersymmetric Standard Model
persymmetric particles, at tree level. We continue by examining, in Sec. 3.3, the DM
candidates in the NMSSM and particularly the neutralino. The processes which determine the neutralino relic density are described in Sec. 3.4. The detection possibilities
of a potential NMSSM neutralino as DM are discussed in (Sec. 3.5). We close this
chapter (Sec. 3.6) by examining possible ways to include non-zero neutrino masses and
the additional DM candidates that are introduced.
3.1 Motivation – The µ-problem of the MSSM
As we saw, the minimal extension of the SM, the MSSM, contains two Higgs SU(2)L
doublets Hu and Hd. The Lagrangian of the MSSM should contain a supersymmetric
mass term, µHuHd, for these two doublets. There are several reasons, which we will
subsequently review, that require the existence of such a term. On the other hand,
the fact that |µ| cannot be very large, actually it should be of the order of the EW
scale, brings back the problem of naturalness. A parameter of the model should be
much smaller than the “natural” scale (the GUT or the Planck scale) before the EW
symmetry breaking. This leads to the so-called µ-problem of the MSSM [146].
The reasons that such a term should exist in the Lagrangian of the MSSM are
mainly phenomenological. The doublets Hu and Hd are components of chiral superfields that also contain fermionic SU(2)L doublets. Their electrically charged components mix with the superpartners of the W± bosons, forming two charged Dirac
fermions, the charginos. The unsuccessful searches for charginos in LEP have excluded
charginos with masses almost up to its kinetic limit (∼ 104 GeV) [147]. Since the µ term
determines the mass of the charginos, µ cannot be zero and actually |µ| >∼ 100 GeV,
independently of the other free parameters of the model. Moreover, µ = 0 would result
in a Peccei-Quinn symmetry of the Higgs sector and an undesirable massless axion.
Finally, there is one more reason for µ 6= 0 related to the mass generation by the Higgs
mechanism. The term µHuHd will be accompanied by a soft SUSY breaking term
BµHuHd. This term is necessary so that both neutral components of Hu and Hd are
non-vanishing at the minimum of the potential.
The Higgs mechanism also requires that µ is not too large. In order to generate
the EW symmetry breaking, the Higgs potential has to be unstable at its origin Hu =
Hd = 0. Soft SUSY breaking terms for Hu and Hd of the order of the SUSY breaking
scale generate such an instability. However, the µ induced squared masses for Hu,
Hd are always positive and would destroy the instability in case they dominate the
negative soft mass terms.
The NMSSM is able to solve the µ-problem by dynamically generating the mass
µ. This is achieved by the introduction of an SU(2)L singlet scalar field S. When S
acquires a vev, a mass term for the Hu and Hd emerges with an effective mass µeff of
the correct order, as long as the vev is of the order of the SUSY breaking scale. This
can be obtained in a more “natural” way through the soft SUSY breaking terms.
3.2 The NMSSM Lagrangian 45
3.2 The NMSSM Lagrangian
All the necessary information for the Lagrangian of the NMSSM can be extracted from
the superpotential and the soft SUSY breaking Lagrangian, containing the soft gaugino and scalar masses, and the trilinear couplings. We begin with the superpotential,
writing all the interactions of the NMSSM superfields, which include the MSSM superfields and the additional gauge singlet chiral superfield2 Sb. Hence, the superpotential
reads
W = λSbHbu · Hbd +
1
3
κSb3
+ huQb · HbuUbc
R + hdHbd · QbDbc
R + heHbd · LbEbc
R.
(3.1)
The couplings to quarks and leptons have to be understood as 3 × 3 matrices and the
quark and lepton fields as vectors in the flavor space. The SU(2)L doublet superfields
are given (as in the MSSM) by
Qb =

UbL
DbL
!
, Lb =

νb
EbL
!
, Hbu =

Hb +
u
Hb0
u
!
, Hbd =

Hb0
d
Hb −
d
!
(3.2)
and the product of two doublets is given by, for example, Qb · Hbu = UbLHb0
u − Hb +
u DbL.
An important fact to note is that the superpotential given by (3.1) does not include all possible renormalizable couplings (which respect R-parity). The most general
superpotential would also include the terms
W ⊃ µHbu · Hbd +
1
2
µ
′Sb2 + ξF s, b (3.3)
with the first two terms corresponding to supersymmetric masses and the third one,
with ξF of dimension mass2
, to a tadpole term. However, the above dimensionful
parameters µ, µ
′ and ξF should be of the order of the SUSY breaking scale, a fact
that contradicts the motivation behind the NMSSM. Here, we omit these terms and
we will work with the scale invariant superpotential (3.1). The Lagrangian of a scale
invariant superpotential possesses an accidental Z3 symmetry, which corresponds to a
multiplication of all the components of all chiral fields by a phase ei2π/3
.
The corresponding soft SUSY breaking masses and couplings are
−Lsof t = m2
Hu
|Hu|
2 + m2
Hd
|Hd|
2 + m2
S
|S|
2
+ m2
Q|Q|
2 + m2
D|DR|
2 + m2
U
|UR|
2 + m2
L
|L|
2 + m2
E|ER|
2
+

huAuQ · HuU
c
R − hdAdQ · HdD
c
R − heAeL · HdE
c
R
+λAλHu · HdS +
1
3
κAκS
3 + h.c.

+
1
2
M1λ1λ1 +
1
2
M2λ
i
2λ
i
2 +
1
2
M3λ
a
3λ
a
3
,
(3.4)
2Here, the hatted capital letters denote chiral superfields, whereas the corresponding unhatted
ones indicate their complex scalar components.
46 The Next-to-Minimal Supersymmetric Standard Model
where we have also included the soft breaking masses for the gauginos. λ1 is the U(1)Y
gaugino (bino), λ
i
2 with i = 1, 2, 3 is the SU(2)L gaugino (winos) and, finally, the λ
a
3
with a = 1, . . . , 8 denotes the SU(3)c gaugino (gluinos).
The scalar potential, expressed by the so-called D and F terms, can be written
explicitly using the general formula
V =
1
2

D
aD
a + D
′2

+ F
⋆
i Fi
, (3.5)
where
D
a = g2Φ
∗
i T
a
ijΦj (3.6a)
D
′ =
1
2
g1YiΦ
∗
i Φi (3.6b)
Fi =
∂W
∂Φi
. (3.6c)
We remind that T
a are the SU(2)L generators and Yi the hypercharge of the scalar
field Φi
. The Yukawa interactions and fermion mass terms are given by the general
Lagrangian
LY ukawa = −
1
2
∂
2W
∂Φi∂Φj
ψiψj + h.c.
, (3.7)
using the superpotential (3.1). The two-component spinor ψi
is the superpartner of
the scalar Φi
.
3.2.1 Higgs sector
Using the general form of the scalar potential, the following Higgs potential is derived
VHiggs =

λ

H
+
u H
−
d − H
0
uH
0
d

+ κS2

2
+

m2
Hu + |λS|
2

H
0
u

2
+

H
+
u

2

+

m2
Hd + |λS|
2

H
0
d

2
+

H
−
d

2

+
1
8

g
2
1 + g
2
2

H
0
u

2
+

H
+
u

2
−

H
0
d

2
−

H
−
d

2
2
+
1
2
g
2
2

H
+
u H
0
d
⋆
+ H
0
uH
−
d
⋆

2
+ m2
S
|S|
2 +

λAλ

H
+
u H
−
d − H
0
uH
0
d

S +
1
3
κAκS
3 + h.c.

.
(3.8)
The neutral physical Higgs states are defined through the relations
H
0
u = vu +
1
√
2
(HuR + iHuI ), H0
d = vd +
1
√
2
(HdR + iHdI ),
S = s +
1
√
2
(SR + iSI ),
3.2.1 Higgs sector 47
where vu, vd and s are, respectively, the real vevs of Hu, Hd and S, which have to be
obtained from the minima of the scalar potential (3.8), after expanding the fields using
Eq. (3.9). We notice that when S acquires a vev, a term µeffHbu · Hbd appears in the
superpotential, with
µeff = λs, (3.10)
solving the µ-problem.
Therefore, the Higgs sector of the NMSSM is characterized by the seven parameters
λ, κ, m2
Hu
, m2
Hd
, m2
S
, Aλ and Aκ. One can express the three soft masses by the three
vevs using the minimization equations of the Higgs potential (3.8), which are given by
vu

m2
Hu + µ
2
eff + λ
2
v
2
d +
1
2
g
2

v
2
u − v
2
d

− vdµeff(Aλ + κs) = 0
vd

m2
Hd + µ
2
eff + λ
2
v
2
u +
1
2
g
2

v
2
d − v
2
u

− vuµeff(Aλ + κs) = 0
s

m2
S + κAκs + 2κ
2σ
2 + λ
2

v
2
u + v
2
d

− 2λκvuvd

− λAλvuvd = 0,
(3.11)
where we have defined
g
2 ≡
1
2

g
2
1 + g
2
2

. (3.12)
One can also define the β angle by
tan β =
vu
vd
. (3.13)
The Z boson mass is given by MZ = gv with v
2 = v
2
u + v
2
d ≃ (174 GeV)2
. Hence, with
MZ given, the set of parameters that describes the Higgs sector of the NMSSM can be
chosen to be the following
λ, κ, Aλ, Aκ, tan b and µeff. (3.14)
CP-even Higgs masses
One can obtain the Higgs mass matrices at tree level by expanding the Higgs potential
(3.8) around the vevs, using Eq. (3.9). We begin by writing3
the squared mass matrix
M2
S
of the scalar Higgses in the basis (HdR, HuR, SR):
M2
S =


g
2
v
2
d + µ tan βBeff (2λ
2 − g
2
) vuvd − µBeff 2λµvd − λ (Aλ + 2κs) vu
g
2
v
2
u +
µ
tan βBeff 2λµvu − λ (Aλ + 2κs) vd
λAλ
vuvd
s + κAκs + (2κs)
2

 ,
(3.15)
where we have defined Beff ≡ Aλ + κs (it plays the role of the B parameter of the
MSSM).
3For economy of space, we omit in this expression the subscript from µ
48 The Next-to-Minimal Supersymmetric Standard Model
Although an analytical diagonalization of the above 3 × 3 mass matrix is lengthy,
there is a crucial conclusion that comes from the approximate diagonalization of the
upper 2 × 2 submatrix. If it is rotated by an angle β, one of its diagonal elements
is M2
Z
(cos2 2β +
λ
2
g
2 sin2
2β) which is an upper bound for its lightest eigenvalue. The
first term is the same one as in the MSSM. The conclusion is that in the NMSSM
the lightest CP-even Higgs can be heavier than the corresponding of the MSSM, as
long as λ is large and tan β relatively small. Therefore, it is much easier to explain
the observed mass of the SM-like Higgs. However, λ is bounded from above in order
to avoid the appearance of the Landau pole below the GUT scale. Depending on the
other free parameters, λ should obey λ <∼ 0.7.
CP-odd Higgs masses
For the pseudoscalar case, the squared mass matrix in the basis (HdI , HuI , SI ) is
M2
P =


µeff (Aλ + κs) tan β µeff (Aλ + κs) λvu (Aλ − 2κs)
µeff
tan β
(Aλ + κs) λvd (Aλ − 2κs)
λ (Aλ + 4κs)
vuvd
s − 3κAκs

 . (3.16)
One eigenstate of this matrix corresponds to an unphysical massless Goldstone
boson G. In order to drop the Goldstone boson, we write the matrix in the basis
(A, G, SI ) by rotating the upper 2 × 2 submatrix by an angle β. After dropping the
massless mode, the 2 × 2 squared mass matrix turns out to be
M2
P =
2µeff
sin 2β
(Aλ + κs) λ (Aλ − 2κs) v
λ (Aλ + 4κs)
vuvd
s − 3Aκs
!
. (3.17)
Charged Higgs mass
The charged Higgs squared mass matrix is given, in the basis (H+
u
, H−
d
⋆
), by
M2
± =

µeff (Aλ + κs) + vuvd

1
2
g
2
2 − λ

cot β 1
1 tan β
!
, (3.18)
which contains one Goldstone boson and one physical mass eigenstate H± with eigenvalue
m2
± =
2µeff
sin 2β
(Aλ + κs) + v
2

1
2
g
2
2 − λ

. (3.19)
3.2.2 Sfermion sector
The mass matrix for the up-type squarks is given in the basis (ueR, ueL) by
Mu =

m2
u + h
2
u
v
2
u −
1
3
(v
2
u − v
2
d
) g
2
1 hu (Auvu − µeffvd)
hu (Auvu − µeffvd) m2
Q + h
2
u
v
2
u +
1
12 (v
2
u − v
2
d
) (g
2
1 − 3g
2
2
)
!
, (3.20)
3.2.3 Gaugino and higgsino sector 49
whereas for down-type squarks the mass matrix reads in the basis (deR, deL)
Md =

m2
d + h
2
d
v
2
d −
1
6
(v
2
u − v
2
d
) g
2
1 hd (Advd − µeffvu)
hd (Advd − µeffvu) m2
Q + h
2
d
v
2
d +
1
12 (v
2
u − v
2
d
) (g
2
1 − 3g
2
2
)
!
. (3.21)
The off-diagonal terms are proportional to the Yukawa coupling hu for the up-type
squarks and hd for the down-type ones. Therefore, the two lightest generations remain
approximately unmixed. For the third generation, the mass matrices are diagonalized
by a rotation by an angle θT and θB, respectively, for the stop and sbottom. The mass
eigenstates are, then, given by
et1 = cos θT
etL + sin θT
etR, et2 = cos θT
etL − sin θT
etR, (3.22)
eb1 = cos θB
ebL + sin θB
ebR, eb2 = cos θB
ebL − sin θB
ebR. (3.23)
In the slepton sector, for a similar reason, only the left- and right-handed staus are
mixed and their mass matrix
Mτ =

m2
E3 + h
2
τ
v
2
d −
1
2
(v
2
u − v
2
d
) g
2
1 hτ (Aτ vd − µeffvu)
hτ (Aτ vd − µeffvu) m2
L3 + h
2
τ
v
2
d −
1
4
(v
2
u − v
2
d
) (g
2
1 − g
2
2
)
!
(3.24)
is diagonalized after a rotation by an angle θτ . Their mass eigenstates are given by
τe1 = cos θτ τeL + sin θτ τeR, τe2 = cos θτ τeL − sin θτ τeR. (3.25)
Finally, the sneutrino masses are
mνe = m2
L −
1
4

v
2
u − v
2
d
g
2
1 + g
2
2

. (3.26)
3.2.3 Gaugino and higgsino sector
The gauginos λ1 and λ
3
2 mix with the neutral higgsinos ψ
0
d
, ψ
0
u
and ψS to form neutral
particles, the neutralinos. The 5 × 5 mass matrix of the neutralinos is written in the
basis
(−iλ1, −iλ3
2
, ψ0
d
, ψ0
u
, ψS) ≡ (B, e W , f He0
d
, He0
u
, Se) (3.27)
as
M0 =


M1 0 − √
1
2
g1vd √
1
2
g1vu 0
M2 √
1
2
g2vd − √
1
2
g2vu 0
0 −µeff −λvu
0 −λvd
2κs


. (3.28)
The mass matrix (3.28) is diagonalized by an orthogonal matrix Nij . The mass eigenstates of the neutralinos are usually denoted by χ
0
i
, with i = 1, . . . , 5, with increasing
masses (i = 1 corresponds to the lightest neutralino). These are given by
χ
0
i = Ni1Be + Ni2Wf + Ni3He0
d + Ni4He0
u + Ni5S. e (3.2
50 The Next-to-Minimal Supersymmetric Standard Model
We use the convention of a real matrix Nij , so that the physical masses mχ
0
i
are real,
but not necessarily positive.
In the charged sector, the SU(2)L charged gauginos λ
− = √
1
2
(λ
1
2 + iλ2
2
), λ
+ =
√
1
2
(λ
1
2 − iλ2
2
) mix with the charged higgsinos ψ
−
d
and ψ
+
u
, forming the charginos ψ
±:
ψ
± =

−iλ±
ψ
±
u
!
. (3.30)
The chargino mass matrix in the basis (ψ
−, ψ+) is
M± =

M2 g2vu
g2vd µeff !
. (3.31)
Since it is not symmetric, the diagonalization requires different rotations of ψ
− and
ψ
+. We denote these rotations by U and V , respectively, so that the mass eigenstates
are obtained by
χ
− = Uψ−, χ+ = V ψ+. (3.32)
3.3 DM Candidates in the NMSSM
Let us first review the characteristics that a DM candidate particle should have. First,
it should be massive in order to account for the missing mass in the galaxies. Second,
it must be electrically and color neutral. Otherwise, it would have condensed with
baryonic matter, forming anomalous heavy isotopes. Such isotopes are absent in nature. Finally, it should be stable and interact only weakly, in order to fit the observed
relic density.
In the NMSSM there are two possible candidates. Both can be stable particles if
they are the LSPs of the supersymmetric spectrum. The first one is the sneutrino (see
[148,149] for early discussions on MSSM sneutrino LSP). However, although sneutrinos
are WIMPs, their large coupling to the Z bosons result in a too large annihilation cross
section. Hence, if they were the DM particles, their relic density would have been very
small compared to the observed value. Exceptions are very massive sneutrinos, heavier
than about 5 TeV [150]. Furthermore, the same coupling result in a large scattering
cross section off the nuclei of the detectors. Therefore, sneutrinos are also excluded by
direct detection experiments.
The other possibility is the lightest neutralino. Neutralinos fulfill successfully, at
least in principle, all the requirements for a DM candidate. However, the resulting
relic density, although weakly interacting, may vary over many orders of magnitude as
a function of the free parameters of the theory. In the next sections we will investigate
further the properties of the lightest neutralino as the DM particle. We begin by
studying its annihilation that determines the DM relic density.
3.4 Neutralino relic density 51
3.4 Neutralino relic density
We remind that the neutralinos are mixed states composed of bino, wino, higgsinos
and the singlino. The exact content of the lightest neutralino determines its pair
annihilation channels and, therefore, its relic density (for detailed analyses, we refer
to [151–154]). Subsequently, we will briefly describe the neutralino pair annihilation
in various scenarios. We classify these scenarios with respect to the lightest neutralino
content.
Before we proceed, we should mention another mechanism that affects the neutralino LSP relic density. If there is a supersymmetric particle with mass close to the
LSP (but always larger), it would be abundant during the freeze-out and LSP coannihilations with this particle would contribute to the total annihilation cross section.
This particle, which is the Next-to-Lightest Supersymmetric Particle (NLSP), is most
commonly a stau or a stop. In the above sense, coannihilations refer not only to the
LSP–NLSP coannihilations, but also to the NLSP–NLSP annihilations, since the latter
reduce the number density of the NLSPs [155].
• Bino-like LSP
In principle, if the lightest neutralino is mostly bino-like, the total annihilation
cross section is expected to be small. Therefore, a bino-like neutralino LSP would
have been overabundant. The reason for this is that there is only one available
annihilation channel via t-channel sfermion exchange, since all couplings to gauge
bosons require a higgsino component. The cross section is even more reduced
when the sfermion mass is large.
However, there are still two ways to achieve the correct relic density. The first one
is using the coannihilation effect: if there is a sfermion with a mass slightly larger
(some GeV) than the LSP mass, their coannihilations can be proved to reduce
efficiently the relic density to the desired value. The second one concerns a binolike LSP, with a very small but non-negligible higgsino component. In this case,
if in addition the lightest CP-odd Higgs A1 is light enough, the annihilation to a
pair A1A1 (through an s-channel CP-even Higgs Hi exchange) can be enhanced
via Hi resonances. In this scenario a fine-tuning of the masses is necessary.
• Higgsino-like LSP
A mostly higgsino LSP is as well problematic. The strong couplings of the higgsinos to the gauge bosons lead to very large annihilation cross section. Therefore,
a possible higgsino LSP would have a very small relic density.
• Mixed bino–higgsino LSP
In this case, as it was probably expected, one can easily fit the relic density to
the observed value. This kind of LSP annihilates to W+W−, ZZ, W±H∓, ZHi
,
HiAj
, b
¯b and τ
+τ
− through s-channel Z or Higgs boson exchange or t-channel
neutralino or chargino exchange. The last two channels are the dominant ones
when the Higgs coupling to down-type fermions is enhanced, which occurs more
commonly in the regime of relatively large tan β. The annihilation channel to a
52 The Next-to-Minimal Supersymmetric Standard Model
pair of top quarks also contributes to the total cross section, if it is kinematically
allowed. However, in order to achieve the correct relic density, the higgsino
component cannot be very large.
• Singlino-like LSP
Since a mostly singlino LSP has small couplings to SM particles, the resulting relic
density is expected to be large. However, there are some annihilation channels
that can be enhanced in order to reduce the relic density. These include the
s-channel (scalar or pseudoscalar) Higgs exchange and the t-channel neutralino
exchange.
For the former, any Higgs with sufficient large singlet component gives an important contribution to the cross section, increasing with the parameter κ (since
the singlino-singlino-singlet coupling is proportional to κ). Concerning the latter
annihilation, in order to enhance it, one needs large values of the parameter λ.
In this case, the neutralino-neutralino-singlet coupling, which is proportional to
λ, is large and the annihilation proceeds giving a pair of scalar HsHs or a pair
of pseudoscalar AsAs singlet like Higgs.
As in the case of bino-like LSP, one can also use the effect of s-channel resonances
or coannihilations. In the latter case, an efficient NLSP can be the neutralino
χ
0
2
or the lightest stau τe1. In the case that the neutralino NLSP is higgsinolike, the LSP-NLSP coannihilation through a (doublet-like) Higgs exchange can
be proved very efficient. A stau NLSP reduces the relic density through NLSPNLSP annihilations, which is the only possibility in the case that both parameters
κ and λ are small. We refer to [156,157] for further discussion on this possibility.
Assuming universality conditions the wino mass M2 has to be larger than the bino
mass M1 (M2 ∼ 2M1). This is the reason that we have not considered a wino-like LSP.
3.5 Detection of neutralino DM
3.5.1 Direct detection
Since neutralinos are Majorana fermions, the effective Lagrangian describing their
elastic scattering with a quark in a nucleon can be written, in the Dirac fermion
notation, as [158]
Leff = a
SI
i χ¯
0
1χ
0
1
q¯iqi + a
SD
i χ¯
0
1γ5γµχ
0
1
q¯iγ5γ
µ
qi
, (3.33)
with i = u, d corresponding to up- and down-type quarks, respectively. The Lagrangian has to be understood as summing over the quark generations.
In this expression, we have omitted terms containing the operator ψγ¯
5ψ or a combination of ψγ¯
5γµψ and ψγ¯
µψ (with ψ = χ, q). This is a well qualified assumption:
Due to the small velocity of WIMPs, the momentum transfer ~q is very small compared
3.5.1 Direct detection 53
to the reduced mass of the WIMP-nucleus system. In the extreme limit of zero momentum transfer, the above operators vanish4
. Hence, we are left with the Lagrangian
(3.33) consisting of two terms, the first one corresponding to spin-independent (SI)
interactions and the second to spin-dependent (SD) ones. In the following, we will
focus again only to SI scattering, since the detector sensitivity to SD scattering is low,
as it has been already mentioned in Sec. 1.5.1.
The SI cross section for the neutralino-nucleus scattering can be written as [158]
(see, also, [159])
σ
SI
tot =
4m2
r
π
[Zfp + (A − Z)fn]
2
. (3.34)
mr is the neutralino-nucleus reduced mass mr =
mχmN
mχ+mN
, and Z, A are the atomic and
the nucleon number, respectively. It is more common, however, to use an expression
for the cross section normalized to the nucleon. In this case, on has for the neutralinoproton scattering
σ
SI
p =
4
π

mpmχ
0
1
mp + mχ
0
1
!2
f
2
p ≃
4m2
χ
0
1
π
f
2
p
, (3.35)
with a similar expression for the neutron.
The form factor fp is related to the couplings a to quarks through the expression
(omitting the “SI” superscripts)
fp
mp
=
X
q=u,d,s
f
p
T q
aq
mq
+
2
27
fT G X
q=c,b,t
aq
mq
. (3.36)
A similar expression may be obtained for the neutron form factor fn, by the replacement
p → n in the previous expression (henceforth, we focus to neutralino-proton scattering).
The parameters fT q are defined by the quark mass matrix elements
hp| mqqq¯ |pi = mpfT q, (3.37)
which corresponds to the contribution of the quark q to the proton mass and the
parameter fT G is related to them by
fT G = 1 −
X
q=u,d,s
fT q. (3.38)
The above parameters can be obtained by the following quantities
σπN =
1
2
(mu + md)(Bu + Bd) and σ0 =
1
2
(mu + md)(Bu + Bd − 2Bs,) (3.39)
with Bq = hN| qq¯ |Ni, which are obtained by chiral perturbation theory [160] or by
lattice simulations. Unfortunately, the uncertainties on the values of these quantities
are large (see [161], for more recent values and error bars).
4While there are possible circumstances in which the operators of (3.33) are also suppressed and,
therefore, comparable to the operators omitted, they are not phenomenologically interesting.
54 The Next-to-Minimal Supersymmetric Standard Model
χ
0
1
χ
0
1
χ
0
1 χ
0
1
qe
q q
q q
Hi
Figure 3.1: Feynman diagrams contributing to the elastic neutralino-quark scalar scattering amplitude at tree level.
The SI neutralino-nucleon interactions arise from t-channel Higgs exchange and
s-channel squark exchange at tree level (see Fig. 3.1), with one-loop contributions from
neutralino-gluon interactions. In practice, the s-channel Higgs exchange contribution
to the scattering amplitude dominates, especially due to the large masses of squarks.
In this case, the effective couplings a are given by
a
SI
d =
X
3
i=1
1
m2
Hi
C
1
i Cχ
0
1χ
0
1Hi
, aSI
u =
X
3
i=1
1
m2
Hi
C
2
i Cχ
0
1χ
0
1Hi
. (3.40)
C
1
i
and C
2
i
are the Higgs Hi couplings to down- and up-type quarks, respectively, given
by
C
1
i =
g2md
2MW cos β
Si1, C2
i =
g2mu
2MW sin β
Si2, (3.41)
with S the mixing matrix of the CP-even Higgs mass eigenstates and md, mu the
corresponding quark mass. We see from Eqs. (3.36) and (3.41) that the final cross
section (3.35) is independent of each quark mass. We write for completeness the
neutralino-neutralino-Higgs coupling Cχ
0
1χ
0
1Hi
:
Cχ
0
1χ
0
1Hi =
√
2λ (Si1N14N15 + Si2N13N15 + Si3N13N14) −
√
2κSi3N
2
15
+ g1 (Si1N11N13 − Si2N11N14) − g2 (Si1N12N13 − Si2N12N14), (3.42)
with N the neutralino mixing matrix given in (3.29).
The resulting cross section is proportional to m−4
Hi
. In the NMSSM, it is possible
for the lightest scalar Higgs eigenstate to be quite light, evading detection due to its
singlet nature. This scenario can give rise to large values of SI scattering cross section,
provided that the doublet components of th

[Demi Habile]

Phenom ´ enologie du Higgs aupr ´ es des collisionneurs hadroniques : `
du Modele Standard a la Supersym etrie. ´
R´esum´e
Cette these, conduite dans le contexte de la recherche du boson de Higgs, derniere pi`ece
manquante du m´ecanisme de brisure de la sym´etrie ´electrofaible et qui est une des plus importantes recherches aupr`es des collisionneurs hadroniques actuels, traite de la pé´enom´enologie
de ce boson a la fois dans le Modele Standard (SM) et dans son extension supersym´etrique
minimale (MSSM). Apres un r´esum´e de ce qui constitue le Modele Standard dans une premi`ere partie, nous pr´esenterons nos pr´edictions pour la section efficace inclusive de production
du boson de Higgs dans ses principaux canaux de production aupr`es des deux collisionneurs
hadroniques actuels que sont le Tevatron au Fermilab et le grand collisionneur de hadrons
(LHC) au CERN, en commen¸cant par le cas du Mod`ele Standard. Le principal r´esultat pr´esent´e est l’´etude la plus exhaustive possible des diff´erentes sources d’incertitudes th´eoriques
qui p`esent sur le calcul : les incertitudes d’´echelles vues comme une mesure de notre ignorance
des termes d’ordre sup´erieur dans un calcul perturbatif `a un ordre donn´e, les incertitudes reli´ees aux fonctions de distribution de partons dans le proton/l’anti–proton (PDF) ainsi que
les incertitudes reli´ees `a la valeur de la constante de couplage fort, et enfin les incertitudes
provenant de l’utilisation d’une th´eorie effective qui simplifie le calcul des ordres sup´erieurs
dans la section efficace de production. Dans un second temps nous ´etudierons les rapports
de branchement de la d´esint´egration du boson de Higgs en donnant ici aussi les incertitudes
th´eoriques qui p`esent sur le calcul. Nous poursuivrons par la combinaison des sections efficaces
de production avec le calcul portant sur la d´esint´egration du boson de Higgs, pour un canal
sp´ecifique, montrant quelles en sont les cons´equences int´eressantes sur l’incertitude th´eorique
totale. Ceci nous ameneraa un r´esultat significatif de la th`ese qui est la comparaison avec l’exp´erience et notamment les r´esultats des recherches du boson de Higgs au Tevatron. Nous irons
ensuite au-dela du Modele Standard dans une troisieme partie ou nous donnerons quelques
ingr´edients sur la supersym´etrie et sa mise en application dans le MSSM o`u nous avons cinq
bosons de Higgs, puis nous aborderons leur production et d´esint´egration en se focalisant sur
les deux canaux de production principaux par fusion de gluon et fusion de quarks b. Nous
pr´esenterons les r´esultats significatifs quant `a la comparaison avec aussi bien le Tevatron que
les r´esultats tr`es r´ecents d’ATLAS et CMS au LHC qui nous permettront d’analyser l’impact
de ces incertitudes sur l’espace des param`etres du MSSM, sans oublier de mentionner quelques
bruits de fond du signal des bosons de Higgs. Tout ceci va nous permettre de mettre en avant
le deuxieme r´esultat tres important de la th`ese, ouvrant une nouvelle voie de recherche pour
le boson de Higgs standard au LHC. La derni`ere partie sera consacr´ee aux perspectives de
ce travail et notamment donnera quelques r´esultats pr´eliminaires dans le cadre d’une ´etude
exclusive, d’un int´erˆet primordial pour les exp´erimentateurs.
Mots-clefs : Mod`ele Standard, Higgs, Supersym´etrie, Chromodynamique quantique, incertitudes th´eoriques.

Abstract
This thesis has been conducted in the context of one of the utmost important searches at
current hadron colliders, that is the search for the Higgs boson, the remnant of the electroweak
symmetry breaking. We wish to study the phenomenology of the Higgs boson in both the
Standard Model (SM) framework and its minimal Supersymmetric extension (MSSM). After
a review of the Standard Model in a first part and of the key reasons and ingredients for
the supersymmetry in general and the MSSM in particular in a third part, we will present the
calculation of the inclusive production cross sections of the Higgs boson in the main channels at
the two current hadron colliders that are the Fermilab Tevatron collider and the CERN Large
Hadron Collider (LHC), starting by the SM case in the second part and presenting the MSSM
results, where we have five Higgs bosons and focusing on the two main production channels that
are the gluon gluon fusion and the bottom quarks fusion, in the fourth part. The main output
of this calculation is the extensive study of the various theoretical uncertainties that affect the
predictions: the scale uncertainties which probe our ignorance of the higher–order terms in a
fixed order perturbative calculation, the parton distribution functions (PDF) uncertainties and
its related uncertainties from the value of the strong coupling constant, and the uncertainties
coming from the use of an effective field theory to simplify the hard calculation. We then
move on to the study of the Higgs decay branching ratios which are also affected by diverse
uncertainties. We will present the combination of the production cross sections and decay
branching fractions in some specific cases which will show interesting consequences on the
total theoretical uncertainties. We move on to present the results confronted to experiments
and show that the theoretical uncertainties have a significant impact on the inferred limits
either in the SM search for the Higgs boson or on the MSSM parameter space, including some
assessments about SM backgrounds to the Higgs production and how they are affected by
theoretical uncertainties. One significant result will also come out of the MSSM analysis and
open a novel strategy search for the Standard Higgs boson at the LHC. We finally present in
the last part some preliminary results of this study in the case of exclusive production which
is of utmost interest for the experimentalists.
Keywords : Standard Model, Higgs, Supersymmetry, QCD, theoretical uncertainties.

Remerciements
Trois ann´ees ont pass´e depuis que j’ai pouss´e pour la premi`ere fois les portes du Laboratoire de Physique Th´eorique d’Orsay, chaleureusement accueilli par son directeur Henk
Hilhorst que je remercie beaucoup. Trois ann´ees d’une activit´e intense, aussi bien dans
mes recherches scientifiques au LPT et au CERN, dans le groupe de physique th´eorique,
ou j’ai pass´e quelques moisa partir de la seconde ann´ee, que dans mes activit´es hors
recherche au sein de l’universit´e Paris-Sud 11. J’ai appris beaucoup et rencontr´e un certain nombre de personnes dont je vais me rappeler pour longtemps, si je ne les ´enum`ere
pas ici qu’elles veuillent bien me pardonner cela ne signifie pas que je les ai pour autant
oubli´ees.
Tout ceci n’aurait pu se faire sans les encouragements, les conseils et les discussions passionn´ees avec Abdelhak Djouadi, mon directeur de th`ese qui a guid´e ainsi mes
premiers pas de professionnel dans ma carri`ere de physicien th´eoricien des particules
´el´ementaires. Je l’en remercie profond´ement et j’esp`ere qu’il aura appr´eci´e notre collaboration autant que moi, aussi bien lors de notre travail qu’en dehors.
Je voudrais aussi remercier Rohini Godbole avec qui j’ai collabor´e sur la passionnante
physique du Higgs au Tevatron. Je ne peux non plus oublier Ana Teixeira pour son
soutien constant et les nombreuses discussions passionnantes aussi bien scientifiques que
personnelles que nous avons eues ensemble. Ma premi`ere ann´ee en tant que doctorant
lui doit beaucoup.
Je remercie aussi tous les membres de mon jury de th`ese et en particulier mes deux
rapporteurs qui m’ont certainement maudit d’avoir ´ecrit autant, non seulement pour le
temps qu’ils auront pris pour assister a ma soutenance et lire ma these, mais aussi pour
toutes leurs judicieuses remarques et questions.
Aussi bien le LPT que le CERN se sont r´ev´el´es des lieux tr`es enrichissants pour
le d´ebut de ma carri`ere scientifique. Je voudrais profiter tout d’abord de ces quelques
mots pour remercier les ´equipes administratives des deux laboratoires pour leur aide au
jour le jour, toujours avec le sourire, et pour toute leur aide dans mes divers voyages
scientifiques. Je remercie aussi tous les chercheurs de ces deux laboratoires pour toutes les
discussions que j’ai eues et qui m’ont beaucoup appris. Je pense tout particuli`erement
a Asmˆaa Abada eta Gr´egory Moreau d’un cˆot´e, `a G´eraldine Servant et Christophe
Grojean qui m’a invit´e `a venir au CERN, de l’autre. Je ne peux bien sur pas oublier les
doctorants et jeunes docteurs du groupe de physique th´eorique du CERN, Sandeepan
Gupta, Pantelis Tziveloglou et tous les autres, ainsi que L´ea Gauthier, doctorante au
CEA, que j’ai rencontr´ee au CERN : les magnifiques randonn´ees autour de Gen`eve
que nous avons faites ont ´et´e salutaires. Enfin je remercie aussi tous mes camarades
doctorants et jeunes docteurs du SINJE `a Orsay pour tous les merveilleux moments que
nous avons pass´es et toutes les discussions passionn´ees et passionnnantes, je ne vous cite
pas tous mais le cœur y est. Je pense quand mˆeme tout particulierementa mes camarades
ayant partag´e mon bureau et bien plus, Adrien Besse et C´edric Weiland, mais aussi `a
Guillaume Toucas, Blaise Gout´eraux et Andreas Goudelis. J´er´emie Quevillon qui va
prendre ma succession aupres de mon directeur de these n’est pas non plus oubli´e. Mes
amis de Toulouse eux aussi sont loin d’avoir ´et´e oubli´es et ont fortement contribu´e non
seulement a rendre exceptionnel mon stage de Master 2 mais aussi ma premiere ann´ee
de these, de loin en loin : mercia Ludovic Arnaud, Gaspard Bousquet, Arnaud Ralko,
Cl´ement Touya, Fabien Trousselet, mais aussi mes deux tuteurs Nicolas Destainville et
Manoel Manghi.
Je ne peux terminer sans exprimer ma profonde gratitude a ma famille eta mes amis
de longue date, qui se reconnaˆıtront. Anne, Charles, Elise, Gaetan, Lionel, Mathieu,
Matthieu, Patrick, Pierre, Rayna, Sophie, Yiting et tous ceux que je n’ai pas cit´es mais
qui sont dans mes pens´ees, ces mots sont pour vous ! Le mot de la fin revient `a ma
fianc´ee, Camille : sans ton profond amour et ton soutien constant, ces trois derni`eres
ann´ees auraient ´et´e bien diff´erentes, et certainement pas aussi f´econdes. Merci pour tout.
Acknowledgments
Three years have now passed since my first steps in the Laboratoire de Physique
Th´eorique at Orsay, where I have been warmly welcomed by its director Henk Hilhorst
that I thank a lot. They have been very intense, both in the laboratory and at the CERN
Theory Group in Geneva, where I spent some months starting from the second year. I
have learnt much, either within these labs or outside, encountered many people that I
will remember for a long time. If some of you are not cited in these acknowledgments,
please be kind with me: that does not mean I have forgotten you.
This would have never been possible without the constant encouragement, advices
and fruitful discussions with Dr. Abdelhak Djouadi, my thesis advisor, who guided my
first steps in theoretical particle physics research. I hope he got as much great time as
I had working with him and more than that.
I also would like to thank Pr. Rohini Godbole whom I worked with from time to
time on Higgs physics at the Tevatron. I cannot also forget Dr. Ana Teixeira for her
constant support and all the great discussions on various topics we had together. My
first year as a PhD candidate was scientifically exciting thanks to her.
I am very grateful to all the members in the jury for my defence, for the time they
would took and the useful comments. In particular I would like to thank my two referees
who certainly have cursed me for the length of the thesis.
The LPT environnement as well as the CERN Theory Group have been proven to be
very fruitful environnements for the beginning of my career. I then would like to thank
the administrative staff from both laboratories for their constant help in day–to–day life
and support when I had to travel for various workshops, conferences or seminars. I would
like to thank all the members of these two groups for the very passionate discussions
we had and where I have learnt a lot. I dedicate special thanks to Asmˆaa Abada and
Gr´egory Moreau on the one side, G´eraldine Servant and also Christophe Grojean, who
invited me to come by, on the other side. I cannot forget the PhD candidates and
post-doctoral researchers from the CERN Theory Group, Sandeepan Gupta, Pantelis
Tziveloglou and all the others, not to forget L´ea Gauthier, who is a PhD candidate
at the CEA and was at CERN at that time: the hiking we did in the Jura and Alps
around Geneva were great. I also would like to thank all my SINJE fellows at the
LPT, with whom I had so many great time and passionate discussions; you are not all
cited but I do not forget you. I dedicate special thanks to my office (and more than
office) friends Adrien Besse and C´edric Weiland, and also to Blaise Gout´eraux, Andreas
Goudelis and Guillaume Toucas. The next PhD candidate, J´er´emie Quevillon, who will
follow my path, is also thanked for the discussions we had. I finally cannot forget my
friends from Toulouse, where I did my Master 2 internship and whom I collaborated with
during my first PhD thesis year from time to time: many thanks to Ludovic Arnaud,
Gaspard Bousquet, Arnaud Ralko, Cl´ement Touya, Fabien Trousselet, and also to my
two internship advisors Nicolas Destainville and Manoel Manghi.
I now end this aknowledgments by expressing my deep gratitude and love to my family and long–time friends who will recognize themselves. Anne, Charles, Elise, Gaetan,
Lionel, Mathieu, Matthieu, Patrick, Pierre, Rayna, Sophie, Yiting and all the others,
these words are for you! The last word is for Camille, my fiancee: without your deep
love and constant support these three years would have been without doubts completely
different and not as fruitful.

Contents
Introduction 1
I A brief review of the Standard Model of particle physics 5
1 Symmetry principles and the zoology of the Standard Model 6
1.1 A brief history of the Standard Model . . . . . . . . . . . . . . . . . . . 6
1.2 Gauge symmetries, quarks and leptons . . . . . . . . . . . . . . . . . . . 12
2 The Brout–Englert–Higgs mechanism 16
2.1 Why do we need the electroweak symmetry breaking? . . . . . . . . . . . 16
2.2 The spontaneous electroweak symmetry breaking . . . . . . . . . . . . . 19
II SM Higgs production and decay at hadron colliders 27
3 Where can the SM Higgs boson be hiding? 29
3.1 Theoretical bounds on the Higgs mass . . . . . . . . . . . . . . . . . . . 29
3.2 Experimental bounds on the Higgs mass . . . . . . . . . . . . . . . . . . 36
4 Higgs production at the Tevatron 43
4.1 The main production channels . . . . . . . . . . . . . . . . . . . . . . . 47
4.2 Scale variation and higher order terms . . . . . . . . . . . . . . . . . . . 58
4.3 The PDF puzzle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.4 EFT and its uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.5 Combination and total uncertainty . . . . . . . . . . . . . . . . . . . . . 81
4.6 Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.A Appendix: analytical expressions for µR–NNLO terms in gg → H . . . . 90
5 Higgs production at the LHC 92
5.1 The main channel at the lHC . . . . . . . . . . . . . . . . . . . . . . . . 93
5.2 The scale uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.3 The PDF+αS uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.4 EFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.5 Total uncertainy at 7 TeV . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.6 LHC results at different center–of–mass energies . . . . . . . . . . . . . 110
5.7 Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6 Higgs decay and the implications for Higgs searches 116
6.1 Important channels for experimental search . . . . . . . . . . . . . . . . 116
6.2 Uncertainties on the branching ratios . . . . . . . . . . . . . . . . . . . . 121
6.3 Combination at the Tevatron . . . . . . . . . . . . . . . . . . . . . . . . 125
6.4 Combination at the LHC . . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.5 The Tevatron exclusion limit . . . . . . . . . . . . . . . . . . . . . . . . 129
6.6 Summary of the results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
III The Minimal Supersymmetric extension of the Standard
Model 137
7 Why Supersymmetry is appealing 138
7.1 The hierarchy problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
7.2 Coupling constants convergence at high energies . . . . . . . . . . . . . 140
7.3 SUSY and Dark Matter searches . . . . . . . . . . . . . . . . . . . . . . 142
8 Formal SUSY aspects 145
8.1 SUSY Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
8.2 Superspace, superfields and superpotential . . . . . . . . . . . . . . . . . 149
8.3 Soft SUSY breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
9 The Minimal Supersymmetric Standard Model 156
9.1 Fields content: Higgs and SUSY sectors of the MSSM . . . . . . . . . . 156
9.2 The Higgs sector and the number of Higgs doublets . . . . . . . . . . . . 161
9.3 The MSSM is not the end of the story . . . . . . . . . . . . . . . . . . . 168
IV MSSM Higgs(es) production and decay 171
10 The MSSM Higgs sector at hadron colliders 173
10.1 SUSY corrections to Higgs couplings to fermions . . . . . . . . . . . . . 173
10.2 Model independence of the results . . . . . . . . . . . . . . . . . . . . . 177
11 MSSM Higgs production at the Tevatron 180
11.1 Gluon–gluon fusion and bottom quarks fusion . . . . . . . . . . . . . . . 181
11.2 The scale uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
11.3 The PDF and αS uncertainties . . . . . . . . . . . . . . . . . . . . . . . 186
11.4 The b–quark mass uncertainty . . . . . . . . . . . . . . . . . . . . . . . 187
11.5 Summary and combination of the different sources of uncertainties . . . . 190
12 MSSM Higgs production at the LHC 192
12.1 Gluon–gluon fusion and bottom quarks fusion channels . . . . . . . . . . 192
12.2 The scale uncertainty at the lHC . . . . . . . . . . . . . . . . . . . . . . 194
12.3 The PDF and αS uncertainties at the lHC . . . . . . . . . . . . . . . . . 195
12.4 The b–quark mass issue . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
12.5 Combination and total uncertainty . . . . . . . . . . . . . . . . . . . . . 198
12.6 The case of the charged Higgs production in association with top quark
at the LHC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
13 Higgs→ τ τ channel and limits on the MSSM parameter space 209
13.1 The main MSSM Higgs branching ratios . . . . . . . . . . . . . . . . . . 209
13.2 Combination of production cross section and Higgs→ τ τ decay . . . . . 212
13.3 Impact of the theoretical uncertainties on the limits on the MSSM parameter space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
13.4 Consequences on the SM H → τ τ search at the LHC . . . . . . . . . . . 224
13.5 Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
V Perspectives 229
14 Exclusive study of the gluon–gluon fusion channel 230
14.1 Exclusive SM Higgs production . . . . . . . . . . . . . . . . . . . . . . . 231
14.2 SM Backgrounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
Conclusion 236
A Appendix : Synopsis 240
A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
A.2 Production et d´esint´egration du boson de Higgs du Mod`ele Standard . . 244
A.3 Le Mod`ele Standard Supersym´etrique Minimal (MSSM) . . . . . . . . . . 252
A.4 Production et d´esint´egration des bosons de Higgs supersym´etriques . . . 256
A.5 Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
References 263
List of Figures
1 Feynman diagrams at the Born level for the process e
+e
− → W+W− . . 17
2 Higgs potential in the case of a real scalar field, depending on the sign of
the mass term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3 Higgs potential in the case of the SM . . . . . . . . . . . . . . . . . . . . 21
4 Tree–level SM Higgs boson couplings to gauge bosons and fermions . . . 25
5 One–loop SM Higgs boson couplings to the photons and the gluons . . . 25
6 Feynman diagrams up to one–loop correction for the Higgs self–coupling 34
7 Theoretical bounds on the Higgs mass in function of the scale of new
physics beyond the SM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
8 Electroweak precision data . . . . . . . . . . . . . . . . . . . . . . . . . . 39
9 Indirect constraints on the SM Higgs boson mass . . . . . . . . . . . . . 40
10 95%CL exclusion limit on the SM Higgs boson mass at the LEP collider . 41
11 95%CL exclusion limit on the SM Higgs boson mass at the Tevatron collider 43
12 Feynman diagrams of the four main SM Higgs production channel . . . . 49
13 Some Feynman diagrams for NLO SM gg → H production . . . . . . . . 50
14 Some Feynman diagrams for NNLO SM gg → H production . . . . . . . 51
15 NLO QCD corrections to pp¯ → V
∗
. . . . . . . . . . . . . . . . . . . . . 55
16 NNLO QCD corrections to pp¯ → V
∗
. . . . . . . . . . . . . . . . . . . . 56
17 Total cross sections for Higgs production at the Tevatron in the four main
channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
18 Scale variation in the gg → H process at the Tevatron . . . . . . . . . . 62
19 Scale variation in the pp¯ → W H process at the Tevatron . . . . . . . . . 67
20 Comparison between different PDFs sets in gg → H at the Tevatron
using CTEQ/ABKM/MSTW PDF sets for 90%CL uncertainties and
MSTW/ABKM/HERA/JR for central predictions comparison . . . . . . 70
21 Comparison between MSTW PDFs set and ABKM PDFs set predictions
in gg → H channel at the Tevatron as for the uncertainties related to
PDF+∆αs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
22 The total PDF, PDF+∆expαs and PDF+∆exp+thαs uncertainties in gg →
H at the Tevatron using the MSTW PDFs set. . . . . . . . . . . . . . . . 75
23 Central predictions for NNLO pp¯ → W H at the Tevatron using the
MSTW, CTEQ and ABKM PDFs sets, together with their 90% CL PDF
uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
24 Comparison between MSTW PDFs set and ABKM PDFs set predictions
in pp¯ → W H channel at the Tevatron as for the uncertainties related to
PDF+∆αs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
25 b–loop uncertainty in gg → H at the Tevatron . . . . . . . . . . . . . . . 79
26 EW uncertainties in gg → H at the Tevatron . . . . . . . . . . . . . . . . 81
27 Production cross sections for gg → H at the Tevatron together with the
total theoretical uncertainties . . . . . . . . . . . . . . . . . . . . . . . . 85
28 Production cross sections for pp¯ → W H and pp¯ → ZH at the Tevatron
together with the total theoretical uncertainties . . . . . . . . . . . . . . 88
29 Total cross sections for SM Higgs production at the lHC . . . . . . . . . 95
30 Scale uncertainty at the lHC in gg → H at NNLO . . . . . . . . . . . . . 98
31 PDF and ∆exp,thαs uncertainties in gg → H at the lHC . . . . . . . . . . 99
32 Comparison between the predictions given by the four NNLO PDF sets
for gg → H at the lHC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
33 Uncertainties due to EFT in the top quark and bottom quark loops of
gg → H at NNLO at the lHC . . . . . . . . . . . . . . . . . . . . . . . . 104
34 Total uncertainty due to the EFT approach in gg → H at NNLO at the
lHC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
35 Central prediction with its total uncertainty for gg → H at NNLO at the
lHC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
36 Central predictions for gg → H at NNLO at the lHC with √
s = 8, 9, 10
TeV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
37 Scale and total EFT uncertainties in gg → H at the LHC with √
s = 14
TeV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
38 PDF+∆exp,thαs uncertainties and the comparison between the 4 NNLO
PDF sets in gg → H at the LHC with √
s = 14 TeV . . . . . . . . . . . . 113
39 Central prediction and total uncertainty in gg → H at NNLO at the LHC
with √
s = 14 TeV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
40 SM Higgs decay channels on the interesting Higgs mass range . . . . . . 117
41 The Higgs decays branching ratios together with the total uncertainty
bands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
42 The production cross section times branching ratio for SM pp¯ → W H →
W b¯b and gg → H → W+W− at the Tevatron together with the total
uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
43 The production cross section times branching ratio for SM gg → H →
W+W− at the lHC together with the total uncertainty . . . . . . . . . . 129
44 The SM Higgs boson production cross section gg → H at the Tevatron
together with the total uncertainty using 4 different ways of adding the
theoretical uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
45 The CDF/D0 95%CL limit on the SM Higgs boson mass confronted to
our theoretical expectations in a naive approach. . . . . . . . . . . . . . . 132
46 The luminosity needed by the CDF experiment to recover their current
claimed sensitivity when compared to our theoretical expectations for the
uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
47 One–loop corrections to the Higgs boson mass within the SM . . . . . . . 139
48 One–loop corrections to gauge couplings . . . . . . . . . . . . . . . . . . 141
49 SU(3)c × SU(2)L × U(1)Y gauge couplings running from the weak scale
up to the GUT scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
50 Possible proton decay in SUSY theories without R–parity conservation . 143
51 The constrained NMSSM parameter space . . . . . . . . . . . . . . . . . 170
52 The impact of main one–loop SUSY corrections to the Φb
¯b coupling in
the MSSM at hadron colliders . . . . . . . . . . . . . . . . . . . . . . . . 178
53 Feynman diagrams for the bottom quark fusion process in the MSSM . . 184
54 The NLO gg → A and NNLO b
¯b→A cross sections at the Tevatron with
tan β = 30 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
55 Scale uncertainty in the gg → Φ and b
¯b → Φ processes at the Tevatron . 186
56 PDF+∆exp,thαs uncertainty in the gg → Φ and bb → Φ processes at the
Tevatron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
57 The comparison between the MSTW, ABKM and JR prediction for the
NNLO bottom quark fusion cross section at the Tevatron . . . . . . . . . 187
58 Specific b–quark mass uncertainties in the gg → Φ and b
¯b → Φ processes
at the Tevatron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
59 The gg → A and b
¯b → A cross sections at the Tevatron together with
their different sources of uncertainty and the total uncertainties . . . . . 191
60 The gg → Φ and b
¯b → Φ at the LHC for different center–of–mass energies 194
61 Scale uncertainty in the gg → Φ and b
¯b → Φ processes at the lHC . . . . 195
62 PDF+∆αs uncertainty in the gg → Φ and bb → Φ processes at the lHC . 196
63 Comparison between the different PDFs sets in the gg → Φ and b
¯b → Φ
processes at the lHC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
64 Specific b–quark mass uncertainties in the gg → Φ and b
¯b → Φ processes
at the lHC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
65 The gg → Φ and b
¯b → Φ cross sections at the lHC together with their
different sources of uncertainty and the total uncertainties . . . . . . . . 199
66 LO σ(gb → tL,RH−) cross section and polarization asymmetry at the lHC
in the MSSM in two benchmark scenarios as a function of tan β . . . . . 205
67 Scale and PDF dependence on top–charged Higgs asymmetry at the lHC 206
68 The impact of the NLO SUSY corrections on the top–charged Higgs asymmetry at the LHC with √
s = 14 TeV . . . . . . . . . . . . . . . . . . . . 208
69 CP–odd A boson production in the pp¯ → A → τ
+τ
− channel at the
Tevatron together with the total uncertainty . . . . . . . . . . . . . . . . 215
70 The total uncertainties on the MSSM Higgs production in the gg → Φ
and b
¯b → Φ channels at the lHC including the impact of the Φ → τ
+τ
−
branching fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
71 CP–odd A boson production in the pp → A → τ
+τ
− channel at the lHC
together with the total uncertainty . . . . . . . . . . . . . . . . . . . . . 219
72 The 95%CL limits on the MSSM parameter space using our theoretical
uncertainties confronted to the Tevatron results . . . . . . . . . . . . . . 221
73 The 95%CL limits on the MSSM parameter space using our theoretical
uncertainties confronted to the lHC results . . . . . . . . . . . . . . . . . 222
74 Expectations at higher luminosity at the lHC for the 95%CL limits on
the MSSM parameter space using our theoretical calculation . . . . . . . 223
75 The MSSM Higgs analysis applied to the SM H → τ
+τ
− search channel
compared to the ATLAS H → γγ limits . . . . . . . . . . . . . . . . . . 226
76 Potentiel de Higgs dans le cas d’un champ scalaire r´eel selon le signe du
terme de masse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
77 Incertitude d’´echelle dans le processus gg → H au Tevatron . . . . . . . . 246
78 Comparaison entre les pr´edictions des diff´erentes collaborations de PDFs
pour le canal gg → H au NNLO en QCD . . . . . . . . . . . . . . . . . . 247
79 Incertitude PDF+∆αs dans les canaux de production gg → H et pp¯ →
HW au Tevatron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
80 Sections efficaces de production inclusives des canaux gg → H et pp¯ →
HV au Tevatron ainsi que les incertitudes th´eoriques totales associ´ees . . 249
81 Sections efficaces de production inclusives du canal gg → H au LHC `a 7
et 14 TeV ainsi que les incertitudes th´eoriques totales associ´ees . . . . . . 250
82 Luminosit´e n´ecessaire `a l’exp´erience CDF afin qu’elle obtienne la sensibilit´e qu’elle pr´etend avoir actuellement, en tenant compte de nos incertitudes th´eoriques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
83 Les sections efficaces de production inclusives du boson de Higgs A du
MSSM au Tevatron dans les canaux gg → A et b
¯b → A accompagn´ees
des incertitudes th´eoriques . . . . . . . . . . . . . . . . . . . . . . . . . . 258
84 Les sections efficaces de production inclusives du boson de Higgs Φ du
MSSM au lHC dans les canaux gg → Φ et b
¯b → Φ accompagn´ees des
incertitudes th´eoriques . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
85 Les limites a 95% de niveau de confiance sur l’espace des parametres du
MSSM en tenant compte de nos incertitudes th´eoriques confront´ees aux
donn´ees du Tevatron et du lHC . . . . . . . . . . . . . . . . . . . . . . . 260
86 L’analyse MSSM des bosons de Higgs neutres appliqu´ee au canal de
recherche H → τ
+τ
− du Mod`ele Standard, compar´ee aux r´esultats
obtenus par ATLAS dans le canal H → γγ . . . . . . . . . . . . . . . . . 261

List of Tables
1 The fermionic content of the Standard Model . . . . . . . . . . . . . . . 13
2 The NNLO total Higgs production cross sections in the gg → H process
at the Tevatron together with the detailed theoretical uncertainties as
well as the total uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . 84
3 The NNLO total cross section for Higgs–strahlung processes at the Tevatron together with the detailed theoretical uncertainties and the total
uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4 The total Higgs production cross sections in the four main production
channels at the lHC with √
s = 7 TeV . . . . . . . . . . . . . . . . . . . . 96
5 The NNLO total Higgs production cross sections in the gg → H process
at the lHC with √
s = 7 TeV together with the associated theoretical
uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6 The NNLO total production cross section in the gg → H channel at the
LHC with √
s = 8, 9, 10 TeV . . . . . . . . . . . . . . . . . . . . . . . . . 112
7 The NNLO total Higgs production cross section in the gg → H process
at the LHC with √
s = 14 TeV together with the associated theoretical
uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
8 The SM Higgs decay branching ratios in the b
¯b and WW modes for representatives Higgs masses together with the different sources of uncertainties as well as the total uncertainty. . . . . . . . . . . . . . . . . . . . . . 124
9 The SM Higgs decay branching ratios together with the total uncertainty
for the most important decay channels . . . . . . . . . . . . . . . . . . . 126
10 The superparticles and Higgs content of the MSSM before EWSB . . . . 157
11 The neutralinos, charginos and Higgs content of the MSSM after EWSB . 158
12 The main MSSM CP–odd like Higgs bosons decay branching fractions
together with their uncertainties . . . . . . . . . . . . . . . . . . . . . . . 211
13 The central predictions in the MSSM gg → Φ channel at the Tevatron
together with the detailed uncertainties and the impact of the Φ → τ
+τ
−
branching fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
14 The central predictions in the MSSM b
¯b → Φ channel at the Tevatron
together with the detailed uncertainties and the impact of the Φ → τ
+τ
−
branching fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
15 The central predictions in the MSSM gg → Φ channel at the lHC together with the detailed uncertainties and the impact of the Φ → τ
+τ
−
branching fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
16 The central predictions in the MSSM b
¯b → Φ channel at the lHC together with the detailed uncertainties and the impact of the Φ → τ
+τ
−
branching fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
17 CMS cuts used in the SM exclusive study gg → H → WW → νν at
the lHC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
18 Results for the gg → H+jet cross sections with MH = 160 GeV at the
lHC with HNNLO and MCFM programs . . . . . . . . . . . . . . . . . . 232
19 Uncertainties on the exclusive production gg → H → WW → νν with
MH = 160 GeV at the lHC with HNNLO program . . . . . . . . . . . . . . 233
20 Uncertainties on the exclusive production gg → H → WW → νν with
MH = 160 GeV at the lHC with MCFM program . . . . . . . . . . . . . . . 234
21 Central values and uncertainties for the H → WW SM backgrounds
exclusive cross sections at the lHC . . . . . . . . . . . . . . . . . . . . . . 235
22 Contenu fermionique du Mod`ele Standard . . . . . . . . . . . . . . . . . 241
23 Les superparticules et champs de Higgs du MSSM avant brisure ´electrofaible254
Liste des publications
Cette page donne la liste de tous mes articles concernant le travail r´ealis´e depuis 3 ans.
This page lists all the papers that I have written for 3 years in the context of my PhD
work.
Articles publi´es (published papers) :
Predictions for Higgs production at the Tevatron and the associated uncertainties,
J. B. et A. Djouadi, JHEP 10 (2010) 064;
Higgs production at the lHC, J. B. et A. Djouadi, JHEP 03 (2011) 055;
The Tevatron Higgs exclusion limits and theoretical uncertainties: A Critical appraisal, J. B., A. Djouadi, S. Ferrag et R. M. Godbole, Phys.Lett.B699 (2011) 368-371;
erratum Phys.Lett.B702 (2011) 105-106;
Revisiting the constraints on the Supersymmetric Higgs sector at the Tevatron, J. B.
et A. Djouadi, Phys.Lett.B699 (2011) 372-376;
The left-right asymmetry of the top quarks in associated top–charged Higgs bosons at
the LHC as a probe of the parameter tan β, J.B et al., Phys.Lett.B705 (2011) 212-216.
Articles non–publi´es (unpublished papers) :
Implications of the ATLAS and CMS searches in the channel pp → Higgs → τ
+τ
−
for the MSSM and SM Higgs bosons, J. B. et A. Djouadi, arXiv:1103.6247 [hep-ph]
(soumis `a Phys.Lett.B);
Clarifications on the impact of theoretical uncertainties on the Tevatron Higgs exclusion limits, J. B., A. Djouadi et R. M. Godbole, arXiv:1107.0281 [hep-ph].
Rapport de collaboration (review collaboration report) :
Handbook of LHC Higgs Cross Sections: 1. Inclusive Observables, LHC Higgs Cross
Section Working Group, S. Dittmaier et al., arXiv:1101:0593 [hep-ph].
Comptes–rendus de conf´erences (proceedings) :
Higgs production at the Tevatron: Predictions and uncertainties, J. B., ICHEP 2010,
Paris (France), PoS ICHEP2010 (2010) 048;
The Supersymmetric Higgs bounds at the Tevatron and the LHC, J.B., XLVIe
Rencontres de Moriond, EW interactions and unified theory, La Thuile (Italie),
arXiv:1105.1085 [hep-ph].

Cette these est d´edi´eea mon pere eta mes deux grand-p`eres, disparus bien
trop tˆot.

(From http://abstrusegoose.com/118)
Et maintenant, apprends les v´erit´es qui me restent `a te d´ecouvrir,
Tu vas entendre de plus claires r´ev´elations.
Je n’ignore pas l’obscurit´e de mon sujet ;
Lucr`ece, dans De rerum natura, v. 902-943 livre I
Les amoureux fervents et les savants aust`eres
Aiment ´egalement, dans leur mˆure saison,
Les chats puissants et doux, orgueil de la maison,
Qui comme eux sont frileux et comme eux s´edentaires.
Charles Baudelaire, dans Les Fleurs du Mal

Introduction 1
Introduction
In this thesis, we wish to present some predictions for the Higgs boson(s) study at the
two largest hadron colliders currently in activity: the Fermilab Tevatron collider and
the CERN Large Hadron Collider (LHC). Our focus will be on the inclusive production
cross sections and the decay branching fractions, first in the Standard Model which in
itself is the topic of part I and then in its minimal supersymmetric extension which is
the topic of part III.
The study of the fundamental mechanisms of Nature at the elementary level has a
long story and has known many milestones in the past sixty years. Physicists have built
a theory, nowadays known as the Standard Model, to describe the elementary particles
and their interactions, that are those of the strong, weak and electromagnetic, the two
last being unified in a single electroweak interaction. It relies on the elegant concept
of gauge symmetry within a quantum field theory framework and has known many
experimental successes: despite decades of effort to surpass this model it is still the one
that describes accurately nearly all the known phenomena1
. One of its key concepts
is the spontaneous breakdown of electroweak symmetry: indeed in order to give mass
to the weak bosons that mediate the weak interaction, a scalar field is introduced in
the theory whose vacuum breaks the electroweak symmetry and gives mass to the weak
bosons. In fact it also gives masses to the fermions and one piece of this mechanism
remains to be discovered: the Higgs boson, the “Holy Grail” of the Standard Model. Its
discovery is one of the main goal of current high energy colliders.
It is then of utmost importance to give theoretical predictions for the production
cross sections and decay branching fractions of the Higgs boson at current colliders to
serve as a guideline for experiments. However, the hadronic colliders are known to be
very difficult experimental environments because of the huge hadronic, that is Quantum
ChromoDynamics (QCD), activity. This is also true on a theoretical side, which means
that an accurate description of all possible sources of theoretical uncertainties is needed:
this is precisely the main output of this thesis. We shall mention that in the very final
stage of this thesis new results have been presented in the HEP–EPS 2011 conference;
our work is to be read in the light of the results that were available before these newest
experimental output which will be briefly commented in the conclusion.
Part I is entirely devoted to a review of the Standard Model. In section 1 we will draw
a short history of the Standard Model and list its main milestones of the past sixty years,
followed by a description of its main concepts. We will go into more details about the
Higgs mechanism, which spontaneously breaks electroweak symmetry, in section 2: we
will review some reasons to believe that either the Higgs mechanism itself or something
which looks like the Higgs mechanism is needed, and then how the Higgs boson emerges
1We leave aside the neutrino mass issue.
2 Introduction
from the electroweak symmetry breaking and what are its couplings to fermions and
bosons of the Standard Model.
Part II is the core of the Standard Model study of this thesis. Indeed the Higgs
boson remains to be discovered and is one of the major research programs at current
high energy colliders. The old CERN Large Electron Positron (LEP) collider has put
some bounds on the possible value of the Higgs boson mass, which is above 114.4 GeV in
the Standard Model at 95%CL. We will review in section 3 the current experimental and
theoretical bounds on the Higgs mass. We then give our predictions for the Standard
Model Higgs boson inclusive production cross section at the Tevatron in the two main
production channels that are the gluon–gluon fusion and the Higgs–strahlung processes,
giving all the possible sources of theoretical uncertainties: the scale uncertainty viewed
as an estimation of the unknown higher–order terms in the perturbative calculation;
the parton distribution functions (PDFs) uncertainties related to the non–perturbative
QCD processes within the proton, and its related strong coupling constant issue; the
uncertainty coming from the use of an effective theory approach to simplify the hard
calculation in the gluon–gluon fusion process. We will specifically address the issue of
the combination of all the uncertainties in section 4.5. We will then move on to the
same study at the LHC, concentrating on its current run at a 7 TeV center–of–mass
energy that we will name as the lHC for littler Hadron Collider; we will still give some
predictions for the designed LHC at 14 TeV. We will finish this part II by the Higgs
boson decay branching fractions predictions in section 6, together with a detailed study
of the uncertainties that affect these predictions. It will be followed by the combination
of the production cross sections and decay branching fractions into a single prediction,
first at the Tevatron in section 6.3 and then at the lHC in section 6.4. We will then
study the impact of our uncertainties on the Tevatron Higgs searches in section 6.5 and
in particular put into question the Tevatron exclusion limits that are debated within the
community.
Even if the Standard Model is a nice theory with great experimental successes, it
suffers from some problems, both on the theoretical and experimental sides. It is known
for example that the Higgs boson mass is not predicted by the Standard Model, and
even not protected: higher order corrections in the perturbative calculation of the Higgs
boson mass have the tendency to drive the mass up to the highest acceptable scale of the
theory which means that we need a highly fine–tuning of the parameters to cancel such
driving. It is known as the naturalness problem of the Standard Model. They are several
ways to solve such a problem, and one of them is particularly elegant and relies on a new
symmetry between bosons and fermions: supersymmetry. This theoretical concept, born
in the 1970s, has many consequences when applied to the Standard Model of particle
physics and is actively searched at current high energy colliders. This will be the topic
of part III in which we will review some of the reasons that drive the theorists to go
Introduction 3
beyond the Standard Model and in particular what makes supersymmetry interesting
in this view in section 7, then move on to the description of the mathematical aspects
of supersymmetry in section 8. We will finish this part III by a very short review of
the minimal supersymmetric extension of the Standard Model, called the MSSM, in
section 9. We will in particular focus on the Higgs sector of the theory and show that
the MSSM needs two Higgs doublets to break the electroweak symmetry breaking and
has thus a rich Higgs sector as five Higgs boson instead of a single one are present in
the spectrum: two neutral CP–even, one CP–odd and two charged Higgs bosons.
After this review of supersymmetry and the MSSM we will reproduce in part IV the
same outlines that have been developed in part II in the Standard Model case. We will
first review the neutral Higgs sector at hadron colliders in section 10 and show that we
can have a quite model–independent description for our predictions in the sense that
they will hardly depend on most of the (huge) parameters of the MSSM but two of
them, the mass of the CP–odd Higgs boson A and the ratio tan β between the vacuum
expectation values of the two Higgs doublets. We will then give in section 11 our
theoretical predictions for the neutral Higgs bosons inclusive production cross section at
the Tevatron in the two main production channels that are the gluon–gluon fusion and
the bottom quark fusions, the bottom quark playing a very important role in the MSSM
at hadron colliders. We will reproduce the same study at the lHC in section 12 before
giving the implications of our study on the [MA,tan β] parameter space in section 13.
We will first give in this last section our predictions for the main MSSM decay branching
fractions and in particular the di–tau branching fraction that is of utmost importance
for experimental searches. We we will then compare our predictions together with their
uncertainties to the experimental results obtained at the Tevatron and at the lHC that
has now been running for more than a year at 7 TeV and given impressive results. We
will see that the theoretical uncertainties have a significant impact on the Tevatron
results, less severe at the lHC. We will finish section 13 by a very important outcome of
our work: the possibility of using the MSSM neutral Higgs bosons searches in the di–
tau channel for the Standard Model Higgs boson in the gluon–gluon fusion production
channel followed by the di–tau decay channel in the low Higgs boson mass range 115–140
GeV.
Finally, we will give an outlook and draw some conclusions in part V together with
some perspectives for future work. These rest on the next step on the road of the
experiments, that is an exclusive study of the Higgs bosons production channels. We
shall give some early results in section 14 on the Standard Model Higgs boson at the
lHC in the gg → H → WW → νν search channel together with an exclusive study of
the main Standard Model backgrounds. This is also the current roadmap of the Higgs
bosons theoretical community and this work is done in the framework of a collaboration
on this topic.

5
Part I
A brief review of the Standard
Model of particle physics
Summary
1 Symmetry principles and the zoology of the Standard Model 6
1.1 A brief history of the Standard Model . . . . . . . . . . . . . . . . . 6
1.2 Gauge symmetries, quarks and leptons . . . . . . . . . . . . . . . . 12
2 The Brout–Englert–Higgs mechanism 16
2.1 Why do we need the electroweak symmetry breaking? . . . . . . . . 16
2.1.1 The unitarity puzzle . . . . . . . . . . . . . . . . . . . . . . . 16
2.1.2 Masses and gauge invariance . . . . . . . . . . . . . . . . . . 18
2.2 The spontaneous electroweak symmetry breaking . . . . . . . . . . . 19
2.2.1 Weak bosons masses and electroweak breaking . . . . . . . . 20
2.2.2 SM Higgs boson couplings . . . . . . . . . . . . . . . . . . . 24
6 Symmetry principles and the zoology of the Standard Model
1 Symmetry principles and the zoology of the Standard Model
The Standard Model (SM) of particle physics is the current description of the fundamental constituents of our universe together with the interactions that occur between them.
The SM was born in its current form in the seventies, after nearly twenty years of many
experiments and theoretical reflexions on how to build a somewhat simple and elegant
model to describe accurately the experimental results on the one hand and to make powerful predictions in order to have a falsifiable theory on the other hand. Its frameworks
are relativistic quantum field theory and group theory to classify the different interactions. It also needs the key concept of spontaneous (electroweak) symmetry breaking in
order to account for the masses of the different fields in the theory, the (weak) bosons
as well as the matter fermions. Other reasons also push for such a theoretical concept
and will be presented in the next sections.
We will in this section present a short review of the major historical points in the
birth of the SM, and present its theoretical fundations. The focus on the electroweak
symmetry breaking, in particular its minimal realization through the Brout–Englert–
Higgs mechanism, will be discussed in the next section.
1.1 A brief history of the Standard Model
This subsection will sketch the different historical steps that have lead to the current
form of the theory that describes the elementary particles and their interactions among
each other, called the Standard Model (SM). This model has a very rich history over
more than fifty years of the XXth century, not to mention all the diverse and fruitful
efforts made before to attain this level of description of the elementary world. We will
only select some (of the) outstanding events, both from the theoretical and experimental
sides, to present the twisted path leading to the current Standard Model of particle
physics.
The birth of modern QED
The first attempt to decribe electromagnetic phenomena in the framework of special
relativity together with quantum mechanics can be traced back in the 1920s. In particular Dirac was the first to describe the quantization of the electromagnetic fields as
an ensemble of harmonic oscillators, and introduced the famous creation–annihilation
operators [1]. In 1932 came Fermi with a first description of quantum electrodynamics [2], but physicists were blocked by the infinite results that did arise in the calculations
beyond the first order in perturbation theory.
1.1 – A brief history of the Standard Model 7
Years after, the difficulty was solved by Bethe in 1947 [3] with the concept of renormalization, that is the true physical quantities are not the bare parameters of the theory,
and thus the infinite that arise are absorbed in the physical quantities, leaving finite results in the end. This leads to the modern Quantum ElectroDynamics (QED) with the
key concept of gauge symmetry and renormalization, that was formulated by Feynman,
Schwinger and Tomonaga [4–6] in the years 1950s and awarded by a Nobel prize in 1965.
This is the first quantum field theory available and has been the root of all the SM ideas
for the key concepts of gauge symmetry and renormalizability.
P violation and V − A weak theory
It was long considered in physics that the parity symmetry was conserved: if we
repeated an experiment with the experimental apparatus mirror reversed, the results
would be the same as for the initial set–up. This assessment is true for any experiment
involving electromagnetism or strong interaction, but that is not the case for weak
interaction.
It was first proposed by Yang and Lee in 1956 that the weak interaction might indeed
not respect P–symmetry [7]. This was observed in 1957 by Chien-Shiung Wu (“Madam
Wu”) in the beta desintegration of cobalt 60 atoms [8]. Yang and Lee were then awarded
the 1957 Nobel prize for their theoretical developments on this concept.
Up until that period, the weak interaction, that shapes the decay of unstable nucleii,
was described by the Fermi theory in which the fermions interact through a four–particles
vertex. The discovery of the P–violation lead to the construction of an effective V − A
theory where the tensor structure of the thory is correct and does respect the charge and
parity violations. This V − A theory was later on replaced by the electroweak theory,
see below.
The quark description
In the first half of the XXth century the pattern of elementary particles was simple: the
electron (and its antiparticle the positron, postulated by Dirac in 1931 and discovered
in 1932 by Anderson), the proton and the neutron were the only known elementary
particles at that time. The neutrino, first postulated by Pauli in its famous letter in
1930 to save the energy–momentum conservation in beta decay reactions2 was discovered
only in 1956.
Experimental particle physicists discovered numerous new particles (the “hadrons”)
in the 1950s and 1960s after the discovery of the pion in 1947, predicted by Yukawa in
1935, thus casting some doubts on the elementary nature both of the “older” particles
2The original name was “neutron” for neutral particle. Chadwick discovered in 1932 what would be
the neutron, thus Fermi proposed the name “neutrino” meaning “little neutral one” in italian.
8 Symmetry principles and the zoology of the Standard Model
such as the neutron and the proton and on the new zoo discovered. Gell–Man and Zweig
proposed in 1964 a model of constituant particles of these hadrons and mesons that
could explain the pattern seen by experimentalists, using only a limited number of new
constituant particles: the quarks [9,10]. They introduce the SU(3) flavor symmetry with
the three up, down and strange quarks. One year later the charm quark was proposed to
improve the description of weak interactions between quarks, and in 1969 deep inelastic
scattering experiments at the Stanford Linear Accelerator Center (SLAC) discovered
point–like objects within the proton [11], an experimental proof of the compositeness of
the hadrons. It is interesting to note that the term used for these new point–like objects
was “parton”, proposed by Feynman, as the community was not entirely convinced that
they were indeed the Gell–Mann’s quarks. Nowadays “parton” is still a word used in
particle physics to name the different constituants of the hadrons (the quarks, antiquarks
and gluons, the later being the bosons of the strong interaction).
The (nearly) final word on the quark model was given in 1974 when the J/Ψ meson
was discovered [12, 13] and thus proved the existence of the charm quark, which was
proposed by Glashow, Iliopoulos and Maiani in the GIM mechanism [14] in 1970 to explain the universality of weak interaction in the quark sector, preventing flavor changing
neutral currents. The heaviest quark, that is the top quark, was finally discovered in
1995 at the Fermilab Tevatron collider [15, 16].
CP violation and the concept of generation
To explain both the universality and the u ←→ d transitions in weak interactions,
Cabibbo introduced in 1963 what is known as the Cabibbo angle [17] and was used
to write in the mass eigenstates basis the weak eigenstate for the down quark d. A
year later, Cronin and his collaborators discovered that not only C and P symmetries
are broken by weak interactions, but also the combined CP symmetry [18], studing the
K0K
0
oscillations: the probability of oscillating from K0
state into K
0
state is different
from that of the K
0
→ K0
, indicating that T time reversal symmetry is violated. As
the combined CPT is assumed to be conserved, this means that CP is violated.
As mentioned a few lines above, the GIM mechanism introduced a fourth quark, the
charm quark c. It then restores universality in the weak coupling for the quarks, as we
have now two weak eigenstates
|d
0
i = cos θc|di + sin θc|si
|s
0
i = − sin θc|di + cos θc|si (1.1)
coupled to respectively the u quark and the c quark. We thus have two generations
in the quark sector, the first one is the (u, d) doublet and the second one is the (c, s)
1.1 – A brief history of the Standard Model 9
doublet. However, as explained in 1973 by Kobayashi and Maskawa extending the work
initiated by Cabibbo, this is not sufficient to explain the CP violation observed by the
1964 experiment. Only with three generations could be introduced some CP violating
effects through a phase angle, and thus extending the Cabbibo angle to what is known
as the Cabibbo–Kobayashi–Maskawa (CKM) matrix [19]. Kobayashi and Maskawa were
awarded the 2008 Nobel prize for this result3
.
Yang–Mills theory and spontaneous symmetry breaking
We have seen a few lines above that the Fermi theory describing the weak interactions
had been refined by the V − A picture to take into account the P violation. Still the
V − A theory was known to be an effective theory as the theory was not renormalizable
and did not allow for calculations beyond the first order in perturbation theory. The only
gauge theory that was available at that time was QED, an abelian gauge theory, which
obviously is not the right description of weak processes as it describes only light–matter
interactions.
The first step toward the solution was set–up in 1954, when Yang and Mills developed a formulation of non–abelian gauge theories [20] in order to provide (initially) an
explanation for the strong interaction at the hadron level (that we call nuclear interaction). Unfortunately the theory was not a success at first, as the gauge bosons must
remain massless to preserve the symmetry of the theory, thus meaning that the weak
interaction should be long–range; experimentally that is not the case.
The key result to solve this contradiction and then still use the elegant description of
gauge theory is given in 1964 by Brout, Englert, Higgs, Guralnik, Hagen and Kibble after
some important work on the concept of symmetry breaking from Nambu and Goldstone:
the spontaneously gauge symmetry breaking [21–24] described by the Brout–Englert–
Higgs mechanism. This will be presented in the following in details, but we can already
remind the reader that the most important result is that it allows for the use of a
Yang–Mills theory together with a description of massive gauge bosons for any gauge
theory.
Interlude: from nuclear force to strong interaction
Before arriving to the final electroweak description that constitutes the heart of the
SM, we recall the road leading to the description of the strong interaction between the
quarks.
As stated above, Yang–Mills theory in 1954 was the first attempt to describe the
interaction between the hadrons, that we call nuclear interaction, in a gauge formulation.
3Unfortunately the Nobel committee failed to recognize the important pionnering work from
Cabibbo.
10 Symmetry principles and the zoology of the Standard Model
After the introduction of the quark model by Gell–Mann in 1964 (see above) and the
discovery of the quarks in 1969 (see above), it has been proposed that the quarks must
have a new quantum charge, called color, to accomodate for the Pauli exclusion principle
within some baryons [25]. This was experimentally observed in the SLAC experiments
in 1969 which discovered point–like objects within the nucleon, as discussed earlier.
With the help of the discovery of asymptotic freedom [26, 27] in 1973 by Wilczek,
Gross and Politzer (who share the 2004 Nobel prize for this result), that states that at
very high energy quarks are free, and with a SU(3) gauge Yang–Mills theory, Quantum
ChromoDynamics (QCD) was firmly established in the 1970s as being the theory of
the strong interactions, with the gluons as the gauge bosons. Evidence of gluons was
discovered in three jet events at PETRA in 1979 [28], giving further credits to QCD.
The nuclear interaction between the hadrons is then a residual force originating from
the strong interaction between quarks (and gluons). However, as the strong coupling
is indeed very strong at large distance (that is the confinement), preventing from the
use of perturbation theory, an analytical description of the strong interaction within the
hadrons at low energies is still to be found. This problem is now studied within the
framework of lattice gauge theories which give spectacular results.
The weak neutral currents and the path to electroweak theory
As stated above it was known that the V − A theory for the weak interaction was
an effective theory, with difficulties calculating beyond the first order in perturbation
theory. With the advent of Yang–Mills theory and the Brout–Englert–Higgs mechanism,
describing the weak interaction with a gauge theory and in the same time allowing for
massive weak bosons as dictated by the experiments, the weak interaction being a short
distance interaction, it would be possible to account for a renormalizable description of
the weak interaction.
During the 1960s there were many attempts to carry on this roadmap, trying lots of
different gauge groups to account for the QED on the one hand, the weak interaction
on the other hand, as both interactions play a role for lepton particles such as the
electron. The gauge theory that did emerge was the SU(2) × U(1) model where the
weak nd electromagnetic interactions are unified in a single gauge theory description4
,
with contributions notabely from Glashow [29], Salam [30] and Weinberg [31]. This
model together with the Brout–Englert–Higgs mechanism predicts in particular that
there should be a neutral weak boson Z
0
to be discovered and thus neutral currents.
4
It is actually not a complete unified theory as the algebra describing the electroweak interaction is a
product of two Lie algebras. Nevertheless as the decription of the weak and electromagnetic interactions
are intimely connected through the pattern of the electroweak symmetry breaking, see below, this can
be viewed as at least a partial unification.

[françois bégaudeau]

Tristan : qu’attends tu pour saluer ce magnifique post de Demi-habile?
Je trouve ton silence humiliant à l’endroit de ce faible.

[Demi Habile]

François: Bah alors, ça te fatigue un peu tout ça?

[françois bégaudeau]

Ce qui me contrarie, c’est qu’on ne salue pas assez ta générosité – serait ce du mépris?
Qui ici peut se targuer de rédiger des posts aussi copieux et nourrissants?

[Demi Habile]

François: Tu veux m’en dire plus?

[maelstrom]

Quelqu’un peut dire a demi habile que la discussion autour de la culture légitime ou entre chaque message il y a trois pavé a scroller pendant trois quart d’heure qu’on a presque l’impression d’avoir monté l’everest quand on trouve la réponse c’est pas très pratique a la longue

[Eden Lazaridis]

C’est involontaire de sa part, excuse le. Il est un peu bavard, c’est tout.

[Demi Habile]

Quelqu’un peut expliquer à maelstrom que c’est voulu?

[maelstrom]

tu peux copier coller autre chose que de la physique théorique sinon, pour que la cage sois au moins un peux plus agréable a regarder

[Eden Lazaridis]

Aucune volonté de nuire de sa part en tout cas, sois rassuré !

[Demi Habile]

Je me doute bien, il a juste oublié de suivre les épisodes.

[maelstrom]

je m’étais arrêter a deleatur qui spam les topics

[Eden Lazaridis]

Concernant l’illégitimité de Nakamura, je me permets de publier un classique :

[maelstrom]

Je recommande pour ma part cette vidéo ou aoc se confronte au réel https://www.youtube.com/watch?v=ezQdoN8qXX0

[maelstrom]

s’il te plait dis moi comment mettre les vidéos sur le topic comme tu le fait

[maelstrom]

[maelstrom]

je galère je vais finir par aussi polluer le forum involontairement

[Eden Lazaridis]

En fait je pense que ta vidéo n’est pas affichée en entier parce que tu n’es pas revenu à la ligne. Il faut que le lien ait une ligne pour lui tout seul !

[maelstrom]

merci pour la réponse j’essaye

[maelstrom]

victoire

[Eden Lazaridis]

Autre classique :

🔴🗣 « Quand je regarde ses textes, on est loin de la représentation de la France. Par exemple avec "Catchaca", cette ode à la levrette… »

Gérard Larcher critique le choix d'Aya Nakamura pour la cérémonie d'ouverture des #JO2024. #Les4V @gerard_larcher pic.twitter.com/Wpifi6WuAq

— Telematin (@telematin) March 14, 2024

[Eden Lazaridis]

En catchaca baby

[SHB]

Jul finira pas passer à Quotidien.
.
Le problème avec toi Francois c’est que tes démonstrations prétendument de bon sens se basent souvent sur des prémisses fausses. Ici, le passage a quotidien.
.
Contre exemple : Philippe Catherine est passé a Quotidien et même On N’est Pas Couché et pourtant tout le monde le considère comme un olibrius un peu dingue et personne dans la bourgeoisie n’accorde sérieusement de légitimité artistique a ce qu’il fait autrement que pour dire que c’est un peu loufoque et en se foutant de sa gueule. J’invite tout le monde a revoir l’émission a ONPC ou tout le monde le méprise gentillement sur le plateau pendant 1 heure.

[maelstrom]

@quotidienofficiel

On a bien crû que France Info avait décroché l’exclu de l’année 🎙️ #yannbarthes #barthes #video #interview #jul #humour #france #titkokfrance #news #tiktoknews #humour #tiktokhumour #tiktokentertainment

♬ son original – Quotidien – Quotidien

[maelstrom]

https://www.tf1.fr/tmc/quotidien-avec-yann-barthes/videos/le-transpi-a-marseille-tout-le-monde-connait-jul-mais-alors-vraiment-tout-le-monde-35003901.html

[Eden Lazaridis]

Concernant ONPC, si tu veux connaître les avis culturels de la bourgeoisie progressiste, il faut écouter Salamé, Pulvar, Caron.
Si tu veux connaître les avis culturels de la bourgeoisie conservatrice, il faut écouter Zemmour, Naulleau Moix, Burggraff, Consigny.
Je n’ai pas vu l’émission avec Katerine mais je suis près à parier que la première catégorie l’a encensé, et la seconde le méprise gentiment. Je vois mal Barthes mépriser Katerine non plus. Pour la bourgeoisie cool, Katerine présente tous les signes extérieurs de l’artiste : tête en l’air, spontané, drôle. Ils le rangent dans la catégorie Joann Sfar, Michel Gondry, M.

Angot c’est un cas à part. Une pure singularité. Absolument insupportable mais incontestablement originale.

[maelstrom]

je n’aurait pas si vite classer salamé dans la catégorie des bourgeois progressiste

[maelstrom]

et pour pulvar je pense toujours a cette photo
https://www.parismatch.com/lmnr/f/webp/r/960,640,FFFFFF,forcex,center-middle/img/var/pm/public/styles/paysage/public/media/image/2022/03/21/05/Audrey-Pulvar-enfourche-une-nouvelle-vie.jpg?VersionId=9mPMrrSHhL2AGhdE1mH1YDAgNGlFWeVw

[maelstrom]

[Eden Lazaridis]

Une photo qui rend hommage à la grande carrière d’Audrey !

Quand je pense à Audrey, je pense à « Libres et insoumis : Portraits d’hommes singuliers », un livre que ma tante m’avait offert (car elle pensait que je me définissais comme un homme libre ahaha) où elle trace le portrait de nombreux hommes. Le passage où elle traduit littérairement la fougue picturale de Basquiat est peut-être l’une des choses les plus lourdes que j’ai lu de ma vie. Et j’ai lu du Yann Moix !

[Eden Lazaridis]

*lues

[Eden Lazaridis]

Attention j’ai dit « avis culturels des bourgeois progressistes ». Salamé est une illustration de ce que le progressisme bourgeois pense en matière culturel. Elle a présenté une émission, « Stupéfiant » qui était extrêmement représentative de cela (je pourrais développer mais je pense que ça coule de source).
Politiquement c’est autre chose.

[Eden Lazaridis]

Culturellement, Salamé est sur la ligne Yann Barthès.

[maelstrom]

Justement en parlant de stupéfiant, celui sur bernard arnault est plutôt intéressant pour plusieurs raisons, sont rapport a l’art et la métaphore sur le golf/tennis ou on comprend bien qu’il marche par gain de productivité ceux qui rejoint la réflexion plus haut, son nonmépris de kanye west et évidemment la scène d’anthologie au début sur l’ambiance socialo-marxiste et le « mais je ne parle pas de vous »

[Florent]

Moi je note aussi son aversion au « risque », justifiant qu’il ne réponde pas aux demandes d’interview. c’est quand même surprenant pour un entrepreneur

[SHB]

Le problème avec la « culture » c’est que c’est beaucoup trop large mais si on prend ses différentes composantes une a une (musique, théâtre, cinéma, performance artistique de rue, peinture, littérature, etc) on pourra aisément distinguer des œuvres culturels qui appartiennent a la culture légitime et d’autres vues comme a la marge ou « impures »

[SHB]

Dans la gene occasionnée, tu nous renseigne d’ailleurs très bien sûr la firme légitime du cinéma actuel (folie narrative, plan très courts, mise en avant de récits extraordinaires, classes laborieuses peu présentes dans les œuvres, etc….)

[SHB]

Pour le rap je suis pas d’accord ce dernier est encore méprisé par une grande partie de la bourgeoisie que tu appelles « hard ».

[françois bégaudeau]

– ton raisonnement est une aberration :  » se basent souvent sur des prémisses fausses. Ici, le passage a quotidien. » Il s’agit ici d’une prédiction. Une réduction ne saurait être fausse en soi. Elle ne le devient que si le temps passant, elle ne se réalise pas. En l’état, le verdict n’a pas eu lieu. Tu dis donc strictement n’importe quoi. Et tu en tires une généralité sur ma tendance à ne penser que sur des prémisses fausses. N’importe quoi bis.
-ton exemple sur Philippe Katerine est très mal choisi. Il est adulé par la bourgeoisie – pas seulement elle, mais aussi elle. Ceci est factuel et vérifiable sur bien des plateaux, dans bien des classements. Il serait même l’égérie de la bourgeoisie cool.
-sur le rap : que des gens continuent à détester le rap (et pas seulement la bourgeoisie hard) n’empêche pas du tout sa légitimation, vérifiable partout. J’ai déjà cité une archive implacable de ça : le rap dans La Sentinelle, de Desplechin. Film de 1991. Je pourrais en citer maints autres, et ce dès les années 90.
(semblablement je pourrais clamer que le punk demeure une musique délégitimée. Mais la vérité du punk, c’est que c’est une musique peu aimée, et par tout un tas de gens, qui vont des prolos aux bourgeois. Pas seulement des arbitres du bon gout, donc. Le punk n’est donc pas illégitime – beaucoup de livres et d’expos lui sont consacrés. Il est juste très marginal.)
-ce qu’il m’arrive de circonscrire dans la GO. ce sont les standards formels, narratifs et thématiques de l’époque. Ceci n’a rien à voir avec la notion de légitimité (ou alors très lointainement). Une norme de production n’est pas forcément légitime. Au 20ème siècle la littérature de gare est vertébrée par des codes stricts, des normes, et elle est délégitimée (ne serait ce que par son nom) Dans les années 60 la forme de chanson dominante c’est la chanson yé-yé, adaptation en français de tubes américains chantées par des jeunes ravis de la crèche. C’est une norme, et elle est hautement méprisée par la bourgeoisie d’alors, et délégitimée. Mille autres exemples possibles.
Ta compréhension de la notion de légitimité est pour le moins brumeuse. J’aurai essayé de clarifier la chose. Je vois que ça ne t’intéresse pas.

[SHB]

Pour la culture je suis d’accord avec toi tes exemples son précis et la différence entre norme et légitimité est intéressante.
.
Mais si tu soutiens qu’une forme hégémonique peut tout de même être illégitime et inversement alors dans ce cas ton argument de dire que le rap est hégémonique donc légitime ne marche pas.
.
Une des confusions de notre dialogue réside dans la définition même de ce qu’est le rap. On ne peut pas mettre dans la même case « rap » Lomepal et Le roi Heenok. Pour moi la nouvelle vague consensuelle lisse chantée toujours sur le même ton avec toujours les mêmes paroles et aucune prise de position au subversive en dehors des chansons c’est au mieux une séquestration du rap mais je sais pas si on peut appeler ça du rap a proprement parler.

[SHB]

Ceci n’a rien à voir avec la notion de légitimité (ou alors très lointainement).
.
Non ça a avoir avec la notion de légitimité d’ailleurs Rougeyron en était une illustration parfaite lorsqu’il disait durant votre échange que le cinéma français était catastrophique car on voyait bien qu’il avait en tête de cinéma légitime du moment a savoir le bon vieux film Francia mainstream de petite comédie avec une trame sociale nulle et des acteurs semblables a chaque film. Alors que le cinéma français c’est bien + que cela notamment des petites productions dont Rougeyron n’a évidemment aucune idée.

[Jeanmonnaie]

Le cinéma français craint.

Films français : seulement 12 sur 574 (ou 2 %) sont rentables

[Eden Lazaridis]

Hallucinant ce chiffre…
Le cinéma rejoint le théâtre et la littérature.

[Eden Lazaridis]

Bientôt il y aura plus d’argent à se faire dans la philatélie que dans le cinéma.

[SHB]

Ceci est factuel et vérifiable sur bien des plateaux.
.
Pas sur celui de ONPC en tout cas.

[SHB]

Si j’ai bien compris ce que tu dis pense que la notion de culture légitime est périmée car aujourd’hui les goûts légitimes ne sont pas forcément hégémoniques et les goûts hégémoniques pas forcément légitimes c’est ça ?

[SHB]

J’avoue ne pas être un spécialiste de la culture en général dans la société mais pour un domaine dont je suis sûr et celui sur lequel je me basait initialement était celui de l’université. Jaffirme que la culture légitime existe au sein de l’université et je peux le defence avec énormément d’exemples concrets et précis.

[Jeanne]

Merci pour cette discussion sur le rapport des bourgeois à la culture et à la distinction. (Merci aux intervenants depuis tout là haut là haut).
La question de savoir comment les Bourgeois, ou les aspirants Bourgeois (et la Bourgeoisie aime, je crois, à s’ouvrir un petit peu, ou a minima elle aime à se le raconter) me travaille beaucoup et depuis longtemps.
Je tente un truc:
Si autrefois les Bourgeois se distinguaient par leur référence à une « culture légitime « , peut-être qu’aujourd’hui, dans notre ère managériale, elle cherche à se distinguer par les compétences et « savoir être  » valorisés dans ses boulots de cadres, à savoir la vitesse et la technicité. Je parle de la technicité au sens large: celle consistant à maîtriser des techniques informatiques, celle au gré de laquelle il existerait des techniques relationnelles (mon cul, je n’y crois pas), des techniques manageriales (généralement magnifiques et hautement recommandables, comme on sait), des techniques de relaxation, des techniques pour s’endormir (afin de reconstituer son capital énergétique), des techniques pour bien cuire les légumes (afin d’être en bonne santé afin d’éviter les congés maladies afin d’entretenir sa réputation de professionnel investi et sur qui on peut compter), des techniques d’analyse des diagnostics de territoire et autres indispensables techniques de communication.
C’est fou comme le Bourgeois est technique, je trouve.
Il y a quelque temps, je me trouvais dans une fête où une jeune femme investie dans son travail, heureuse de mettre en œuvre la politique de réduction des dépenses de personnel pour laquelle elle était (quel bonheur!) missionnée, aimant l’union européenne et en parlant, (une bourgeoise, quoi), m’a expliqué qu’elle aimait beaucoup tel prénom et pour attester ça elle a dit:
 » Techniquement, ma fille le porte, ce prénom « .
(J’ai rigolé intérieurement, je sais pas trop pourquoi).
Autrefois le Bourgeois avait le temps de lire, aujourd’hui il a juste le temps de ne pas réfléchir et d’appliquer les formidables techniques professionnelles au gré desquelles son métier, régulièrement, ne sert à rien.
Ou à rien d’intéressant.

[Jeanne]

Je voulais dire « la question de savoir comment les Bourgeois se distinguent ».

[Emile Novis]

@Jeanne
D’accord avec toi pour la prédominance, dans la bourgeoisie, du paradigme « technique », mot qu’elle sort désormais à toutes les sauces. Et la phrase de la jeune femme à propos du prénom de sa fille (« Techniquement, ma fille le porte, ce prénom ») t’as fait rire parce que, à mon avis, cette phrase est complètement débile.
.
Et c’est peut-être là le problème de la bourgeoisie ou du devenir historique de celle-ci : un dépérissement de la culture au profit d’un mode d’existence technique où le moindre recoin du réel et de la vie est contrôlable et soumis au dogme de l’efficience.
.
Si on entend pas culture les pratiques et les productions par lesquelles des individus humanisent leur existence et affirment une puissance de la vie, on peut en effet se demander si le mode d’être bourgeois garde encore une place pour cela. C’est le philosophe Michel Henry qui disait que l’hégémonie de la technique en Occident, pure application plus ou moins illusoire de la science moderne vouée à encadrer la totalité de l’existence, était en train de se retourner contre la vie elle-même et de produire une barbarie collective sous des airs parfois souriants et bien présentables. Car la science moderne élimine a priori la sensibilité et tout ce qui est lié, pour le dire vite, à l’existence vécue, pour ne retenir que ce qui est objectivable et mesurable par l’entendement calculateur. La technique s’enracine désormais dans cette science moderne et quand elle s’applique à la vie concrète, elle nie le vécu des individus et écrase l’existence sous le poids de ses catégories désincarnées (tu parles bien de « capital énergétique », et le bourgeois ajoutera sa petite montre connectée qui surveille ses quantités de calories ou autre conneries du genre). Ainsi elle supprime la condition de possibilité d’une culture. Exemplairement : le travailleur n’est plus un individu vivant et sensible qui s’éprouve lui-même concrètement dans son action sur le monde et avec les autres, puisque ce n’est plus qu’une ressource offerte à la gestion managériale, ressource qu’on pourra dégraisser, compresser, virer, réduire, etc. en fonction des lois « objectives » du marché et de l’intensité de l’effort fournit que des appareils de mesure sont censés objectiver. C’est cela, la barbarie : le manager en costard cravate qui nie la vie au nom d’une technique gestionnaire qui nie l’homme en son entier. C’est une négation de la culture et, par conséquent, une déshumanisation.
.
La marchandisation totalitaire du monde, comme le rappelle FB, fait peut-être partie intégrante de cette barbarie. Dans les produits dits « culturels », on consomme plus des marchandises que des œuvres, et la bourgeoisie est tout particulièrement concernée par cela.
.
Dans ce cadre, je ne suis pas certain qu’on puisse encore parler de « culture » à propos de la bourgeoisie, et ainsi le problème de la « culture légitime » est peut-être, en effet, totalement périmé. Périmé pour des raisons qui tiennent à ce processus historique qui aboutit à cette forme de barbarie. Ou alors il faudrait parler de « barbarie » plus légitime qu’une autre, mais ce serait avoir les idées mal placées que de prendre au sérieux une telle question… Peut-être que la culture est devenue un « underground » qui se fraie des chemins comme elle peut au milieu de cette barbarie technique et marchande qui devient hégémonique.

[Jeanne]

Émile tu abondes dans mon sens et j’abonde dans le tien. A 100%.

[I.G.Y]

Cette prétendue technicité bourgeoise est un vaste écran de fumée. La réalité est que de très vastes pans de la « bourgeoisie » — je mets des guillemets car on utilise là le mot dans un sens très large — n’ont aucun appétit pour la technique, que celle-ci soit intellectuellement puissante ou ridiculement pseudo-scientifique. On peut entre autres le contempler de façon spectaculaire à travers les médias : les pans parmi les plus influents de la bourgeoisie et des classes dirigeantes se contentent avant tout de donner leur avis, de commander et de posséder. La technique, même à ces fins là, ils la voient de très loin.
.
C’est en parfait accord avec cette fumisterie qu’est singulièrement délaissée la recherche universitaire, y compris très largement dans les sciences « de la nature » (quoique ces dernières ne soient pas les plus mal loties). C’est particulièrement vrai en France. Et l’on peut aussi constater matériellement que dans la hiérarchie des revenus, les métiers les plus techniques voire les plus scientifiques ne sont pas du tout les plus rémunérés (c’est encore une fois plus vrai en France que dans d’autres pays capitalo-bougeois) — étant entendu que les métiers très techniques, quel que soit le sens du mot, restent souvent mieux payés que la moyenne/médiane (même chez les ouvriers).
.
La tendance à donner un vernis technique à tout et n’importe quoi dans la « bourgeoisie » est cela dit parfaitement vraie. Mais elle l’est de façon tout aussi inconséquente que quand, par le passé, elle ripolinait son discours à grand renforts de citations de Camus, Plutarque ou Jésus. Seule l’ère du temps a changé.

Et même parmi les outils techniques intellectuellement puissants, je déplore avec vous et tant d’autres qu’ils soient utilisés à ce point comme pure rationalisation d’un besoin de commandement. Mais au fond, puisque la principale caractéristique de la bourgeoisie au sens strict est de posséder et diriger, il est bien logique qu’elle utilise tous les outils accessibles.

[SHB]

Le lien que tu fais entre dépérissement de la recherche universitaire et le manque d’intérêt pour la technique de la bourgeoisie qui ordonne elle même la société n’est pas.si évident.
.
Ce n’est pas ici une question de technique. La recherche universitaire, c’est la recherche parfois pour la recherche sans forcément d’objectif de rentabilité dernière. C’est ça que la bourgeoisie veut tuer. Elle ne veut pas tuer la technique, elle veut tuer la technique si elle ne s’inscrit pas dans la rentabilité.
.
J’en veux pour preuve le nombre florissant de start-up ultra technologisées avec des cerveaux brillants au service du capital qui chaque jour inventent une nouvelle technique pour optimiser la rentabilité de la production.

[JeanMonnaie]

Ce n’est pas ici une question de technique. La recherche universitaire, c’est la recherche parfois pour la recherche sans forcément d’objectif de rentabilité dernière. C’est ça que la bourgeoisie veut tuer

1)La recherche à explosé sous le capitalisme. Cuba ne dépose aucun brevet
2) Le capital à justement tellement d’argents qu’elle peut permettre des projets non rentable.
SpaceX – Mission Mars.
IBM Q Experience : IBM investit dans le développement de l’informatique quantique .

Il est vrai que le Japon à fermé 26 facs de sciences humaines et sociales. Simplement la question de l’utilité est consubstantiel de la rentabilité.
Qu’est ce que geoffroy lagasnerie à apporté à l’humanité ?

[I.G.Y.]

@SHB tout à fait d’accord. A ceci près : la différence importante entre la technicité (même haute) en entreprise et la technicité dans la recherche. Beaucoup de techniques utilisées dans les entreprises sont basées sur des travaux d’un niveau technique encore beaucoup plus pointu (qui ont demandé beaucoup de temps / d’errements) et qui est issu … de la recherche. Il y a certes des entreprises qui pratiquent de la recherche, techniquement très charpentée, cela existe, mais un peu moins en France qu’aux US par exemple. Cela dit, il y a des chercheurs techniquement faibles, bref, il faut toujours dialectiser et être concret.

Je ne voulais pas dire que la bourgeoisie veut tuer la technique, sûrement pas (cf. plus bas). Je voulais faire remarquer que ce délaissement de la recherche entre en parfaite résonance avec le fait que le vernis technique que se donne une partie bruyante et puissante de la bourgeoisie n’est qu’un vernis. Par exemple, François a dit plus haut que la bourgeoisie n’envoie plus sa progéniture à Normale Sup. Il pensait très probablement aux sections Lettres, mais cela peut valoir pour la section scientifique, alors même que l’on y trouve depuis longtemps les plus brillants scientifiques du pays. Compte tenu de la valorisation par la société et par ses classes dominantes de la recherche et de l’enseignement, il va de soi que, par exemple, choisir Normale Sup (même Ulm) à la place de Polytechnique (quand on a le niveau stratosphérique requis pour avoir droit à ce choix royal) relève de nos jours quasiment du choix militant.
.
Dans beaucoup de domaines scientifiques les meilleurs techniciens sont encore (pour combien de temps?) dans la recherche publique. Pour l’informatique par exemple, c’est un peu différent, dans la mesure où ce savoir est tellement appliqué que la plupart de ses grands techniciens sont directement au sein de l’appareil de production.

[Emile Novis]

@I.G.Y
Je suis en grande partie d’accord avec toi, mais je ne sais pas si nous parlons exactement de la même chose. Je ne crois pas que les salaires d’untel ou untel soient au cœur de la question. Il me semble que c’est un processus à la fois historique et structurel qui peut très bien s’accommoder d’une relative prolétarisation des métiers plus spécifiquement dévoués à la technique – la banalisation de ces métiers, logique au vu de l’extension indéfinie de cette approche de la vie, tend nécessairement à niveler les salaires.
.
La belle expression que tu proposes en parlant d’une « rationalisation d’un besoin de commandement » me paraît s’inscrire dans cet esprit technicien qui ne perçoit dans ce qui est que son efficience, c’est-à-dire la production d’un maximum d’effets avec un minimum de moyens. Le fait de voir, dans tout ce qui est, une ressource dont on peut extraire le maximum d’effets (économiques, pouvoir, force, communication, etc.) colonise la totalité du mode d’être dominant, jusque dans le langage, qui est inondé d’un lexique technicien plus ou moins pertinent selon les cas. Dans ce processus, l’existence vécue n’apparaît même plus, les voies d’expression qui permettent à la vie de s’humaniser (la culture) n’ont plus leur place : seule compte une organisation « objective » et désincarnée du monde.
.
Exemple anecdotique mais tellement symptomatique : le marcheur ne s’appuie plus sur le sentiment intime et vécu qu’il a de lui-même, mais il se fie à un dispositif technique qui calcule le nombre de ses pas, ce qui signifie que son rapport au monde et à lui-même est complètement désincarné et dévitalisé, avec, à la place, des données dites « objectives » parce que quantifiables et mesurables. Autre exemple : le monde du sport, et notamment le football, est lui aussi complètement colonisé par les datas, les statistiques en tout genre, etc., à tel point que cela devient le plus souvent ridicule, d’autant plus qu’il semble y avoir là une croyance obstinée dans la validité d’une telle approche du réel. La qualité disparaît progressivement au profit d’une approche quantitative de l’existant.
.
C’est en ce sens qu’on peut parler de barbarie, c’est-à-dire d’une négation de la culture. Car il n’y a pas de culture en dehors d’une chair vivante qui s’exprime à travers des pratiques et des œuvres qui lui permettent de croître et d’exister humainement. La vie est malade de cela, frustrée dans sa capacité à se retrouver elle-même dans le monde. Dernier exemple : l’architecture. Les immenses centres commerciaux qui entourent les villes n’ont plus grand chose à voir avec les halles de nos centres ville historiques. Dans le second cas, il y avait bien une fonction utilitaire, mais elle n’était pas décorrélée d’un souci esthétique; dans le premier cas, il n’y a que des plateformes logistiques conçues par des ingénieurs et permettant une optimisation maximum de l’organisation de l’espace. La vie ne peut pas se sentir chez elle dans de tels lieux. Il est frappant de voir que bien des bâtiments publics sont désormais fabriqués sur ce modèle à la fois transparent et logistique.
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Si tout est appréhendé de cette manière, y compris les « productions artistiques », alors il devient difficile de parler encore de « culture » à propos de la « culture dominante ». Nous sommes peut-être devant un dépérissement de la culture dont la pointe avancée serait la bourgeoisie, à tel point que parler de « culture bourgeoise » devient problématique. Le « temps de cerveau disponible pour Coca Cola » est une expression qui résume à elle seule l’approche bourgeoise des « productions » diverses et variées qu’on appelle encore, par habitude, « culture ».
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Ce n’est pas vraiment une approche sociologique, mais on peut en tirer des conséquences relativement aux discussions sociologiques portant sur la « culture légitime ».
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@SHB
Il me semble aussi que les historiens font de plus en plus appel aux datas dans leur recherche, comme par exemple l’orientation de leur recherche en fonction de la fréquence de certains mots ou expressions dans les documents, avec, par conséquent, une approche purement quantitative intégrée au cœur de la méthodologie de la recherche historique. Mais tu confirmeras ou infirmeras mon propos en fonction de ce que tu peux voir au quotidien en histoire.

[SHB]

Non tu as tout a fait raison, en témoigne mon récent cours d’informatique appliqué a l’histoire où les outils proposés par le prof rejoignaient ce que tu décris

[Claire N]

Oui , je remarque aussi cela dans la recherche
Avec un effet «  tinder »
Les paramètres et les objectifs de rentabilité présidents au financement et donc à la mise en marche du dispositif technique
Ainsi la «  rencontre « avec l’imprévu de la vie est de plus en plus écartée ; j’avais bien aimé la façon dont Foucault parlait de ses recherches lorsqu’en substance il expliquait avoir découvert quelque chose qu’il n’était pas parti chercher, c’était justement sur cet imprévu qu’il avait ensuite travaillé

[françois bégaudeau]

Bien des chercheurs savent qu’on ne trouve réellement quelque chose que lorsqu’on ne le cherche pas. La recherche a donc besoin de cheminer dans l’aléatoire, sans financiers au cul
Mais ajoutons tout de même par honnêteté, et par souci dialectique, que le capitalisme est aussi parfois très performant pour densifier certaines recherches.

[Claire N]

Effectivement force est de le constater
Les laboratoires les plus puissants financent la recherche dans la sclérose en plaques et oui les traitements sont plus performants

[I.G.Y.]

Ayant connu (connaissant) les deux faces de ce que tu décris François, je ne saurais qu’approuver.

A noter aussi les financements d’État pour la recherche privée, dont une part relève de la blague. Notamment le CIR (crédit impôt recherche), 7 Mds par an. Je peux en témoigner dans la mesure où j’ai moi même eu à remplir des demandes de CIR dans deux entreprises différentes. Or dans la première, le métier avait beau être très technique et un peu scientifique, les travaux en question n’avaient purement et simplement rien à voir avec de la « recherche », ni même vraiment avec de la « recherche et développement ». C’est purement de la subvention publique aux entreprises (notamment aux très grosses)

[I.G.Y.]

@Emile vu que l’on prend le phénomène de façon assez large, je pense tout de même qu’on parle de la même chose prise sous des angles différents. La raison pour laquelle je réinscrivais la donnée matérielle du salaire dans l’affaire, c’est qu’elle représente d’une certaine façon la valorisation agrégée des agents (de la société toute entière, et spécialement bien sûr des classes « bourgeoises »). Je trouve ce hiatus entre technicisation de façade et considération matérielle concrète intéressant.

[SHB]

Dans mon modeste domaine qu’est l’histoire, j’ai par exemple observé que dans les centres d’archives, un des critères de sélection avant même la pertinence, la valeur historique, philosophique, etc.. c’est est-ce que l’archive a des chances maximales d’être consultée pour que le centre soit rentable et continue d’être subventionné.

[Jeanne]

La technicité dont je parlais, et dont je disais qu’elle relevait peut-être d’une (récente) stratégie de distinction de la part de la bourgeoisie, ne doit en effet pas être ramenée à la technicité des techniciens. Des « vraies » techniciens. Mais plutôt, oui, à un vernis. Une petite couleur technique apposée sur toute chose, sur tout domaine de la vie, et sans ancrage dans le réel.
Comme par exemple les techniques de team building que l’on a vues récemment ici dans un court métrage de fiction très drôle posté par François. Les techniques en question ne build aucune team puisqu’au contraire, à la fin tout le monde est (encore) plus mal à l’aise qu’au début.
On revient là à l’idée, portée par François, que le Bourgeois et le réel des fois ça fait deux.
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Encore une petite illustration de ce tropisme techniciste (ou pseudo techniciste) caractérisant la bourgeoisie et s’étendant donc au-delà d’elle puisque en tant que dominante elle oriente régulièrement les propos et les esprits :
Depuis 1995 à peu près, le monsieur du métro espère que nous avons « effectué  » un bon voyage.
« Fait » , ce serait trop peu. Ça ne dirait pas bien tout l’effort et la dextérité que nous avons investi dans l’opération (bon, assis tout simplement sur nos sièges) de voyager.

[Jeanne]

J’ajouterais que de mon côté, je ne pense pas que la bourgeoisie veuille tuer quelque chose. (SHB tu dis que la bourgeoisie veut tuer la technique si la technique ne s’inscrit pas dans la rentabilité).
C’est, de mon point de vue, suffisant que la bourgeoisie soit à côté de ses pompes. Il ne me semble pas nécessaire d’en plus lui prêter des intentions destructrices.
Ce serait presque, et d’une certaine manière, lui faire trop d’honneur.

[I.G.Y.]

Je suis bien d’accord qu’il y a une « technicisation » de la novlangue, c’est frappant. Autre effet de ce nouvel air du temps.

Quant à savoir s’il s’agit d’une stratégie de distinction, c’est une vraie question. Ça me semble beaucoup plus profond. La « technique » est en effet un nouveau « paradigme » (pour reprendre Émile), dans un sens fort. C’est le cadre général du monde moderne « sécularisé », qui devient par la même religion séculière. Elle imprègne tout. Que la bourgeoisie se moule dans ce cadre et en fasse usage, c’est certain. Nombreux sont ceux qui ont compris qu’il y avait certes là beaucoup de « bullshit », mais une grande puissance concrète de compréhension et de maîtrise du monde. Mais imagine-t-on la bourgeoisie délaisser ce cadre une fois la « technicisation » imposée partout (ce qui serait typique d’une stratégie de distinction)? Certains secteurs de la bourgeoisie le feront (et le font déjà), c’est certain (ceux qui précisément ne sont pas à l’aise avec la technique et les sciences : on trouve déjà aujourd’hui sans peine des discours bourgeois à droite qui sont très critiques de la « rationalisation technique ». Il y en a plein Youtube, les titres des vidéos suffisent). Assistera-t-on à une recrudescence de « l’aristocratisme » (au sens non-étymologique) dans la bourgeoisie en réaction à la victoire du paradigme « technique »? Possible !

[Emile Novis]

@I.G.Y
« Assistera-t-on à une recrudescence de « l’aristocratisme » (au sens non-étymologique) dans la bourgeoisie en réaction à la victoire du paradigme « technique »? Possible ! »
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C’est une question que je me pose aussi. De fait, cette partie de la droite existe encore aujourd’hui, mais elle est extrêmement minoritaire et impuissante. C’est la droite contre révolutionnaire qui s’inscrit dans la critique de la modernité, souvent issue des franges du catholicisme traditionnaliste – même si elle va au-delà de ce cercle désormais -, avec le rejet du modernisme et, plus fondamentalement, la conviction que les conditions de la foi – telle qu’ils se représentent la foi – et le mode d’existence « authentique » qu’ils louent sont rendus impossible par le rationalisme scientifique et technicien qui advient au XVIIème siècle et poursuit son cours encore aujourd’hui. Cette droite dont tu parles est l’héritière de ce courant, semble-t-il – on les a vu ressortir dans leur critique du vaccin contre le covid, d’ailleurs. Il s’agit de la droite contre-révolutionnaire, souvent ancrée dans un naturalisme naïf et très hostile aux conditions de vie moderne.
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Mais je pense que ce courant ne peut pas dépasser le stade de la posture très minoritaire et impuissante à modifier le cours des choses, et qu’il est impossible qu’elle incarne un courant politique réel. Tout simplement parce que leur condition d’existence objective dépend intégralement de ce paradigme technique qu’ils conchient par ailleurs, que ce soit pour la sécurisation concrète de leur privilège, le fonctionnement de la finance, la productivité dans le cadre de la compétition économique mondiale, etc., autant de domaines qui contraignent la bourgeoisie à épouser totalement les dispositifs techniques modernes sous peine de périr immédiatement. Ils pourront bien louer la petite manufacture ancestrale et fantasmée qui laisse au travail la beauté du geste et la vie déconnectée, ils changeront assez vite d’avis quand ils se feront écraser par une entreprise connectée hyper-moderne et ultra-connectée qui produit 20 fois plus en 3 fois moins de temps.
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Dans le cirque électoral, je crois qu’une partie de la droite filloniste incarnait un peu cette contradiction comique entre la « conservation des valeurs traditionnelles qui ne se réduisent pas à l’argent » (nos paysages, notre mode de vie des années 50, etc.) et la nécessité de composer avec le devenir historique et technique de nos sociétés. Inutile de dire que la technique et la finance l’emportent toujours dans cette histoire, les « valeurs conservatrices » n’étant plus qu’une carte postale touristique – car les mêmes qui veulent « protéger nos paysages » de la laideur moderne sont les premiers à optimiser le tourisme de masse et l’extension illimité des zones commerciales qui bousillent tout sur leur passage, organisant ainsi un aménagement technique du territoire entièrement dévoué à la rentabilité financière.
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On remarquera d’ailleurs que Fillon a intégré un fond de pension financier qui spécule sur tout ce qui existe. Comme quoi…
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Je crois donc que cette position restera nécessairement marginale : une sorte de droite esthétique pour des petits parcours individuels mis en spectacle. Un maintien de la biodiversité, en somme, propre à cet écosystème de droite, en somme, mais rien de plus à mon sens.

[Emile Novis]

Pas le temps de me relire. Pas mal de fautes et de mots répétés, mais ça reste intelligible.

[françois bégaudeau]

« Dans le cirque électoral, je crois qu’une partie de la droite filloniste incarnait un peu cette contradiction comique entre la « conservation des valeurs traditionnelles qui ne se réduisent pas à l’argent » (nos paysages, notre mode de vie des années 50, etc.) et la nécessité de composer avec le devenir historique et technique de nos sociétés. Inutile de dire que la technique et la finance l’emportent toujours dans cette histoire, les « valeurs conservatrices » n’étant plus qu’une carte postale touristique – car les mêmes qui veulent « protéger nos paysages » de la laideur moderne sont les premiers à optimiser le tourisme de masse et l’extension illimité des zones commerciales qui bousillent tout sur leur passage, organisant ainsi un aménagement technique du territoire entièrement dévoué à la rentabilité financière.
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On remarquera d’ailleurs que Fillon a intégré un fond de pension financier qui spécule sur tout ce qui existe. Comme quoi… »

Pas mieux.
La distinction entre bourgeoisie hard et cool s’abolit totalement dans le concret de leurs vies respectives, dans le concret de leurs affaires. Cette distinction subjective (au sens : la façon dont ils se perçoivent, se nomment, se décrivent, s’ornent) n’a aucune réalité objective.
Pour complément : Jesus les bourgeois et nous.

[Emile Novis]

@FB
J’ai acheté, il y a quelques mois, le livre Jésus, les bourgeois et nous, mais je n’ai pas encore eu le temps de le lire. Je vais me pencher dessus.
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@Jeanne (par rapport au message en dessous)
Je ne pense pas trahir ta pensée en disant que le processus de distinction sociale n’opère plus, désormais, au sein de la culture, que la bourgeoisie déserte de plus en plus, mais au sein de l’intégration plus ou moins réussie de leur existence aux dispositifs techniques et à la consommation des marchandises produites par le système économique dominant. Intégration jamais terminée, d’ailleurs, puisque ces dispositifs sont en constantes « évolution », et il y a toujours une énième innovation pour venir exiger un effort d’intégration supplémentaire.
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@I.G.Y
Je pense que nous sommes d’accord sur l’essentiel, en effet.

[Jeanne]

Merci Émile pour cette juste restitution.

[I.G.Y.]

Je crois tout à fait que cette frange restera minoritaire. Mais pas sans influence. Son rôle est bien connu et sera toujours de tirer la tendance capitaliste dans un sens plus réactionnaire et plus illibéral (avec des conséquences très concrètes, mais pas sur le fond du mode de production). Je pense qu’ils ne sont pas du tout assez forts pour infléchir le paradigme technique (d’autant que bien entendu, et comme vous l’avez rappelé, ils en profitent).
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Rejeter la technique quand on fait de l’art, par exemple, peut éventuellement rendre puissant. Dans le concret de la production (et dans le contrôle de l’État), ce rejet est voué à les rendre impuissants et clownesques.

[I.G.Y.]

C’est typiquement le genre de bourgeoisie qui pourrait si besoin se faire déposséder par une sorte de « nouvelle révolution bourgeoise », de la même manière que la noblesse avait été sortie du jeu (noblesse qui d’ailleurs était aussi très largement… bourgeoise, cf. noblesse de charges etc…)

[Jeanne]

On pourrait dire que le paradigme technique (ou technologique) est partout et donc déborde (largement) la bourgeoisie comme aussi ses stratégies de distinction. Oui.
Alors je vais essayer d’étayer autrement cette intuition (certes un peu floue) que j’amenais.
Les bourgeois ont toujours justifié leur domination par l’idée qu’ils possédaient plus d’intelligence et plus de savoir.
(D’ailleurs possédaient-ils, possèdent-ils encore plus d’intelligence et plus de savoir ? C’est une vraie question, et qu’il faudrait ne pas éluder, même si là tout de suite on va le faire quand-même).
Autrefois, plus d’intelligence et plus de savoir signifiait: J’ai lu des livres, j’ai une culture générale (et légitime) avec aussi éventuellement une culture scientifique.
Maintenant, cela signifie plutôt : Je maîtrise PowerPoint, je connais les noms des acteurs français en vogue, je vois plein de séries (car ça me plaît et en plus j’aime me rapprocher du peuple, en tout cas de cette manière-là et qui n’engage à rien), j’ai décidé, au boulot, de me plonger dans la refonte de l’organigramme afin de passer d’un organigramme fonctionnel (fonctionnel = caca, comme fonctionnaire) à un organigramme stratégique (stratégique = la classe, la mienne, justement, celle dont on parle).
Les bourgeois d’aujourd’hui sont globalement tout aussi nullards, voire plus nullards, en matière d’assise technique de leurs vies. (Ils ne visitent pas les centrales nucléaires présidant à leurs existences connectées ; personne – je crois – ne visite les centrales nucléaires, mis à part ceux qui y travaillent), mais ils conçoivent des organigrammes stratégiques, des PCSES: projet culturel, scientifique, éducatif et social, et consomment intelligent.

[I.G.Y.]

L’intuition me paraît juste dans l’ensemble, mon intervention ne visait qu’à apporter quelques précisions qui m’importent, pas à la délégitimer.
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« possèdent-ils encore plus d’intelligence et plus de savoir ? C’est une vraie question, et qu’il faudrait ne pas éluder ».

Vraie question en effet. On n’y entrera pas vraiment, sauf à trop digresser. De larges de pans de la bourgeoisie ne possèdent soit pas tellement de savoirs, soit des savoirs très puissants mais très spécialisés/partiels, sans véritablement élargir leur focale. Mais le reproche de l’élargissement de la focale est à manier avec précaution car il est réversible, et ne concerne pas que la bourgeoisie.

Sur plusieurs aspects de la bêtise bourgeoise, un certain maître des clés ici présent a griffonné deux trois trucs là dessus ^^

[SHB]

Pour le côté « illiberal » de la bourgeoisie de droite dure je pense que François a déjà démasqué la supercherie.

[Jeanne]

C’est une question tellement compliquée, celle de l’intelligence. Et tellement pavée d’embûches.
Rien qu’à définir ce mot ça nous prendrait bien 3 semaines de forum.
(J’ai bien sûr lu et adoré Histoire de ta bêtise, I.G.Y).

[I.G.Y]

Oh je n’en doutais pas 😉

[SHB]

D’ailleurs est-ce que se serait pas une bonne idée comme nouveau livre de FB, que reste-il d’une culture de la bourgeoisie française en 2020 ?

[françois bégaudeau]

Sujet qui ne vaut pas en soi un livre, mais pourrait s’insérer dans un livre à sujet plus vaste.

[SHB]

Bon du coup vous la voulez ma théorie sur la pertinence d’une culture bourgeoise au sein des sciences sociales universitaires ou?

[SHB]

Persistance*

[SHB]

Je vous la donne quand même :
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Selon moi, la notion de « culture légitime bourgeoise » reste pertinente dans le strict cadre des sciences sociales universitaires d’après plusieurs points que j’ai pu remarquer cette année :
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1) Absence presque totale de référence a des auteurs/chercheurs communistes ou anarchistes par les professeurs, quand bien même les cours traitent de sujets explicitement liés a des courants intellectuels. Au contraire, les professeurs privilégient des lectures assez consensuelles et libérales.
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2) Conception du rôle du chercheur : on nous répète a longueur de cours qu’il est mal de prendre position dans un travail universitaire tout en faisant la promotion constante d’une certaine neutralité déontologique assez abstraite et faisant fi des affects.
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3) Méthode de rédaction des travaux : les dissertations où une hypothèse concise et clairement dirigé vers une option ou une autre en début de travail font en sorte que la possibilité de rendre au réel sa complexité est atténuée. Bien souvent, le travail se conclu donc par une analyse qui omet toute une partie du réel, contradictoire par essence. On nous encourage au contraire a établir des vérités tout en nous disant que cette dernière n’existe pas vraiment. Contradiction classique du professeur d’université qui peu tout autant nous parler de l’importance de ne pas tomber dans le mépris social face aux « non universitaires » qui s’intéressent a l’histoire tout en faisant une distinction entre histoire professionelle et histoire amatrice dans la même phrase.
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ETC…..

[SHB]

Voici déjà 3 éléments constitutifs a mon sens de la présence d’une culture légitime bourgeoise des sciences sociales en particulier en ce qui concerne la distinction (de classe pourrais-je dire) entre histoire professionelle (légitime, diplômée) et histoire dite amatrice (passionnés non diplômés, youtubeurs, etc).

[SHB]

4) la forme plutôt que le fond : la licence et le master en histoire sont des exercices de méthodologie davantage que de réflexion pure d’historien (ce sont les mots de mes profs). Cela veut dire qu’un mec qui enseigne au lycée ou autre avec un master a passé L’ENTIÈRETÉ DE SES 5-6 ANNÉES UNIVERSITAIRES a travailler la forme plutôt que le fond de sa pensée. Conclusion, la formation en science sociales a l’université sert d’abord a se distinguer par la forme très académique que prend sa réflexion plutôt qu’au fond tangible de la pensée, ce qui est un autre marqueur de distinction social qui révèle la présence d’une culture légitime bourgeoise des sciences sociales face a des travaux (avant même de savoir de.quoi ils traitent) considérés comme « pas aux normes ».

[Titouan R]

@SHB
Je souscris à une partie de ton propos pour avoir moi-même rencontré ces pratiques ailleurs (faculté de droit) : promotion éhontée de la neutralité axiilogique, acharnement sur la méthodologie et la forme de ce qui est rendu plus que le fond.
Mais le problème, c’est que tu es confus en parlant de « culture bourgeoise légitime ». Ici, tu ne parles pas de culture (au sens d’art, par exemple) mais d’un problème de « champ » universitaire, qui cherche à maintenir sa distinction d’avec d’autres champs (médiatique, scolaire,…) – il y a une logique de clôture externe du champ (caractérisée sans doute en science humaine par une fétichisation du discours scientifique, de l’usage d’une forme et d’un jargon rendant le champ inaccessible de l’extérieur). Il y a aussi des logiques de division internes (en disciplines ; en écoles théoriques au sein d’une discipline) relevant moins directement d’une question de légitimité que de pouvoir (être publié, être en vue sur son domaine de recherche, occuper telle place administrative dans la fac,…) et les acteur.rices dominant.es du champ recouvrent leur domination, dans un 2nd temps, par la promotion d’une soi-disant légitimité académique. Ces jeux de pouvoirs se retrouvent partout ailleurs qu’à la fac.
Tu parles ensuite d’options politiques tues ou écartées par la fac : oui, certes. Cela est connu et relève la encore de logiques de champ, que viennent influencer les affets politiques des universitaires (là où d’autres champs sont peu perméables à ces questions : un maçon construira un mur à peu près de la même façon qu’il soit de gauche ou de droite)

[SHB]

Les points que tu soulève sont intéressants mais tu élude la question de la scission entre histoire professionelle et histoire amatrice.
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Je pense qu’à la vue de la longévité de l’école dans l’histoire de l’humanité (premières universités au moyen-âge) et les mécaniques constantes de distinction sociale dans l’apprentissage pour se distinguer des savoirs dits « non légitimes » (car non porteurs de diplomes ou vus comme pas pertinents car appris par soi même donc c’est bien connu on a tous besoin d’un professeur (lol)) on peut réellement parler d’une culture universitaire qui pour moi est bourgeoise et d’une culture vue comme légitime si l’on parle du sujet de comment on acquiert des connaissances. Dans notre société, le seul mode d’acquisition de connaissances légitime, c’est a dire qui peut déboucher sur quelque chose de socialement valorisant (salaire, diplôme, capital social, etc) est l’école (universitaire)

[I.G.Y]

Accord plus que modéré avec ce qui est écrit au-dessus @SHB (mais c’est encore une fois une question de flou sur les termes, en particulier la différence entre légitimation « sociale » et légitimité « académique » de travaux de recherche. J’ai presque l’impression que tu mélanges consciemment les deux. Mauvaise impression? ).
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Soyons très concrets afin de dissiper le brouillard : un docteur en histoire du XIXè Français dont les travaux (par exemple d’HDR) portent sur Commune de Paris a-t-il autant de légitimité a priori (ou statistiquement) pour parler de la Commune de Paris que Jean-Michel J’aime-L’Histoire? La réponse permettra de savoir si la suite de la discussion est utile ou non. (je précise que je pose la question sachant que l’on est en France, pas en URSS sous Staline).
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D’autre part, je n’ai aucun doute qu’il existe une tendance générale à survalorisation de la « neutralité axiologique », strictement aucun. Même si je m’empresse d’ajouter qu’un effort maximal de neutralité dans la lecture des sources est évidemment la base de tout travail de recherche (croire à l’absence totale de neutralité dans la pratique d’une science sociale est exactement aussi faux que de croire à la neutralité axiologique complète ou presque complète).

Quand je lis ce que tu dis sur les profs d’histoire, j’ai vraiment le sentiment que tu n’as surtout vraiment pas de chance (ou alors que tu déformes ce qu’il dise, ou encore que tu décris surtout une « tendance »). J’écoute beaucoup les historiens et notamment sur leurs problèmes de méthode, et ce que j’entends assez souvent est beaucoup plus fin que ça

[Titouan R]

Accord avec IGY. Je pense qu’il y a un problème de confusion terminologique. Problème dont je ne suis pas exempt, ayant rabattu, dans mon message de ce matin, la culture sur « l’art », quand SHB évoquait plutôt la « culture » au sens de savoir. Il n’en demeure pas moins que je maintiens mon propos de ce matin sur les logiques de champ universitaires.
Et SHB, je pense que ton appréciation de la « culture légitime bourgeoise » est très datée. L’université s’est prolétarisée, dans son recrutement et son prestige externe. Il faut avoir bonne vue pour y voir « l’élite » intellectuelle qu’elle était aux yeux du corps social il y a longtemps.
Ainsi, sérieux doute quand tu écris « Faut vraiment avoir de la merde dans les yeux pour pas voir que l’université est faite pour modeler le savoir aux codes de la bourgeoisie et former une élite sociale seule dépositaire de la connaissance légitime »
Doute renforcé par le fait que les enfants de la bourgeoisie désertent pas mal la fac (au profit d’écoles privées, prépas, séjours Erasmus).
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A la vérité, pour sortir du strict terrain universitaire, ne faudrait-il pas creuser l’idée que :
– la « culture bourgeoise légitime » a changé dans ses formes et contenus privilégiés (ex : série aujourd’hui art légitimé, autant que consensuel…) ;
– la bourgeoisie se pique moins de culture qu’avant (ce qui n’est pas forcément positif). Hypothèse esquissée dans Histoire de ta bêtise et Boniments.

[I.G.Y]

Tout à fait @Titouan, je ne nie pas l’existence (et l’importance!) de ces logiques de champ. Elles existent même hors des « sciences humaines ». Quant au problème du silotage par sous-discipline et par « écoles », c’est certain aussi.

Là où tout se mélange, c’est qu’on peut tout à fait avoir des chercheurs « à juste titre » reconnus et sérieux qui, du fait des positions de pouvoir qu’ils acquièrent, finissent par « faire chapelle » et par scléroser la réflexion (plus le chercheur en question a un tempérament autoritaire, pire c’est). Et ceci est potentiellement vrai partout, à des degrés divers (les effets de chapelle sont moindres dans lesdites « sciences dures », c’est sûr, mais ils existent. C’est aussi, d’une certaine façon, ce qui fait avancer la recherche…).

[SHB]

un docteur en histoire du XIXè Français dont les travaux (par exemple d’HDR) portent sur Commune de Paris a-t-il autant de légitimité a priori (ou statistiquement) pour parler de la Commune de Paris que Jean-Michel J’aime-L’Histoire?
.
Dit nous plutôt ce qui te permet de penser que le doctorant est plus a même de parler dudit sujet que « jean Michel » si ce dernier se documente sérieusement sur le sujet.
.
Quand je lis ce que tu dis sur les profs d’histoire, j’ai vraiment le sentiment que tu n’as surtout vraiment pas de chance.
.
A te lire on croirait presque que tu te sens personnellement touché par la remarque. De plus tu confonds l’écoute de chercheurs qui parlent de leur recherches avec l’exercice de discipline académique qu’est l’enseignement qui découle sur des mentalités et des pratiques qui induisent d’ailleurs la forme et le fond que prenne les recherches. Je pense que le fait que la plupart des professeurs d’université soient des humanistes réformistes moi du genou a pas complètement rien a voir avec la manière dont ils travaillent et conçoivent leurs recherches à travers leur belle méthodologie toute bien ficelée.

[SHB]

Mou*

[I.G.Y]

« Dit nous plutôt ce qui te permet de penser que le doctorant est plus a même de parler dudit sujet que « jean Michel » si ce dernier se documente sérieusement sur le sujet. »

Pour des raisons évidentes : statistiquement, globalement, un docteur en histoire sur la Commune de Paris (tu parles de « doctorant », je n’ai pas parlé de doctorant, mais peu importe) a travaillé beaucoup plus profondément le sujet, il a généralement, entre autres, travaillé sur archives (dont bon nombre ne sont tout simplement pas accessibles aux non doctorants et docteurs). Il a aussi statistiquement plus de chances de ne pas tomber dans des pièges de lecture de sources, il bénéficie de meilleures conditions pour accéder à une mise en perspective d’un certain nombre de faits (ce que j’appelle faits en histoire est très précis : untel a dit ça à telle date, untel est mort tué par untel à tel endroit tel jour, untel a écrit ceci à tel endroit, etc… Je ne parle pas de conclusions générales). Bref, il a toutes les chances d’être plus précis sur ce qu’il dit, plus de chances d’avoir de s’être « sérieusement documenté sur le sujet » comme tu dis. Et dire cela n’empêche en rien que sur des points locaux précis, des non-historiens contredisent l’état de l’art (notamment, par exemple, s’ils ont été des témoins et qu’ont peut établir que leur témoignage est corroboré par des preuves matérielles qu’ils apportent). Ça n’empêche pas non plus des divergences d’interprétation ou de volonté politique pour l’avenir. Et ça n’empêche par non plus des historiens amateurs de travailler si sérieusement qu’ils finissent par être reconnus par les historiens eux-mêmes (comme tu le sais, Philippe Ariès par exemple). Pas davantage que cela ne prémunit contre l’existence d’historiens très mauvais. On parle de tendances. Les gens qui n’apportent rien à la recherche seront juste oubliés (et parfois, il arrivera que certains soient oubliés alors qu’ils n’auraient pas dû l’être, etc…).
.
« A te lire on croirait presque que tu te sens personnellement touché par la remarque »

Je ne suis ni prof ni historien. Je constate simplement que, manque de chance, les chercheurs que j’écoute chaque semaine parler des sujets sur lesquels ils ont passé 10 ans, parfois 20 ans, ou même leur vie entière, sont aussi souvent… enseignants (et minoritairement macroniens ou hollandistes). Je dois donc tomber sur les seules et uniques enseignants sérieux du pays? Peu probable. Et s’il s’agit de plaider pour le fait qu’un enseignant soit possiblement meilleur s’il est enseignant-chercheur, je veux bien te suivre sans problème.

D’autre part, tu me contredis en disant parler de « la plupart » des enseignants, alors même que je t’ai concédé qu’il y a une tendance bien réelle à la promotion bébête de la neutralité axiologique (la palme revenant peut-être à ladite science économique?). Le fait qu’il existe des biais sociologiques dans la recherche et l’enseignement à l’université n’a jamais été contesté ici.

[SHB]

On devrait se méfier de tout propos qui commence par « statistiquement ».

[I.G.Y]

D’accord, donc donc toi tu parles de « tendances » (mot que je reprends) et on ne devrait pas se méfier? Quand je dis statistiquement c’est bien sûr un synonyme de « tendances » majoritaires. Je ne vais pas utiliser 100 fois le mot tendance

[SHB]

Il a aussi statistiquement plus de chances de ne pas tomber dans des pièges de lecture de sources.
.
Tu nous assène des vérités a la Jean Monnaie là, ton affirmation se base sur rien si ce n’est l’idée que tu te fais du chercheur, si seulement tu voyais comment ils travaillent je pense que tu changerais d’avis.

[I.G.Y]

C’est bizarre que tu ne te sois pas aperçu au vu de ce que j’ai écrit plus haut et de tous les détails que je donne, que justement je connais vraiment bien la recherche. Et c’est exactement parce que je la connais que contrairement à ce que tu essaies de me faire dire, et en conformité avec tout ce que j’ai écrit, il ne me vient pas une seule seconde à l’idée de sanctifier et idéaliser la position du chercheur. Raison pour laquelle je passe autant de ligne à faire en sorte qu’on ne croie pas que je l’idéalise.

Quant à ce « comment ils travaillent », je renvoie donc à Jean Monnaie. Puisque la discussion prend cette tournure

[SHB]

Même si je m’empresse d’ajouter qu’un effort maximal de neutralité dans la lecture des sources est évidemment la base de tout travail de recherche.
.
Ça tombe bien j’ai jamais dis le contraire.

[SHB]

L’université est un crime contre la radicalité.
.
Tout est toujours en nuance pour la nuance, en peur de prendre position, en écrits platoniques, en forme méthodologique incompréhensibles pour les non initiés avec des profs qui parlent de l’importance de l’histoire et du manque de culture historique des gens c’est quand même un foutage de gueule assez sensationnel pour des mecs pas lus et illisibles pour 80% de la population.
.
Faut vraiment avoir de la merde dans les yeux pour pas voir que l’université est faite pour modeler le savoir aux codes de la bourgeoisie et former une élite sociale seule dépositaire de la connaissance légitime (en témoigne d’ailleurs ton mépris pour « jean Michel histoire » qui traduit bien ce que je dis, a savoir que les sciences sociales en dehors de l’université c’est bien mignon mais sous entendu c’est pas assez rigoureux. À quoi je réponds lol mdr).

[I.G.Y]

« Faut vraiment avoir de la merde dans les yeux pour pas voir que l’université est faite pour modeler le savoir aux codes de la bourgeoisie et former une élite sociale seule dépositaire de la connaissance légitime » : mais qui a dit que cette tendance là n’existait pas? On mélange tout.

Quant à mon mépris pour Jean-Michel J’aime-L’Histoire, je suis ravis d’apprendre, donc, que je me méprise moi-même. C’est vrai que j’ai une certaine tendance à l’auto-dévalorisation (je trouve cela amusant), mais tu m’apprends que c’est plus grave que prévu !

[Charles]

SHB, je connais pas le sujet de l’intérieur comme toi mais n’est-ce pas un peu caricatural quand même? Beaucoup de grands historiens français sont issus du sérail universitaire ou grande école type Normale sup, de l’Ecole des annales à Ingrao et Chapoutot, pas exactement des Jean-Michel faisant de l’Histoire. Cela ne veut pas dire qu’ils n’ont pas du affronter en leur temps des résistances académiques – je pense également à Foucault qui était pour le coup en marge, marge relative car il a quand même été adoubé par les institutions. même si les historiens ont du mal à l’accepter. Et on peut être un socedem et produire un travail d’historien intéressant, dernière exemple en date André Loenz qui nous régale avec ses Paroles d’histoire alors qu’à côté il pense que Tondelier pourrait être la solution pour 2027.

[SHB]

Dire qu’il y a une tendance ne dit pas que des exceptions ne sont pas possibles (la preuve Barbara Stiegler, Lordon, Friot, même Aurélien Barrau) mais dès exceptions n’infirment pas le fait majoritaire de l’université.

[Charles]

Mais alors comment l’expliquer? Est-ce que l’institution n’est pas capable de produire les deux, une approche conformiste et une plus profonde voire radicale? Si les historiens amateurs donnent rarement autre chose que des vulgarisateurs, c’est parce que faire de l’Histoire demande du temps (donc de l’argent) et l’accès aisé à des sources, des travaux de recherche et la discussion entre chercheurs, ce que permet l’institution, bien qu’avec les différentes réformes touchant l’université cela doive être moins vrai.

[SHB]

ce que j’appelle faits en histoire est très précis : untel a dit ça à telle date, untel est mort tué par untel à tel endroit tel jour, untel a écrit ceci à tel endroit, etc… Je ne parle pas de conclusions générales.
.
C’est pas si précis que ça justement, tu conscrit ta vision de l’histoire aux temps modernes et a l’époque contemporaine. Les vieux écrits sont souvent rédigés par des nobles au service du pouvoir et les « faits » sont parfois inventés de toute pièce, il est donc difficile de démêler le vrai du faux. J’ai comme exemple la plus connue des falsification historique démasquée « récemment » : le faux edit de Constantin

[I.G.Y]

C’est exactement ce que je dis, donc je ne comprends même pas le désaccord. C’est très précis. Untel a écrit un truc à tel heure, ça demande à être vérifié. Le simple fait que tu dises « le faux édit de Constantin » montre que bien sûr tu considères qu’il y a des faits en histoire du genre de ce dont je parle. Ça n’empêche absolument pas l’existence de « faux » (il peut être un fait qu’il existe un faux).

[I.G.Y]

Et il y a des faits plus simples que d’autres, bien sûr. Après, de toute manière, on peut toujours tout mettre en doute puisque la certitude absolue au sens strict ça n’existe pas. On peut toujours nier qu’un type nommé Macron soit président, on peut toujours rejeter le témoignage de 4786 personnes qui ne se sont pas concertées et qui on vu un truc, on peut toujours, en effet

[SHB]

T’a BAC +2 en sophisme ?

[I.G.Y]

Bac + beaucoup plus que ça. Comme tu sais, il y a aussi des gens qui disent qu’on vit dans une simulation. Et ils pourront toujours s’en sortir! (Ça ne change rien au fait qu’ils me fassent subjectivement pitié)

[SHB]

Après si t’a de gros diplômes universitaires je comprends mieux pourquoi tu défends bec et ongles l’institution qui te fait socialement exister

[I.G.Y]

Tu parlais de bac en sophismes, je rebondis sur ta blague et tu oublies ta propre blague. Bien

[I.G.Y]

Par ailleurs cette institution ne me fait pas exister, tu ne sais pas ce que tu racontes, tu commences définitivement à te décrédibiliser, c’est dommage.

[SHB]

Après dans la catégorie je ne sais pas ce que je racontes je pense que j’ai trouvé un maître là.

[SHB]

La fausse donation de Constantin**

[SHB]

Si vous considérez qu’un chercheur peut s’émanciper des structures qui le mette au monde vous êtes un peu naïf (en témoigne par exemple la quantité infinitésimale de travaux qui critiquent l’institution scolaire, domaine pourtant ultra important et massif dans notre société actuelle qui devrait attirer la curiosité et la critique?

[Charles]

Personne ne dit ça (en tout cas pas comme ça) mais bon t’es pas venu pour discuter mais pour asséner des vérités que toi seul as pu constater, pas vrai?

[SHB]

Je te retourne le compliment

[Dune]

Bon, c’est une discussion bloquée. Dommage elle était intéressante. Que l’université provoque de l’élitisme augmenté d’une forme de conformisme (un centrisme mou, une apologie du raisonnable, etc.) est indéniable, mais il faudrait s’attaquer non à l’université en tant que pourvoyeuse, par nature, de doctrine molle produisant des petits soldats centristes mais à l’institution, sa structure même au sein du système dans lequel nous vivons. Préciser les positions relatives en son sein et peut-être alors décrire un embourgeoisement conduisant les mieux placés dans le carcan de la dite pensée (quand ils n’en sont pas issus). Comme indiqué dans le titre, Bourdieu l’a en partie traité (ça a peut-être été évoqué ici mais dans le fatras du forum, ça m’a échappé…) dans Homo academicus et de nombreux articles.
Je confirme cependant Charles, la pensée molle peut cotoyer une rigueur d’analyse, voire une approche très radicale des mous dans leurs disciplines respectives. Je pense typiquement à un Offenstadt ou Serna pour l’Histoire (ce dernier centriste assumé s’est d’ailleurs fait le pourfendeur de son « bord » en partant de son objet : la Révolution Française) ou une Florence Weber en sociologie. Elle était à mes yeux affligeante de banalité dans ses commentaires d’actu mais possédait une audace bluffante dans l’art de se faire accepter sur un terrain d’enquête et une écoute de grande qualité des enquêtés. Sa « bêtise » politique s’accompagnait d’une finesse d’analyse du réel directement observé.
Quant au mépris il existe, s’étale, se plastronne même, mais je n’ai jamais été témoin de mépris ouvertement assumé pour les « incultes ». Ce qui est courant par contre c’est du mépris interne. Plus encore quand la concurrence est matérialisée géographiquement : Paris 1 vs Paris 4, l’Histoire sociale contre les petits bricoleurs de l’histoire culturelle jugés sympathiques, parfois brillants, mais scientifiquement médiocres. Je ne peux écouter Chapoutot et ses formules chocs sans un arrière-goût de cette forme de mépris inculquée.

[I.G.Y]

Peut être qu’à force que des personnes reformulent, ça finira par se débloquer. Je suis encore une fois d’accord avec à peu près tout. Dommage en effet cette discussion est passionnante

[SHB]

Après pour être honnête les réactions que suscite mes dires sur ce forum en particulier ne m’étonne qu’à moitié.
.
Comprendra qui pourra.

[Papo2ooo]

Peut être faut il partir, comme François Bégaudeau l’a fait pour les séries, des caractéristiques des « objets culturels »
Certains objets culturels pointus entrant en contradiction de plus en plus forte avec la temporalité et le paysage capitaliste, ils n’intéresse plus tellement de monde. En cela ils sont délégitimés, car ils n’ont que peu de rendements et offrent peu de débouchés professionnels. On ne compte plus les experts d’un domaine pointu qui sont complètement fauchés. Certaines pratiques culturelles s’accompagne d’une forme de marginalité au sens fort, culturelle et sociale. Un exemple typique serait le punk ou des trucs de rock ultra pointu comme Mr Bungle ou autres. Ils suscitent pour ces raisons la méfiance des famille, la méfiance des étudiants et des profs qui visent un minimum de centralité dans l’espace culturel au sens large.
Ces pratiques culturelles minortaires sont sauvées par quelques systèmes de solidarités (bénévolat, argent public, abonnement à des médias) qui permettent un peu de visibilité, un peu d’échange. La minorité s’organise pour survivre, parfois avec beaucoup d’énergie.
Des petites salles proposent de faire des concerts le week end, on organise soit même une tournée dans un bus.
Veron peut travailler sur les transclasses et la littérature grâce à l’argent public, elle fait quelques interview pour des « petits médias », elle touche quelques lecteurs. Ca survit comme cela.
–
Et d’autres objets culturels, largement « consommés », qui captent plus facilement l’attention, deviennent légitimes, car ils permettent de s’élever ou de conserver une place sociale de dominant. Là se niche la vraie légitimité.
L’argent comme facteur principal, qui pèse plus sur la balance que le prestige. Ce qui rapporte est légitime. Certains peuvent le déplorer, se moquer, vouloir se distinguer de la plèbe, mais au final ça ne change pas grand chose à la situation d’ensemble. On peut aussi débattre pendant 10 000 ans de Aya Nakamura ou de Jul, mais ces discussions sont automatiquement légitimées par l’audience qu’elles suscitent.
Ce qui fait débat est plus légitime socialement, car il y a une économie autour, que ce qui n’intéresse personne.
Il y a encore une économie autour des Philharmonie, autour du théâtre, autour du cinéma d’auteur. Donc ça reste plus légitime comme parcours professionnel que le punk rock. Mais si c’était amené à se casser la gueule, les auditeurs de musique philharmonique deviendraient des créatures marginales. On parlerait d’opéra au PMU lol et de littérature dans les bars défoncés.

[SHB]

Soit dit par parenthèse je ne sais pas qui a dit plus haut que il écoutait des historiens qui savaient justement prendre du recul sur des événements historiques et n’étaient pas soumis aux structures de l’université. La même personne disait elle même qu’elle n’avait pas de connaissances poussées en histoire et qu’elle aimait écouter des chercheurs. Je me demande donc comment ladite personne peut savoir si les chercheurs en question lui raconte des cracs vu qu’elle ne connait pas pointilleusement les évènements historiques abordés. Cela m’amène a une autre critique de l’élite bourgeoise universitaire. On arrive a un tel degré de mise en forme de la pensée, de codes linguistiques élitistes, de débats dans les débats entre historiens, etc.. qu’il devient impossible pour le commun des mortels de se positionner face aux dires d’un historien universitaire. L’histoire, comme la politique, doit être l’affaire de tous. Il est curieux que cela ne dérange personne que 80% des gens soient tributaires de la bonne parole des chercheurs sans jamais pouvoir vérifier les analyses historiques avancées pour des raisons que l’on connait et qui tiennent a la vie d’un prolétaire qui va pas passer 5heures a éplucher un texte après le travail. L’histoire ce n’est pas aussi complexe que ça en a l’air. Comme en politique, l’opération qui vise a faire paraître quelque chose comme compliqué conforte le fait qu’on attribue la responsabilité historique et politique a des professionnels des domaines ici cités. Dans ce jeu en vase clos, les classes populaires ne peuvent pas s’inviter

[I.G.Y]

« Je me demande donc comment ladite personne peut savoir si les chercheurs en question lui raconte des cracs ».

Et tu penses qu’on ne se pose pas la question ici? Ce que tu dis est valable pour absolument tout. La seule et unique réponse, c’est « travailler, écouter l’éventail le plus varié de personnes possibles, voir où il y a consensus, où il y a dissensus » etc… Ce que tu dis est entièrement réversible contre ton argumentation.

La seule et unique chose que j’ai dite en définitive c’est que « en tendance un historien spécialiste de son sujet va raconter moins de cracs qu’un type comme toi ou moi ». Chose qui est une évidence totale et qui semble te fâcher. Pourquoi n’y aurait-il strictement aucun rapport entre un doctorat en médecine et un doctorat en histoire ? Vraiment aucun ?

Je fâche du monde si je dis qu’en tendance je préfère me faire soigner la gorge par un docteur en médecine plutôt que par mon père ? (Et pitié qu’on ne se lance pas dans la discussion de « mais oui mais y’a plein de médecins qui disent des bêtises », je le sais mieux que personne, de vécu)

[SHB]

en tendance un historien spécialiste de son sujet va raconter moins de cracs qu’un type comme toi ou moi.
.
Sauf qu’on parlait pas de ça on parlait d’historien dits amateurs pas de mecs qui ne font pas d’histoire donc ton truc ne marche pas a moins que tu fasse l’amalgame entre historien amateur et « mec comme toi et moi » car tu croirais qu’en dehors de l’université les connaissances ne valent pas grand chose.

[I.G.Y]

Mec comme toi et moi ou historien amateur, c’est bien ce que je voulais dire

[SHB]

Je sais que c’est ça que tu voulais dire, et tu as donc ma réponse au dessus.

[SHB]

Encore une fois, l’université contribue à réduire l’apprentissage a l’enseignement, le savoir aux travaux universitaires et la véracité historiques aux professionnels de l’histoire.
.
L’historien universitaire est avant une position sociale qui dicte les grandes lignes du vrai, du faux, de ce qui est pertinent d’apprendre et de ce qui ne l’est pas, savoirs qui seront ensuite transmis dans les écoles des le plus jeune âge.
.
Si on a appris que les rois a l’école c’est bien parce que les universitaires considéraient que c’était le savoir légitime a apprendre au contraire de l’histoire des femmes par exemple.

[SHB]

Vous voudriez que la structure scolaire universitaire qui est la même depuis Grosso modo 600 ans dans les grandes lignes produise d’un coup des effets différents parce que vous avez 10 chercheurs universitaires radicaux en France.

[Charles]

Personne n’a dit ça, bis.

[SHB]

Charles
SHB, je connais pas le sujet de l’intérieur comme toi mais n’est-ce pas un peu caricatural quand même? Beaucoup de grands historiens français sont issus du sérail universitaire ou grande école type Normale sup, de l’Ecole des annales à Ingrao et Chapoutot, pas exactement des Jean-Michel faisant de l’Histoire. Cela ne veut pas dire qu’ils n’ont pas du affronter en leur temps des résistances académiques – je pense également à Foucault qui était pour le coup en marge, marge relative car il a quand même été adoubé par les institutions. même si les historiens ont du mal à l’accepter. Et on peut être un socedem et produire un travail d’historien intéressant, dernière exemple en date André Loenz qui nous régale avec ses Paroles d’histoire alors qu’à côté il pense que Tondelier pourrait être la solution pour 2027.
.
Je traduis : nous pouvons produire des travaux radicaux même si on a été façonné par la structure de l’école (conclusion la structure n’a en définitive pas autorité et.donc tu pense les individus plus fort que les structures).
.
Si vous voulez qu’on discute va falloir a un moment que vous assumiez ce que vous penser et dite

[françois bégaudeau]

Sur la culture comme sur l’université, tes prises de parole s’affaiblissent, et se déjugent, d’être systématiquement abstraites, jamais illustrées, d’exemples (je passe sur ta tonalité rageuse non moins pénible). Or la question de l’université, comme de la culture, ne peut se penser qu’au travers d’exemples précis, et au prix d’un examen détaillé de la situation.
Seule manière d’affronter cette difficulté que ta pensée à gros sabots évite allègrement : qu’il peut y avoir, au sein du capitalisme (pas si) absolutisé (que ça) des espaces de moindre-marchandise, car précisément ils portent moins d’enjeux économiques. N’oublions pas, PAR EXEMPLE, que le champ culturel est quasi entièrement laissé au Parti communiste pendant les années 50-60-70. Regardons , PAR EXEMPLE, ce qui se passe dans l’édition (en tout cas l’édition anté-Bolloré)
Je connais mal l’université, et je n’ai rien à y défendre, mais il me parait structurellement possible qu’y émergent des travaux radicaux. Il est PAR EXEMPLE bien connu que la pensée décoloniale est plutot active à Paris 8. Et que la gauche radicale est très bien représentée à Rennes 2 (j’y ai d’ailleurs été invité l’an dernier)

[SHB]

Je connais mal l’université.
.
Je pense que l’on peut s’arrêter là.

[JeanMonnaie]

Je pense que l’on peut s’arrêter là.

SHB continue dans sa mauvaise foi avec cette fois un échappatoire des plus minables. Il arrive à faire passer François, et ce n’est pas une mince prouesse, pour une personne d’une probité et d’une justesse dans ses analyses. On ne pourra pas m’accuser de parti pris, les échanges qui durent depuis longtemps désignent sans contestation le même perdant.

[Charles]

Il faudrait s’entendre sur savoir radical car j’ai l’impression que tu mets des choses très différentes derrière. Est-ce que Lordon, Bourdieu et Rancière dont le travail critique est hermétique à 90% de la population c’est radical ou pas? Est-ce que l’Ecole des annales, plus lisible mais moins critique et pourtant fondamentale, ça l’est?
Ce que je dis c’est que les institutions de type universitaire peuvent avoir des effets ambivalents. Elles produisent naturellement un certain conformisme à l’égard de l’institution mais pas seulement car elles permettent aussi à des chercheurs d’avoir du temps pour précisément faire un travail de recherche et de s’extraire un peu du monde marchand, ce qui ne peut que favoriser (différent de produire mécaniquement et nécessairement) un savoir critique. Je pense aussi que Normale sup favorise davantage cela que l’université. C’est ce qui explique que toutes les semaines tu vois défiler sur Hors-série des intellectuels, chercheurs critiques, de Grégoire Chamayou à Sylvie Laurent en passant par Loic Wacquant.
Je pense aussi que la teneur critique ou non du travail de recherche ne dépend pas que de l’institution mais aussi de l’humeur de l’époque (qui dépend des conditions matérielles etc.). Sinon on ne comprend pas pourquoi on a eu tant d’intellectuels critiques dans les années 60-70 et nettement moins depuis. Enfin, et c’est le prolongement du même argument, il n’est pas anormal que les travaux critiques ne soient pas majoritaires alors que la gauche radicale est elle-même très minoritaire dans la population. On a déjà une surreprésentation de la gauche dans le champ de la recherche – c’est qu’un Lagasnerie reconnait lui-même alors qu’il est très critique de l’université – en attendre plus de l’université est illusoire.

[SHB]

Quelle est la situation sociale d’un chercheur et professeur d’université.
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France : un peu plus de 4000€/mois
Québec : environ 200 000$/an.
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Donc déjà on a affaire socialement a des petits bourgeois. On sait très bien que le confort ramolli un peu et qu’on est beaucoup moins apte a prôner la radicalité (notamment sur l’école) quand on pourrait directement en payer les frais.
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Être professeur c’est une statut social et c’est aussi une habitus qui vient avec (tu va pas faire tes courses chez Liddle, tu regardes d’ailleurs pas le prix de ses dernières, tu peux partir en vacances, etc.).
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Ensuite, être professeur, c’est avoir une perception extrêmement restreinte de la jeunesse puisque tu ne côtoie en fait presque jamais de prolos.
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Être professeur c’est donner des conférences a des gens qui globalement partagent tes idées.
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En fait, être professeur, c’est nager dans un entre soi intellectuel et social qui, peu à peu, nous étiole, nous rend mou.
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Maintenant des exemples :
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1) pour les travaux universitaires, les sources qui ne sont pas produites par des chercheurs seront méprisées (je me suis déjà tapé une sale note pour avoir utilisé des journaux anarchistes pour mettre en contexte…….des mouvements anarchistes). On pourrait se dire ah bah oui mais parler des anarchistes avec les anarchistes c’est pas objectif. A noter que le même professeur n’a rien trouvé a redire quand j’ai utilisé l’encyclopédie canadienne (produite par les instances.du gouvernement du Canada) pour parler des autochtones, quand bien même le.gouvernement du Canada a commis des.crimes pouvant s’apparenter a un génocide contre les autochtones jusque dans les.annes 60 ! (Voir l’histoire des pensionnats autochtones). Le deux poids deux mesures est flagrant et n’a donc rien a voir avec la supposée objectivité.
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2) Dans les travaux universitaires, on m’a déjà rabroué pour des mots grossiers pourtant mis.entre parenthèses en précisant que de tels termes « n’avaient pas.leur place a.luniversité ».
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3) les professeurs nous disent eux même que durant la licence puis le master, on travail + la forme méthodologique (intro, mise en page, formulation d’une hypothèse, d’une conclusion, etc.). Pour beaucoup (.et.jen suis témoin) on se retrouve avec des élèves super forts pour faire.un travail parfaitement aux normes mais incapable de réfléchir sur le sujet pourtant travaillé, incapable de remettre en cause. La construction de la docilité dont parle François a l’école se produit aussi a la FAC.

[JeanMonnaie]

SHB est triste de ne pas pouvoir dire FDP et enculé de sa mère dans les travaux universitaires. Il ne comprend pas non plus que si l’université exige une méthode de travail précise, c’est justement pour aider les étudiants à développer des compétences analytiques et critiques. Ils apprennent à formuler des hypothèses, à mener des recherches, à analyser des données et à tirer des conclusions fondées. Voyons voir comment l’Encyclopédie canadienne traite le génocide canadien. Mince, elle le traite comme la gauche radicale le ferait !
https://www.thecanadianencyclopedia.ca/fr/article/genocide-and-indigenous-peoples-in-canada

[SHB]

Voir a quel point Jean Monnaie défend l’université dans son apport de science rationnelle et complexe devrait ici alerter tout le monde (en général quand l’autre débile est d’accord avec vous faut se poser des questions).

[JeanMonnaie]

Johann Chapoutot, dont vous me parlez tous les jours sur ce forum, est bien plus proche de la doxa dominante sur la Seconde Guerre mondiale que je ne le suis. Il en va de même pour de nombreux sujets où peu de mes positions sont en accord avec le milieu universitaire. Montre-moi ce qui est de droite dans l’article sur le génocide indien dans l’Encyclopédie canadienne, j’ai peut-être manqué le passage.

[JeanMonnaie]

Je vois les JO, et il y a le groupe des réfugiés qui participe aux jeux. En ce moment, un trouple homo métis s’embrasse. Pas de doute les idées de Zemmour sont hégémoniques, et les tiennes pourraient être passibles de prison, tant tes idées sont dangereuses.

[K. comme mon Code]

En anglais pour signifier son mépris face à la débilité d’autrui, on dit ironiquement : bless your soul.

[JeanMonnaie]

La les JO le passage sont sur les statues féministes dont l’anarchiste Louise Michel l’anarchiste.
Pas de toute la, françois à raison, la France est de droite !

[K. comme mon Code]

Oui, la réalité sociale de la France, c’est un symbole de mise en scène. Tu ne viendras pas à Paris en septembre, mais j’aimerais bien lire ton compte rendu de mauvaise foi de Notre Joie du père François qui a l’air de tant t’obséder. Tu pourras présenter comme alibi le fait que tu étais trop occupé à lire un livre.

[K. comme mon Code]

Philippe Katerine ! La France adore le cinéma de Sophie Letourneur !

[SHB]

François toi qui parle de la construction de l’élection avec des gens qui voudraient que les mêmes causes qui prévalent depuis 200 ans produisent des effets différents je t’invite à te renseigner sur la constitution de l’université au Moyen-âge et de la professionnalisation progressive de l’enseignement.

[françois bégaudeau]

Je connais un peu ces segments d’histoire, et ce que j’en connais n’objecte en rien à ce que je disais précédemment. Il y aurait objection si je prétendais que l’université soit un lieu de radicalité. Or je n’ai pas dit ça, sauf dans ta tête. J’ai dit que cette institution, vouée en tant que telle à préserver l’ordre et à le garantir, peut, par un jeu complexe de déterminations, abriter des chercheurs radicaux.
Tes déboires avec certains profs chiens de garde ne sont pas surprenantes – j’ai connu des épisodes semblables, notamment aux oraux d’agreg. Ils ne sont pas non plus une objection à cette idée simple et riche d’exemples que, je le répète pour que ce soit clair, cette institution, vouée en tant que telle à préserver l’ordre et à le garantir, peut, par un jeu complexe de déterminations, abriter des chercheurs radicaux.
Incidemment je ne crois pas que le problème principal de l’université soit idéologique. Il est avant tout économique. La mainmise du capital sur l’université ne se traduit pas par des injonctions idéologiques, mais par une paupérisation qui fait de l’université, je le redis aussi, la poubelle du supérieur.

[SHB]

J’ai dit que cette institution, vouée en tant que telle à préserver l’ordre et à le garantir, peut, par un jeu complexe de déterminations, abriter des chercheurs radicaux.
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1) Je n’ai jamais contesté ce phénomène, j’ai d’ailleurs énoncé plusieurs chercheurs et professeurs d’université français particulièrement radicaux et passionnants (Stiegler, etc.). Je peux même vous donner le nom de mon professeur d’histoire du Canada que j’affectionne particulièrement : Léon Robichaud. Comme certains disent qu’ils ont des amis bourgeois ou des amis noirs, je peux dire que j’ai des amis profs. Comme tu le disais si bien a André Compte-Sponville, les individus sont peu de choses face aux structures. La question de savoir s’il existe des exceptions à la règle n’infirme pas pour autant cette dernière. Le fait que des gens comme Édouard Louis existent n’infirment pas la réalité de la reproduction de classe. Le fait que Barbara Stiegler existe n’infirme pas la réalité de cette machine a créer des penseurs centristes et mou qu’est l’Université.
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Je maintiens donc que l’Université présente, oui, une structure culturel en son sein (et qui date de bien longtemps, on pourrait en examiner le détail) qui est de l’ordre d’une certaine proclamation de ce qui serait une scission entre un savoir légitime et un savoir qui ne le serait pas (je reviens a mon exemple des lectures anarchistes mais aussi du mépris qu’on les universitaires pour le parlé populaire vu comme « indigne de la recherche », je pourrais donner des exemples toute la journée comme j’en ais déjà donné 5-6 extrêmement précis et concrets plus haut dans le topic.

[Froulano]

Tu as des exemples de mots précis qui t’ont valu des remarques ? Le registre académique est soutenu, c’est un fait, tu peux décider de ne pas t’y plier mais on te le reprochera, en effet. À fortiori si tu ne maîtrises pas très bien la grammaire et la conjugaison

[SHB]

dit moi d’abord ce que tu pense du fait que « Le registre académique est soutenu, c’est un fait ».

[Froulano]

J’en pense que la capacité à écrire dans un registre formel voire soutenu est un marqueur de classe et désavantage très certainement les étudiants issus de milieux populaires. Mais ce que je dis là est très banal.

[SHB]

pour les mots je pourrais fouiller dans mes travaux corrigés et te revenir avec ça si ça t’intéresse.

[SHB]

Autrement, j’ai vu quelqu’un dire que finalement, l’Université est simplement à l’image de la société minoritairement de gauche radicale et que c’est pour cela que l’on trouverait peu de chercheurs de gauche. Je pense que c’est une inversion du stigmate. L’Université n’est pas une victime sociale, elle est l’un des agents de production de cette hégémonie libérale (comme les médias par exemple) à travers la création d’une corporation molle qui croit réfléchir tout en nuance mais qui est radicale dans son centrisme.

[Charles]

C’était moi. Les deux se nourrissent, l’université est victime et autrice pour reprendre tes termes. Et tu n’as pas répondu à ma remarque sur la plus grande radicalité des intellectuels issus de l’université dans les années 70 par rapport aux années 90 ou maintenant. Comment expliquer cela si les institutions n’ont pas changé ? Tu n’expliques pas non plus comment on peut avoir des chercheurs très en vue assez à gauche.

[SHB]

Les deux se nourrissent, l’université est victime et autrice pour reprendre tes termes.
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Davantage autrice que victime de mon point de vu.
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Et tu n’as pas répondu à ma remarque sur la plus grande radicalité des intellectuels issus de l’université dans les années 70 par rapport aux années 90 ou maintenant.
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C’est assez logique de manière générale la gauche radicale était plus en vogue à cette époque et ça se reflétait à l’Université.
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Là encore vous êtes limités dans votre réflexion. Dire qu’une institution ordonne des choses ne veut pas dire qu’elle est hermétique à tous les phénomènes sociaux. On dirait vraiment, pardon, des arguments de CE1 du genre de ceux qui prennent l’élection de 81 pour dire que la gauche a une chance dans les élections. L’institution électorale est impérieuse et pourtant, enregistre les mouvements sociaux qui ont lieux dans la société. Pareil pour l’Université, qui ne perd pas pour autant son caractère impérieux. Le tout est de voir les tendances sur le temps long. Les conséquences de l’institution universitaire qui prévalent depuis, on va être gentil, 200 ans, ne sauraient être bafouées par 20 années de trève.

[SHB]

Le tout est de voir les tendances sur le temps long. Les conséquences de l’institution universitaire qui prévalent depuis, on va être gentil, 200 ans, ne sauraient être bafouées par 20 années de trêve.
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La preuve, depuis les années 90, on est reparti comme en 40….

[Charles]

J’abandonne à mon tour ce non-échange car tu ne prends pas la peine de lire ou à tout le moins de comprendre ce qu’on te répond, tu refuses de préciser les termes que tu emploies et tout ça sur un ton de cour de récré que rien ne vient justifier. Tu veux juste exprimer ta frustration liées à tes déconvenues à la fac.

[SHB]

Bah abandonne si tu veux c’est dommage que le vrai désaccord de fond entre nous ne soit pas apparu du fait de ta malhonnêteté intellectuelle, à savoir que tu tiens en haute estime l’Université qui pour toi est le lieu saint du savoir et que tu ne supportes pas qu’on vient éreinter ce qui est pour toi un édifice intellectuel. Ce ressort affectif de ta part a été très visible durant toute la durée de notre échange. À bon entendeur, bonne journée.

[JeanMonnaie]

C’est assez logique de manière générale la gauche radicale était plus en vogue à cette époque et ça se reflétait à l’Université.

Ta réponse est d’une faiblesse. Dire que tu fréquentes la faculté.
D’ailleurs, le RN est très en vogue et pourtant quasi absent de l’université.
Tu restes dans la même mélasse habituelle, les structures sont capitalistes et que par conséquent, tout ce qui serait de gauche ne peut être que des exceptions à la règle, des cailloux dans les chaussures ou une fausse gauche.

[Charles]

Comment peux-tu aboutir à cette conclusion alors que tu as lu en diagonale ce que j’ai écrit ? Tu as fatigué tout le monde sur cette page, remets-toi un peu en question cinq minutes au lieu de considérer que tout le monde est débile.

[SHB]

J’abandonne à mon tour ce non-échange.
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Tu es encore là?

[Charles]

Définitivement la cour de récré, merci. Bonne continuation.

[SHB]

Oui oui la cour de récré tkt t’a tout compris.

[Froulano]

Je confirme, tu es pénible et à te lire tu sais tout mieux que tout le monde du haut de tes deux années de fac en L2 histoire. On te laisse donc enfoncer des portes ouvertes et ventiler ton ressentiment en paix, nous tu casses les pieds.

[SHB]

Ah ouai tu me dis que j’ai que 2 années a la FAC fichtre….. ça me touche vraiment moi qui tiens en si haute estime le savoir universitaire.
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nous tu casses les pieds.
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J’avais remarqué et je m’en réjouis.

[SHB]

tu refuses de préciser les termes que tu emploies.
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Je maintiens donc que l’Université présente, oui, une structure culturel en son sein (et qui date de bien longtemps, on pourrait en examiner le détail) qui est de l’ordre d’une certaine proclamation de ce qui serait une scission entre un savoir légitime et un savoir qui ne le serait pas (je reviens a mon exemple des lectures anarchistes mais aussi du mépris qu’on les universitaires pour le parlé populaire vu comme « indigne de la recherche ».
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Et après c’est moi qui sait pas lire. Quel toupet.

[SHB]

Tu n’expliques pas non plus comment on peut avoir des chercheurs très en vue assez à gauche.
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Je n’ai jamais contesté ce phénomène, j’ai d’ailleurs énoncé plusieurs chercheurs et professeurs d’université français particulièrement radicaux et passionnants (Stiegler, etc.). Je peux même vous donner le nom de mon professeur d’histoire du Canada que j’affectionne particulièrement : Léon Robichaud. Comme certains disent qu’ils ont des amis bourgeois ou des amis noirs, je peux dire que j’ai des amis profs. Comme tu le disais si bien a André Compte-Sponville, les individus sont peu de choses face aux structures. La question de savoir s’il existe des exceptions à la règle n’infirme pas pour autant cette dernière. Le fait que des gens comme Édouard Louis existent n’infirment pas la réalité de la reproduction de classe. Le fait que Barbara Stiegler existe n’infirme pas la réalité de cette machine a créer des penseurs centristes et mou qu’est l’Université.

[Emile Novis]

Sur la non-radicalité des professeurs à l’Université, j’ai entendu de nombreux témoignages sur le mécanisme de sélection des professeurs avec le cursus de la thèse et de la recherche des directeurs de thèse. Des individus qui veulent faire quelque chose de leur thèse au sein de l’université (en gros, espérer avoir un poste stable) sentent bien qu’il faut choisir le bon cheval, le directeur qui aura de l’influence et qui est bien installé, et qu’il ne faut pas trop brusquer ses vues sur un sujet donné pour espérer obtenir ce qu’on veut. Ce système de cooptation porte en lui, semble-t-il, des effets pervers : s’il faut plaire au directeur de thèse influent, et que ce même directeur influent est quelqu’un de très bien installé et dont les financements de recherche dépendent plus ou moins du pouvoir en place, alors il est clair que c’est une machine à produire du conformisme. Et si la majorité des professeurs influents ne sont pas radicaux, alors il faudra ne pas être radical pour espérer trouver une place, et si on veut être radical, on sera mécaniquement écarté dans une grande partie des cas.
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D’autres témoignages disaient qu’on est véritablement libre dans une thèse quand on n’attend rien de l’université puisqu’on a déjà un boulot stable ailleurs. Plus besoin de plaire aux gens importants, de chercher le bon cheval, de trouver des réseaux pour pouvoir être publié et lu (certains qui briguent un poste confessent que ça occupe une partie importante de leur temps), etc. Mais la conséquence, c’est que l’individu radical en question n’intègrera pas l’université.
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Barbara Stiegler, dont il a été question plus haut, explique bien ce problème. Elle témoigne, dans certaines vidéos, comment elle s’est faite relativement ostracisée au sein de l’université en raison de ses positions sur la politique sanitaire : des collègues mettaient en garde contre elle, disaient qu’il fallait faire attention, que faire une thèse avec elle n’était une bonne idée, etc.
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Peut-être que les réformes ont miné encore un peu plus les statuts protecteurs tout en approfondissant cet esprit de courtisanerie et de recherche de place pour la survie économique, ce qui pourrait expliquer le fait qu’il y a encore moins de professeurs radicaux aujourd’hui qu’avant.

[SHB]

Attention Émile tu va te faire rabrouer par le Père Charles et ses acolytes.

[SHB]

Sinon pour être sérieux tu as évidemment raison j’ai des amis en master et ce que tu dis est vrai. Cela ne va pas forcément se traduire par des affrontements directes du genre « tu ne peux pas faire ce sujet c’est trop radical » mais on va recommander de mieux réfléchir, de changer d’approche, d’inclure plus de thèmes, d’être plus nuancé dans l’énoncé de recherche, de varier les sources (alors qu’elles le sont déjà mais pas de sources qui plaisent au professeur), etc….).

[SHB]

Tu te retrouves à la fin avec un travail qui porte sur un sujet différent de ce que tu voulais aborder.

[Emile Novis]

De nombreux témoignages vont dans ce sens en effet. Et je tiens à préciser que pour ma part, j’ai beaucoup apprécié être à la fac mais je n’ai jamais désiré y passer ma vie non plus.
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Après, ce système peut se retourner contre lui-même : il suffit qu’un professeur-chercheur radical obtienne un poste confortable et influent, et ça peut vite faire un « bastion de radicalité » dans un lieu donné, avec des étudiants qui vont reproduire cette radicalité pour être dans les petits papiers. Le système de sélection reste pourri à mes yeux, mais c’est peut-être en partie pour ça que lors des manifestations, on entend que certaines facs bougent systématiquement, tandis qu’on entend jamais parler de certaines.
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Néanmoins, pour avoir discuté avec quelques chercheurs il y a quelques temps, j’étais aussi très étonné de leur condition de travail. Je veux dire que les conditions de production des articles et des écrits sont difficilement conciliables avec une certaine radicalité qualitative. Ils doivent écrire de nombreux textes, contribuer au bon classement de leur unité de recherche, ils doivent être cités un grand nombre de fois pour les mêmes raisons, etc. (à ce propos, il paraît que dans les tableaux officiels qui récapitulent la production des unités de recherche, les enseignants-chercheurs ne sont plus désignés sous ce nom, mais sous le nom de « produisant », ce qui en dit long sur l’idéologie sous-jacente à cette manière d’évaluer les choses). Ainsi il faut prendre des sujets en vogue, dire des choses susceptibles d’être reprises et citées, être bien placé auprès d’une revue bien en vue, etc. Une telle structure semble produire du mimétisme et du tout-fait, et je pense que ce critère tend à tuer la pensée, même si certains professeurs valeureux s’efforcent quand même de tenir le cap à l’intérieur d’un milieu assez hostile de ce point de vue là. Mais j’ai aussi remarqué que les quelques chercheurs qui m’ont raconté ça étaient bien conscients du problème, et ils ne cessaient de pester contre ce système qu’ils subissent, bien qu’au quotidien, ils sont obligés de s’y plier et de jouer le jeu pour pouvoir garder leur place (ou obtenir la place qu’ils désirent). Je me suis dit qu’au fond, il y a peut-être aussi des radicaux qui se cachent à la fac.

[SHB]

Oui, la logique productiviste rejoins ce que je disais plus haut sur le domaine archivistique qui est lié au domaine historique car les diplômés en histoire qui ne deviennent pas profs rejoignent parfois/souvent les archives.
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Dans mon modeste domaine qu’est l’histoire, j’ai par exemple observé que dans les centres d’archives, un des critères de sélection avant même la pertinence, la valeur historique, philosophique, etc.. c’est est-ce que l’archive a des chances maximales d’être consultée pour que le centre soit rentable et continue d’être subventionné.
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Ils doivent écrire de nombreux textes, contribuer au bon classement de leur unité de recherche, ils doivent être cités un grand nombre de fois pour les mêmes raisons.
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C’est d’ailleurs l’un des critères explicite de sélection au poste de professeur d’Université (source : 3 professeurs qui m’ont dit cela).

[SHB]

Épisode 1289 de François a raison sur l’école (même si il a pas encore poussé la radicalité a aussi enterrer l’Université) :
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Je lis un texte sur la Révolution française et c’est littéralement écrit mot pour mot par l’Assemblée nationale post 1789 que l’instruction obligatoire par l’État a été faite pour éviter que les oisifs s’Organisent entre eux dans un esprit Contre-Révolutionnaire.

[françois bégaudeau]

Je suis pour le maintien de l’Université, qui permet à certains d’ajourner de trois ou quatre ans l’entrée sur le marché du travail en faisant valoir à leurs parents une inscription dans une fac dont ils se foutent

[SHB]

À ce titre je suis alors pour l’école obligatoire jusqu’à 16 ans qui empêche le travail des enfants.

[françois bégaudeau]

L’étudiant inscrit en fac peut très bien n’y aller qu’un jour sur 12, ou seulement pour les examens.. Le collégien non
(je signale que je n’irai pas plus avant dans ce débat que je trouve tout à fait vain)

[SHB]

Quand on envisage la fin d’une institution telle que l’Université on l’envisage dans le cadre plus large d’un changement profond de société. Au même titre que l’abolition de l’école ne peut intervenir dans la Société actuelle. Apposé à la volonté de détruire l’institution universitaire le fait que, dans le cadre actuel, des jeunes puissent trouver refuge à l’Université pour ne pas travailler quand on parle d’institutions qui ont plusieurs siècles d’existence, cela n’a aucun sens.

[françois bégaudeau]

Sur ce sujet qui ne m’intéresse pas, mon argument se voulait fantaisiste
Ce n’est pas la première fois qu’ici tu ne piges rien.

[SHB]

Ce sujet ne m’intéresse pas dixit le mec qui a fait 200 émissions où il déglingue l’école.
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Un peu de sérieux François.

[françois bégaudeau]

Le sujet de l’université
Une incompréhension de plus de ta part.

[SHB]

Ah d’accord je ne savais pas que l’université ne faisait pas partie du phénomène général de « l’école ». C’est vrai qu’à l’université on est pas assis dans des classes déterminées, durant un temps déterminé et avec des enseignants eux aussi déterminés a l’avance. Et puis y’a pas de notes a l’université, et ça permet pas la reproduction sociale, etc. etc.
.
C’est pas comme si Illich, dans son livre, parle tout à fait en même temps de l’école et de l’université en les rassemblant dans la problématique plus large de la conception très particulière de l’éducation dans nos sociétés.
.
François tu t’enfonce. La problématique de l’école est liée en tous points avec celle de l’université il s’agit de la même chose a quelques subtilités près.
.
Après tu disais toi même que tu connaissais une amie de gauche incapable de se rebeller contre l’école car étant prof. Peut-etre qu’en tant que bon élève tu as du mal a te rebeller contre l’université qui t’a semblé un lieu plus propice que le lycée et le collège pour apprendre.

[thierry]

Ou pour paraphraser Renaud et faire un lien avec l’autre sujet :
Étudiant en que dalle, tu glandes dans les facultés. T’as jamais lu l’capital mais y’a longtemps que t’as pigé qu’il faut jamais travailler….

[Tenmoquet]

Pour poursuivre le sujet évoqué plus haut sur Jul:

[Tchitchikov]

Comme il a été rappelé ici la question urgente en sciences humaines n’est pas de savoir si les profs sont encore des bourgeois mais s’ils pourront le rester. À l’époque où Bourdieu écrit La Distinction il parle déjà de « dominés parmi les dominants » à propos des intellectuels. Il n’y a plus d’argent pour faire de la recherche en SHS dans les universités. Même quand on a le cursus honorum il faut parfois rouler sa bosse en tant qu’attaché temporaire de recherche (qui est le Graal des contrats précaires, 1700-1900 euros/mois, variation qui dépend de la dispense de cours ou non) dans plusieurs universités avant de devenir maître de conf’. Et puis pour être plus précis il faut spécifier en fonction de la discipline. En philosophie il est certain que c’est compliqué d’enseigner à la fac quand on ne dispose ni de l’agrég’ ni d’un doctorat (qu’il soit en cours de rédaction ou non). Mais en anthropologie, par exemple, on emploie facilement des intervenants extérieurs ; notamment anglo-saxons. Plus généralement on estime le taux d’enseignants précaires jusqu’à 40% dans les facs. Je crois que ce taux est sous-estimé par endroit. Sans compter que la massification scolaire a fait débarquer des enfants de prolo ou de la « classe moyenne », particulièrement en sociologie (Coquard vient de là). Si on appelle bourgeois, avec Bourdieu, un détenteur de capitaux (culturel, économique etc.), on peut dire qu’il y en a beaucoup évidemment. Peu peuvent se permettre d’enchaîner les post-doc’ mal rémunérés. Mais les héritiers ne sont plus légion.

Croire que ces supposés bourgeois méprisent la culture dite populaire c’est avoir vingt ans de retard. Il suffit de regarder Comment je me suis disputé ma vie sexuelle de Desplechin. Où on écoute Kanye West et non Schubert. Où est le cool ? était déjà la question de cette petite bourgeoisie intellectuelle. Effectivement la race des esthètes disparaît. Celle de ceux qui s’adonnent à l’art parce que cela les affecte vraiment. Parce que les injonctions administratives, à la disponibilité numérique etc. croissent. Et puis comme le disait François dans Histoire de ta bêtise : la bite de Bigard n’est pas celle de Lemercier. Bourdieu dans la Distinction le disait à propos de la Deux chevaux : celle de l’ouvrier agricole n’équivaut pas celle de l’intellectuel. Même objet, signification différente, effet distinctif divergeant. Le goût est une notion relative, mouvante, et non d’essence.

Quoi de mieux que le brio synthétique de Chamayou pour savoir où en est la recherche ? Et ce qui n’en enlève rien à son intelligence, son écriture trimballe une belle drôlerie, une joyeuse ironie. https://shs.cairn.info/revue-du-mauss-2009-1-page-208?lang=fr

[Anna H]

Merci !

[essaisfragiles]

Merci aussi !!

[Tchitchikov]

Avec plaisir, l’article date un peu mais reste actuel.

[Tchitchikov]

Je le relis aussi et je me marre. « Hors de l’anglais, point de salut. Si la dernière révision de vos verbes irréguliers remonte à votre classe de première, achetez une méthode Assimil. Sur le marché mondialisé de l’article, il vous faut écrire dans la langue de Bill Gates. Vos doctorants préférés traduiront en français vos chefs-d’œuvre. Si vous êtes nul en langue, utilisez votre fille au pair britannique pour la version anglaise. À défaut, renouez le contact sur Facebook avec votre correspondant anglais du collège et salariez-le via Paypal. » Quel cynisme, quel réalisme !

[essaisfragiles]

@ Tchitchikov
Je te remerciais pour ton propos.

[SHB]

Ton propos évacue complètement la souffrance des élèves et la machine a créer du conformisme intellectuel qu’est l’école et l’université. C’était ça le sujet de base du topic.

[Auteur]

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