2.4 The Minimal Supersymmetric Standard Model
2.4.1 Foundations of the MSSM
If a consistent picture of DM signals emerges, the observed dark matter mass and direct detection scattering cross section will put severe constraints on the BSM parameter space.
Moreover, the existence of the suspected DM candidate(s) needs to be confirmed by complementary searches at collider experiments.
It should be kept in mind that with the large number of available experimental tests of the SM from the various areas, more or less significant deviations between observation and the SM expectation are naturally expected for a few of these observables already for purely statistical reasons, i.e. even if the expectation is based on the correct theory. And indeed, currently observed deviations are found for instance in someB-flavor observables: The branching ratio of the decay B →D(∗)τ ν measured by BaBar [179] currently disagrees with the SM expectation at the 3.4σ level23, and another discrepancy is observed in a kinematical distribution of the decayB0→K∗0µ+µ− by the LHCb experiment [181], which amounts to a local discrepancy of 3.7σ.
Another (less significant and rather speculative) discrepancy is found at both ATLAS [182]
and CMS [183] in the measurement of the W W production cross section, which appears to be slightly larger than expected in the SM. Possible explanations can e.g. be found within the MSSM [184, 185]. Last but not least, a ∼ 2σ discrepancy is observed at the Tevatron experiments CDF [186] and DØ [187] in the forward-backward asymmetry inpp→t¯tevents. A possible explanation of this phenomenon may be the existence of a heavy neutral spin-1 gauge boson Z0 [188].
In the supersymmetric extension of the SM (SSM), each SM particle has a superpartner with its spin differing by 1/2. Together they form asupermultiplet, which is an irreducible representation of the SUSY algebra and contains equal number of bosonic and fermionic degrees of freedom.
The SUSY generatorQ commutes both with the generators of the gauge interactions and with P2 = M2, hence, in exact supersymmetry, the superpartners of the SM fields have the same quantum numbers and mass M. As we have seen in Section 2.3, these equalities lead to the exact cancellation of problematic radiative corrections to the Higgs mass. Thus, supersymmetry protects the light Higgs mass and makes the theory technically natural. However, we have not observed any superpartners of SM particles yet, so SUSY must be a broken symmetry if it indeed exists in nature. We will discuss below how this breaking can be accommodated without spoiling the naturalness.
Two types of supermultiplets are needed to assemble the SM fields and their superpartners:
A chiral (or matter) supermultiplet, which contains a single two-component Weyl fermion (spin 1/2) and a complex scalar field (spin 0), and a vector (or gauge) supermultiplet with a massless vector boson (spin 1) and a two-component Weyl fermion. In order to include gravity an additional supermultiplet with a graviton (spin 2) and its fermionic superpartner, the gravitino (spin 3/2), is needed. Supermultiplets are most conveniently written using the superfield formalism, see e.g. Refs. [15,22] for an introduction.
The MSSM particle content
The chiral supermultiplets of the MSSM, i.e. the supersymmetric extension of the SM with minimal particle content, are given in Tab.2.6. They contain the SM fermions and their scalar superpartners, the sfermions. We have three generations (i = 1,2,3) of left-handed (right-handed) up- and down-typesquarks, ˜uLi, ˜dLi (˜uRi, ˜dRi), which are the spin-0 superpartners of the left-handed (right-handed) up-and down-type quarks, respectively; the superpartners of the left-handed (right-handed) charged leptons are the left-handed (right-handed) chargedsleptons,
`˜Li(˜eRi); and thesneutrinos, ˜νi, which are the superpartners of the neutrinos. Conventionally, all SM fermions are described by left-handed Weyl spinors, hence the conjugates of the right-handed quarks and leptons are given in Tab.2.6.
An important feature of supersymmetric models is the necessity of having at least two Higgs doublet fields. The fermionic partner of a single Higgs doublet, the Higgsino, would lead to a gauge anomaly of the electroweak symmetry [20, 190]. With two Higgs doublets of opposite hypercharge, cf. Tab.2.6, the contributions to the anomaly cancel. Moreover, the superpotential (see below) must be an analytic function of chiral superfields, hence we need (at least) two Higgs doublets to give mass to both the up- and down-type quarks via the Higgs mechanism. The Higgs sector will be discussed in more detail in Section2.4.3.
The SM gauge bosons and their fermionic superpartners, thegauginos, are assembled in the vector supermultiplets presented in Tab. 2.7. We have the bino, ˜B, the superpartner of the U(1)Y gauge boson; three winos, ˜WA (A = 1,2,3), the superpartners of the SU(2)L gauge bosons; and the gluinos, ˜ga (a= 1, . . . ,8), the superpartners of the SU(3)C gauge bosons, the gluons.
After EWSB, some of the fields in Tab.2.6 and 2.7will have the same remaining conserved quantum numbers and can therefore mix. The bino, the third wino component, ˜W3, and the neutral Higgsinos mix and their mass eigenstates are calledneutralinos, ˜χ0n, wheren= 1, . . . ,4 denotes an ordering according to an increasing mass hierarchy (i.e. ˜χ01is the lightest neutralino).
The mixing of the winos ˜W1 and ˜W2 and the charged Higgsinos results in the charginos, ˜χ±m
Field names superfield spin 0 spin 1/2 SU(3)C,SU(2)L,U(1)Y
squarks, quarks (×3 families)
Qi q˜Li≡ u˜Li d˜Li
!
qLi= uLi dLi
!
(3,2,+1/6) U¯i u˜∗Ri u†Ri (¯3,1,−2/3) D¯i d˜∗Ri d†Ri (¯3,1,+1/3) sleptons, leptons Li `˜Li ≡ ν˜i
`˜Li
!
`Li ≡ νi
`Li
!
(1,2,−1/2)
(×3 families) E¯i e˜∗Ri e†Ri (1,1, +1)
Higgs, higgsinos H1 h1≡ h01 h−1
!
˜h1≡ h˜01
˜h−1
!
(1,2,−1/2)
H2 h2≡ h+2 h02
!
˜h2≡ ˜h+2
˜h02
!
(1,2,+1/2)
Table 2.6: Chiral (ormatter) supermultiplets of the MSSM.i= 1,2,3 is the generation index. The last column gives the gauge representation under the SM gauge groupGSM.
Field names superfield spin 1/2 spin 1 SU(3)C,SU(2)L,U(1)Y
bino,B boson V1 B˜ Bµ (1,1,0)
winos,W bosons V2 W˜A WµA (1,3,0)
gluinos, gluons V3 g˜a Gaµ (8,1,0)
Table 2.7:Vector (orgauge) supermultiplets (with their gauge representation) of the MSSM.
(m= 1,2). Again, ˜χ±1 denotes the lighter mass eigenstate.
The squarks and sleptons in Tab.2.6are given in the flavor basis. In general, these can mix between different generations and right- and left-handed fields after EWSB, leading to 6×6 mass matrices for each the sleptons, down- and up-type squarks. However, the generational mixing is highly restricted by experimental bounds on FCNC processes [191, 192] (see Section 2.4.2 for more details). Furthermore, the mass terms and trilinear scalar interactions24, which lead to the mixing of left- and right-handed states within one generation, are usually assumed to be proportional to the corresponding SM fermion mass and Yukawa couplings, respectively.
In this work, we therefore only consider left-right mixing for the third generation squarks and sleptons. The mass eigenstates are labeled by ˜tk, ˜bk, ˜τk, (k = 1,2), respectively, following the same mass ordering as above.
Note, that so far we have been assuming that lepton-number,L, and baryon number,B, are conserved quantities. In the SM, this is indeed the case, albeit for no apparent deeper reason25. In general, supersymmetric models can feature violation of lepton and baryon number (see below). In that case, the mixing of superfields will be more complicated [193].
24These terms break supersymmetry explicitly as will be discussed later.
25The lepton number and baryon number conservation in the SM are regarded asaccidental symmetries. It is simply not possible to write down Lorentz- and gauge-invariant renormalizable operators that violateLorB in the SM.
The superpotential
In a renormalizable supersymmetric field theory26, all interactions and masses of the particles are solely determined by their gauge transformation properties and thesuperpotential W, which is a holomorphic function of the chiral superfields. With the particle content given in Tab.2.6, the most general gauge invariant and renormalizable superpotential of the SSM is27 [194,195]
WSSM=WRPC+WRPV, (2.51)
WRPC=Yij(`)H1·LiE¯j+Yij(d)H1·QiD¯j−Yij(u)H2·QiU¯j −µH1·H2, (2.52) WRPV = 1
2λijkLi·LjE¯k+λ0ijkLi·QjD¯k+1
2λ00ijkU¯iD¯jD¯k−κiLi·H2, (2.53) where i, j, k = 1,2,3 are generation indices. We use the notation A·B ≡ abAaBb for the contraction of two SU(2)-doublet superfields A and B, with the SU(2) indicesa, b = 1,2 and the total anti-symmetric tensorab with12=−21= +1. We omitted the SU(3) color indices.
The first part of the superpotential, WRPC, includes supersymmetric generalizations of the Yukawa couplings for the leptons, down- and up-type quarks superfields. These contain the Yukawa terms of the SM, cf. Eq. (2.22), which give mass to the fermions after EWSB. Note, that H1 gives mass to the down-type quarks and charged leptons, while H2 gives mass to the up-type quarks28. The last term in Eq. (2.52) is a supersymmetric generalization of a Higgsino mass term. The parameterµ has the dimension of mass and is required to be ∼ O(102 GeV) for consistent EWSB.
The second part of the superpotential, WRPV, contains baryon-number violating (BNV) and lepton-number violating (LNV) operators. If both types are present simultaneously in the theory the proton decays rapidly [198–200]. This is in contradiction with experiments [201] and the lower limit on the proton lifetime [74] severely constrains the product of LNV and BNV couplings to unnaturally small values [199].
The problem of rapid proton decay can be cured be imposing adiscrete gauge symmetry[202]
that prohibits either the BNV or LNV terms, or both. A very popular choice isR-parity [203], Rp, (also calledmatter parity) which prohibits all terms inWRPV, Eq. (2.53). It can be defined as
Rp ≡(−1)2S+3B+L=
( +1 for SM particles
−1 for SUSY particles , (2.54) whereB and L are the baryon and lepton number of the particle, respectively, and S denotes its spin. This discrete symmetry is usually assumed to be conserved in the MSSM. It features the interesting phenomenological consequence that SUSY particles can only be produced (and annihilated) pairwise. Once SUSY particles are produced they (cascade) decay into the lightest supersymmetric particle (LSP), which cannot further decay due to the symmetry. As it is stable, the LSP is a good candidate for dark matter [12, 129, 130, 204]. For the same reason it is restricted to be color-, flavor- and charge-neutral. In many considered SUSY models the lightest neutralino ˜χ01 is the LSP.
26 If SUSY is broken there will be additional soft SUSY-breaking terms in the Lagrangian (see below).
27 The subscripts RPC and RPV are abbreviations for R-parity conservation and R-parity violation and are explained below.
28 This pattern of (tree-level) Yukawa interactions corresponds to a Type II Two-Higgs-Doublet-Model (2HDM) [196,197].
In fact, the phenomenologically equivalent discrete symmetryproton hexality [205,206], P6, is better suited for protecting the proton from its untimely demise, because it also forbids dangerous dimension-5 proton decay operators.
Another possible discrete symmetry is baryon triality [206], B3, which prohibits the BNV operator ¯UD¯D¯, but allows the LNV terms inWRP V, Eq. (2.53). It also forbids the aforemen-tioned dimension-5 proton decay operators. Phenomenologically, the LSP is unstable in this case and therefore generally not a good DM candidate. Any SUSY particle can now be the LSP, which potentially leads to interesting and unexpected collider signatures [207]. Furthermore, SUSY particles can be produced singly, possibly at resonance [207–210]. This signature will be investigated in Chapter 7 in order to derive bounds on the R-parity violating couplings from the non-observation in LHC searches. Note also, that the LNV operators are well-suited to generate neutrino masses [143,211].
Soft SUSY breaking
A realistic weak-scale SUSY model must contain SUSY breaking, since superpartners with equal mass as the SM particles are experimentally excluded. Similarly as in the electroweak sector, the mechanism of SUSY breaking is anticipated to be spontaneous. However, there is no consensus on how exactly the SUSY breaking should be done and many models of spontaneous symmetry breaking have been proposed. For phenomenological studies, it is very useful to parametrize this ignorance by adding extra interaction terms to the supersymmetrized SM Lagrangian, which break SUSY explicitly. The SUSY-breaking couplings should be soft (i.e. of positive mass dimension) in order not to reintroduce quadratic divergencies in the radiative corrections to scalar masses spoiling the natural solution of the hierarchy problem, cf. Section 2.3. The most general form of explicit SUSY-breaking consistent with R-parity conservation and the minimal particle content of the SSM is [212]
−Lsoft=1 2
M1B˜B˜+M2W˜AW˜A+M3g˜ag˜a+ h.c. +m2H1h†1h1+m2H2h†2h2−(Bµ h1·h2+ h.c.)
+ (M`˜2)ij`˜†Li`˜Lj+ (Me˜2)ije˜∗Rie˜Rj+ (Mq˜2)ijq˜†Liq˜Lj+ (Mu˜2)iju˜∗Riu˜Rj+ (Md2˜)ijd˜∗Rid˜Rj
+h(A˜eY(`))ijh1·`˜Lie˜∗Rj+ (Ad˜Y(d))ijh1·q˜Lid˜∗Rj−(Au˜Y(u))ijh2·q˜Liu˜∗Rj+ h.c.i. (2.55) The first row contains three (complex) Majorana mass terms for the gauginos. Three soft-breaking mass parameters for the Higgs fields appear in the second row. The third row holds soft-breaking mass terms for the squarks and sleptons, given by five hermitian 3×3 squared-mass matrices. The last row contains the scalar interactions that correspond to the Yukawa couplings in the superpotential, Eq. (2.52). These trilinear scalar interaction terms contribute to the slepton and squark masses and are given by complex 3×3 matrixA-parameters. After EWSB, these terms mix the left- and right-handed sleptons and squarks. It is apparent that Lsoft breaks supersymmetry as it involves only fields without their superpartners.
With the supersymmetrization of the SM, no additional parameters (apart from new paramet-ers in the extended Higgs sector, cf. Section 2.4.3) are introduced. However, Lsoft, Eq. (2.55), contains 105 new parameters, which have no counterpart in the SM. Thus, the MSSM contains 124 independent parameters (including the SM parameters) [23]. In this extensive parameter
space, phenomenological studies are very difficult. Thus we need a guiding principle to reduce the amount of free parameters. We address this issue in the next section.