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arX
iv:1
010.
5419
v1 [
hep-
ph]
13
Oct
201
0 SUPERSYMMETRY ON THE RUN: LHC AND DARK MATTER
D. I. Kazakova
aBLTP, JINR, Dubna and ITEP, Moscow
Supersymmetry, a new symmetry that relates bosons and fermions in particle physics, still escapes observation.
Search for SUSY is one of the main aims of the recently launched Large Hadron Collider. The other possible
manifestation of SUSY is the Dark Matter in the Universe. The present lectures contain a brief introduction to
supersymmetry in particle physics. The main notions of supersymmetry are introduced. The supersymmetric
extension of the Standard Model - the Minimal Supersymmetric Standard Model - is considered in more detail.
Phenomenological features of the MSSM as well as possible experimental signatures of SUSY at the LHC are
described. The DM problem and its possible SUSY solution is presented.
1. Introduction: What is supersymmetry
Supersymmetry is a boson-fermion symmetrythat is aimed to unify all forces in Nature includ-ing gravity within a singe framework [1]-[5]. Mod-ern views on supersymmetry in particle physicsare based on string paradigm, though the lowenergy manifestations of SUSY can be possiblyfound at modern colliders and in non-acceleratorexperiments.Supersymmetry emerged from the attempts to
generalize the Poincare algebra to mix represen-tations with different spin [1]. It happened tobe a problematic task due to the no-go theoremspreventing such generalizations [6]. The way outwas found by introducing the so-called graded Liealgebras, i.e. adding the anti-commutators to theusual commutators of the Lorentz algebra. Sucha generalization, described below, appeared to bethe only possible one within relativistic field the-ory.IfQ is a generator of SUSY algebra, then acting
on a boson state it produces a fermion one andvice versa
Q|boson>= |fermion>, Q|fermion>= |boson>.
Since bosons commute with each other andfermions anticommute, one immediately findsthat SUSY generators should also anticommute,they must be fermionic, i.e. they must changethe spin by a half-odd amount and change the
statistics. The key element of SUSY algebra is
Qα, Qα = 2σµα,αPµ, (1.1)
where Q and Q are SUSY generators and Pµ isthe generator of translation, the four-momentum.In what follows we describe SUSY algebra
in more detail and construct its representationswhich are needed to build a SUSY generalizationof the Standard Model of fundamental interac-tions. Such a generalization is based on a softlybroken SUSY quantum filed theory and containsthe SM as a low energy theory.Supersymmetry promises to solve some prob-
lems of the SM and of Grand Unified Theories.In what follows we describe supersymmetry as anearest option for the new physics on a TeV scale.
2. Motivation of SUSY in particle physics
2.1. Unification with gravityThe general idea is a unification of all forces
of Nature including quantum gravity. However,the graviton has spin 2, while the other gaugebosons (photon, gluons, W and Z weak bosons)have spin 1. Therefore, they correspond to differ-ent representations of the Poincare algebra. Tomix them one can use supersymmetry transfor-mations. Starting with the graviton state of spin2 and acting by SUSY generators we get the fol-lowing chain of states:
spin 2 → spin3
2→ spin 1 → spin
1
2→ spin 0.
1
2
Thus, a partial unification of matter (fermions)with forces (bosons) naturally arises from an at-tempt to unify gravity with other interactions.
Taking infinitesimal transformations δǫ =ǫαQα, δǫ = Qαǫ
α, and using eq.(1.1) one gets
δǫ, δǫ = 2(ǫσµǫ)Pµ, (2.1)
where ǫ is a transformation parameter. Choosingǫ to be local, i.e. a function of a space-time pointǫ = ǫ(x), one finds from eq.(2.1) that an anticom-mutator of two SUSY transformations is a localcoordinate translation. And a theory which isinvariant under local coordinate transformationis General Relativity. Thus, making SUSY lo-cal, one naturally obtains General Relativity, ora theory of gravity, or supergravity [2].
2.2. Unification of gauge couplingsAccording to the Grand Unification hypothe-
sis, gauge symmetry increases with energy [7].All known interactions are different branches of aunique interaction associated with a simple gaugegroup. The unification (or splitting) occurs athigh energy. To reach this goal one has to con-sider how the couplings change with energy. Thisis described by the renormalization group equa-tions. In the SM the strong and weak couplingsassociated with non-Abelian gauge groups de-crease with energy, while the electromagnetic oneassociated with the Abelian group on the contraryincreases. Thus, it becomes possible that at someenergy scale they become equal.
After the precise measurement of the SU(3)×SU(2) × U(1) coupling constants, it has becomepossible to check the unification numerically. Thethree coupling constants to be compared are
α1 = (5/3)g′2/(4π) = 5α/(3 cos2 θW ),
α2 = g2/(4π) = α/ sin2 θW , (2.2)
α3 = g2s/(4π)
where g′, g and gs are the usual U(1), SU(2) andSU(3) coupling constants and α is the fine struc-ture constant. The factor of 5/3 in the definitionof α1 has been included for proper normalizationof the generators.
In the modified minimal subtraction (MS)scheme, the world averaged values of the coup-lings at the Z0 energy are obtained from a fit to
the LEP and Tevatron data [8]:
α−1(MZ) = 128.978± 0.027
sin2 θMS = 0.23146± 0.00017 (2.3)
αs = 0.1184± 0.0031,
that gives
α1(MZ) = 0.017,
α2(MZ) = 0.034, (2.4)
α3(MZ) = 0.118± 0.003.
Assuming that the SM is valid up to the unifica-tion scale, one can then use the known RG equa-tions for the three couplings. In the leading orderthey are:
dαi
dt= biα
2i , αi =
αi
4π, t = log(
Q2
µ2), (2.5)
where for the SM the coefficients are bi =(41/10,−19/6,−7).The solution to eq.(2.5) is very simple
1
αi(Q2)=
1
αi(µ2)− bilog(
Q2
µ2). (2.6)
The result is demonstrated in Fig.1 showing theevolution of the inverse of the couplings as a func-tion of the logarithm of energy. In this presenta-tion, the evolution becomes a straight line in firstorder. The second order corrections are small anddo not cause any visible deviation from a straightline. Fig.1 clearly demonstrates that within theSM the coupling constant unification at a singlepoint is impossible. It is excluded by more than 8standard deviations. This result means that theunification can only be obtained if new physicsenters between the electroweak and the Planckscales.In the SUSY case, the slopes of the RG evo-
lution curves are modified. The coefficients bi ineq.(2.5) now are bi = (33/5, 1,−3). The SUSYparticles are assumed to effectively contribute tothe running of the coupling constants only forenergies above the typical SUSY mass scale. Itturns out that within the SUSY model a perfectunification can be obtained as is shown in Fig.1.From the fit requiring unification one finds for
3
10log Q
1/α i
1/α1
1/α2
1/α3
MSSM
10log Q
1/α i
Unification of the Coupling Constants in the SM and the minimal MSSM
0
10
20
30
40
50
60
0 5 10 150
10
20
30
40
50
60
0 5 10 15
Figure 1. Evolution of the inverse of the threecoupling constants in the Standard Model (left)and in the supersymmetric extension of the SM(MSSM) (right).
the break pointMSUSY and the unification pointMGUT [9]
MSUSY = 103.4±0.9±0.4 GeV,
MGUT = 1015.8±0.3±0.1 GeV, (2.7)
α−1GUT = 26.3± 1.9± 1.0,
The first error originates from the uncertainty inthe coupling constant, while the second one is dueto the uncertainty in the mass splittings betweenthe SUSY particles.This observation was considered as the first
”evidence” for supersymmetry, especially sinceMSUSY was found in the range preferred by thefine-tuning arguments.
2.3. Solution of the hierarchy problemThe appearance of two different scales V ≫ v
in a GUT theory, namely, MW and MGUT , leadsto a very serious problem which is called the hi-erarchy problem. There are two aspects of thisproblem.The first one is the very existence of the hier-
archy. To get the desired spontaneous symmetrybreaking pattern, one needs
mH ∼ v ∼ 102 GeVmΣ ∼ V ∼ 1016 GeV
mH
mΣ∼ 10−14 ≪ 1, (2.8)
whereH and Σ are the Higgs fields responsible forthe spontaneous breaking of the SU(2) and the
GUT groups, respectively. The question arises ofhow to get so small number in a natural way.The second aspect of the hierarchy problem is
connected with the preservation of a given hi-erarchy. Even if we choose the hierarchy likeeq.(2.8) the radiative corrections will destroy it!To see this, consider the radiative correction tothe light Higgs mass given by the Feynman di-agram shown in Fig.2. This correction, pro-
2
" light (m)
. heavy (M)
=) Æm
2
2
M
2
o o o
10
2
10
1
10
16
1
Figure 2. Radiative correction to the light Higgsboson mass
portional to the mass squared of the heavy par-ticle, obviously, spoils the hierarchy if it is notcancelled. This very accurate cancellation witha precision ∼ 10−14 needs a fine tuning of thecoupling constants.The only known way of achieving this kind of
cancellation of quadratic terms (also known asthe cancellation of the quadratic divergencies) issupersymmetry. Moreover, SUSY automaticallycancels quadratic corrections in all orders of PT.This is due to the contributions of superpartnersof ordinary particles. The contribution from bo-son loops cancels those from the fermion ones be-cause of an additional factor (-1) coming fromFermi statistics, as shown in Fig.3. One can seehere two types of contribution. The first line isthe contribution of the heavy Higgs boson and itssuperpartner. The strength of interaction is givenby the Yukawa coupling λ. The second line rep-resents the gauge interaction proportional to thegauge coupling constant g with the contributionfrom the heavy gauge boson and heavy gaugino.In both the cases the cancellation of quadratic
terms takes place. This cancellation is true up tothe SUSY breaking scale, MSUSY , which shouldnot be very large (≤ 1 TeV) to make the fine-tuning natural. Indeed, let us take the Higgs bo-
4
g
2
gauge
. boson
+
. gaugino
= 0
g g
2
. boson
+
. fermion
= 0
Figure 3. Cancellation of quadratic terms (diver-gencies)
son mass. Requiring for consistency of perturba-tion theory that the radiative corrections to theHiggs boson mass do not exceed the mass itselfgives
δM2h ∼ g2M2
SUSY ∼M2h . (2.9)
So, if Mh ∼ 102 GeV and g ∼ 10−1, one needsMSUSY ∼ 103 GeV in order that the relation(2.9) is valid. Thus, we again get the same roughestimate of MSUSY ∼ 1 TeV as from the gaugecoupling unification above.
That is why it is usually said that supersymme-try solves the hierarchy problem. We show belowhow SUSY can also explain the origin of the hi-erarchy.
2.4. Astrophysics and CosmologyThe shining matter is not the only one in the
Universe. Considerable amount consists of theso-called dark matter. The direct evidence forthe presence of the dark matter are the rotationcurves of galaxies [10] (see Fig.4). To explainthese curves one has to assume the existence ofgalactic halo made of non-shining matter whichtakes part in gravitational interaction. There aretwo possible types of the dark matter: the hotone, consisting of light relativistic particles andthe cold one, consisting of massive weakly inter-acting particles (WIMPs) [11]. The hot dark mat-ter might consist of neutrinos, however, this has
Figure 4. Roration curves for the solar systemand galaxy
problems with galaxy formation. As for the colddark matter, it has no candidates within the SM.At the same time, SUSY provides an excellentcandidate for the cold dark matter, namely neu-tralino, the lightest superparticle [12].
2.5. Beyond GUTs: superstringAnother motivation for supersymmetry follows
from even more radical changes of basic ideasrelated to the ultimate goal of construction ofconsistent unified theory of everything. At themoment the only viable conception is the su-perstring theory [13]. In the superstring the-ory, strings are considered as fundamental ob-jects, closed or open, and are nonlocal in nature.
5
Ordinary particles are considered as string excita-tion modes. String interactions, which are local,generate proper interactions of usual particles, in-cluding gravitational ones.To be consistent, the string theory should be
conformal invariant in D-dimensional target spaceand have a stable vacuum. The first requirementis valid in classical theory but may be violatedby quantum anomalies. Cancellation of quantumanomalies takes place when space-time dimensionof a target space equals to a critical one which isDc = 26 for bosonic string and Dc = 10 for afermionic one.The second requirement is that the massless
string excitations (the particles of the SM) arestable. This assumes the absence of tachyons, thestates with imaginary mass, which can be guar-anteed only in supersymmetric string theories!
3. Basics of supersymmetry
3.1. Algebra of SUSYCombined with the usual Poincare and internal
symmetry algebra the Super-Poincare Lie alge-bra contains additional SUSY generators Qi
α andQi
α [3]
[Pµ, Pν ] = 0,
[Pµ,Mρσ] = i(gµρPσ − gµσPρ),
[Mµν ,Mρσ]= i(gνρMµσ−gνσMµρ−gµρMνσ+gµσMνρ),
[Br, Bs] = iCtrsBt,
[Br, Pµ] = [Br,Mµσ] = 0, (3.1)
[Qiα, Pµ] = [Qi
α, Pµ] = 0,
[Qiα,Mµν ]=
1
2(σµν)
βαQ
iβ , [Q
iα,Mµν ]=
−1
2Qi
β(σµν)
βα,
[Qiα, Br] = (br)
ijQ
jα, [Qi
α, Br] = −Qjα(br)
ij ,
Qiα, Q
j
β = 2δij(σµ)αβPµ,
Qiα, Q
jβ = 2ǫαβZ
ij , Zij = arijbr, Zij = Z+ij ,
Qiα, Q
j
β = −2ǫαβZ
ij , [Zij , anything] = 0,
α, α = 1, 2 i, j = 1, 2, . . . , N.
Here Pµ and Mµν are four-momentum and an-gular momentum operators, respectively, Br arethe internal symmetry generators, Qi and Qi arethe spinorial SUSY generators and Zij are the so-
called central charges; α, α, β, β are the spinorial
indices. In the simplest case one has one spinorgenerator Qα (and the conjugated one Qα) thatcorresponds to an ordinary or N=1 supersymme-try. When N > 1 one has an extended supersym-metry.A natural question arises: how many SUSY
generators are possible, i.e. what is the value ofN? To answer this question, consider masslessstates. Let us start with the ground state labeledby energy and helicity, i.e. projection of a spinon the direction of momenta, and let it be anni-hilated by Qi
Vacuum = |E, λ >, Qi|E, λ >= 0.
Then one and more particle states can be con-structed with the help of a creation operators as
State Expression # of States
vacuum |E, λ> 11−particle Qi|E, λ>= |E, λ+1/2>i N
2−particle QiQj |E, λ>= |E, λ+1>ijN(N−1)
2... ... ...N−particle Q1...QN |E, λ>= |E, λ+N
2> 1
Total # of states:
N∑
k=0
(
Nk
)
= 2N = 2N−1
bosons + 2N−1 fermions. The energy E is notchanged, since according to (3.1) the operatorsQi commute with the Hamiltonian.Thus, one has a sequence of bosonic and
fermionic states and the total number of bosonsequals that of fermions. This is a generic propertyof any supersymmetric theory. However, in CPTinvariant theories the number of states is dou-bled, since CPT transformation changes the signof helicity. Hence, in CPT invariant theories, onehas to add the states with opposite helicity to theabove mentioned ones.Consider some examples. Let us take N = 1
and λ = 0. Then one has the following set ofstates:
N = 1 λ = 0helicity 0 1
2 helicity 0 − 12
CPT=⇒
# of states 1 1 # of states 1 1
6
Hence, a complete N = 1 multiplet is
N = 1 helicity −1/2 0 1/2# of states 1 2 1
which contains one complex scalar and one spinorwith two helicity states.
This is an example of the so-called self-conjugated multiplet. There are also self-conjugated multiplets with N > 1 correspondingto extended supersymmetry. Two particular ex-amples are the N = 4 super Yang-Mills multipletand the N = 8 supergravity multiplet
N = 4 SUSY YM λ = −1
helicity −1 −1/2 0 1/2 1# of states 1 4 6 4 1
N = 8 SUGRA λ = −2
−2 −3/2 −1 −1/2 0 1/2 1 3/2 21 8 28 56 70 56 28 8 1
One can see that the multiplets of extended su-persymmetry are very rich and contain a vastnumber of particles.
The constraint on the number of SUSY gen-erators comes from a requirement of consistencyof the corresponding QFT. The number of super-symmetries and the maximal spin of the particlein the multiplet are related by
N ≤ 4S,
where S is the maximal spin. Since the theorieswith spin greater than 1 are non-renormalizableand the theories with spin greater than 5/2 haveno consistent coupling to gravity, this imposes aconstraint on the number of SUSY generators
N ≤ 4 for renormalizable theories (YM),N ≤ 8 for (super)gravity.
In what follows, we shall consider simple super-symmetry, or N = 1 supersymmetry, contrary toextended supersymmetries with N > 1. In thiscase, one has the following types of supermulti-plets which are used in the construction of SUSYgeneralization of the SM
(φ, ψ) (λ, Aµ)Spin = 0, Spin = 1/2 Spin = 1/2, Spin = 1scalar chiral majorana vector
fermion fermion
each of them contains two physical states, oneboson and one fermion. They are called chiraland vector multiplets, respectively. Constructionthe generalization of the SM one has to put allthe particles into these multiplets. For instance,quarks should go into chiral multiplet and photoninto vector multiplet.
3.2. Superspace and supermultipletsAn elegant formulation of supersymmetry
transformations and invariants can be achievedin the framework of superspace [4]. Superspacediffers from the ordinary Euclidean (Minkowski)space by adding of two new coordinates, θα andθα, which are Grassmannian, i.e. anticommuting,variables
θα, θβ = 0, θα, θβ = 0, θ2α = 0, θ2α = 0,
α, β, α, β = 1, 2.
Thus, we go from space to superspace
Space ⇒ Superspacexµ xµ, θα, θα
A SUSY group element can be constructed in su-perspace in the same way as an ordinary transla-tion in the usual space
G(x, θ, θ) = ei(−xµPµ + θQ+ θQ). (3.2)
It leads to a supertranslation in superspace
xµ → xµ + iθσµε− iεσµθ,θ → θ + ε, θ → θ + ε,
(3.3)
where ε and ε are Grassmannian transformationparameters. From eq.(3.3) one can easily obtainthe representation for the supercharges (3.1) act-ing on the superspace
Qα=∂
∂θα−iσµ
ααθα∂µ, Qα=
−∂∂θα
+iθασµαα∂µ.(3.4)
Working in superspace all the fields becomefunctions of not only the space-time point xµ butalso the Grassmanian coordinates θ, i.e. they be-come superfields. The superfield contains insidethe whole supermultiplet. We will not describethe superfields here and refer the reader to exist-ing literature. What is important for us is thatthis formalizm is straightforward and allows oneto construct a SUSY generalization of any theory.
7
4. SUSY generalization of the StandardModel. The MSSM
As has been already mentioned, in SUSY the-ories the number of bosonic degrees of freedomequals that of fermionic. At the same time, inthe SM one has 28 bosonic and 90 fermionic de-grees of freedom (with massless neutrino, other-wise 96). So the SM is to a great extent non-supersymmetric. Trying to add some new parti-cles to supersymmetrize the SM, one should takeinto account the following observations:• There are no fermions with quantum numbers
of the gauge bosons;• Higgs fields have nonzero v.e.v.s; hence they
cannot be superpartners of quarks and leptonssince this would induce spontaneous violation ofbaryon and lepton numbers;• One needs at least two complex chiral Higgs
multiplets to give masses to Up and Down quarks.The latter is due to the form of a superpoten-
tial and chirality of matter superfields. Indeed,the superpotential should be invariant under theSU(3)× SU(2)× U(1) gauge group. If one looksat the Yukawa interaction in the Standard Model,one finds that it is indeed U(1) invariant since thesum of hypercharges in each vertex equals zero.For the up quarks this is achieved by taking theconjugated Higgs doublet H = iτ2H
† instead ofH . However, in SUSY H is a chiral superfieldand hence a superpotential, which is constructedout of chiral fields, can contain only H but not Hwhich is an antichiral superfield.Another reason for the second Higgs doublet
is related to chiral anomalies. It is known thatchiral anomalies spoil the gauge invariance and,hence, the renormalizability of the theory. Theyare canceled in the SM between quarks and lep-tons in each generation [14]
TrY 3 = 3 ( 127 + 1
27 − 6427 + 8
27 )− 1− 1 + 8 = 0color uL dL uR dR νL eL eR
However, if one introduces a chiral Higgs su-perfield, it contains higgsinos, which are chiralfermions, and contain anomalies. To cancel themone has to add the second Higgs doublet withthe opposite hypercharge. Therefore, the Higgssector in SUSY models is inevitably enlarged, it
contains an even number of doublets.Conclusion: In SUSY models supersymmetry
associates known bosons with new fermions andknown fermions with new bosons.
4.1. The field contentConsider the particle content of the Minimal
Supersymmetric Standard Model [15,16]. Ac-cording to the previous discussion, in the minimalversion we double the number of particles (intro-ducing a superpartner to each particle) and addanother Higgs doublet (with its superpartner).Thus, the characteristic feature of any super-
symmetric generalization of the SM is the pres-ence of superpartners (see Fig.5) [17]. If super-symmetry is exact, superpartners of ordinary par-ticles should have the same masses and have tobe observed. The absence of them at modern en-ergies is believed to be explained by the fact thattheir masses are very heavy, that means that su-persymmetry should be broken. Hence, if the en-ergy of accelerators is high enough, the superpart-ners will be created.
Figure 5. The shadow world of SUSY particles
The particle content of the MSSM then appearsas shown in the table. Hereafter, tilde denotes asuperpartner of an ordinary particle.The presence of an extra Higgs doublet in
SUSY model is a novel feature of the theory. Inthe MSSM one has two doublets with the quan-
8
Particle Content of the MSSM
Superfield Bosons Fermions SU(3)SU(2 UY (1)
GaugeGa gluon ga gluino ga 8 0 0Vk Weak W k (W±, Z) wino, zino wk (w±, z) 1 3 0
V′ Hypercharge B (γ) bino b(γ) 1 1 0
Matter
Li
Ei
sleptons
Li = (ν, e)LEi = eR
leptons
Li = (ν, e)LEi = ecR
11
21
−12
Qi
Ui
Di
squarks
Qi = (u, d)LUi = uRDi = dR
quarks
Qi = (u, d)LUi = ucRDi = dcR
33∗
3∗
211
1/3−4/32/3
Higgs
H1
H2
Higgses
H1
H2higgsinos
H1
H2
11
22
−11
tum numbers (1,2,-1) and (1,2,1), respectively:
H1 =
(
H01
H−1
)
=
(
v1 +S1+iP1√
2
H−1
)
,
H2 =
(
H+2
H02
)
=
(
H+2
v2 +S2+iP2√
2
)
,
where vi are the vacuum expectation values of theneutral components.
Hence, one has 8=4+4=5+3 degrees of free-dom. As in the case of the SM, 3 degrees of free-dom can be gauged away, and one is left with 5physical states compared to 1 in the SM. Thus, inthe MSSM, as actually in any of two Higgs dou-blet models, one has five physical Higgs bosons:two CP-even neutral, one CP-odd neutral andtwo charged. We consider the mass eigenstatesbelow.
4.2. Lagrangian of the MSSMTo construct a SUSY Lagrangian one has to
follow the following three steps:
• 1st step: Take your favorite Lagrangianwritten in terms of fields
• 2nd step: Replace the fields (φ, ψ,Aµ) bysuperfields Φ, V
• 3rd step: Replace the Action by superAc-tion
A =
∫
d4xL(x) ⇒ A =
∫
d4x d4θL(x, θ, θ)
At the last step one has to perform the integrationover the Grassmannian variables. The rules ofintegration are very easy [18]:
∫
dθα = 0,
∫
θαdθβ = δα,β .
Now we can construct the Lagrangian of theMSSM. It consists of two parts; the first part isthe SUSY generalization of the Standard Model,while the second one represents the SUSY break-ing as mentioned above.
L = LSUSY + LBreaking , (4.1)
where
LSUSY = LGauge + LY ukawa. (4.2)
We will not describe the gauge part since itis essentially the gauge invariant kinetic termsbut rather concentrate on Yukawa terms. Theyare given by the so-called superpotential which is
9
nothing else but the usual Yukawa terms with thefields replaced by superfields as explained above.
LY ukawa = ǫij(yUabQ
jaU
cbH
i2 + yDabQ
jaD
cbH
i1
+yLabLjaE
cbH
i1 + µHi
1Hj2), (4.3)
where i, j = 1, 2 are the SU(2) and a, b = 1, 2, 3are the generation indices; colour indices are sup-pressed. This part of the Lagrangian almost ex-actly repeats that of the SM. The only differenceis the last term which describes the Higgs mix-ing. It is absent in the SM since there is only oneHiggs field there.However, one can write down the other Yukawa
terms
LY ukawa = ǫij(λLabdL
iaL
jbE
cd + λL′
abdLiaQ
jbD
cd
+µ′aL
iaH
j2) + λBabdU
caD
cbD
cd. (4.4)
These terms are absent in the SM. The reason isvery simple: one can not replace the superfieldsin eq.(4.4) by the ordinary fields like in eq.(4.3)because of the Lorentz invariance. These termshave a different property, they violate either lep-ton (the first 3 terms in eq.(4.4)) or baryon num-ber (the last term). Since both effects are not ob-served in Nature, these terms must be suppressedor excluded. One can avoid such terms introduc-ing a special symmetry called R-symmetry[19].This is the global U(1)R invariance
U(1)R : θ → eiαθ, Φ → einαΦ, (4.5)
which is reduced to the discrete group Z2, calledthe R-parity. The R-parity quantum number isgiven by R = (−1)3(B−L)+2S for particles withspin S. Thus, all the ordinary particles havethe R-parity quantum number equal to R = +1,while all the superpartners have R-parity quan-tum number equal to R = −1. The first part ofthe Yukawa Lagrangian is R-symmetric, while thesecond part is R-nonsymmetric. The R-parity ob-viously forbids the terms. However, it may wellbe that these terms are present, though experi-mental limits on the couplings are very severe
λLabc, λL′abc < 10−4, λBabc < 10−9.
4.3. Properties of interactionsIf one assumes that the R-parity is preserved,
then the interactions of superpartners are essen-tially the same as in the SM, but two of three
particles involved into an interaction at any ver-tex are replaced by superpartners. The reason forit is the R-parity. Conservation of the R-parityhas two consequences• the superpartners are created in pairs;• the lightest superparticle (LSP) is stable.
Usually it is photino γ, the superpartner of a pho-ton with some admixture of neutral higgsino.Typical vertices are shown in Figs.6. The tilde
above a letter denotes the corresponding super-partner. Note that the coupling is the same in allthe vertices involving superpartners.
yU
Rigid
AU
Soft Rigid
yU
Q UL R
H2
_
H2
H2
~
Q QU UL R L R
+_ _~ ~~
Figure 6. Gauge-matter interaction, Gauge self-interaction and Yukawa-type interaction
4.4. Creation and decay of superpartnersThe above-mentioned rule together with the
Feynman rules for the SM enables one to draw di-agrams describing creation of superpartners. Oneof the most promising processes is the e+e− an-nihilation (see Fig.7).The usual kinematic restriction is given by the
c.m. energy mmaxsparticle ≤
√s2 . Similar processes
10
e
+
e
=Z
~
+
1
;
~
l
+
L;R
;
~
t
i
;
~
b
i
; ~
0
i
~
1
;
~
l
L;R
;
~
t
i
;
~
b
i
; ~
0
j
~
+
1
; ~
0
i
~
1
; ~
0
j
~
e
(~e
L;R
)
~e
+
L;R
(
~
e
)
~e
L;R
(~
e
)
~
0
i
(~
1
)
e
+
e
e
+
e
Figure 7. Creation of superpartners at electron-positron colliders
take place at hadron colliders with electrons andpositrons being replaced by quarks and gluons.
Experimental signatures at hadron colliders aresimilar to those at e+e− machines; however, hereone has much wider possibilities. Besides theusual annihilation channel, one has numerousprocesses of gluon fusion, quark-antiquark andquark-gluon scattering (see Fig.8).
Creation of superpartners can be accompaniedby creation of ordinary particles as well. Weconsider various experimental signatures below.They crucially depend on SUSY breaking patternand on the mass spectrum of superpartners.
The decay properties of superpartners also de-pend on their masses. For the quark and leptonsuperpartners the main processes are shown inFig.9.
5. Breaking of SUSY in the MSSM
Usually it is assumed that supersymmetry isbroken spontaneously via the v.e.v.s of somefields. However, in the case of supersymmetryone can not use the scalar fields like the Higgsfield, but rather the auxiliary fields present inany SUSY multiplet. There are two basic mecha-nisms of spontaneous SUSY breaking: the Fayet-Iliopoulos (or D-type) mechanism [20] based on
Figure 8. Gluon fusion, qq scattering, quark-gluon scattering
Figure 9. Decay of superpartners
the D auxiliary field from a vector multiplet andthe O’Raifeartaigh (or F-type) mechanism [21]based on the F auxiliary field from a chiral mul-tiplet. Unfortunately, one can not explicitly usethese mechanisms within the MSSM since noneof the fields of the MSSM can develop non-zerov.e.v. without spoiling the gauge invariance.Therefore, a spontaneous SUSY breaking shouldtake place via some other fields.The most common scenario for producing low-
energy supersymmetry breaking is called the hid-den sector one [22]. According to this scenario,
11
there exist two sectors: the usual matter belongsto the ”visible” one, while the second, ”hidden”sector, contains fields which lead to breaking ofsupersymmetry. These two sectors interact witheach other by exchange of some fields called mes-sengers, which mediate SUSY breaking from thehidden to the visible sector. There might be var-ious types of messenger fields: gravity, gauge,etc. The hidden sector is the weakest part of theMSSM. It contains a lot of ambiguities and leadsto uncertainties of the MSSM predictions consid-ered below.So far there are known four main mechanisms
to mediate SUSY breaking from a hidden to avisible sector:
• Gravity mediation (SUGRA) [23];• Gauge mediation [24];• Anomaly mediation [25];• Gaugino mediation [26].
All four mechanisms of soft SUSY breaking aredifferent in details but are common in results.Predictions for the sparticle spectrum depend onthe mechanism of SUSY breaking. In what fol-lows, to calculate the mass spectrum of superpart-ners, we need an explicit form of SUSY breakingterms. For the MSSM and without the R-parityviolation one has in general
−LBreaking = (5.1)
=∑
i
m20i|ϕi|2 +
(
1
2
∑
α
Mαλαλα +BH1H2
+ AUabQaU
cbH2 +AD
abQaDcbH1 +AL
abLaEcbH1
)
,
where we have suppressed the SU(2) indices.Here ϕi are all scalar fields, λα are the gauginofields, Q, U , D and L, E are the squark and slep-ton fields, respectively, and H1,2 are the SU(2)doublet Higgs fields.Eq.(5.1) contains a vast number of free param-
eters which spoils the prediction power of themodel. To reduce their number, we adopt theso-called universality hypothesis, i.e., we assumethe universality or equality of various soft param-eters at a high energy scale, namely, we put allthe spin 0 particle masses to be equal to the uni-versal value m0, all the spin 1/2 particle (gaug-ino) masses to be equal to m1/2 and all the cu-bic and quadratic terms, proportional to A and
B, to repeat the structure of the Yukawa super-potential (4.3). This is an additional require-ment motivated by the supergravity mechanismof SUSY breaking. Universality is not a neces-sary requirement and one may consider nonuni-versal soft terms as well. However, it will notchange the qualitative picture presented below; sofor simplicity, in what follows we consider the uni-versal boundary conditions. In this case, eq.(5.1)takes the form
−LBreaking = (5.2)
= m20
∑
i
|ϕi|2 +(
m1/2
2
∑
α
λαλα +B[µH1H2]
+ A[yUabQaUcbH2 + yDabQaD
cbH1 + yLabLaE
cbH1]
)
.
Thus, we are left with five free parameters,namely, m0,m1/2, A,B and µ versus two param-eters of the SM coming from the Higgs potential,m2 and λ. In supersymmetry the Higgs poten-tial is not arbitrary but is calculated from theYuakawa and gauge terms as we shall see below.The soft terms explicitly break supersymme-
try. As will be shown later, they lead to the massspectrum of superpartners different from that ofordinary particles. Remind that the masses ofquarks and leptons remain zero until SU(2) in-variance is spontaneously broken.
5.1. The soft terms and the mass formulasThere are two main sources of the mass terms
in the Lagrangian: the D terms and soft ones.With given values of m0,m1/2, µ, Yt, Yb, Yτ , A,and B one can construct the mass matrices forall the particles. Knowing them at the GUTscale, one can solve the corresponding RG equa-tions, thus linking the values at the GUT andelectroweak scales. Substituting these parame-ters into the mass matrices, one can predict themass spectrum of superpartners [27,28].
5.1.1. Gaugino-higgsino mass termsThe mass matrix for gauginos, the superpart-
ners of the gauge bosons, and for higgsinos, thesuperpartners of the Higgs bosons, is nondiago-nal, thus leading to their mixing. The mass termslook like
LGaugino−Higgsino = (5.3)
12
= −1
2M3λaλa − 1
2χM (0)χ− (ψM (c)ψ + h.c.),
where λa, a = 1, 2, . . . , 8, are the Majorana gluinofields and
χ =
B0
W 3
H01
H02
, ψ =
(
W+
H+
)
(5.4)
are, respectively, the Majorana neutralino andDirac chargino fields.
The neutralino mass matrix is
M (0)=
M1 0 -MZcβsW MZsβsW0 M2 MZcβcW -MZsβcW
-MZcβsW MZcβcW 0 -µMZsβsW -MZsβcW -µ 0
,
where tanβ = v2/v1 is the ratio of two Higgsv.e.v.s and sinW = sin θW is the usual sinus ofthe weak mixing angle. The physical neutralinomasses Mχ0
i
are obtained as eigenvalues of thismatrix after diagonalization.
For charginos one has
M (c) =
(
M2
√2MW sinβ√
2MW cosβ µ
)
. (5.5)
This matrix has two chargino eigenstates χ±1,2
with mass eigenvalues
M21,2 =
1
2
[
M22 + µ2 + 2M2
W∓ (5.6)√
(M22−µ2)2+4M4
W c22β+4M
2W (M2
2 +µ2+2M2µs2β)
]
.
5.1.2. Squark and slepton massesNon-negligible Yukawa couplings cause a mix-
ing between the electroweak eigenstates and themass eigenstates of the third generation particles.The mixing matrices for m2
t , m2b and m2
τ are
(
m2tL mt(At − µ cotβ)
mt(At − µ cotβ) m2tR
)
, (5.7)
(
m2bL mb(Ab − µ tanβ)
mb(Ab − µ tanβ) m2bR
)
, (5.8)
(
m2τL mτ (Aτ − µ tanβ)
mτ (Aτ − µ tanβ) m2τR
)
(5.9)
with
m2tL = m2
Q +m2t +
1
6(4M2
W −M2Z) cos 2β,
m2tR = m2
U +m2t −
2
3(M2
W −M2Z) cos 2β,
m2bL = m2
Q +m2b −
1
6(2M2
W +M2Z) cos 2β,
m2bR = m2
D +m2b +
1
3(M2
W −M2Z) cos 2β,
m2τL = m2
L +m2τ − 1
2(2M2
W −M2Z) cos 2β,
m2τR = m2
E +m2τ + (M2
W −M2Z) cos 2β
and the mass eigenstates are the eigenvalues ofthese mass matrices. For the light generationsthe mixing is negligible.The first terms here (m2) are the soft ones,
which are calculated using the RG equationsstarting from their values at the GUT (Planck)scale. The second ones are the usual masses ofquarks and leptons and the last ones are the Dterms of the potential.
5.2. The Higgs potentialAs has already been mentioned, the Higgs po-
tential in the MSSM is totally defined by superpo-tential (and the soft terms). Due to the structureof LY ukawa the Higgs self-interaction is given bythe D-terms while the F -terms contribute only tothe mass matrix. The tree level potential is
Vtree = m21|H1|2 +m2
2|H2|2−m23(H1H2+h.c.)
+g2 + g
′2
8(|H1|2 − |H2|2)2 +
g2
2|H+
1 H2|2, (5.10)
where m21 = m2
H1+ µ2,m2
2 = m2H2
+ µ2. At theGUT scale m2
1 = m22 = m2
0 + µ20, m
23 = −Bµ0.
Notice that the Higgs self-interaction coupling ineq.(5.10) is fixed and defined by the gauge inter-actions as opposed to the SM.The potential (5.10), in accordance with super-
symmetry, is positive definite and stable. It hasno nontrivial minimum different from zero. In-deed, let us write the minimization condition forthe potential (5.10)
1
2
δV
δH1=m2
1v1−m23v2+
g2+g′2
4(v21−v22)v1=0,(5.11)
1
2
δV
δH2=m2
2v2−m23v1+
g2+g′2
4(v21−v22)v2=0,(5.12)
13
where we have introduced the notation
< H1 >≡ v1 = v cosβ, < H2 >≡ v2 = v sinβ,
v2 = v21 + v22 , tanβ ≡ v2v1.
Solution of eqs.(5.11),(5.12) can be expressed interms of v2 and sin 2β
v2=4(m2
1−m22 tan
2 β)
(g2+g′2)(tan2 β−1), sin 2β =
2m23
m21+m
22
.(5.13)
One can easily see from eq.(5.13) that if m21 =
m22 = m2
0 + µ20, v
2 happens to be negative, i.e.the minimum does not exist. In fact, real posi-tive solutions to eqs.(5.11),(5.12) exist only if thefollowing conditions are satisfied:
m21 +m2
2 > 2m23, m2
1m22 < m4
3, (5.14)
which is not the case at the GUT scale. Thismeans that spontaneous breaking of the SU(2)gauge invariance, which is needed in the SM togive masses for all the particles, does not takeplace in the MSSM.This strong statement is valid, however, only at
the GUT scale. Indeed, going down with energy,the parameters of the potential (5.10) are renor-malized. They become the “running” parameterswith the energy scale dependence given by theRG equations.
5.3. Radiative electroweak symmetrybreaking
The running of the Higgs masses leads to theremarkable phenomenon known as radiative elec-troweak symmetry breaking. Indeed, one can seein Fig.10 that m2
2 (or both m21 and m2
2) decreaseswhen going down from the GUT scale to the MZ
scale and can even become negative. As a result,at some value of Q2 the conditions (5.14) are sat-isfied, so that the nontrivial minimum appears.This triggers spontaneous breaking of the SU(2)gauge invariance. The vacuum expectations ofthe Higgs fields acquire nonzero values and pro-vide masses to quarks, leptons and SU(2) gaugebosons, and additional masses to their superpart-ners.In this way one also obtains the explanation of
why the two scales are so much different. Due to
the logarithmic running of the parameters, oneneeds a long ”running time” to get m2
2 (or bothm2
1 and m22) to be negative when starting from a
positive value of the order of MSUSY ∼ 102÷ 103
GeV at the GUT scale.
5.4. The mass spectrumThe mass spectrum is defined by low energy
parameters. To calculate the low energy valuesof the soft terms, we use the corresponding RGequations [29]. Having all the RG equations, onecan now find the RG flow for the soft terms. Tak-ing the initial values of the soft masses at theGUT scale in the interval between 102÷ 103 GeVconsistent with the SUSY scale suggested by uni-fication of the gauge couplings (2.7) leads to theRG flow of the soft terms shown in Fig.10. [27,28]
0
100
200
300
400
500
600
700
2 4 6 8 10 12 14 16
m1/2
m0
√(µ02+m0
2)
Wino
Bino
GluinoqL~
tL~
tR~
lL~
lR~
m1
m2
tan β = 1.65Yb = Yτ
log10 Q
mas
s [G
eV]
0
100
200
300
400
500
600
700
2 4 6 8 10 12 14 16
m1/2
m0
√(µ02+m0
2)
Wino
Bino
Gluino
qL~
tL~
tR~
lL~
lR~
m1 m2
tan β = 50Yt = Yb = Yτ
log10 Q
mas
s [G
eV]
Figure 10. An example of evolution of sparticlemasses and soft supersymmetry breaking param-eters m2
1 = m2H1
+µ2 and m22 = m2
H2+µ2 for low
(left) and high (right) values of tanβ
14
One should mention the following general fea-tures common to any choice of initial conditions:
i) The gaugino masses follow the running of thegauge couplings and split at low energies. Thegluino mass is running faster than the others andis usually the heaviest due to the strong interac-tion.
ii) The squark and slepton masses also split atlow energies, the stops (and sbottoms) being thelightest due to relatively big Yukawa couplings ofthe third generation.
iii) The Higgs masses (or at least one of them)are running down very quickly and may even be-come negative.
Typical dependence of the mass spectra on theinitial conditions (m0) is also shown in Fig.11[30,31]. For a given value of m1/2 the massesof the lightest particles are practically indepen-dent ofm0, while the heavier ones increase with itmonotonically. One can see that the lightest neu-tralinos and charginos as well as the stop squarkmay be rather light.
Provided conditions (5.14) are satisfied, onecan also calculate the masses of the Higgs bosons.The mass matrices at the tree level areCP-odd components P1 and P2 :
Modd =∂2V
∂Pi∂Pj
∣
∣
∣
∣
Hi=vi
=
(
tanβ 11 cotβ
)
m23, (5.15)
CP-even neutral components S1 and S2:
Mev =∂2V
∂Si∂Sj
∣
∣
∣
∣
Hi=vi
=
(
tanβ −1−1 cotβ
)
m23
+
(
cotβ −1−1 tanβ
)
MZsin 2β
2, (5.16)
Charged components H− and H+:
Mch =∂2V
∂H+i ∂H
−j
∣
∣
∣
∣
∣
Hi=vi
(5.17)
=
(
tanβ 11 cotβ
)
(m23 +MW
sin 2β
2).
Diagonalizing the mass matrices, one gets the
χ01
χ+1
χ02
χ+2
χ03
χ04
0
100
200
300
400
500
600
700
m0[GeV]
0 100 200 300 400 500
mas
s [G
eV]
800
m1/2 = 150 GeV
µ < 0
m g ~
m ν ~ m τ 1
~
t m ~ 1
t m ~ 2
b m ~
1
0
100
200
300
400
500
600
700
800
m 0 [GeV]
0 100 200 300 400 500
mas
s [G
eV]
– me1 ~ ~
m1/2 = 150 GeV
µ < 0
Figure 11. The masses of sparticles as functionsof the initial value m0
mass eigenstates:
G0 = − cosβP1 + sinβP2, Goldst boson→ Z0,A = sinβP1 + cosβP2, Neutral CPodd Higgs,
G+=−cosβ(H−1 )∗+sinβH+
2 , Goldst boson→W+,H+ = sinβ(H−
1 )∗ + cosβH+2 , Charged Higgs,
h = − sinαS1 + cosαS2, SM CP even Higgs,H = cosαS1 + sinαS2, Extra heavy Higgs,
where the mixing angle α is given by
tan 2α = tan 2β
(
m2A +M2
Z
m2A −M2
Z
)
.
The physical Higgs bosons acquire the followingmasses [16]:
CP-odd neutral Higgs A : m2A = m2
1+m22, (5.18)
15
Charge Higgses H± : m2H± = m2
A+M2W ,
CP-even neutral Higgses H, h:
m2H,h=
1
2
[
m2A+M
2Z±√
(m2A+M
2Z)
2−4m2AM
2Zc
22β
]
,
(5.19)
where, as usual,
M2W =
g2
2v2, M2
Z =g2 + g′2
2v2.
This leads to the once celebrated SUSY mass re-lations
mH± ≥MW , mh ≤ mA ≤MH ,
mh ≤MZ | cos 2β| ≤MZ , (5.20)
m2h +m2
H = m2A +M2
Z
Thus, the lightest neutral Higgs boson happensto be lighter than the Z boson, which clearly dis-tinguishes it from the SM one. Though we donot know the mass of the Higgs boson in the SM,there are several indirect constraints leading tothe lower boundary of mSM
h ≥ 135 GeV. Afterincluding the radiative corrections, the mass ofthe lightest Higgs boson in the MSSM, mh, reads
m2h=M
2Z cos2(2β)+
3g2m4t
16π2M2W
logm2
t1m2t2
m4t
+...(5.21)
which leads to about 40 GeV increase [32]. Thesecond loops correction is negative but small [33].
6. Constrained MSSM
6.1. Parameter space of the MSSMThe Minimal Supersymmetric Standard Model
has the following free parameters: i) three gaugecouplings αi; ii) three matrices of the Yukawacouplings yiab, where i = L,U,D; iii) the Higgsfield mixing parameter µ; iv) the soft supersym-metry breaking parameters. Compared to the SMthere is an additional Higgs mixing parameter,but the Higgs self-coupling, which is arbitrary inthe SM, is fixed by supersymmetry. The mainuncertainty comes from the unknown soft terms.With the universality hypothesis one is left
with the following set of 5 free parameters defin-ing the mass scales
µ, m0, m1/2, A and B ↔ tanβ =v2v1.
While choosing parameters and making predic-tions, one has two possible ways to proceed:i) take the low-energy parameters like su-
perparticle masses mt1, mt2,mA, tanβ, mixingsXstop, µ, etc. as input and calculate cross-sectionsas functions of these parameters.ii) take the high-energy parameters like the
above mentioned 5 soft parameters as input, runthe RG equations and find the low-energy values.Now the calculations can be carried out in termsof the initial parameters. A typical range of theseparameters is
100 GeV ≤ m0,m1/2, µ ≤ 1− 2 TeV,
−3m0 ≤ A0 ≤ 3m0, 1 ≤ tanβ ≤ 70.
The experimental constraints are sufficient to de-termine these parameters, albeit with large un-certainties.
6.2. The choice of constraintsWhen subjecting constraints on the MSSM,
perhaps, the most remarkable fact is that all ofthem can be fulfilled simultaneously. In our anal-ysis we impose the following constraints on theparameter space of the MSSM:• Gauge coupling constant unification;
This is one of the most restrictive constraints,which we have discussed in Sect 2. It fixes thescale of SUSY breaking of an order of 1 TeV.• MZ from electroweak symmetry breaking;
Radiative EW symmetry breaking (see eq.(5.13))defines the mass of the Z-boson
M2Z
2=m2
1−m22 tan
2 β
tan2 β−1=−µ2+
m2H1
−m2H2
tan2 β
tan2 β−1.
This condition determines the value of µ for givenvalues of m0 and m1/2.• Precision measurement of decay rates;
We take the branching ratio BR(b → sγ) whichhas been measured by the CLEO [34] collabora-tion and later by ALEPH [35] and the branch-ing ration BR(Bs → µ+µ−) measured recentlyby CDF collaboration [36]. Susy contributionsshould not destroy the agreement with the SMand in some cases can improve it. This require-ment imposes severe restrictions on the parameterspace, especially for the case of large tanβ.
16
• Anomalous magnetic moment of muon;Recent measurement of the anomalous magneticmoment indicates small deviation from the SM ofthe order of 2 σ. The deficiency may be easilyfilled with SUSY contribution.
• The lightest superparticle (LSP) should beneutral, otherwhise we would have charged cloudsof stable particles in the Universe which is notobserved.
• Experimental lower limits on SUSY masses;SUSY particles have not been found so far andthe searches at LEP impose the lower limit on thecharged lepton and chargino masses of about halfof the centre of mass energy [37]. The lower limiton the neutralino masses is smaller. There existalso limits on squark and gluino masses from theTevatron collider [38]. These limits restrict theminimal values for the SUSY mass parameters.
• Dark Matter constraint;Recent very precise astrophysical data restrict theamount of the Dark matter in the Universe upto 23%. Assuming h0 ∼ 0.7 one finds that thecontribution of each relic particle species χ hasto obey Ωχh
20 ∼ 0.13 ± 0.03.. This serves as a
very severe bound on SUSY parameters [39].Having in mind the above mentioned con-
straints one can find the most probable region ofthe parameter space by minimizing the χ2 func-tion [28]. Since the parameter space is 5 di-mensional one can not plot it explicitly and isbounded to use various projections. We will ac-cept the following strategy: We first choose thevalue of the Higgs mixing parameter µ from therequirement of radiative EW symmetry breakingand then take the set of values of tanβ. Parame-ter A happens to be less important and we will fixit typically like A0 = 0. Then we are left with twoparameters m0 and m1/2 and we present the re-strictions coming from various constraints in them0,m1/2 plane.
6.3. The excluded regions of parameterspace
• We start with the Higgs mass constraint. Ex-perimental lower limit on the Higgs mass fromLEP2: mh ≥ 114.7 GeV cuts the part of the pa-rameter space as shown in Fig.12.
• The next two constrains are related to rare
decays where SUSY may contribute. The firstone is b → sγ decay which in the SM given bythe diagrams shown on top of Fig.13 and leads to
BRSM (b→ sγ) = (3.28± 0.33) · 10−4
while experiment gives [34,35]
BREX(b→ sγ) = (3.43± 0.36) · 10−4.
These two values almost coincide but still leavesome room for SUSY. SUSY contribution comesfrom the diagrams shown in the bottom of Fig.13and is enhanced by tanβ[40]
BRSUSY(b→ sγ) ∝ µAtmb tanβf(m2t1 , m
2t1 ,mχ±)
The obtained constraints are shown in Fig.12.
Figure 13. The diagrams contributing to b → sγdecay in the SM and in the MSSM.
The second decay is Bs → µ+µ−. In the SMit is given by the diagrams shown in Fig.14. Thebranching ratio is BRSM (Bs → µ+µ−) = 3.5 ·10−9, while the recent experiment gives only thelower bound BREx(Bs → µ+µ−) < 4.5 ·10−8[36].In the MSSM one has several diagrams but themain contribution enhanced by (tanβ)6 (!) comesfrom the one shown in the bottom of Fig.14. It isproportional to [41]
BRSUSY (bs → µµ) ∝ tan6 βm2
bm2tm
2µµ
2
M4Wm4
A
×
17
×
m2t1 log
m2
t1
µ
mu2 −m2t1
−m2
t2 logm2
t2
µ
mu2 −m2t2
2
As a result for large tanβ one comes in a con-tradiction with experiment. The values of thebranching ratio for various parameters are shownin Fig.15 [41] and the restrictions on the param-eter space in Fig.12
−W −µ
+µ+W
s
b
t υl
−W
−µ
+µ
Z
+W
s
b
t
b µ
µs
+
H,A
~t_
~i
.. .i
χ+_
Figure 14. The diagrams contributing to Bs →µµ decay in the SM and in the MSSM.
• Anomalous magnetic moment of muon. Re-cent measurement of the anomalous magnetic mo-ment indicates small deviation from the SM of theorder of 2.5 σ[42]:
aexpµ = 11 659 202 (14)(6) · 10−10
aSMµ = 11 659 159.6 (6.7) · 10−10
∆aµ = aexpµ − atheorµ = (27± 10) · 10−10,
where the SM contribution comes from
aQEDµ = 11 658 470.56 (0.29) · 10−10
aWeakµ = 15.1 (0.4) · 10−10
ahadronµ = 673.9 (6.7) · 10−10,
1
10
10 2
250 500 750 1000m1/2[GeV]
Br[
Bs→
µµ] ×
108
A0=0, µ>0m0=300 GeV
tanβ=50tanβ=45
tanβ=40
tanβ=30
Figure 15. The values of the branching ratioBs → µµ decay in the MSSM.
so that the accuracy of the experiment finallyreaches the order of the weak contribution. Thecorresponding diagrams are shown in Fig.16.
µ µ
ν
γ
W W
µ µ
Z
γ
µµ µ
H
γ
µ
µ µ
∼ν
γ
∼χ ∼χµ µ
∼χ0
γ
∼µ ∼µ
Figure 16. The diagrams contributing to aµ inthe SM and in the MSSM.
The deficiency may be easily filled with SUSYcontribution coming from the diagrams shown inthe bottom of Fig.16. They are similar to that ofthe weak interactions after replacing the vectorbosons by charginos and neutralinos.The total contribution to aµ can be approxi-
mated by [43]
|aSUSYµ | ≃ α(MZ)
8π sin2 θW
m2µ tanβ
m2SUSY
(
1−4α
πlnmSUSY
mµ
)
18
-400
-200
0
200
400
600
0 10 20 30 40 50 60 70
a µSU
SY *
1011
tanβ
(m 0,m 1/2)
µ0 < 0(
200,200)
(300,300)
(400,400)
µ 0 > 0
(200,200)
(300,300)
(400,400)
Figure 17. The dependence of aSUSYµ versus tanβ
for various values of the SUSY breaking param-eters m0 and m1/2. The horizontal band showsthe discrepancy between the experimental dataand the SM estimate.
≃ 140 · 10−11
(
100 GeV
mSUSY
)2
tanβ,
where mµ is the muon mass, mSUSY is an aver-age mass of supersymmetric particles in the loop(essentially the chargino mass). It is proportionalto µ and tanβ as shown in Fig.17. This requirespositive sign of µ that kills a half of the parameterspace of the MSSM [44].
If SUSY particles are light enough they give thedesired contribution to the anomalous magneticmoment. However, if they are too light the con-tribution exceeds the gap between the experimentand the SM. For too heavy particles the contribu-tion is too small. This defines the allowed regionsas shown in Fig.12.
• The requirement that the lightest supersym-metric particle (LSP) is neutral also restricts theparameter space. This constraint is a conse-quence of R-parity conservation. The regions ex-cluded by the LSP constraint are shown in Fig.12.
Summarizing all together we have the allowedregion in parameter space as shown at the lastplots in Fig.12 [44,45]. Some requirementsare complimentary being essential for smaller orlarger values of tanβ. One can see that a) all re-quirements are consistent and b) they still leave alot of freedom for the choice of parameters. Anal-
ogous analysis has been performed in a numberof papers [46] with similar results.• Astrophysical constraints. One can also im-
pose the constraint that comes from astrophysics.The accuracy of measurement of the amount ofthe Dark Matter in the Universe defines withhigh precision the cross-section of DM annihila-tion. This in its turn requires the adjustment ofparameters. We consider this problem in moredetail in the last section. As a result one findsthat this constraint is fulfilled in a narrow bandin m0,m1/2 plane for any fixed value of tanβ asshown in Fig.18 [47]. This plot corresponds totanβ = 50. With decreasing values of tanβ thecurve moves to the left and the funnel disappears.
Figure 18. The light (lbue) band is the regionallowed by the WMAP data. The excluded re-gions where the LSP is stau (red up left corner),where the radiative electroweak symmetry break-ing mechanism does not work (red low right cor-ner), and where the Higgs boson is too light (yel-low lower left corner) are shown with dots.
In the narrow allowed region one fulfills allthe constraints simultaneously and has the suit-able amount of the dark matter. Phenomenologyessentially depends on the region of parameterspace and has direct influence on the strategy ofSUSY searches. Each point in this plane corre-sponds to a fixed set of parameters and allows oneto calculate the spectrum, the cross-sections, etc.
19
Figure 12. Regions excluded by various constraints for tanβ = 35 and 50 shown in blue( dark). The lastplots show the combination of all constraints leaving the allowed region of parameter space
20
6.4. Experimental signatures at e+e− col-liders
Experiments are finally beginning to push intoa significant region of supersymmetry parameterspace. We know the sparticles and their cou-plings, but we do not know their masses and mix-ings. Given the mass spectrum one can calculatethe cross-sections and consider the possibilities ofobserving new particles at modern accelerators.Otherwise, one can get restrictions on unknownparameters.
We start with e+e− colliders. In the leadingorder creation of superpartners is given by thediagrams shown in Fig.7 above. For a given cen-ter of mass energy the cross-sections depend onthe mass of created particles and vanish at thekinematic boundary. Experimental signatures aredefined by the decay modes which vary with themass spectrum. The main ones are summarizedbelow.
Production Decay Modes Signatures
• lL,R lL,R l±R → l±χ0i acompl pair of
l±L → l±χ0i charged lept+
/
ET
• νν ν → l±χ01
/
ET
• χ±1 χ
±1 χ±
1 → χ01l
±ν isol lept+2jets+/
ET
χ±1 → χ0
2f f′ pair of acompl
χ±1 → lνl leptons +
/
ET
→ lνlχ01
χ±1 → νl l 4 jets +
/
ET
→ νllχ01
• χ0i χ
0j χ0
i → χ01X X = νlνl invisible
= γ, 2l, 2 jets
2l+/
ET , l+2j+/
ET
• ti tj t1 → cχ01 2 jets+
/
ET
t1 → bχ±1 2b jets+2lept+
/
ET
→ bf f ′χ01
2 b jets+lept+/
ET
• bibj bi → bχ01 2 b jets +
/
ET
bi → bχ02 2 b jets+2 lept+
/
ET
→ bf f ′χ01 2 b jets+2 jets+
/
ET
A characteristic feature of all possible signaturesis the missing energy and transverse momenta,which is a trade mark of a new physics.Numerous attempts to find superpartners at
LEP II gave no positive result thus imposing thelower bounds on their masses [37]. Typical LEPII limits on the masses of superpartners are
mχ0
1
> 40 GeV, me > 105 GeV, mt > 90 GeV
mχ±
1
> 100 GeV, mµ > 100 GeV, mb > 80 GeV
mτ > 80 GeV
6.5. Experimental signatures at hadroncolliders
Experimental SUSY signatures at the Tevatronand LHC are similar. The strategy of SUSYsearch is based on an assumption that the massesof superpartners indeed are in the region of 1 TeVso that they might be created on the mass shellwith the cross section big enough to distinguishthem from the background of the ordinary parti-cles. Calculation of the background in the frame-work of the Standard Model thus becomes essen-tial since the secondary particles in all the caseswill be the same.There are many possibilities to create su-
perpartners at hadron colliders. Besides theusual annihilation channel there are numerousprocesses of gluon fusion, quark-antiquark andquark-gluon scattering. The maximal cross sec-tions of the order of a few picobarn can beachieved in the process of gluon fusion.As a rule all superpartners are short lived and
decay into the ordinary particles and the light-est superparticle. The main decay modes of su-perpartners which serve as the manifestation ofSUSY are
Production Decay Modes Signatures
• gg, qq, gqg → qqχ0
1
qq′χ±1
gχ01
/
ET +multijets
(+leptons)
q → qχ0i
q → q′χ±i
• χ±1 χ
02 χ±
1 → χ01l
±ν Trilepton +/
ET
χ02 → χ0
1ll
21
χ±1 → χ0
1qq′ Dilept+ jet+
/
ET
χ02 → χ0
1ll
• χ+1 χ
−1 χ+
1 → lχ01l
±ν Dilepton +/
ET
• χ0i χ
0i χ0
i → χ01X
/
ET+Dilept+jets
χ0i → χ0
1X′
• t1t1 t1 → cχ01 2 acollin jets+
/
ET
t1 → bχ±1 sing lept+
/
ET+b′s
χ±1 → χ0
1qq′
t1 → bχ±1 Dilept+
/
ET+b′s
χ±1 → χ0
1l±ν
• ll, lν, νν l± → l ± χ0i Dilepton +
/
ET
l± → νlχ±i Single lept + /ET
ν → νχ01 /ET
Noe again the typical events with missing en-ergy and transverse momentum that is the maindifference from the background processes of theStandard Model. Contrary to e+e− colliders,at hadron machines the background is extremelyrich and essential. The missing energy is car-ried away by the heavy particle with the massof the order of 100 GeV that is essentially differ-ent from the processes with neutrino in the finalstate. In hadron collisions the superpartners arealways created in pairs and then further quicklydecay creating a cascade with the ordinary quarks(i.e. hadron jets) or leptons at the final state plusthe missing energy. For the case of gluon fusionwith creation of gluino it is presented in Table 1.Chargino and neutralino can also be produced
in pairs through the Drell-Yang mechanism pp→χ±1 χ
02 and can be detected via their lepton de-
cays χ±1 χ
02 → ℓℓℓ +
/
ET . Hence the main signal
of their creation is the isolated leptons and miss-ing energy (Table 2). The main background intrilepton channel comes from creation of the stan-dard particles WZ/ZZ, tt, Zbb bb. There mightbe also the supersymmetric background from thecascade decays of squarks and gluino into multi-lepton modes.Numerous SUSY searches have been already
performed at the Tevatron. Pair-producedsquarks and gluinos have at least two large-ET
jets associated with large missing energy. The fi-
nal state with lepton(s) is possible due to leptonicdecays of the χ±
1 and/or χ02.
In the trilepton channel the Tevatron is sensi-tive up to m1/2 ≤ 250 GeV if m0 ≤ 200 GeV andup to m1/2 ≤ 200 GeV if m0 ≥ 500 GeV. Theexisting limits on the squark and gluino massesat the Tevatron are [48] : mq ≥ 300 GeV, mg ≥195 GeV.The LHC has an advantage of higher energy
and bigger luminosity. The cross sections forvarious superpartner production at the LHC inm0,m1/2 plane are shown in Fig. 19. One can seethat the biggest cross-section reaching 100 pb inthe maximum is achieved for gluino production.And though it strongly depends on the gluonmass, with a planned luminosity of LHC one mayhave a reliable detection. It should be mentioned,however, that being produced in collisions the su-perpartners follow the cascade decay chain andthe cross section at the final stage is essentiallysmaller being multiplied by the branching ratiosof the corresponding processes. The resultingcross-sections for particular final states are in thefb region. They are higher for hadron final stateswhere one has jets with missing energy and lowerfor lepton ones which are cleaner for detection.The cross-sections for chargino production are al-most one order of magnitude lower reaching 10pb in the maximum and those for squark produc-tion are below 1 pb. In some regions of parame-ter space with light neutralino and chargino theproduction cross-sections can reach those of thestrongly interacting particles [49].In most of the cases the superpartners are very
short lived and decay practically at the collisionpoint without leaving a secondary vertex. Onethen has hadron jets (mostly b-jets) and leptonsflying outside. The typical process of gluino pro-duction is presented in Fig.20 where the cascadedecay of one of the gluinos is shown [50]. For agiven choice of soft SUSY breaking parametersthe cross-section at the first stage reaches 13 pbbut with the 4-lepton + 4-jet final state is reducedto a few fb. To distinguish this reaction from thebackground one has to perform the analysis ofthe missing energy and consider the peculiaritiesin the invariant mass distribution of the muonpair, free pass of B-mesons, etc [50].
22
Process final states
g
gg
g
g
b
bt
χ+
1
W−
W+
b
bχ0
2
b
Z
χ0
1 q
q
lν
lν
b
χ0
12ℓ2ν6j/
ET
g
gg
g
g
b
b
b
χ02
Z
χ0
1l
lχ0
1χ±1
W∓
W±qi
qkqiqk
2ℓ6j/
ET
g
gg
g
g
b
b
b
χ02
Z
χ0
1l
lb
b
χ02
Z
χ0
1
q
qb
2ℓ6j/
ET
g
gg
g
g
b
bt
χ+
1
W−
W+
b
b
χ+
1
t
W−
W+
di
ui
di
ui
lν
ν
l
b
b
χ0
1
χ0
1
2ℓ2ν8j/
ET
g
gg
g
g
q
q
q
χ±i
W±
χ0
1qi
qkq
q
χ±i
W±
χ0
1
qi
qkq
8j/
ET
Table 1Creation of the pair of gluino with further cascadedecay
Process final states
p (q)
p (q)
Z
1
1
W
W
l
l
0
1
0
1
2ℓ2ν/
ET
p (q)
p (q)
Z
1
1
W
W
q
i
q
j
l
0
1
0
1
ℓν2j/
ET
p (q
i
)
p ( q
j
)
W
0
2
1
W
Z
l
l
l
0
1
0
1
3ℓν/
ET
p (q
i
)
p ( q
j
)
W
0
2
1
W
Z
l
0
1
0
1
ℓ3ν/
ET
p (q
i
)
p ( q
j
)
W
0
2
1
W
Z
q
q
l
0
1
0
1
ℓν2j/
ET
Table 2Creation of the lightest chargino and the secondneutralino with further cascade decay.
23
Process final states
2ℓ2j/
ET
ℓ2j/
ET
Table 3The background processes at hadron colliders(weak interactions).
Process final states
2ℓ6j/
ET
4ℓ4j/
ET
Table 4The background processes at hadron colliders(strong interactions).
[GeV]0m
0500
10001500
2000
[GeV]1/2m
050100150200250300350400
pb
1
10
210
g~ g~cross section p-p to
[GeV]0m
0500
10001500
2000
[GeV]1/2m
050100150200250300350400
pb
-210
-110
1
10
20χ -1χcross section p-p to
[GeV]0m
0500
10001500
2000
[GeV]1/2m
050100150200250300350400
pb
-310
-210
-110
1
uR~ uL~cross section p-p to
[GeV]0
m
0500
10001500
2000
[GeV]1/2m
050100150200250300350400
pb
-210
-110
1
g~ uL~cross section p-p to
Figure 19. The cross sections of superpartnerscreation as functions of m1/2 and m0 for tanβ =51, A0 = 0 and positive sign of µ.
24
Figure 20. Gluino production at the LHC accom-panied with cascade decays
6.6. The long-lived superpartnersWithin the framework of Constrained MSSM
with gravity mediated soft supersymmetry break-ing mechanism there exists an interesting pos-sibility to get long-lived next-to-lightest super-symmetric particles (staus or stops or charginos).Their production cross-sections crucially dependon a single parameter, the mass of the superpar-ticle, and for light staus can reach a few % ofpb. This might be within the LHC reach. Thestop production cross-section can achieve evenhundreds pb. Decays of long-lived superpartnerswould have an unusual signature if heavy chargedparticles decay with a considerable delay produc-ing secondary vertices inside the detector, or even
escape the detector. This possibility can be real-ized at the boundary regions of allowed parameterspace where one can have a mass degeneracy be-tween the LSP and the NLSP. The life time ofthe NLSP is inversely proportional to the massdegeneracy.
Figure 21. The regions of the parameters spaceof mSUGRA where the long-lived sparticles mightexist
We show in Fig.21 the regions where this mighthappen. The first region is the so-called co-annihilation region which is interesting from thepoint of view of existence of long-lived chargedsleptons. Their life-time may be large enough tobe produced in proton-proton collisions and tofly away from the detector area or to decay insidethe detector at a considerable distance from thecollision point. Clearly that such an event cannot be unnoticed. However, to realize this possi-bility one need a fine-tuning of parameters of themodel [51]. The second region is the border of thebulk region where light long-lived stops can exist.It appears only for large negative trilinear softsupersymmetry breaking parameter A0. On theborder of this region, in full analogy with the stauco-annihilation region, the top squark becomesthe LSP and near this border one might get thelong-lived stops. Stops can form the so-called R-hadrons (bound states of SUSY particles) if theirlifetime is bigger than the hadronisation time.
25
The last interesting region of parameter spaceis a narrow band along the line where the radia-tive electroweak symmetry breaking fails. On theborder of this region the Higgs mixing parameterµ, which is determined from the requirement ofelectroweak symmetry breaking via radiative cor-rections, tends to zero. This leads to existence oflight and degenerate states: the second charginoand two neutralinos, all of them being essentiallyhiggsinos.Experimental Higgs and chargino mass limits
as well as WMAP relic density limit can be easilysatisfied in these scenarios. However, the strongfine-tuning is required. The discussed scenariosdiffer from the GMSB scenario [52] with the grav-itino as the LSP, and next-to-lightest supersym-metric particles typically live longer.Searches for long-lived particles were already
made by LEP collaborations [53]. It is alsoof great interest at the moment since the firstphysics results of the coming LHC are expectedin the nearest future. Light long-lived sparticlescould be produced already during first monthsof its operation [54]. Since staus and stops arerelatively light in this scenario, their productioncross-sections are quite large and may achieve afew per cent of pb for stau production, and tensor even hundreds of pb for light stops, mt < 150GeV. The cross-section then quickly falls downwhen the mass of stop is increased. However, evenfor very large values of |A| when stops becomeheavier than several hundreds GeV, the produc-tion cross-section is of order of few per cent of pb,which is enough for detection with the high LHCluminosity.
6.7. The reach of the LHCThe LHC hadron collider is the ultimate ma-
chine for new physics at the TeV scale. Its c.m.energy is planned to be 14 TeV with very highluminosity up to a few hundred fb−1. The LHCis supposed to cover a wide range of parametersof the MSSM (see Figs. below) and discover thesuperpartners with the masses below 2 TeV [56].This will be a crucial test for the MSSM and thelow energy supersymmetry. The LHC potentialto discover supersymmetry is widely discussed inthe literature [56]-[57].
1000
800
600
400
200
0 400 800 1200 1600 2000m0 GeV
m1/
2, G
eV
q (2000)
~
q (1500)~
q (1000)
~
g (1500)
g (2000)~
~
g (1000)~
g (500)~
105pb-1
D_D
_925
c
Expected sparticle reach in various channels
m SUGRA; tg β = 2 (about the same up to ~ 5), A0 = 0, µ < 0
5 σ contours (σ = Nsig / Nsig + Nbkgd) for 105pb-1
h(90)
q (500)~
(400)
4 3 2 SS 2 OS 1
CMS;
12χ0 χ– ~ ~
~ ~L
Figure 22. Expected range of reach for superpart-ners in various channels at LHC [55].
To present the region of reach for the LHC indifferent channels of sparticle production it is use-ful to take the same plane of soft SUSY breakingparametersm0 andm1/2. In this case one usuallyassumes certain luminosity which will be presum-ably achieved during the accelerator operation.Thus, for instance, in Fig. 22 it is shown the
regions of reach in different channels. The linesof a constant squark mass form the arch curves,and those for gluino are almost horisontal. Thecurved lines show the reach bounds in differentchannel of creation of secondary particles. Thetheoretical curves are obtained within the MSSMfor a certain choice of the other soft SUSY break-ing parameters. As one can see, for the fortunatecircumstances the wide range of the parameterspace up to the masses of the order of 2 Tev willbe examined.The other example is shown in Fig. 23 where
the regions of reach for squarks and gluino areshown for various luminosities. One can see thatfor the maximal luminosity the discovery rangefor squarks and gluino reaches 3 TeV for the cen-ter of mass energy of 14 TeV and even higher forthe double energy.
26
CMS100 fb-1
2 OS
1
2 OS
2 SS
EmissT
EmissT
(300 fb-1)
no leptons
~g (2000)
~g (3000)
~g (1500)
~g (1000)
g (500)~
~g (2500)
q (2500)~
q (2000)
~
q (1500)
~
q (1000)
~
h (123)
h(110)
EX
1400
1200
1000
800
600
400
200
0 500 1000 1500 2000m0 (GeV)
m1/
2 (GeV
)
Explorable domain in q, g seaches
m SUGRA, A0 = 0, tan β= 35, µ > 05 σ contours ; non - isolated muons
in n leptons + Emiss + > 2 jets final statest
~ ~
D_D
_126
6c.m
od
TH
visibility of dilepton structure
q (500)
~
Figure 23. Expected domain of searches forsquarks and gluions at LHC [55].
The LHC will be able to discover SUSY withsquark and gluino masses up to 2 ÷ 2.5 TeV forthe luminosity Ltot = 100 fb−1. The most power-ful signature for squark and gluino detection aremultijet events; however, the discovery potentialdepends on relation between the LSP, squark, andgluino masses, and decreases with the increase ofthe LSP mass.
The same is true for the sleptons as shown inFig. 24. The slepton pairs can be created viathe Drell-Yang mechanism pp → γ∗/Z∗ → ℓ+ℓ−
and can be detected through the lepton decaysℓ → ℓ + χ0
1. The typical signal used for sleptondetection is the dilepton pair with the missingenergy without hadron jets. For the luminosity ofLtot = 100 fb−1 the LHC will be able to discoversleptons with the masses up to 400 GeV [56]. Thediscovery reach for sleptons in various channels isshown in Fig.24.
6.8. The lightest superparticleOne of the crucial questions is the properties of
the lightest superparticle. Different SUSY break-ing scenarios lead to different experimental signa-tures and different LSP.
• Gravity mediationIn this case, the LSP is the lightest neutralino
χ01, which is almost 90% photino for a low tanβ
100
100
50
150
200
250
300
350
400
450
150 50 0 0
200 250 300 350 400 450 500
m0 (GeV)
m1/2
(G
eV
)
D_
D_
20
62
.c.1
TH
TH
LEP2 + Tevatron
CMS reach
mSUGRA-MSSM
Slepton mapping of parameter space
Significance of expected excess of events in 2 leptonfinal state over SM + SUSY bkgd with 105 pb-1
tg β = 35, A0 = 0, µ > 0
4.5 σ
5 σ
5.1 σ
5.1 σ
4.2 σ
5.3 σ
5.5 σ
4.3 σ
3.9 σ
5.3 σ
tg β = 35, A0 = 0 µ > 0
mlL= 350 GeV~
mlR = 300 GeV~
5σ contour, σ = S / √S+B
tan β= 2 A0= 0; µ < 0
TevatronLEP2
TH
TH
EX
0 100 200 300 400 500
100
200
300
400
500
600
m0 (GeV)
m1
/2 (
Ge
V)
D_
D_
10
16
c
CMS reachat 105 pb-1
R (250)~
L (250)
~
L (150)
R (400)~
L (400)~
~
R (150)~
χ0 (200)1
~
χ0 (100)
1~
tg β = 2, A0 = 0, µ < 0
Figure 24. Expected range of reach for sleptonsat LHC [55].
solution and contains more higgsino admixturefor high tanβ. The usual signature for LSP ismissing energy; χ0
1 is stable and is the best can-didate for the cold dark matter in the Universe.Typical processes, where the LSP is created, end
up with jets +/
ET , or leptons +/
ET , or both jets
+ leptons + /ET .• Gauge mediationIn this case the LSP is the gravitino G which
also leads to missing energy. The actual questionhere is what the NLSP, the next lightest particle,is. There are two possibilities:i) χ0
1 is the NLSP. Then the decay modes are:χ01 → γG, hG, ZG. As a result, one has two hard
photons + /ET , or jets + /ET .ii) lR is the NLSP. Then the decay mode is
lR → τG and the signature is a charged leptonand the missing energy.• Anomaly mediationIn this case, one also has two possibilities:i) χ0
1 is the LSP and wino-like. It is almostdegenerate with the NLSP.ii) νL is the LSP. Then it appears in the decay
of chargino χ+ → νl and the signature is thecharged lepton and the missing energy.• R-parity violationIn this case, the LSP is no longer stable and
27
decays into the SM particles. It may be charged(or even colored) and may lead to rare decays likeneutrinoless double β-decay, etc.Experimental limits on the LSP mass follow
from non-observation of the corresponding events.Modern lower limit is around 40 GeV .
7. Supersymmetric Dark Matter
7.1. The problem of the dark matter in theUniverse
As was already mentioned the shining matterdoes not compose all the matter in the Universe.According to the latest precise data [58] the mat-ter content of the Universe is the following:
Ωtotal = 1.02± 0.02,
Ωvacuum = 0.73± 0.04,
Ωmatter = 0.23± 0.04,
Ωbaryon = 0.044± 0.004,
so that the dark matter prevails the usual matterby factor of 6.Besides the rotation curves of stars the dark
matter manifests itself in the observation of grav-itational lensing effects [59] and the large struc-ture formation. It is believed that the dark mat-ter played the crucial role in formation of largestructures like clusters of galaxies and the usualmatter just fell down in a potential well attractedby gravitational interaction afterwards. The darkmatter can not make compact objects like theusual matter since it does not take part in stronginteraction and can not lose energy by photonemission since it is neutral. For this reason thedark matter can be trapped in much larger scalestructures like galaxies.In general one may assume two possibilities:
either the dark matter interacts only gravitation-ally, or it participates also in the weak interac-tion. The latter case is preferable since then onemay hope to detect it via the methods of particlephysics. What makes us to believe that the darkmatter is probably the Weakly Interacting Mas-sive Particle (WIMP)? This is because the cross-section of DM annihilation which can be figuredout of the amount of the DM in the Universe isclose to a typical weak interaction cross-section.
Indeed, let us assume that all the DM is madeof particles of a single type. Then the amountof the DM can be calculated from the Boltzmanequatio [60,61]
dnχ
dt+ 3Hnχ = − < σv > (n2
χ − n2χ,eq), (7.1)
where H = R/R is the Hubble constant and nχ,eq
is the equilibrium concentration. The relic abun-dance is expressed in terms of nχ as
Ωχh2 =
mχnχ
ρc≈ 2 · 1027 cm3 sec−1
< σv >. (7.2)
Having in mind that Ωχh2 ≈ 0.113 ± 0.009 and
v ∼ 300 km/sec one gets
σ ≈ 10−34 cm2 = 100 pb, (7.3)
which is a typical EW cross-section.
7.2. Detection of the Dark matterThere are two methods of the DM detection:
direct and indirect. In direct detection one as-sumes that the particles of DM come to Earthand interact with the nuclei of a target. In un-derground experiments one can hope to observesuch events measuring the recoil energy. Thereare several experiments of this type: DAMA,Zeplin, CDMS and Edelweiss. Among them onlyDAMA collaboration claims to observe a positiveoutcome in annual modulation of the signal withthe fitted DM particle mass around 50 GeV [62].
All the other experiments do not see it thoughCDMS collaboration recently announced abouta few events of a desired type [63]. The rea-son of this disagreement might be in differentmethodology and the targets used since the cross-section depends on a spin of a target nucleus.The collected statistics is also essentially differ-ent, DAMA has accumulated by far more dataand this is the only collaboration which studiesthe modulation of the signal that may be crucialfor reducing the background.The typical exclusion plots for spin-
independent and spin-dependent cross-sectionsare shown in Fig.25 where one can see DAMA al-lowed region overlapping with the other exclusionones. Still today we have no convincing evidencefor direct DM detection or exclusion.
28
Figure 25. The exclusion plots from direct DMdetection experiments. Spin-independent case(top) from Chicagoland Observatory for Under-ground Particle Physics (COUPP) and spind-dependent case (bottom) from Cryogenic DarkMatter Search (CDMS)
Indirect detection of the DM is aimed to theregistration of a signal from the DM annihilationin the form of additional gamma rays and chargedparticles (antiprotons and positrons) in cosmicrays. These particles should have the energeticspectrum reflecting their origin from annihilationof heavy massive particles which is different fromthe background coming from the known sources.Hence one may expect the appearance of a ”shoul-der“ in the cosmic ray spectrum. There areseveral experiments of this kind: EGRET (dif-fuse gamma rays) which is followed by FERMI;HEAT and AMS1 (positrons) which is followed by
PAMELA; BESS (antiprotons) which will be fol-lowed by PAMELA and AMS2. All these exper-iments see some deviation from the backgroundthough the uncertainties are large and the back-ground is not known very accurately especiallyfor charged particles.From this point of view the most detailed in-
formation was obtained by EGRET cosmic tele-scope [64] which orbited the Earth for 9 years andmeasured the spectrum and intensity of diffusegamma rays over the whole celestial sphere withthe 4 degree bins. The form of the spectrum wasmeasured in the region of 0.1-10 GeV. It allowsone to perform the independent analysis in dif-ferent directions of the celestial sphere. Gammarays have the advantage that they point back tothe source and do not suffer energy losses, so theyare the ideal candidates to trace the dark matterdensity. The charged components interact withGalactic matter and are deflected by the Galac-tic magnetic field, so they do not point back tothe source.The diffuse component shows a clear excess
for the energy above 1 GeV about a factor twoover the expected background from known nu-clear interactions, inverse Compton scatteringand bremsstrahlung as shown in Fig.26 [65]. Dif-ferent plots correspond to different regions in thesky: A - inner galaxy, B - outer disk, C - outergalaxy, D- low attitude, E- intermediate latitude,F -galactic poles.As one can see the excess of a signal above the
background is isotropic in celestial sphere thatsuggests the common source which might be theDM. It was shown that the observed excess inthe spectrum of diffuse gamma reays, if taken se-riously, has all the features of the decay of π0
mesons produced by monoenergetic quarks com-ing from the DM annihilation. Fitting the back-ground together with the signal from the DM an-nihilation one can get remarkable agreement forall directions if the mass of the DM particle isaround 60 GeV. A detailed picture for the regionof the sky in the direction of the galactic centeris shown in Fig.28. Here one can see the allowedbackground variations and the variations of theDM particle mass used for fitting the data. Pos-sible background variations are not enough to ex-
29
Figure 26. Excess in diffuse gamma rays as measured by EGRET in various regions in the sky. Thesolid(blue) line is the background as calculated by the GALPROP code. Discrete slashes representEGRET data. Also shown are the contributions of the known background sources
Figure 27. Excess in diffuse gamma rays as measured by FERMI - dark (blue) slashes in comparisonwith EGRET - light (red) slashes in the same regions in the sky
30
plain EGRET data while the variation of theWIMP mass within 50-70 GeV does not contra-dict these data.
10-5
10-4
10-1
1 10 102
E [GeV]
E2 *
flux
[GeV
cm
-2 s
-1sr
-1]
EGRETbackgroundsignal
mWIMP=50-70 GeV
Dark MatterPion decayInverse ComptonBremsstrahlung
Figure 28. The spectrum of diffuse gamma raysmeasured by EGRET in the direction of thegalaxy center and the fit to the data. The lightshaded (yellow) areas indicate the backgroundusing the shape of the conventional GALPROPmodel [66], while the dark shaded (red) areas arethe signal contribution from DMA for a 60 GeVWIMP mass.
It is instructive to compare EGRET data withrecently released FERMI data which are muchmore precise. This comparison is shown inFi.27 [67]. One can see that the new data arenot in contradiction with the old ones. The ex-cess is still visible though is smaller compared toEGRET. On has to admit, however, that the in-
terpretation of the data in favour of backgroundmodification is also possible. So, taking the opti-mistic point of view, one can interpret these dataas a signal from the DM annihilation, otherwiseeverything is sinked in the error bars.The experimental data with the charge parti-
cles looks more contradictory. We present theantiproton and positron data in Figs. 29 [68] and30 [69], respectively. While there is no excessobserved in antiproton data, the positron spec-trum measured by PAMELA is quite unusual.It strongly contradicts the expectations from theGALPROP. Possible interpretations of PAMELAdata include: background from hadronic show-ers with large electromagnetic component; astro-physical sources like pulsars, positron accelera-tion in SNR, locality of sources; leptophilic DMannihilation, very heavy (∼ 1 TeV) WIMPs, etc.The situation is still to be clarified.
kinetic energy (GeV)1 10 210
/pp
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4-310¥
=500MV)¥Donato 2001 (D, =500MV)¥Simon 1998 (LBM,
=550MV) ¥Ptuskin 2006 (PD,
PAMELA
Figure 29. Antiproton/proton ratio as measuredby PAMELA. No excess is found
One should mention here, that interpreting theexcess in diffuse gamma rays data as the WIMPannihilation one has to enhance the intensity of asignal by factor of 10-100 that is usually achievedby assumption of clumpiness of the DM. This al-most obvious property of the DM has no exper-imental confirmation so far. The same enhance-ment, however, is not needed for antiprotonswhere one seems to have an agreement with the
31
Energy (GeV)
0.1 1 10 100
))
-(e
)+
+
(e
) /
(+
(e
Po
sit
ron
fra
cti
on
0.01
0.02
0.1
0.2
0.3
0.4
Muller & Tang 1987
MASS 1989
TS93
HEAT94+95
CAPRICE94
AMS98
HEAT00
Clem & Evenson 2007
PAMELA
Figure 30. Positron fraction as measured byPAMELA in comparison with the background
data. This contradiction might be attributed todifferent behaviour of charged particles in galacticmagnetic fields.
7.3. Supersymmetric Interpretation of theDark Matter
Supersymmetry offers several candidates forthe role of the cold dark matter. If one looksat the particle content of the MSSM from thepoint of view of a heavy neutral particle, one findsseveral such particles, namely: a superpartner ofthe photon (photino γ), a superpartner of the Z-boson (zino z), a superpartner of neutrino (sneu-trino ν) and superpartners of the Higgs bosons(higgsino H). The DM particle can be the light-est of them, the LSP. The others decay on theLSP and the SM particles, while the LSP is sta-ble and can survive since the Big Bang. As arule the lightest supersymmetric particle is theso-called neutralino, the spin 1/2 particle whichis the combination of photino, zino and two neu-tral higgsinos and is the eigenstate of the massmatrix
|χ01〉 = N1|B0〉+N2|W 3
0 〉+N3|H1〉+N4|H2〉.
Thus, supersymmetry actually predicts the ex-istence of the dark matter. Moreover, we haveshown above that one can choose the parametersof a soft supersymmetry breaking in such a waythat one gets the right amount of the DM. This
requirement serves as a constraint for these pa-rameters and is consistent with the requirementscoming from particle physics.The search for the LSP was one of the tasks
of LEP. They were supposed to be produced asa result of chargino decays and be detected viamissing transverse energy and momentum. Neg-ative results defined the limit on the LSP mass asshown in Fig.31.
Preliminary DELPHI LSP limit at 189 GeV
0
5
10
15
20
25
30
35
40
45
50
1 2 3 4 5 6 7 8 9 10
tanβ
χ∼
Mas
s (G
eV/c
2 )any m0
31.2 GeV/c2
10
20
30
40
50
60
1 10
tan β
Mχ~
10 (G
eV)
m0=500 GeV
L3 preliminary
<-- 30.1 GeV
Excluded at 95% C.L.
Figure 31. Exclusion limit on the LSP mass fromDelphi coll. and L3 coll. (LEP) [70]
The DM particles which form the halo of thegalaxy annihilate to produce the ordinary parti-cles in the cosmic rays. Identifying them withthe LSP from a supersymmetric model one cancalculate the annihilation rate and to study the
32
secondary particle spectrum. The dominant an-nihilation diagrams of the lightest supersym-metric particle (LSP) neutralino are shown inFig.32. The usual final states are either thequark-antiquark pairs or the W and Z bosons.Since the cross sections are proportional to thefinal state fermion mass, the heavy fermion finalstates, i.e. third generation quarks and leptons,are expected to be dominant. The W- and Z-finalstates from t-channel chargino and neutralino ex-change have usually a smaller cross section.
Figure 32. The dominant annihilation diagramsfor the lightest neutralino in the MSSM
The dominant contribution comes from A-boson exchange: χ+χ→ A→ bb. The sum of thediagrams should yield < σv >= 2 ·10−26 cm3/secto get the correct relic density.
The spectral shape of the secondary particlesfrom DMA is well known from the fragmenta-tion of mono-energetic quarks studied at electron-positron colliders, like LEP at CERN, which hasbeen operating up to centre-of-mass energies ofabout 200 GeV, i.e. it corresponds to the neu-tralino masses up to 100 GeV (see Fig.33). Thedifferent quark flavours all yield similar gammaspectra at high energies. Hence, the specrta
of positrons, gammas and antiprotons is known.The relative ammount of γ, p− and e+ is alsoknown. One expects around 37 gammas per col-lision.
Figure 33. The final states in the process of e+e−-annihilation at colliders in the SM
The gamma rays from the DM annihilation canbe distinguished from the background by theircompletely different spectral shape: the back-ground originates mainly from cosmic rays hit-ting the gas of the disc and producing abun-dantly π0 mesons, which decay into two photons.The initial cosmic ray spectrum is a steep powerlaw spectrum, which yields a much softer gammaray spectrum than the fragmentation of the hardmono-energetic quarks from the DM annihilation.The spectral shape of the gamma rays from thebackground is well known from fixed target ex-periments given the known cosmic ray spectrum.
7.4. SUSY interpretation of EGRET ex-cess
If one takes the EGRET excess in diffusegamma rays seriously then one can try to identifythe DM particle responsible for this excess withthe LSP. The mass of this hypothetical WIMPas it follows from EGRET data is in the range
33
of 50 to 100 GeV and is fully compatible withthe neutralino. Since in the MSSM all the cou-plings are known one can calculate the annihila-tion rate given by the diagrams in Fig.32. Theonly unknown parameters are the SUSY masses(and mixings) which one can choose to fit thedata.Combining various requirements on soft SUSY
parameters together with the assumed EGRETenergy range for the mass of neutralino one getsan essentially constrained allowed region shownin Fig.34 [71] One can see that the ”EGRET” re-
200
400
2000 4000m0 [GeV]
m1/
2 [G
eV]
••••••••••••••••••••••••••••••••••••••••
••••••••••••••••••••••••••
•••••••
g-2
bsγ
mh
EGRET
excl.
excl.
tan β = 52.295 % C.L.
• excl. LSP no EWSB
Figure 34. The allowed region of parameterspace with account of the EGRET data on dif-fuse gamma rays. The star indicates the best firvalue.
gion of parameter space corresponds to high val-ues of tanβ, low m1/2 and high m0. This is therange of the so-called focus point region wherechrarginos and neutralinos, whose mass is gov-erned by the value of m1/2, are light and squarksand sleptons, whose mass is governed by m0, areheavy. The lightest neutralino in the this region is95% photino being the superpartner of a photonof the cosmic microwave background. Choosingthe point in this allowed region one can calcu-
late the whole mass spectrum of superpartners.We present in the Table5 the sample mass spec-trum corresponding to the best fit point in the”EGRET” region. [71].
Particle Mass [GeV]χ01,2,3,4 64, 113, 194, 229χ±1,2, g 110, 230, 516
u1,2 = c1,2 1519, 1523
d1,2 = s1,2 1522, 1524t1,2 906, 1046
b1,2 1039, 1152e1,2 = µ1,2 1497, 1499
τ1,2 1035, 1288νe, νµ, ντ 1495, 1495, 1286h,H,A,H± 115, 372, 372, 383Observable ValueBr(b → sγ) 3.02 · 10−4
∆aµ 1.07 · 10−9
Ωh2 0.117
Table 5The mass spectrum of superpartners in theEGRET point: m0 = 1500 ,m1/2 = 170 , A0 = 0,tanβ = 52.2, µ > 0
As one can see from the table, in the ”EGRET”point one has considerable splitting between therelatively light superpartners of the gauge fieldsand heavy squarks and sleptons. The masses ofneutralinos and charginos are almost at the lowerboundary of experimentally allowed range. Thesame is true for the lightest Higgs boson. Exper-imental lower limit on the SM Higgs boson masstoday is 114.7 GeV as follows from the negativeresults of the search at LEP. This bound is alsotrue for the MSSM for large tanβ.Thus, taking the optimistic point of view that
the excess in diffuse gamma rays actually existsand accepting the supersymmetric interpretationof this excess one can simultaneously give answerto the following questions:• In Cosmology: What is CDM made of?• In Astrophysicists: What is the origin of excessof diffuse Galactic Gamma Rays?
34
• In Particle physicists : Where are the Super-symmetric Particles?And the answer is:• DM is made of WIMPs which are SUSY par-ticles distributed in Halo of our Galaxy with amass around 70 GeV.
What is important, supersymmetric interpre-tation of the DM is testable since it predicts themass spectrum which can be directly checked atthe LHC in the nearest future.
8. Conclusion
Supersymmetry is now the most popular ex-tension of the Standard Model. Comparison ofthe MSSM with precision experimental data forthe MSSM is as good as for the SM and some-times even better. For example the branchingratio BR(b → sγ) and the anomalous magneticmoment of muon are fitted better in the MSSMthan in the SM. The relic density of the DM is notdescribed in the SM but is naturally predicted bythe MSSM. One can see this comparison for mainobservables in Fig.35 [45].
Still today after 30 years since the invention ofsupersymmetry we have no single convincing ev-idence that supersymmetry is realized in particlephysics. It remains very popular in quantum fieldtheory and in string theory due to its exceptionalproperties but needs experimental justification.
Let us remind the main pros and contras forsupersymmetry in particle physics
Pro:• Provides natural framework for unification withgravity• Leads to gauge coupling unification (GUT)• Solves the hierarchy problem• Is a solid quantum field theory• Provides natural candidate for the WIMP coldDM• Predicts new particles and thus generates newjob positions
Contra:• Does not shed new light on the problem of
∗ Quark and lepton mass spectrum∗ Quark and lepton mixing angles∗ the origin of CP violation∗ Number of flavours
LEP:
SLC:
MZ
ΓZ
σhad
Rl
AFB
l
Rb
Rc
AFB
b
AFB
c
Mt
sin2Θeff
lept
MW(LEP)
sin2Θeff
lept(ALR)
b → Xsγ
aµSUSY
pulls=(data-theo)/error
SM: χ2/d.o.f = 27.1/16
MSSM: χ2/d.o.f = 16.4/12
Figure 35. The SM versus the MSSM in compar-ison with precision experimental data
∗ Baryon assymetry of the Universe• Doubles the number of particlesLow energy supersymmetry promises us that
new physics is round the corner at a TeV scaleto be exploited at colliders and astroparticle ex-periments of this decade. If our expectations arecorrect, very soon we will face new discoveries,the whole world of supersymmetric particles willshow up and the table of fundamental particleswill be enlarged in increasing rate. This wouldbe a great step in understanding the microworld.The future will show whether we are right in ourexpectations or not.
35
Acknowledgements
The author would like to express his gratitudeto the organizers of the School for their effort increating a pleasant atmosphere and support. Thiswork was partly supported by RFBR grant # 08-02-00856 and Russian MIST grant # 1027.2008.2.I would also like to thank A.Gladyshev for hishelp in preparation of the manuscript.
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