88. Supersymmetry, Part I (Theory)1

88. Supersymmetry, Part I (Theory)
Revised August 2023 by B.C. Allanach (DAMTP, Cambridge U.) and H.E. Haber (UC Santa Cruz).

88.1
88.2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Structure of the MSSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88.2.1 R-parity and the lightest supersymmetric particle . . . . . . . . . . . . . . . .
88.2.2 The goldstino and gravitino . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88.2.3 Hidden sectors and the structure of SUSY breaking . . . . . . . . . . . . . .
88.2.4 SUSY and extra dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88.2.5 Split-SUSY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88.3 Parameters of the MSSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88.3.1 The SUSY-conserving parameters . . . . . . . . . . . . . . . . . . . . . . . . .
88.3.2 The SUSY-breaking parameters . . . . . . . . . . . . . . . . . . . . . . . . . .
88.3.3 MSSM-124 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88.4 The supersymmetric-particle spectrum . . . . . . . . . . . . . . . . . . . . . . . . .
88.4.1 The charginos and neutralinos . . . . . . . . . . . . . . . . . . . . . . . . . .
88.4.2 The squarks and sleptons . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88.5 The supersymmetric Higgs sector . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88.5.1 The tree-level Higgs sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88.5.2 The radiatively-corrected Higgs sector . . . . . . . . . . . . . . . . . . . . . .
88.6 Restricting the MSSM parameter freedom . . . . . . . . . . . . . . . . . . . . . . .
88.6.1 Gaugino mass relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88.6.2 Constrained versions of the MSSM: mSUGRA, CMSSM, etc. . . . . . . . . .
88.6.3 Gauge-mediated SUSY breaking . . . . . . . . . . . . . . . . . . . . . . . . .
88.6.4 The phenomenological MSSM . . . . . . . . . . . . . . . . . . . . . . . . . . .
88.6.5 Simplified models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88.7 Experimental data confronts the MSSM . . . . . . . . . . . . . . . . . . . . . . . .
88.7.1 Naturalness constraints and the little hierarchy . . . . . . . . . . . . . . . . .
88.7.2 Indirect constraints on supersymmetric models . . . . . . . . . . . . . . . . .
88.8 Massive neutrinos in weak-scale SUSY . . . . . . . . . . . . . . . . . . . . . . . . .
88.8.1 The supersymmetric seesaw . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88.8.2 R-parity-violating SUSY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88.9 Extensions beyond the MSSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1
2
4
5
5
6
7
7
7
8
9
9
10
11
12
13
14
16
17
17
19
20
20
21
21
24
26
26
26
28

88.1 Introduction
Supersymmetry (SUSY) is a generalization of the space-time symmetries of quantum field theory that transforms fermions into bosons and vice versa [1]. The existence of such a non-trivial
extension of the Poincaré symmetry of ordinary quantum field theory was initially surprising, andits form is highly constrained by theoretical principles [2]. SUSY also provides a framework for theunification of particle physics and gravity [3–5] at the Planck energy scale, MP ∼ 1019 GeV, where
the gravitational interactions become comparable in strength to the gauge interactions. Moreover,supersymmetry can stabilize the hierarchy between the energy scale that characterizes electroweak
symmetry breaking, MEW ∼ 100 GeV, and the Planck scale [6–9] against large radiative corrections. The stability of this large gauge hierarchy with respect to radiative quantum corrections is
S. Navas et al. (Particle Data Group), Phys. Rev. D 110, 030001 (2024)
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88. Supersymmetry, Part I (Theory)

not possible to maintain in the Standard Model (SM) without an unnatural fine-tuning of the parameters of the fundamental theory at the Planck scale. In contrast, in a supersymmetric extensionof the SM, it is possible to maintain the gauge hierarchy while providing a natural framework forelementary scalar fields.
If supersymmetry were an exact symmetry of nature, then particles and their superpartners,
which differ in spin by half a unit, would be degenerate in mass. Since superpartners have not(yet) been observed, supersymmetry must be a broken symmetry. Nevertheless, the stability of thegauge hierarchy can still be maintained if the SUSY breaking is soft [10,11], and the corresponding
SUSY-breaking mass parameters are no larger than a few TeV. Whether this is still plausible inlight of recent SUSY searches at the LHC (see Sec. 89) will be discussed in Sec. 88.7.
In particular, soft-SUSY-breaking terms of the Lagrangian involve combinations of fields withtotal mass dimension of three or less, with some restrictions on the dimension-three terms as
elucidated in Ref. [10]. The impact of the soft terms becomes negligible at energy scales muchlarger than the size of the SUSY-breaking masses. Thus, a theory of weak-scale supersymmetry,
where the effective scale of supersymmetry breaking is tied to the scale of electroweak symmetry
breaking, provides a natural framework for the origin and the stability of the gauge hierarchy [6–9].
At present, there is no unambiguous experimental evidence for the breakdown of the SM ator below the TeV scale. The expectations for new TeV-scale physics beyond the SM are based
primarily on three theoretical arguments. First, in a theory with an elementary scalar field of
mass m and interaction strength λ (e.g., a quartic scalar self-coupling, the square of a gauge
coupling or the square of a Yukawa coupling), the stability with respect to quantum corrections
requires the existence of an energy cutoff roughly of order (16π 2 /λ)1/2 m, beyond which new physics
must enter [12]. A significantly larger energy cutoff would require an unnatural fine-tuning ofparameters that govern the effective low-energy theory. Applying this argument to the SM leads
to an expectation of new physics at the TeV scale [9].
Second, the unification of the three SM gauge couplings at a very high energy close to the
Planck scale is possible if new physics beyond the SM (which modifies the running of the gauge
couplings above the electroweak scale) is present. The minimal supersymmetric extension of theSM, where superpartner masses lie below a few TeV, provides an example of successful gauge
coupling unification [13].
Third, the existence of dark matter that makes up approximately one quarter of the energy
density of the universe, cannot be explained within the SM of particle physics [14]. Remarkably, astable weakly-interacting massive particle (WIMP) whose mass and interaction rate are governed by
new physics associated with the TeV-scale can be consistent with the observed density of dark matter(this is the so-called WIMP miracle, which is reviewed in Ref. [15]). The lightest supersymmetricparticle (LSP), if stable, is a promising (although not the unique) candidate for the dark matter[16–20]. Further aspects of dark matter can be found in Sec. 27.

88.2 Structure of the MSSM
The minimal supersymmetric extension of the SM (MSSM) consists of the fields of the twoHiggs-doublet extension of the SM and the corresponding superpartner fields [21–25]. A particle
and its superpartner together form a supermultiplet. The corresponding field content of the supermultiplets of the MSSM and their gauge quantum numbers are shown in Table 88.1. The electriccharge Q = T3 + 12 Y is determined in terms of the third component of the weak isospin (T3 ) andthe U(1) weak hypercharge (Y ).
The gauge supermultiplets consist of the gluons and their gluino fermionic superpartners andthe SU(2)×U(1) gauge bosons and their gaugino fermionic superpartners. The matter supermultiplets consist of three generations of left-handed quarks and leptons and their scalar superpartners
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88. Supersymmetry, Part I (Theory)

3

Table 88.1: The fields of the MSSM and their representations under the SU(3)×SU(2)×U(1)gauge group are listed. For simplicity, only one generation of quarks and leptons is exhibited. For
each lepton, quark, and Higgs supermultiplet (each denoted by a hatted upper-case letter), there
is a corresponding antiparticle multiplet of charge-conjugated fermions and their associated scalarpartners [26].
Field Content of the MSSM
Supermultiplets

Superfield

Bosonic
fieldsFermionic
partners

SU(3)

SU(2)

U(1)

gluon/gluino

V̂8

g

ge

8

1

0

gauge boson/
gauginoV̂
V̂ 0

W± , W0B

f± , W
f0
W
e
B

1
1

3
1

0
0

slepton/leptonL̂
Ê c(νeL , ee−
L)
+
ẽR

(ν, e− )L
ecL

1
1

2
1

−1
2

squark/
quark

Q̂
Û c
D̂c

eL , deL )
(u
e∗R
u
de∗R

(u, d)L
ucL
dcL

3
3̄
3̄

2
1
1

1/3
−4/3
2/3

Higgs boson/Ĥd(Hd0 , Hd− )e −)e 0, H
(H
d
d

1

2

−1

higgsinoĤu(Hu+ , Hu0 )e 0)e +, H
(H
u
u

1

2

1

(squarks and sleptons, collectively referred to as sfermions), and the corresponding antiparticles.
The Higgs supermultiplets consist of two complex Higgs doublets, their higgsino fermionic superpartners, and the corresponding antiparticles. The enlarged Higgs sector of the MSSM constitutes
the minimal structure needed to guarantee the cancellation of gauge anomalies [27] generated by
the higgsino superpartners that can appear as internal lines in triangle diagrams with three external
electroweak gauge bosons. Moreover, without a second Higgs doublet, one cannot generate massfor both “up”-type and “down”-type quarks (and charged leptons) in a way consistent with theunderlying SUSY [28–30].
In the most elegant treatment of SUSY, spacetime is extended to superspace which consists ofthe spacetime coordinates and new anticommuting fermionic coordinates θ and θ† [25,31,32]. Eachsupermultiplet is represented by a superfield that is a function of the superspace coordinates. The
fields of a given supermultiplet (which are functions of the spacetime coordinates) are coefficientsof the θ and θ† expansion of the corresponding superfield.
Vector superfields contain the gauge-boson fields and their gaugino partners. Chiral superfields
contain the spin-0 and spin-1/2 fields of the matter or Higgs supermultiplets. A general supersymmetric Lagrangian is determined by three functions of the chiral superfields [4]: the superpotential,
the Kähler potential, and the gauge kinetic function (which can be appropriately generalized to accommodate higher derivative terms [33]). Minimal forms for the Kähler potential and gauge kinetic
function, which generate canonical kinetic energy terms for all the fields, are required for renormalizable globally supersymmetric theories. A renormalizable superpotential, which is at most cubicin the chiral superfields, yields supersymmetric Yukawa couplings and mass terms. A combinationof gauge invariance and SUSY produces couplings of gaugino fields to matter (or Higgs) fields andtheir corresponding superpartners. The (renormalizable) MSSM Lagrangian is then constructedby including all possible supersymmetric interaction terms (of dimension four or less) that sat31st May, 2024

488. Supersymmetry, Part I (Theory)

isfy SU(3)×SU(2)×U(1) gauge invariance and B−L conservation (where B = baryon number and
L = lepton number). Finally, the most general soft-supersymmetry-breaking terms consistent withthese symmetries are added [10, 11, 34].
Although the MSSM is the focus of much of this review, there is some motivation for considering
non-minimal supersymmetric extensions of the SM [35]. For example, extra structure is needed togenerate non-zero neutrino masses as discussed in Sec. 88.8. In addition, in order to address sometheoretical issues and tensions associated with the MSSM, it has been fruitful to introduce one
additional singlet Higgs superfield. The resulting next-to-minimal supersymmetric extension of
the Standard Model (NMSSM) [36] is briefly considered in Sec. 88.4–88.7 and 88.9. Finally, one isalways free to add additional fields to the SM along with the corresponding superpartners. However,
only certain choices for the new fields (e.g., the addition of complete SU(5) multiplets) will preserve
the successful gauge coupling unification of the MSSM. Some examples will be briefly mentioned
in Sec. 88.9.
88.2.1 R-parity and the lightest supersymmetric particle
The (renormalizable) SM Lagrangian possesses an accidental global B−L symmetry due tothe fact that B and L-violating operators composed of SM fields must have dimension d = 5 orlarger [37]. Consequently, B and L-violating effects are suppressed by (MEW /M )d−4 , where M isthe characteristic mass scale of the physics that generates the corresponding higher dimensional operators. Indeed, values of M of order the grand unification scale or larger may be responsible for theobserved (approximate) stability of the proton and suppression of neutrino masses. Unfortunately,
these results are not guaranteed in a generic supersymmetric extension of the SM. For example, it
is possible to construct gauge invariant supersymmetric dimension-four B and L-violating operators made up of fields of SM particles and their superpartners. Such operators, if simultaneously
present in the theory, would typically yield a proton decay rate many orders of magnitude larger
than the current experimental bound. It is for this reason that B−L conservation is imposed on
the supersymmetric Lagrangian when defining the MSSM, which is sufficient for eliminating all B
and L-violating operators of dimension d ≤ 4.
As a consequence of the B−L symmetry, the MSSM possesses a multiplicative R-parity invariance, where R = (−1)3(B−L)+2S for a particle of spin S [38]. This implies that all the particlesof the SM have even R-parity, whereas the corresponding superpartners have odd R-parity. The
conservation of R-parity in scattering and decay processes has a critical impact on supersymmetricphenomenology. For example, any initial state in a scattering experiment will involve ordinary(R-even) particles. Consequently, it follows that supersymmetric particles must be produced inpairs. In general, these particles are highly unstable and decay into lighter states. Moreover, Rparity invariance also implies that the LSP is absolutely stable, and must eventually be produced
at the end of a decay chain initiated by the decay of a heavy unstable supersymmetric particle. In
order to be consistent with cosmological constraints, a stable LSP is almost certainly electricallyand color neutral [18]. Consequently, the LSP in an R-parity-conserving theory is weakly interacting with ordinary matter, i.e., it behaves like a stable heavy neutrino and will escape collider
detectors without being directly observed. Thus, the canonical signature for conventional R-parityconserving supersymmetric theories is missing (transverse) momentum, due to the escape of the
LSP. Moreover, as noted in Sec. 88.1 and reviewed in Refs. [19] and [20], the stability of the LSP
in R-parity-conserving SUSY makes it a promising candidate for dark matter.
The possibility of relaxing the R-parity invariance of the MSSM (which would generate new B
and/or L-violating interactions) will be addressed in Sec. 88.8.2. However, note that in R-parity
violating (RPV) models, the LSP is no longer stable and thus would not be a viable candidate forthe dark matter (unless its lifetime was significantly longer than the age of the universe). In such
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scenarios, one has to look elsewhere to explain the origin of dark matter.
88.2.2 The goldstino and gravitino
In the MSSM, SUSY breaking is accomplished by including the most general renormalizable softSUSY-breaking terms consistent with the SU(3)×SU(2)×U(1) gauge symmetry and R-parity invariance. These terms parameterize our ignorance of the fundamental mechanism of supersymmetry
breaking. If supersymmetry breaking occurs spontaneously, then a massless Goldstone fermione ) must exist. The goldstino would then be the LSP, and could play ancalled the goldstino (G1/2
important role in supersymmetric phenomenology [39].
However, the goldstino degrees of freedom are physical only in models of spontaneously-broken
global SUSY. If SUSY is a local symmetry, then the theory must incorporate gravity; the resulting theory is called supergravity [5, 40]. In models of spontaneously-broken supergravity, thee the spin-3/2 superpartner of the graviton, via thegoldstino is “absorbed” by the gravitino (G),super-Higgs mechanism [41]. Consequently, the goldstino is removed from the physical spectrum
and the gravitino acquires a mass (denoted by m3/2 ). If m3/2 is smaller than the mass of thelightest superpartner of the SM particles, then the gravitino is the LSP.
In processes with center-of-mass energy E  m3/2 , one can employ the goldstino–gravitino
equivalence theorem [42], which implies that the interactions of the helicity ± 12 gravitino (whose
properties approximate those of the goldstino) dominate those of the helicity ± 32 gravitino. The interactions of gravitinos with other light fields can be described by a low-energy effective Lagrangianthat is determined by fundamental principles [43].
88.2.3 Hidden sectors and the structure of SUSY breaking
It is very difficult (perhaps impossible) to construct a realistic model of spontaneously-broken
weak-scale supersymmetry where the supersymmetry breaking arises solely as a consequence of theinteractions of the particles of the MSSM. A more successful scheme posits a theory with at least
two distinct sectors: a visible sector consisting of the particles of the MSSM [34] and a sector where
SUSY breaking is generated. It is often (but not always) assumed that particles of the hidden sector
are neutral with respect to the SM gauge group. The effects of the hidden sector supersymmetry
breaking are then transmitted to the MSSM by some mechanism (often involving the mediation
by particles that comprise an additional messenger sector). Two theoretical scenarios that exhibit
this structure are gravity-mediated and gauge-mediated SUSY breaking.
Supergravity models provide a natural mechanism for transmitting the SUSY breaking of thehidden sector to the particle spectrum of the MSSM. In models of gravity-mediated SUSY breaking,gravity is the messenger of supersymmetry breaking [44–48]. More precisely, supersymmetry breaking is mediated by effects of gravitational strength (i.e. suppressed by inverse powers of the Planckmass). The soft-SUSY-breaking parameters with dimensions of mass arise as model-dependent
multiples of the gravitino mass m3/2 . In this scenario, m3/2 is of order the electroweak-symmetrybreaking scale, while the gravitino couplings are roughly gravitational in strength [3, 49].1
Under certain theoretical assumptions that govern the structure of the Kähler potential (theso-called sequestered form introduced in Ref. [51]), SUSY breaking is due entirely to the superconformal (super-Weyl) anomaly, which is common to all supergravity models [51]. In particular,
gaugino masses are radiatively generated at one-loop, and squark and slepton squared-mass matrices
are flavor-diagonal. In sequestered scenarios, sfermion squared-masses arise at two-loops, which
implies that gluino and sfermion masses are of the same order of magnitude. This approach iscalled anomaly-mediated SUSY breaking (AMSB). Indeed, anomaly mediation is more genericthan originally conceived, and provides a ubiquitous source of SUSY breaking [52]. However in
1
However, such a gravitino typically plays no direct role in supersymmetric phenomenology at colliders (except
perhaps indirectly in the case where the gravitino is the LSP [50]).

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688. Supersymmetry, Part I (Theory)the simplest formulation of AMSB as applied to the MSSM, the squared-masses of the sleptons
are negative (known as the tachyonic slepton problem). It may be possible to cure this otherwise
fatal flaw in non-minimal extensions of the MSSM [53]. Alternatively, one can assert that anomaly
mediation is not the sole source of SUSY breaking in the sfermion sector. In non-sequesteredscenarios, sfermion squared-masses can arise at tree-level, in which case squark masses would be
parametrically larger than the loop-suppressed gaugino masses [54].
In gauge-mediated supersymmetry breaking (GMSB), gauge forces transmit the supersymmetry
breaking to the MSSM. A typical structure of such models involves a hidden sector where SUSY isbroken, a messenger sector consisting of particles (messengers) with nontrivial SU(3)×SU(2)×U(1)quantum numbers, and the visible sector consisting of the fields of the MSSM [55–58]. The direct
coupling of the messengers to the hidden sector generates a supersymmetry-breaking spectrum inthe messenger sector. Supersymmetry breaking is then transmitted to the MSSM via the virtual
exchange of the messenger fields. In models of direct gauge mediation, there is no separate hidden
sector. In particular, the sector in which the SUSY breaking originates includes fields that carry
nontrivial SM quantum numbers, which allows for the direct transmission of SUSY breaking to the
MSSM [59].
In models of gauge-mediated SUSY breaking with a minimal Kähler potential, the gravitino is
the LSP [16], as its mass can range from a few eV (in the case of low SUSY breaking scales) up toa few GeV (in the case of high SUSY breaking scales). In particular, the gravitino is a potential
dark matter candidate (for a review and guide to the literature, see Ref. [20]). The couplings ofe to the particles of the MSSM (which approximate those of thethe helicity ± 12 components of G
goldstino as previously noted in Sec. 88.2.2) are significantly stronger than gravitational strengthand amenable to experimental collider analyses.
The mass ranges of the gravitino in either gravity-mediated or gauge-mediated SUSY breaking
are further constrained by cosmological considerations [60, 61]. In particular, there is a dangerof over-abundance of gravitinos if it is the dark matter or, if it decays before nucleosynthesis,
modifications to the successful predictions of light element abundances. Avoiding these cosmological gravitino problems imposes strong constraints on gravity-mediated and gauge-mediated SUSY
breaking models.
The concept of a hidden sector is more general than SUSY. Hidden valley models [62] posit
the existence of a hidden sector of new particles and interactions that are very weakly coupled toparticles of the SM. The impact of a hidden valley on supersymmetric phenomenology at colliders
can be significant if the LSP lies in the hidden sector [63].
88.2.4 SUSY and extra dimensions
Approaches to SUSY breaking have also been developed in the context of theories in which
the number of spatial dimensions is greater than three. In particular, a number of SUSY-breaking
mechanisms have been proposed that are inherently extra-dimensional [64]. The size of the extra
dimensions can be significantly larger than MP−1 ; in some cases of order (TeV)−1 or even larger(see, e.g., Sec. 85 and Ref. [65]).
For example, in one approach the fields of the MSSM live on some brane (a lower-dimensional
manifold embedded in a higher-dimensional spacetime), while the sector of the theory that breaks
SUSY lives on a second spatially-separated brane. Two examples of this approach are AMSB [51]and gaugino-mediated SUSY breaking [66]. In both cases, SUSY breaking is transmitted throughfields that live in the bulk (the higher-dimensional space between the two branes). This setup has
some features in common with both gravity-mediated and gauge-mediated SUSY breaking (e.g.,hidden and visible sectors and messengers).
Since a higher dimensional theory must be compactified to four spacetime dimensions, one can
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88. Supersymmetry, Part I (Theory)

also generate a source of SUSY breaking by employing boundary conditions on the compactified
space that distinguish between fermions and bosons. This is the so-called Scherk-Schwarz mechanism [67]. The phenomenology of such models can be strikingly different from that of the usual
MSSM [68].
88.2.5 Split-SUSY
If SUSY is not connected with the origin of the electroweak scale, it may still be possible thatsome remnant of the superparticle spectrum survives down to the TeV-scale or below. This is theidea of split-SUSY [69,70], in which scalar superpartners of the quarks and leptons are significantly
heavier than 1 TeV, whereas the fermionic superpartners of the gauge and Higgs bosons have masseson the order of 1 TeV or below. With the exception of a single light neutral scalar whose properties
are practically indistinguishable from those of the SM Higgs boson, all other Higgs bosons are also
assumed to be very heavy. Among the supersymmetric particles, only the fermionic superpartners
may be kinematically accessible at the LHC.
In models of split SUSY, the top squark masses cannot be arbitrarily large, as these parameters
enter in the radiative corrections to the mass of the observed Higgs boson [71–73]. In the MSSM,
a Higgs boson mass of 125 GeV (see Sec. 11) implies an upper bound on the top squark mass scalein the range of 10 to 108 TeV [74–76], depending on the value of the ratio of the two neutral Higgs
field vacuum expectation values, although this mass range can be somewhat extended by varying
other relevant MSSM parameters. In some approaches, gaugino masses are one-loop suppressed
relative to the sfermion masses, corresponding to the so-called mini-split SUSY spectrum [72, 77].
The higgsino mass scale may or may not be likewise suppressed depending on the details of themodel [78].
The SUSY breaking required to produce such a split-SUSY spectrum would destabilize thegauge hierarchy, and thus would not provide an explanation for the scale of electroweak symmetry
breaking. Nevertheless, models of split-SUSY can account for the dark matter (which is assumed
to be the LSP gaugino or higgsino) and gauge coupling unification, thereby preserving two of
the desirable features of weak-scale SUSY. Finally, as a consequence of the very large squarkand slepton masses, neutral flavor changing and CP-violating effects, which can be problematicin models with TeV-scale SUSY-breaking masses, are sufficiently reduced to avoid conflict withexperimental observations.

88.3 Parameters of the MSSM
The parameters of the MSSM are conveniently described by considering separately the supersymmetry-conserving and the supersymmetry-breaking sectors. A careful discussion of the conventions used here in defining the tree-level MSSM parameters can be found in Refs. [25, 79, 80]. For
simplicity, consider first the case of one generation of quarks, leptons, and their scalar superpartners.
88.3.1 The SUSY-conserving parameters
The parameters of the supersymmetry-conserving sector consist of: (i) gauge couplings, gs ,g, and g 0 , corresponding to the SM gauge group SU(3)×SU(2)×U(1) respectively; (ii) a supersymmetry-conserving higgsino mass parameter µ; and (iii) Higgs-fermion Yukawa couplings, λu ,λd , and λe , of one generation of left- and right-handed quarks and leptons, and their superpartners
to the Higgs bosons and higgsinos. Because there is no right-handed neutrino/sneutrino in the
MSSM as defined here, a Yukawa coupling λν is not included. The complex µ parameter andYukawa couplings enter via the most general renormalizable R-parity-conserving superpotential,
WMSSM = λd Ĥd Q̂D̂c − λu Ĥu Q̂Û c + λe Ĥd L̂Ê c + µĤu Ĥd ,(88.1)

where the superfields are defined in Table 1 and the gauge group indices are suppressed. Moreexplicitly, the so-called “µ-term” can be written out as µab (Ĥu )a (Ĥd )b with an implicit sum over
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888. Supersymmetry, Part I (Theory)

repeated indices, where ab is used to tie together the SU(2) weak isospin indices a, b ∈ {1, 2} in agauge-invariant way (where 12 = −21 = 1 and 11 = 22 = 0). Likewise, the term Ĥu Q̂Û c can be
written out as ab (Ĥu )a Q̂ib (Û c )i , where i ∈ {1, 2, 3} is the SU(3) color index. Finally, Ĥd Q̂D̂c canbe written out as ab (Ĥd )a Q̂ib (D̂c )i , with an analogous expression for Ĥd L̂Ê c .
88.3.2 The SUSY-breaking parameters
The supersymmetry-breaking sector contains the following sets of parameters: (i) three complex
gaugino Majorana mass parameters, M3 , M2 , and M1 , associated with the SU(3), SU(2), andU(1) subgroups of the SM; (ii) five sfermion squared-mass parameters, M 2e , M e2 , M 2e , M e2 , and
Q

U

D

L

M 2e , corresponding to the five electroweak gauge multiplets, i.e., superpartners of the left-handedEfields (u, d)L , ucL , dcL , (ν, e− )L , and ecL , where the superscript c indicates a charge-conjugatedfermion field [26]; and (iii) three Higgs-squark-squark and Higgs-slepton-slepton trilinear interaction
terms, with complex coefficients TU ≡ λu AU , TD ≡ λd AD , and TE ≡ λe AE (which define the “Aparameters”), following the notation employed in Ref. [80]. It is conventional to separate out thefactors of the Yukawa couplings in defining the A-parameters [3, 25] (originally motivated by asimple class of gravity-mediated SUSY-breaking models). If the A-parameters are parametricallyof the same order (or smaller) relative to other SUSY-breaking mass parameters, then in most cases
only the third generation A-parameters will be phenomenologically relevant.
Finally, we have (iv) two real squared-mass parameters, m2Hd and m2Hu (also called m21 and m22 ,respectively, in the literature), and one complex squared-mass parameter, m212 ≡ µB (the latter
defines the “B-parameter”), which appear in the MSSM tree-level scalar Higgs potential [30],V = (m2Hd + |µ|2 )Hd† Hd + (m2Hu + |µ|2 )Hu† Hu + (m212 Hu Hd + h.c.)11+ (g 2 + g 0 2 )(Hd† Hd − Hu† Hu )2 + g 2 |Hd† Hu |2 ,8
2

(88.2)

where the SU(2)-invariant combination of the complex doublet scalar fields Hu and Hd that appears
in Eq. (88.2) is given by Hu Hd ≡ ab (Hu )a (Hd )b = Hu+ Hd− − Hu0 Hd0 . Note that the quartic Higgs
couplings are related to the gauge couplings g and g 0 as a consequence of SUSY. The breaking of
the SU(2)×U(1) electroweak symmetry group to U(1)EM is only possible after incorporating the
SUSY-breaking Higgs squared-mass parameters m2Hd , m2Hu (which can be negative) and m212 . After
minimizing the Higgs scalar potential, these three squared-mass parameters
can be re-expressedin√√terms of the two Higgs vacuum expectation values, hHd0 i ≡ vd / 2 and hHu0 i ≡ vu / 2, and theCP-odd Higgs mass mA [cf. Eqs. (88.4) and (88.5) below]. One is always free to rephase the Higgs
doublet fields such that vd and vu (also called v1 and v2 , respectively, in the literature) are both
real and positive.
The quantity, vd2 + vu2 = 4m2W /g 2 = (2G2F )−1/2 ' (246 GeV)2 , is fixed by the Fermi constant,
GF , whereas the ratiotan β = vu /vd(88.3)
is a free parameter such that 0 < β < π/2. By employing the tree-level conditions resulting
from the minimization of the scalar potential, one can eliminate the diagonal and off-diagonal
Higgs squared-masses in favor of m2Z = 41 (g 2 + g 0 2 )(vd2 + vu2 ), the CP-odd Higgs mass mA and theparameter tan β,
2m212
2m212
sin 2β = 2=,(88.4)
mHd + m2Hu + 2|µ|2
m2A
m2Hd − m2Hu tan2 β
1 2
mZ = −|µ|2 +.
2
tan2 β − 1
31st May, 2024

(88.5)

9

88. Supersymmetry, Part I (Theory)

One must also guard against the existence of charge and/or color breaking global minima due tonon-zero vacuum expectation values for the squark and charged slepton fields. This possibility can
be avoided if the A-parameters are not unduly large [45, 81, 82]. Additional constraints must also
be respected to avoid the possibility of directions in scalar field space in which the full tree-level
scalar potential can become unbounded from below [82]. A computer program has been developed
to calculate vacuum stability bounds in general models at the one-loop level [83], and has been
applied to the MSSM in Ref. [84].
Note that SUSY-breaking mass terms for the fermionic superpartners of the scalar fields andnon-holomorphic trilinear scalar interactions (i.e., interactions that mix scalar fields and theircomplex conjugates) have not been included above in the soft-SUSY-breaking sector. These terms
can potentially destabilize the gauge hierarchy [10] in models with a gauge-singlet superfield. The
latter is not present in the MSSM; hence as noted in Ref. [11], these so-called non-standard softSUSY-breaking terms are benign. The phenomenological impact of non-holomorphic soft SUSYbreaking terms has been reconsidered in Refs. [85–87]. However, in the most common approaches
to constructing a fundamental theory of SUSY-breaking, the coefficients of these terms (which havedimensions of mass) are significantly suppressed compared to the TeV-scale [88]. Consequently, we
follow the usual approach and omit these terms from further consideration.
88.3.3 MSSM-124
The total number of independent physical parameters that define the MSSM (in its most generalform) is quite large, primarily due to the soft-supersymmetry-breaking sector. In particular, in thecase of three generations of quarks, leptons, and their superpartners, M 2e , M e2 , M 2e , M e2 , and M 2e
E
L
D
U
Q
are hermitian 3 × 3 matrices, and AU , AD , and AE are complex 3 × 3 matrices. In addition, M1 ,M2 , M3 , B, and µ are in general complex parameters. Finally, as in the SM, the Higgs-fermion
Yukawa couplings, λf (f = u, d, and e), are √complex 3 × 3 matrices that are related to the quark
and lepton mass matrices via: Mf = λf vf / 2, where ve = vd [with vu and vd as defined above
Eq. (88.3)].
However, not all these parameters are physical. Some of the MSSM parameters can be eliminated by expressing interaction eigenstates in terms of the mass eigenstates, with an appropriate
redefinition of the MSSM fields to remove unphysical degrees of freedom. The analysis of Ref. [89]shows that the MSSM possesses 124 independent real degrees of freedom. Of these, 18 correspondto SM parameters (including the QCD vacuum angle θQCD ), one corresponds to a Higgs sector parameter (the analogue of the SM Higgs mass), and 105 are genuinely new parameters of the model.
The latter include: five real parameters and three CP-violating phases in the gaugino/higgsino
sector, 21 squark and slepton (sfermion) masses, 36 real mixing angles to define the sfermion mass
eigenstates, and 40 CP-violating phases that can appear in sfermion interactions. The most generalparameterization of the R-parity-conserving MSSM (without additional theoretical assumptions)will be denoted henceforth as MSSM-124 [90].

88.4 The supersymmetric-particle spectrum
The supersymmetric particles (sparticles) differ in spin by half a unit from their SM partners.
The superpartners of the gauge and Higgs bosons are fermions, whose names are obtained byappending “ino” to the end of the corresponding SM particle name. The gluino is the color-octet
Majorana fermion partner of the gluon with mass Meg = |M3 |. The superpartners of the electroweak
gauge and Higgs bosons (the gauginos and higgsinos) can mix due to SU(2)×U(1) breaking effects.
As a result, the physical states of definite mass are parameter dependent linear combinations of thecharged or neutral gauginos and higgsinos, called charginos and neutralinos, respectively (sometimes
collectively called electroweakinos). The neutralinos are Majorana fermions, which can lead to somedistinctive phenomenological signatures [91, 92]. The superpartners of the quarks and leptons are
31st May, 2024

88. Supersymmetry, Part I (Theory)10

spin-zero bosons: the squarks, charged sleptons, and sneutrinos, respectively. A complete set of
Feynman rules for the sparticles of the MSSM can be found in Refs. [93, 94]. The MSSM Feynman
rules also are implicitly contained in a number of amplitude generation and Feynman diagramsoftware packages (see e.g., Refs. [95–97]).
It should be noted that all mass formulae quoted below in this Section are tree-level results.
Radiative loop corrections will modify these results and must be included in any precision studyof supersymmetric phenomenology [98]. Beyond tree level, the definition of the supersymmetricparameters becomes convention-dependent. For example, one can define physical couplings or
running couplings, which differ beyond the tree level. This provides a challenge to any effort
that attempts to extract supersymmetric parameters from data. The SUSY Les Houches Accord(SLHA) [80, 99] has been adopted, which establishes a set of conventions for specifying generic filestructures for supersymmetric model specifications and input parameters, supersymmetric massand coupling spectra, and decay tables. These provide a universal interface between spectrum
calculation programs, decay packages, and high energy physics event generators.
88.4.1 The charginos and neutralinos
f ± ) and charged higgsinos (He − ) is describede + and H
The mixing of the charged gauginos (Wud(at tree-level) by a 2 × 2 complex mass matrix [100, 101],
MC ≡

M2
1
√ gvd
2

√1 gvu2!

µ.

(88.6)

To determine the physical chargino states and their masses, one must perform a singular value
decomposition [102] of the complex matrix MC [25, 103]:
U ∗ MC V −1 = diag(Mχ̃+ , Mχ̃+ ) ,1(88.7)

2

where U and V are unitary matrices, and the right-hand side of Eq. (88.7) is the diagonal matrixof (real non-negative) chargino masses. Explicit formulae for the singular value decomposition of±±
MC can be found in Ref. [104]. The physical chargino states are denoted by χ̃1 and χ̃2 . These arelinear combinations of the charged gaugino and higgsino states determined by the matrix elementsof U and V [100, 101] The chargino masses correspond to the singular values [102] of MC , i.e., thepositive square roots of the eigenvalues of MC† MC :1
Mχ̃2+ χ̃+ =
,
1
2
2



2

2

|µ| + |M2 |+ 2m2W ∓q2|µ|2 + |M2 |2 + 2m2W − 4|µM2 − m2W sin 2β|2,(88.8)
in a convention where vu and vd are real and positive, and where the states are ordered such thatMχ̃+ ≤ Mχ̃+ . The relative phase of µ∗ and M2 is physical and potentially observable [105].
12e and Wf 0 ) and neutral higgsinos (He 0 and H
e 0 ) is described
The mixing of the neutral gauginos (Bud(at tree-level) by a 4 × 4 complex symmetric mass matrix [100, 101],


M1


 0

MN ≡  1 0
 − g vd 2

1 0
2 g vu0− 12 g 0 vd

M2

1
2 gvd

1
2 gvd
− 21 gvu

0−µ

1 0
2 g vu− 12 gvu 



.

−µ 


(88.9)

0

To determine the physical neutralino states and their masses, one must perform an Autonne-Takagi
factorization [102, 106] (also called Takagi diagonalization in Refs. [25, 103, 107]) of the complex
31st May, 2024

1188. Supersymmetry, Part I (Theory)

symmetric matrix MN :W T MN W = diag(Mχ̃0 , Mχ̃0 , Mχ̃0 , Mχ̃0 ) ,
1

2

3

4

(88.10)where W is a unitary matrix (which is called N −1 in Refs. [21, 30]) and the right-hand side ofEq. (88.10) is the diagonal matrix of (real non-negative) neutralino masses. The physical neutralinoe0i (for i = 1, . . . , 4), where the states are ordered such that Mχ̃0 ≤ Mχ̃0 ≤states are denoted by χ
12
e0i are the linear combinations of the neutral gaugino and higgsino statesMχ̃0 ≤ Mχ̃0 . The χ
3
4determined by the matrix elements of W . The neutralino masses correspond to the singular values†of MN , i.e., the positive square roots of the eigenvalues of MN
MN . Exact formulae for these masses
can be found in Refs. [108, 109]. A numerical algorithm for determining the mixing matrix W has
been given in Ref. [110].
If a chargino or neutralino state approximates a particular gaugino or higgsino state, it isconvenient to employ the corresponding nomenclature. Specifically, if |M1 | and |M2 | are small
e01 would be nearly a pure photino, γe , thecompared to mZ and |µ|, then the lightest neutralino χsuperpartner of the photon. If |M1 | and mZ are small compared to |M2 | and |µ|, then the lightest
e the superpartner of the weak hypercharge gauge boson.
neutralino would be nearly a pure bino, B,
If |M2 | and mZ are small compared to |M1 | and |µ|, then the lightest chargino pair and neutralinof ± , and W
f 0 , the superpartners
would constitute a triplet of roughly mass-degenerate pure winos, W
3of the weak SU(2) gauge bosons. Finally, if |µ| and mZ are small compared to |M1 | and |M2 |, thenthe lightest chargino pair and neutralino would be nearly pure higgsino states, the superpartnersof the Higgs bosons. Each of the above cases leads to a strikingly different phenomenology.
In the NMSSM, an additional Higgs singlet superfield is added to the MSSM. This superfieldcomprises two real Higgs scalar degrees of freedom and an associated neutral higgsino degree of
freedom. Consequently, there are five neutralino mass eigenstates that are obtained by a Takagidiagonalization of the 5 × 5 neutralino mass matrix. In many cases, the fifth neutralino stateis dominated by its SU(2)×U(1) singlet component, and thus is very weakly coupled to the SMparticles and their superpartners.
88.4.2 The squarks and sleptons
For a given Dirac fermion f , there are two superpartners, feL and feR , where the L and R
subscripts simply identify the scalar partners that are related by SUSY to the left-handed andright-handed fermions, fL,R ≡ 12 (1 ∓ γ5 )f , respectively. (There is no νeR in the MSSM.) However,
feL –feR mixing is possible, in which case feL and feR are not mass eigenstates. For three generations ofsquarks, one must diagonalize 6 × 6 matrices corresponding to the basis (qeiL , qeiR ), where i = 1, 2, 3
are the generation labels. For simplicity, only the one-generation case is illustrated in detail below.
Using the notation of the third family, the one-generation tree-level squark squared-mass matrix
is given by [25, 111],!
M 2e + m2q + Lq
mq Xq∗
2
Q
M =,(88.11)2 + m2 + R
mq Xq
Me
q
q
R

where
Xq ≡ Aq − µ∗ (cot β)2T3q ,(88.12)and T3q = 12 [− 12 ] for q = t [b]. The diagonal squared-masses are governed by soft-SUSY-breaking
2 ≡ M 2 [M 2 ] for q = t [b], the corresponding quark masses m [m ]squared-masses M 2e and M e
t
b
e
e
Q
R
U
D
and the electroweak correction terms:
Lq ≡ (T3q − eq sin2 θW )m2Z cos 2β ,Rq ≡ eq sin2 θW m2Z cos 2β ,31st May, 2024

(88.13)

88. Supersymmetry, Part I (Theory)12

where eq = 23 [− 13 ] for q = t [b]. The off-diagonal squark squared-masses are proportional to thecorresponding quark masses and depend on tan β, the soft-SUSY-breaking A-parameters and thehiggsino mass parameter µ. Assuming that the A-parameters are parametrically of the same order(or smaller) relative to other SUSY-breaking mass parameters, it then follows that the first andsecond generation qeL –qeR mixing is smaller than that of the third generation where mixing can beenhanced by factors of mt and mb tan β.
In the case of third generation qeL –qeR mixing, the squark mass eigenstates (usually denotedby qe1 and qe2 , with mq̃1 < mq̃2 ) are determined by diagonalizing the 2 × 2 matrix M2 given by
Eq. (88.11). The corresponding squared-masses and mixing angle are given by [111]:1
m2q̃1,2 =2

sin 2θq̃ =

2

Tr M ∓

q(TrM2 )2 − 4 det M2

,(88.14)2mq |Xq |.
m2q̃2 − m2q̃1

The one-generation results above also apply to the charged sleptons, with the obvious substitutions:q → ` with T3` = − 12 and e` = −1, and the replacement of the SUSY-breaking parameters:
M 2e → M e2 , M 2e → M 2e , and Aq → Aτ . For the neutral sleptons, νeR does not exist in the MSSM,
E
DL
Q
so νeL is a mass eigenstate.
In the case of three generations, the SUSY-breaking scalar-squared masses [M 2e , M e2 , M 2e , M e2 ,
Q

U

D

L

and M 2e ] and the A-parameters [AU , AD , and AE ] are now 3 × 3 matrices as noted in Sec. 88.3.3.
E

The diagonalization of the 6 × 6 squark mass matrices yields feiL –fejR mixing. In practice, since thefeL –feR mixing is appreciable only for the third generation, this additional complication can often
be neglected (although see Ref. [112] for examples in which the mixing between the second andthird generation squarks is relevant).

88.5 The supersymmetric Higgs sector
Consider first the MSSM Higgs sector [29, 30, 113]. Despite the large number of potential CPviolating phases among the MSSM-124 parameters, the tree-level MSSM Higgs potential given
by Eq. (88.2) is automatically CP-conserving. This follows from the fact that the only potentiallycomplex parameter (m212 ) of the MSSM Higgs potential can be chosen real and positive by re-phasing
the Higgs fields, in which case tan β is a real positive parameter. Consequently, the physical neutral
Higgs scalars are CP-eigenstates (at tree-level). The MSSM Higgs sector contains five physical spinzero particles: a charged Higgs boson pair (H ± ), two CP-even neutral Higgs bosons (denoted by h0
and H 0 where mh < mH ), and one CP-odd neutral Higgs boson (A0 ). The discovery of a SM-like
Higgs boson at the LHC with a mass of 125 GeV (see Sec. 11) strongly suggests that this stateshould be identified with h0 , although the possibility that the 125 GeV state should be identified
with H 0 cannot yet be completely ruled out [114].
In the NMSSM [36], the scalar component of the singlet Higgs superfield adds two additional
neutral states to the Higgs sector. In this model, the tree-level Higgs sector can exhibit explicit
CP-violation. If CP is conserved, then the two extra neutral scalar states are CP-even and CP-odd,
respectively. These states can potentially mix with the neutral Higgs states of the MSSM. If scalarstates exist that are dominantly singlet, then they are weakly coupled to SM gauge bosons andfermions through their small mixing with the MSSM Higgs scalars. Consequently, it is possiblethat one (or both) of the singlet-dominated states is considerably lighter than the Higgs boson thatwas observed at the LHC.
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88. Supersymmetry, Part I (Theory)13

88.5.1 The tree-level Higgs sector
The tree-level properties of the Higgs sector are determined by the Higgs potential given by
Eq. (88.2). The quartic interaction terms are manifestly supersymmetric (although these are modified by SUSY-breaking effects at the loop level). In general, the quartic couplings arise from twosources: (i) the supersymmetric generalization of the scalar potential (the so-called “F -terms”),
and (ii) interaction terms related by SUSY to the coupling of the scalar fields and the gauge fields,
whose coefficients are proportional to the corresponding gauge couplings (the so-called “D-terms”).
In the MSSM, F -term contributions to the quartic Higgs self-couplings are absent. As a result,the strengths of the MSSM quartic Higgs interactions are fixed in terms of the gauge couplings,
as noted below Eq. (88.2). Consequently, all the tree-level MSSM Higgs-sector parameters depend
only on two quantities: tan β [defined in Eq. (88.3)] and one Higgs mass usually taken to be mA .
For example, the tree-level squared mass of the charged Higgs boson is given by
m2H ± = m2A + m2W ,(88.15)where m2A = m2Hd + m2Hu + 2|µ|2 [cf. Eq. (88.5)] andH ± = Hd± sin β + Hu± cos β ,

A=


√ 
2 Im Hd0 sin β + Im Hu0 cos β .

(88.16)The CP-even scalar mass eigenstate fields h and H are identified by diagonalizing the 2×2 squaredmass matrix!
m2A sin2 β + m2Z cos2 β−(m2A + m2Z ) sin β cos β2.
(88.17)M =−(m2A + m2Z ) sin β cos β
m2A cos2 β + m2Z sin2 β
In particular,√√
h = −( 2 Re Hd0 − vd ) sin α + ( 2 Re Hu0 − vu ) cos α ,√√H = ( 2 Re Hd0 − vd ) cos α + ( 2 Re Hu0 − vu ) sin α ,(88.18)(88.19)with corresponding tree-level squared masses,
m2H,h =q
1
m2A + m2Z ± (m2A + m2Z )2 − 4m2Z m2A cos2 2β
2




,

(88.20)and mixing angle α given byscos α =

m2A sin2 β + m2Z cos2 β − m2h,
m2H − m2h

(88.21)in a convention where |α| ≤ π/2. However, because the off-diagonal elements of M2 are negative, itfollows that −π/2 ≤ α ≤ 0 [104]. In light of Eq. (88.20), the tree-level mass of the lighter CP-even
Higgs boson is bounded [29, 30],mh ≤ mZ | cos 2β| ≤ mZ .
(88.22)This bound can be substantially modified when radiative corrections are included, as discussed inSec. 88.5.2.
In the NMSSM, we set µ = 0 in Eq. 88.1 and then add two additional terms to the superpotential,
1
WNMSSM ⊃ λĤu Ĥd Ŝ + κŜ 3 ,3(88.23)where Ŝ is a singlet Higgs superfield. In the NMSSM as defined here, all terms in WNMSSM are
cubic in the superfields due to the presence of a discrete Z3 symmetry. An effective µ-term is
31st May, 2024

14

88. Supersymmetry, Part I (Theory)

generated, µeff = λhSi, where hSi is the vacuum expectation value of the scalar field component ofŜ. Moreover, due to the term proportional to λ in Eq. 88.23, there is now an F -term contribution
to the quartic Higgs self-couplings. Consequently, the tree-level bound for the mass of the lightest
CP-even MSSM Higgs boson is modified [115],1m2h ≤ m2Z cos2 2β + λ2 v 2 sin2 2β ,2(88.24)

where v ≡ (vu2 + vd2 )1/2 = 246 GeV. By requiring that λ remain finite after renormalization-groupevolution up to the Planck scale, one finds that λ is constrained to lie below about 0.7–0.8 at theelectroweak scale [36] (although larger values of λ have also been considered in Ref. [116]).
The tree-level Higgs couplings to gauge bosons and the Higgs boson self-couplings are governedby the electroweak gauge couplings and the parameter cos(β − α). Explicitly,cos(β − α) = q

m2Z sin 2β cos 2β(m2H − m2h )(m2H − m2Z cos2 2β).

(88.25)Note that cos(β−α) → 0 in the limit of mH  mh , mZ . In this decoupling limit [117], the propertiesof h coincide with those of the SM Higgs boson. In light of the LHC Higgs data [118, 119], which
are compatible with the SM predictions (see Sec. 11), one can conclude that if h is identified withthe observed Higgs boson then H, A and H ± must be substantially heavier (most likely of order
500 GeV or larger [120]).
The tree-level Higgs-quark and Higgs-lepton interactions of the MSSM are derived from thesuperpotential given in Eq. (88.1). The corresponding Higgs-fermion Yukawa couplings can be
expressed in terms of the fermion masses and the separate parameters cos(β − α) and tan β. In
particular, the Higgs sector of the MSSM is a Type-II two-Higgs doublet model [121], in which
one Higgs doublet (Hd ) couples exclusively to the right-handed down-type quark (or lepton) fields
and the second Higgs doublet (Hu ) couples exclusively to the right-handed up-type quark fields.
Consequently, the diagonalization of the fermion mass matrices simultaneously diagonalizes thematrix of Yukawa couplings, resulting in flavor-diagonal tree-level couplings of the neutral Higgs
bosons h0 , H 0 and A0 to quark and lepton pairs. One can again check that in the decoupling limit
where cos(β − α) → 0, the couplings of h reduce to those of the SM.
88.5.2 The radiatively-corrected Higgs sector
When radiative corrections are incorporated, additional parameters of the supersymmetricmodel enter via virtual supersymmetric particles that appear in loops. The impact of these corrections can be significant [122]. The qualitative behavior of these radiative corrections can be mosteasily seen in the large top-squark mass limit, where in addition, both the splitting of the two diagonal entries and the off-diagonal entries of the top-squark squared-mass matrix [Eq. (88.11)] aresmall in comparison to the geometric mean of the two top-squark squared-masses, MS2 ≡ Met1 Met2 .
In this case (assuming mA > mZ ), the predicted upper bound for mh is approximately givenby [123]
"
!
!#
MS2
Xt2
Xt2
3g 2 m4t
2
2
2+ 2 1−,(88.26)
mh . mZ cos 2β + 2 2 ln
8π mW
m2t
MS
12MS2
where Xt ≡ At − µ cot β [cf. Eq. (88.12)] is proportional to the off-diagonal entry of the top-squark
squared-mass matrix (where for simplicity, At and µ are taken to be real). The Higgs mass√upper
limit specified by Eq. (88.26) is saturated when tan β is large (i.e., cos2 2β ∼ 1) and Xt = 6 MS ,which defines the so-called maximal mixing scenario.
31st May, 2024

15

88. Supersymmetry, Part I (Theory)

In applying the radiatively corrected MSSM Higgs sector to the analysis of LHC Higgs data, theauthors of Refs. [124,125] suggested that a reasonable approximation would consist of retaining theleading corrections employed in deriving Eq. (88.26), while discarding all other subleading terms.
This procedure was implemented by simply replacing M222 → M222 + ∆M222 in the 22 element of
the CP-even Higgs squared-mass matrix given in Eq. (88.17), since ∆M222 contains the leading
contributions that govern the Higgs mass radiative corrections. One can now re-diagonalize theCP-even Higgs squared-mass matrix and determine ∆M222 in terms of the parameters mA , tan β
and the measured Higgs mass (e.g., mh ' 125 GeV, if h is identified with the observed Higgs boson).
This framework was dubbed the hMSSM in Ref. [124].
Although the hMSSM can be readily applied to LHC data to derive interesting constraints
on the MSSM Higgs sector, it can lead to results that are not robust in a more general MSSM
parameter scan. Indeed, a more complete treatment of the radiative corrections can yield resultsthat cannot be accounted for by the hMSSM framework. Examples of benchmark points of the
MSSM parameter space that cannot be reproduced by the hMSSM analysis are discussed in 11.6.1.1.
The set of approximations employed in obtaining Eq. (88.26) somewhat overestimates the valueof mh . The most complete treatment of the MSSM Higgs mass radiative corrections, which incorporate renormalization group improvement, the two loop, and the leading three-loop contributions [73, 126], yields a predicted value of mh shown in Fig. 88.1, as a function of Xt (assumed forsimplicity to be real).

Figure 88.1: The lighter CP-even Higgs mass in the MSSM as a function of a common SUSY mass
parameter MS and of the stop mixing parameter Xt (normalized to MS ). Figure taken from Ref. [73].

In addition, one-loop radiative corrections can introduce CP-violating effects in the Higgs sector
that depend on some of the CP-violating phases among the MSSM-124 parameters [127]. Thisphenomenon is most easily understood in a scenario where mA  MS (i.e., all five physical Higgs
states are significantly lighter than the SUSY breaking scale). In this case, one can integrate out theheavy superpartners to obtain a low-energy effective theory with two Higgs doublets. The resulting
effective two-Higgs doublet model will now contain all possible Higgs self-interaction terms (bothCP-conserving and CP-violating) and Higgs-fermion interactions (beyond those of Type-II) thatare consistent with electroweak gauge invariance [128].
In the NMSSM with mh ' 125 GeV, the dominant radiative correction to Eq. (88.24) is thesame as the one given in Eq. (88.26). However, in contrast to the MSSM, one does not need as
31st May, 2024

88. Supersymmetry, Part I (Theory)

16

large a boost from radiative corrections to achieve a Higgs mass of 125 GeV in certain regimes of
the NMSSM parameter space (e.g., tan β ∼ 2 and λ ∼ 0.7 [129]).

88.6 Restricting the MSSM parameter freedom
In Sections 88.4 and 88.5, we surveyed the parameters that comprise the MSSM-124. However,the MSSM-124 is not a phenomenologically viable theory over much of its parameter space. In
particular, a generic point of the MSSM-124 parameter space exhibits: (i) no conservation of
the separate lepton numbers Le , Lµ , and Lτ ; (ii) unsuppressed flavor-changing neutral currents(FCNCs); and (iii) new sources of CP violation that are inconsistent with the experimental bounds.
For example, the MSSM contains new sources of CP violation [130]. Indeed, for TeV-scalesfermion and gaugino masses, some combinations of the complex phases of the gaugino-mass parameters, the A-parameters, and µ must be less than about 10−2 –10−3 to avoid generating electric dipole
moments for the neutron, electron, and atoms [131–133] in conflict with observed data [134, 135].
The rarity of FCNCs [136–138] places additional constraints on the off-diagonal matrix elements ofthe squark and slepton soft-SUSY-breaking squared-masses and A-parameters (see Sec. 88.3.3).
The MSSM-124 is also theoretically incomplete as it provides no explanation for the fundamental
origin of the supersymmetry-breaking parameters. The successful unification of the MSSM gauge
couplings at a grand unified mass scale MGUT , close to the Planck scale [7, 70, 139–141],r

gs (MGUT ) = g(MGUT ) =5 0
g (MGUT ),3(88.27)suggests that the high-energy structure of the theory may be considerably simpler than its lowenergy realization.2 In a top-down approach, the dynamics that governs the theory at high energies
is used to derive the effective broken-supersymmetric theory at the TeV scale.
In this Section, we examine a number of theoretical frameworks that potentially yield phenomenologically viable regions of the MSSM-124 parameter space. The resulting supersymmetricparticle spectrum is then a function of a relatively small number of input parameters. This isaccomplished by imposing a simple structure on the soft SUSY-breaking parameters at a common
high-energy scale MX (typically chosen to be the Planck scale, MP , the grand unified theory scale,
MGUT , or the messenger scale, Mmess ). These serve as initial conditions for the MSSM renormalization group equations (RGEs), which are given in the two-loop approximation in Ref. [142]. An
automated program to compute RGEs for the MSSM and other supersymmetric models of new
physics has been developed in Ref. [143]. Solving these equations numerically, one can then derive
the low-energy MSSM parameters relevant for phenomenology. A number of software packages exist that numerically calculate the spectrum of supersymmetric particles, consistent with theoreticalconditions on SUSY breaking at high energies and some experimental data at low energies [73,144].
Examples of viable frameworks are provided by models of gravity-mediated, anomaly-mediated,and gauge-mediated SUSY breaking. In some of these approaches, one of the diagonal Higgs
squared-mass parameters is driven negative by renormalization group evolution [145]. In suchmodels, electroweak symmetry breaking is generated radiatively, and the resulting electroweak
symmetry-breaking scale is intimately tied to the scale of low-energy SUSY breaking.
2

Generically, the normalization of the U(1) hypercharges exhibited in Table 88.1 is a matter of convention. In
particular, the U(1) hypercharges can be rescaled by absorbing the scaling factor into a redefinition of the hypercharge
gauge coupling g 0 . However, the embedding of the hypercharge U(1) generator into the Lie algebra of a grand unified
simple gauge group fixes the normalization of the U(1) hypercharges and results in the rescaled hypercharge gauge
coupling shown in Eq. (88.27).

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88.6.1 Gaugino mass relations
One prediction of many supersymmetric grand unified models is the unification of the (tree-level)
gaugino mass parameters3 at some high-energy scale, MX ,M1 (MX ) = M2 (MX ) = M3 (MX ) = m1/2 .

(88.28)Due to renormalization group running, in the one-loop approximation the effective low-energy
gaugino mass parameters (at the electroweak scale) are related,
M3 = (gs2 /g 2 )M2 ' 3.5M2 ,

M1 = (5g 0 2 /3g 2 )M2 ' 0.5M2 .

(88.29)Eq. (88.29) can arise more generally in gauge-mediated SUSY-breaking models where the gaugino
masses are generated at the messenger scale Mmess (which typically lies significantly below theunification scale where the gauge couplings unify). In this case, the gaugino mass parameters areproportional to the corresponding squared gauge couplings at the messenger scale.
When Eq. (88.29) is satisfied, the chargino and neutralino masses and mixing angles depend
only on three unknown parameters: the gluino mass, µ, and tan β. It then follows that the lightest
neutralino must be heavier than 46 GeV due to the non-observation of charginos at LEP [147]. If in
addition |µ|  |M1 | & mZ , then the lightest neutralino is nearly a pure bino, an assumption often
made in supersymmetric particle searches at colliders. Although Eq. (88.29) is often assumed inmany phenomenological studies, a truly model-independent approach would take the gaugino mass
parameters M1 , M2 , and M3 to be independent parameters to be determined by experiment. Indeed,
an approximately massless neutralino cannot be ruled out at present by a model-independent
analysis [148].
It is possible that the tree-level masses of the gauginos are zero. In this case, the gaugino mass
parameters arise at one-loop and do not satisfy Eq. (88.29). For example, the gaugino masses
in AMSB models arise entirely from a model-independent contribution derived from the superconformal anomaly [51, 149]. In this case, Eq. (88.29) is replaced (in the one-loop approximation)by:bi gi2Mi 'm ,(88.30)16π 2 3/2
where m3/2 is the gravitino mass and the bi are the coefficients of the MSSM gauge beta-functions
corresponding to the corresponding U(1), SU(2), and SU(3) gauge groups, (b1 , b2 , b3 ) = ( 33
5 , 1, −3).
Eq. (88.30) yields M1 ' 2.8M2 and M3 ' −8.3M2 , which implies that the lightest chargino pairf ±, Wf 0 (cf. Table 1), overand neutralino comprise a nearly mass-degenerate triplet of winos, W
most of the MSSM parameter space. For example, if |µ|  mZ , |M2 |, then Eq. (88.30) implies thatMχ̃± ' Mχ̃0 ' M2 [150]. Alternatively, one can construct an AMSB model where |µ|, mZ  M2 ,1which yields an LSP that is an approximate higgsino state [151]. In both cases, the corresponding
supersymmetric phenomenology differs significantly from the standard phenomenology based on
Eq. (88.29) [152, 153].
Finally, it should be noted that the unification of gaugino masses (and scalar masses) can be
accidental. In particular, the energy scale where unification takes place may not be directly related
to any physical scale. One version of this phenomenon has been called mirage unification and canoccur in certain theories of fundamental SUSY breaking [154].
88.6.2 Constrained versions of the MSSM: mSUGRA, CMSSM, etc.
In the minimal supergravity (mSUGRA) framework [3–5, 25, 44–46], the minimal form of the
Kähler potential is employed, which yields standard kinetic energy terms for the MSSM fields
3

Non-universal gaugino mass parameters can also be a viable option in grand unified models with non-minimal
gauge kinetic functions [146].

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88. Supersymmetry, Part I (Theory)[48]. As a result, the soft supersymmetry-breaking parameters at the high-energy scale MX takea particularly simple form in which the scalar squared-masses and the A-parameters are flavordiagonal and universal [46]:
2
2
2
2
MQ
e (MX ) = m0 1 ,e (MX ) = MUe (MX ) = MD

MLe2 (MX ) = ME2e (MX ) = m20 1 ,
m21 (MX ) = m22 (MX ) = m20 ,(88.31)

AU (MX ) = AD (MX ) = AE (MX ) = A0 1 ,where 1 is a 3×3 identity matrix in generation space. As in the SM, this approach exhibits minimal
flavor violation (see e.g. Refs. [155, 156]), whose unique source is the nontrivial flavor structure of
the Higgs-fermion Yukawa couplings. The gaugino masses are also unified according to Eq. (88.28).
Renormalization group evolution is then used to derive the values of the supersymmetric parameters at the low-energy (electroweak) scale. For example, to compute squark masses, one should
use the low-energy values for M 2e , M e2 , and M 2e in Eq. (88.11). Through the renormalization group
Q

U

D

running with boundary conditions specified in Eq. (88.29) and Eq. (88.31), one can show that thelow-energy values of M 2e , M e2 , and M 2e depend primarily on m20 and m21/2 . A number of usefulDUQ
approximate analytic expressions for superpartner masses in terms of the mSUGRA parameters
can be found in Ref. [157].
One can count the number of independent parameters in the mSUGRA framework. In addition
to 18 SM parameters (excluding the Higgs mass), one must specify m0 , m1/2 , A0 , the Planck-scalevalues for µ and B-parameters (denoted by µ0 and B0 ), and the gravitino mass m3/2 . Without additional model assumptions, m3/2 is independent of the parameters that govern the mass spectrumof the superpartners of the SM [46]. In principle, A0 , B0 , µ0 , and m3/2 can be complex, although
in the mSUGRA approach, these parameters are taken to be real for simplicity.
As previously noted, renormalization group evolution is used to compute the low-energy valuesof the mSUGRA parameters, which then fixes all the parameters of the low-energy MSSM. In particular, the two Higgs vacuum expectation values (or equivalently, mZ and tan β) can be expressed
as a function of the Planck-scale supergravity parameters. In light of Eq. (88.4) and Eq. (88.5), a
common procedure is to determine µ0 and B0 in terms of mZ and tan β [the sign of µ0 , denoted
sgn(µ0 ) below, is not fixed in this process]. In this case, the MSSM spectrum and its interaction
strengths are fixed by five parameters:
m0 , A0 , m1/2 , tan β , and sgn(µ0 ) ,(88.32)and an independent gravitino mass m3/2 (in addition to the 18 parameters of the SM). In Ref. [158],
this framework was dubbed the constrained minimal supersymmetric extension of the SM (CMSSM).
Additional relations such as B0 = A0 − m0 and m3/2 = m0 comprise the original mSUGRA proposal [44, 48, 159].
One can also relax the universality of scalar masses by decoupling the squared-masses ofthe Higgs bosons and the squarks/sleptons. This leads to the non-universal Higgs mass models(NUHMs), thereby adding one or two new parameters to the CMSSM depending on whether the
diagonal Higgs scalar squared-mass parameters (m2Hd and m2Hu ) are set equal (NUHM1 [160]) or
taken to be independent (NUHM2 [161]) at the high energy scale MX . Clearly, this modification
preserves the minimal flavor violation of the mSUGRA approach. Nevertheless, the mSUGRA
approach and its NUHM generalizations are probably too simplistic. Theoretical considerations
suggest that the universality of Planck-scale soft SUSY-breaking parameters is not generic [162].
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In particular, effective operators at the Planck scale exist that do not respect flavor universality,
and it is difficult to find a theoretical principle that would forbid them.
In the framework of supergravity, if anomaly mediation is the sole source of SUSY breaking, thenthe gaugino mass parameters, diagonal scalar squared-mass parameters, and the SUSY-breaking
trilinear scalar interaction terms (proportional to λf AF ) are determined in terms of the beta functions of the gauge and Yukawa couplings and the anomalous dimensions of the squark and slepton
fields [51, 149, 153]. As noted in Sec. 88.2.3, this approach yields tachyonic sleptons in the MSSM
unless additional sources of SUSY breaking are present. In the minimal AMSB (mAMSB) scenario,
a universal squared-mass parameter, m20 , is added to the AMSB expressions for the diagonal scalarsquared-masses [153]. Thus, the mAMSB spectrum and its interaction strengths are determinedby four parameters, m20 , m3/2 , tan β and sgn(µ0 ).
The mAMSB scenario appears to be ruled out based on the observed value of the Higgs boson
mass, assuming an upper limit on MS of a few TeV, since the mAMSB constraint on AF implies
that the maximal mixing scenario cannot be achieved [cf. Eq. (88.26)]. Indeed, under the statedassumptions, the mAMSB Higgs mass upper bound lies below the observed Higgs mass value [163].
Thus within the AMSB scenario, either an additional SUSY-breaking contribution to λf AF and/or
new ingredients beyond the MSSM are required.
88.6.3 Gauge-mediated SUSY breaking
In contrast to models of gravity-mediated SUSY breaking, the flavor universality of the fundamental soft SUSY-breaking squark and slepton squared-mass parameters is guaranteed in gaugemediated SUSY breaking (GMSB) because the supersymmetry breaking is communicated to thesector of MSSM fields via gauge interactions [56, 58]. In GMSB models, the mass scale of themessenger sector (or its equivalent) is sufficiently below the Planck scale such that the additional
SUSY-breaking effects mediated by supergravity can be neglected.
In the minimal GMSB approach, there is one effective mass scale, Λ, that determines all lowenergy scalar and gaugino mass parameters through loop effects, while the resulting A-parameters
are suppressed. In addition, the minimal form of the Kähler potential is employed. In order that theresulting superpartner masses be of order 1 TeV, one must have Λ ∼ O(100 TeV). The origin of theµ and B-parameters is model-dependent, and lies somewhat outside the purview of gauge-mediated
SUSY breaking [164].
The simplest GMSB models appear to be ruled out based on the observed value of the Higgs
boson mass. Due to suppressed A parameters, it is difficult to boost the contributions of theradiative corrections in Eq. (88.26) to obtain a Higgs mass as large as 125 GeV. However, this conflictcan be alleviated in more complicated GMSB models [165]. To analyze these generalized GMSB
models, it has been especially fruitful to develop model-independent techniques that encompass all
known GMSB models [166]. These techniques are well-suited for a comprehensive analysis [167] of
the phenomenological profile of gauge-mediated SUSY breaking.
The gravitino is the LSP in minimal GMSB models, as noted in Sec. 88.2.3. As a result, thenext-to-lightest supersymmetric particle (NLSP) now plays a crucial role in the phenomenologyof supersymmetric particle production and decays. Note that unlike the LSP, the NLSP can be0charged. In GMSB models, the most likely candidates for the NLSP are χ̃1 and τeR± . The NLSP will0
e χ̃01 → Z G,e χ̃01 → h0 G
e or τe± → τ ± G),edecay into its superpartner plus a gravitino (e.g., χ̃1 → γ G,
1with lifetimes and branching ratios that depend on the model parameters. There are also GMSB
scenarios in which there are several nearly degenerate co-NLSPs, any one of which can be produced
at the penultimate step of a supersymmetric decay chain [168]. For example, in the slepton coNLSP case, all three right-handed sleptons are close enough in mass and thus can each play therole of the NLSP.
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20

Different choices for the identity of the NLSP and its decay rate lead to a variety of distinctive
0supersymmetric phenomenologies [58,169]. For example, a long-lived χ̃1 -NLSP that decays outsidecollider detectors leads to supersymmetric decay chains with missing energy in association withleptons and/or hadronic jets (this case is indistinguishable from the standard phenomenology of00
e is the dominant decay mode, and the decay occurs
the χ̃1 -LSP). On the other hand, if χ̃1 → γ G
inside the detector, then nearly all supersymmetric particle decay chains would produce a photon.
In contrast, in the case of a τe1± -NLSP, the τe1± would either be long-lived or would decay inside thedetector into a τ -lepton plus missing energy.
A number of attempts have been made to address the origins of the µ and B-parameters in
GMSB models based on the field content of the MSSM (see, e.g., Refs. [164, 170]). An alternative
approach is to consider GMSB models based on the NMSSM [171]. The vacuum expectation valueof the additional singlet Higgs superfield can be used to generate effective µ and B-parameters [172].
Such models provide an alternative GMSB framework for achieving a Higgs mass of 125 GeV, whilestill being consistent with LHC bounds on supersymmetric particle masses.
88.6.4 The phenomenological MSSM
Any of the theoretical assumptions described in the previous three subsections must be tested
experimentally and could turn out to be wrong. To facilitate the exploration of MSSM phenomena
in a more model-independent way while respecting the constraints noted at the beginning of thisSection, the phenomenological MSSM (pMSSM) has been introduced [173].
The pMSSM is governed by 19 independent real supersymmetric parameters: the three gaugino
mass parameters M1 , M2 and M3 , the Higgs sector parameters mA and tan β, the Higgsino mass
parameter µ, five sfermion squared-mass parameters for the degenerate first and second generations(M 2e , M e2 , M 2e , M e2 and M 2e ), the five corresponding sfermion squared-mass parameters for the
Q

U

D

L

E

third generation, and three third-generation A-parameters (At , Ab and Aτ ). The first and second
generation A-parameters are typically neglected in pMSSM studies, as their phenomenologicalconsequences are negligible in most applications. One counterexample arises when considering theAµ dependence of the anomalous magnetic moment of the muon, which can be as significant as othercontributions due to superpartner mediated radiative corrections [174]. Since its initial proposal,
the pMSSM approach has been extended to include a 20th parameter, Aµ [175]. It also has beenfurther extended to include CP-violating SUSY-breaking parameters in Ref. [176].
The 19-parameter pMSSM is often further constrained to expedite scans over the parameter
space. For example, in Ref. [177], the number of pMSSM parameters is reduced to ten by assuming
one common squark squared-mass parameter for the first two generations, a second common squark
squared-mass parameter for the third generation, a common (charged) slepton squared-mass parameter and a common third generation A parameter. In Ref. [178] an eleven parameter pMSSM isdefined by allowing for a different stau squared-mass parameter from that of the first two generation
charged sleptons. Other applications of the pMSSM approach (with a reduced pMSSM parameter
space) to supersymmetric particle searches, and a discussion of the implications for past and future
LHC and dark matter studies can be found in Refs. [177, 179, 180].
88.6.5 Simplified models
As Sec. 89 demonstrates, experiments present their searches for supersymmetric particles primarily in terms of simplified models. Simplified models for supersymmetric searches [181] aredefined mostly by the empirical objects and kinematic variables involved in the search. Their interpretation by an experimental collaboration usually involves only a small number of supersymmetricparticles (often two or three). Other supersymmetric particles are assumed to play no role (thismay happen by virtue of them being too heavy to be produced). Experimental bounds from non31st May, 2024

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88. Supersymmetry, Part I (Theory)

observation of a signal are usually presented in terms of the physical masses of the supersymmetricparticles involved. Bounds may be presented on the relevant supersymmetric particle masses assuming certain values for the branching ratio of certain supersymmetric particle decays, or as an
upper bound on the signal production cross-section as a function of the relevant supersymmetricparticle masses.
For example, consider a search for hadronic jets plus missing transverse momentum. One canmatch such a search to the simplified model of squark pair production followed by the subsequent
decay of each squark into a quark (which appears as a jet) and a neutralino LSP that produces00
the missing transverse momentum, i.e. q̃ q̃ → (q χ̃1 )(q χ̃1 ). Excluded cross-sections resulting fromthe non-observation of a signal (which in this case could consist of some specified minimum valueof missing transverse momentum and at least two hard jets) may be exhibited in the squark mass
versus LSP mass plane.
Simplified models have the apparent advantage that they have fewer free parameters than more
complete supersymmetric models, whose greater number of free parameters makes it difficult topresent excluded regions in any generality. If limits are quoted on supersymmetric particle masses
without reference to the signal production cross-section from a simplified model analysis, then there
is a potential pitfall—namely, mass limits can differ from those obtained in full models because there
may be contributions to the signal coming from processes involving supersymmetric particles other00than those assumed. For example, in the q̃ q̃ → q χ̃1 q χ̃1 process mentioned above, the simplified
model analysis does not account for the interference with tree-level t-channel gluino contributions.
Nevertheless, simplified model bounds quoted purely in terms of supersymmetric particle masses
may still approximately hold over sizable regions of parameter space of more complete models,
within which the simplified model is embedded. When simplified model limits are phrased as bounds
on signal cross-sections, the aforementioned pitfall is sidestepped. Simplified models thus remain
an efficient tool for organizing and presenting the results of supersymmetric particle searches. A
comparison between supersymmetric particle search constraints in the context of simplified models
and the corresponding constraints obtained in the more complete pMSSM can be found in Ref. [182].

88.7 Experimental data confronts the MSSM
At present, there is no significant evidence for weak-scale SUSY from the data analyzed by the
LHC experiments. Recent LHC data have been employed in ruling out the existence of coloredsupersymmetric particles (primarily the gluino and the first generation of squarks) with masses
below about 2 TeV (see Fig. 89.14). Moreover, given that the mass of the observed Higgs boson is 125 GeV, the results exhibited in Fig. 88.1 tend to favor a mass scale of the top squarks
somewhat above 2 TeV. However, the precise mass limits are very model dependent. For example, as Fig. 89.13 demonstrates, regions of the pMSSM parameter space can be identified in whichlighter squarks and gluinos below 1 TeV cannot be definitely ruled out. Under the assumptionof gaugino mass unification [cf. Eq. (88.29)], LHC searches result in a lower bound on neutralino
and chargino masses of roughly 200 GeV. It is difficult to place general bounds on neutralino andchargino masses, since the limit in terms of masses from direct searches tends to be particularly
model dependent. Nevertheless, one must confront the tension that exists between the theoretical expectations for the magnitude of the SUSY-breaking parameters and the non-observation ofsupersymmetric phenomena at colliders.
88.7.1 Naturalness constraints and the little hierarchy
In Sec. 88.1, weak-scale SUSY was motivated as a natural solution to the hierarchy problem,
which could provide an understanding of the origin of the electroweak symmetry-breaking scale
without a significant fine-tuning of the fundamental parameters that govern the MSSM. In thiscontext, the weak scale soft supersymmetry-breaking masses must be generally of the order of
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88. Supersymmetry, Part I (Theory)

1 TeV or below [183]. This requirement is most easily seen in the determination of mZ by thescalar potential minimum condition. In light of Eq. (88.5), to avoid the fine-tuning of MSSM
parameters, the soft SUSY-breaking squared-masses m2Hd and m2Hu and the higgsino squared-mass|µ|2 should all be roughly of O(m2Z ). Many authors have proposed quantitative measures of finetuning [183–188]. One of the simplest measures is the one advocated by Barbieri and Giudice [183](which was also introduced previously in Ref. [184]),∆i ≡∂ ln m2Z,∂ ln pi∆ ≡ max ∆i ,(88.33)where the pi are the MSSM parameters at the high-energy scale MX , which are set by the fundamental SUSY-breaking dynamics. The theory is more fine-tuned as ∆ becomes larger. However,different measures of fine-tuning yield quantitatively different results; in particular, calculating minimal fine-tuning based on the high-scale parameters [as defined in Eq. (88.33)] yields a difference
by a factor ∼ 10 to fine-tuning based on TeV-scale parameters [189, 190].
One can apply the fine-tuning measure to any explicit model of SUSY breaking. For example, in the approaches discussed in Sec. 88.6, the pi are parameters of the model at the energy
scale MX where the soft SUSY-breaking operators are generated by the dynamics of SUSY breaking. Renormalization group evolution then determines the values of the parameters appearing in
Eq. (88.5) at the electroweak scale. In this way, ∆ is sensitive to all the SUSY-breaking parametersof the model (see e.g. Ref. [191]). The computation of ∆ is often based on Eq. (88.5), which is atree-level condition. However, the fine-tuning measure obtained at tree level can be somewhat reduced in value when loop corrections are included while remaining consistent with all experimental
constraints [87, 192].
One way of taking fine-tuning into account in fits to data using Bayesian statistics is to have aprior probability distribution proportional to 1/∆ [193] so that fine-tuning is balanced against thefit to empirical data. In such a Bayesian approach, it is important to choose the prior probability
distribution carefully, since prior probability densities that are flat in one set of variables may
not be flat in another, more fundamental set. One can in fact derive a different measure of finetuning resulting from a Jacobian factor when transforming to other variables.4 By comparing theresults of several Bayesian fits with different (but reasonable) prior probability distributions, one
can assess the robustness of the fit with respect to their variation, mitigating for subjectivity in theinterpretation of the fine-tuning measure.
As anticipated, there is a tension between the present experimental lower limits on the massesof colored supersymmetric particles [195, 196] and the expectation that supersymmetry-breaking
is associated with the electroweak symmetry-breaking scale. Moreover, this tension is exacerbated[197] by the observed value of the Higgs mass (mh ' 125 GeV), which is not far from the MSSM
upper bound (mh . 135 GeV) [which depends on the top-squark mass and mixing as noted inSec. 88.5.2]. If MSUSY characterizes the scale of supersymmetric particle masses, then one would
2
crudely expect ∆ ∼ MSUSY/m2Z . For example, if MSUSY ∼ 1 TeV then one expects a ∆−1 ∼ 1%
fine-tuning of the MSSM parameters to achieve the observed value of mZ . This separation of theelectroweak symmetry-breaking and SUSY-breaking scales is an example of the little hierarchy
problem [198, 199].
The fine-tuning parameter ∆ can depend quite sensitively on the structure of the SUSY-breaking
dynamics, such as the value of MX and relations among SUSY-breaking parameters in the fundamental high energy theory [200]. For example, in so-called focus point SUSY models [187, 201], all4
For example, one may consider the parameters µ and m212 to be more fundamental than tan β and MZ . In thiscase, one would choose a flat prior probability distribution in µ and m212 rather than in tan β and MZ [186, 194]. The
Jacobian factor is then obtained from Eq. (88.4) and Eq. (88.5).

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88. Supersymmetry, Part I (Theory)squark masses can be as heavy as 5 TeV without significant fine-tuning. This can be attributed
to a focusing behavior of the renormalization group evolution when certain relations hold among
the high-energy values of the scalar squared-mass SUSY-breaking parameters. Although the focus
point region of the CMSSM still yields an uncomfortably high value of ∆ due to the observed
Higgs mass of 125 GeV, one can achieve moderate values of ∆ in models with NUHM2 boundary
conditions for the scalar masses [197].
Among the colored superpartners, the third generation squarks typically have the most significant impact on the naturalness constraints [202], while their masses are the least constrainedby the LHC data. Hence, in the absence of any relation between third generation squarks andthose of the first two generations, the naturalness constraints due to present LHC data can beconsiderably weaker than those obtained in the CMSSM. Indeed, models with first and second
generation squark masses in the multi-TeV range do not necessarily require significant fine tuning.
Such models have the added benefit that undesirable FCNCs mediated by squark exchange arenaturally suppressed [203]. Other MSSM mass spectra that are compatible with moderate finetuning have been considered in Refs. [200] and [204].
The lower bounds on squark and gluino masses may not be as large as suggested by the experimental analyses based on the CMSSM or simplified models. For example, mass bounds forthe gluino and the first and second generation squarks based on the CMSSM can often be evaded
in alternative or extended MSSM models, e.g., compressed SUSY [205] and stealth SUSY [206].
Moreover, the experimental upper limits for the third generation squark masses (which have a more
direct impact on the fine-tuning measure) are weaker than the corresponding mass limits for othercolored supersymmetric states.
Among the uncolored superpartners, the higgsinos are typically the most impacted by thenaturalness constraints. Eq. (88.5) suggests that the masses of the two neutral higgsinos andcharged higgsino pair (which are governed by |µ|) should not be significantly larger than mZ toavoid an unnatural fine-tuning of the supersymmetric parameters, which would imply the existenceof light higgsinos (whose masses are not well constrained, as they are difficult to detect directly at
the LHC due to their soft decay products). However, it may be possible to avoid the conclusionthat µ ∼ O(mZ ) if additional correlations among the SUSY breaking mass parameters and µ arepresent. Such a scenario can be realized in models in which the boundary conditions for SUSY
breaking are generated by approximately conformal strong dynamics. For example, in the so-called
scalar-sequestering model of Ref. [207], values of |µ| > 1 TeV can be achieved while naturally
maintaining the observed value of mZ .
Finally, one can also consider extensions of the MSSM in which the degree of fine-tuning isrelaxed. For example, it has already been noted in Sec. 88.5 that it is possible to accommodate
the observed Higgs mass more easily in the NMSSM due to contributions to m2h proportional
to the parameter λ2 . This means that one does not have to rely on a large contribution fromradiative corrections to boost the Higgs mass sufficiently above its tree-level bound. This allows
for smaller top squark masses, which are more consistent with the demands of naturalness. The
reduction of the fine-tuning in various NMSSM models was initially advocated in Ref. [208], andsubsequently treated in more detail in Refs. [116,209]. Naturalness can also be relaxed in extended
supersymmetric models with vector-like quarks [210] and in gauge extensions of the MSSM [211].
The experimental absence of any new physics beyond the SM at the LHC suggests that the
principle of naturalness is presently under significant stress [212]. Nevertheless, one must be very
cautious when drawing conclusions about the viability of weak-scale SUSY to explain the originof electroweak symmetry breaking, since different measures of fine-tuning noted above can leadto different assessments [189, 190]. Moreover, the maximal value of ∆ that determines whether
weak-scale SUSY is a fine-tuned model (should it be ∆ ∼ 10? 100? 1000?) is ultimately subjective.
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Thus, it is premature to conclude that weak-scale SUSY is on the verge of exclusion. It might
be possible to sharpen the upper bounds on superpartner masses based on naturalness arguments,
which ultimately will either confirm or refute the weak scale SUSY hypothesis [213]. Of course,
if evidence for supersymmetric phenomena in the multi-TeV regime were to be established at afuture collider facility (with an energy reach beyond the LHC [214]), it would be viewed as aspectacularly successful explanation of the large gauge hierarchy between the (multi-)TeV scale
and Planck scale. In this case, the remaining little hierarchy, characterized by the somewhat large
value of the fine-tuning parameter ∆ discussed above, would be regarded as a less pressing issue.
88.7.2 Indirect constraints on supersymmetric models
While direct empirical searches for supersymmetric particles provide various limits on theirproperties, indirect constraints can depend more sensitively on details of the whole model. The
cold dark matter relic density inferred from cosmological fits to observational data is one such
example of an indirect constraint. In supersymmetric models where the LSP is stable (and thus
is a dark matter candidate), its thermally-produced relic density depends upon the scattering ofvarious supersymmetric particles into dark matter particles and SM particles. The resulting relicdensity can depend sensitively on the masses of the non-LSP supersymmetric particles as well as on
the mass of the LSP. In a typical model, an appreciable region of the parameter space is ruled out
because it yields an overabundance of dark matter (see for example Ref. [215] for a fit to a sevenparameter version of the pMSSM). However, subsequent tweaks to the supersymmetric model thatyield an unstable LSP, such as the introduction of R-parity violating effects, can mean that therelic density no longer constrains the parameter space.
There are a number of indirect constraints based on low-energy measurements that are sensitive
to the effects of new physics via supersymmetric loop effects. For example, the virtual exchange ofsupersymmetric particles can contribute to the muon anomalous magnetic moment, aµ ≡ 12 (g − 2)µ ,as reviewed in Ref. [216]. The SM prediction for aµ , which employs dispersion relations and low
energy e+ e− scattering data to determine the hadronic corrections [217], exhibits a deviation of 5σfrom the experimentally observed value [218]. The deviation of Ref. [217] from the measured valueof aµ [218] is difficult to accommodate in the constrained models of Sec. 88.6.2 and 88.6.3 given
the present sparticle mass bounds [196]. Nevertheless, such a deviation can be consistent in thecontext of a more general version of the MSSM [219].
However, the Budapest-Marseille-Wupertal (BMW) lattice determination of the leading hadronic
contribution to the muon magnetic moment obtained in Ref. [220] yields a significantly smaller deviation from the SM prediction [221]. Indeed, there is a tension of almost 4σ between the BMW
lattice calculation and the corresponding dispersive estimate based on the same e+ e− data, as
stated in Ref. [222] where the relevant references are cited. Moreover, Ref. [222] notes that crucial
parts of the BMW calculation have recently been cross checked by other lattice groups. In addition,
various tensions exist among the relevant sets of e+ e− scattering data that are used in the dispersive
estimate, some of which have appeared after the publication of Ref. [217]. Thus, the deviation of
the measured value of aµ from the corresponding SM prediction still needs to be clarified.
The precision of the measured value of aµ is not sensitive to the experimental error associatedwith the measured value of the fine structure constant, α. In contrast, the comparison of the SM
prediction with the experimental measurement of the anomalous magnetic moment of the electron,
ae , depends critically on the value of α. Using the experimentally determined value of α given in
Ref. [223] yields a SM prediction for ae that is 2.4σ above its measured value [224]. However, thisprevious determination of α is in tension at the 5σ level with a more recent measurement of the finestructure constant [225]. The latter yields a SM prediction for ae that is 1.6σ below its measured
value [225].
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88. Supersymmetry, Part I (Theory)

Measurements of the fine structure constant, ae and aµ jointly constrain the pMSSM parameter space [226] due to shifts originating from supersymmetric loop effects. In particular, if thesupersymmetric interpretation of the deviation in the measured value of aµ from its SM prediction is combined with the experimental limits on the electron electric dipole moment [135], thenthe resulting upper bounds on MSSM CP-violating phases are even more constraining [227] thanpreviously noted at the beginning of Sec. 88.6.
Flavor transitions in radiative, leptonic and semi-leptonic b quark decays [228] provide a fertile
ground for physics beyond the SM. For example, the rare inclusive decay b → sγ is a sensitive
probe of the virtual effects of new physics beyond the SM. The experimental measurements of
B → Xs + γ [229] are in agreement with the theoretical SM predictions of Ref. [230]. Since
supersymmetric loop corrections can contribute an observable shift from the SM predictions, theabsence of any significant deviation places useful constraints on the MSSM parameter space [231].
The rare decays Bs → µ+ µ− and Bd → µ+ µ− are especially sensitive to supersymmetricloop effects, with some loop contributions scaling as tan6 β when tan β  1 [232]. At present, acombination [233] of the measurements of these rare decay modes [234] is in slight tension at the1.6σ level with the predicted SM rates [235]. Such a tension can be resolved by the aforementioned
supersymmetric loop effects [232].
Several tensions exist between SM predictions and measurements of some other experimental
observables that probe b → sµ+ µ− transitions, although the level of tension depends upon thetheoretical treatment of the SM analysis. In a certain angular distribution parameter (denotedby P50 ) extracted from B 0 → K ∗0 µ+ µ− decays, the tension is around the 4σ level [236]. An evenlarger discrepancy is observed in a combination of angular distributions and rates derived from
B ± → K ± µ+ µ− and B 0 → K 0 µ+ µ− [237]. Finally, there is a 3.6σ deviation in the branching ratioof Bs → φµ+ µ− for di-muon invariant mass squared values between 1.1 GeV2 and 6.0 GeV2 [238].
Finally, a recent measurement of B + → K + ν ν̄ by the Belle II Collaboration [239] obtained abranching fraction that is roughly four times larger than the SM prediction [240], corresponding to
2.7σ above the SM expectation. However, it is unlikely that this result can be attributed to newcontributions from the MSSM [241], as the latter are expected to be negligible in comparison with
the SM.
The decays B ± → τ ± ντ and B → D(∗) τ − ν τ are noteworthy, since in models with extended
Higgs sectors such as the MSSM, these processes possess tree-level charged Higgs exchange contributions that can compete with the dominant W -exchange. As Sec. 72 shows, experimental
measurements of B ± → τ ± ντ are currently consistent with SM expectations [242]. The BaBar Collaboration measured values of the rates for B → Dτ − ν τ and B → D∗ τ − ν τ [243] which exhibiteda combined 3.4σ discrepancy from the SM predictions, which was also not compatible with theType-II Higgs Yukawa couplings employed by the MSSM. Subsequent measurements by the LHCb
and Belle Collaborations were compatible with the BaBar measurements although they displayed
less deviation from the SM expectations; the combined difference between the measured values of
the B → Dτ − ν τ and B → D∗ τ − ν τ decay rates relative to the corresponding SM values has asignificance of 3.3 standard deviations [244].
In summary, although there are a few hints of possible deviations from the SM in B decays,
none of the discrepancies by themselves are significant enough to conclusively imply the existenceof new physics beyond the SM. Moreover, the absence of evidence for sizable deviations in other Bphysics observables from their SM predictions can place useful constraints on the MSSM parameter
space [138, 195, 245, 246].
The CDF Collaboration has reported a measurement of the W boson mass [247] that is 7σ
above the SM prediction. Supersymmetric models have been shown to be able to accommodate
the measured value by the CDF Collaboration at the 2σ level [248]. However, other measurements,
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88. Supersymmetry, Part I (Theory)

including a more recent determination by ATLAS Collaboration [249], are consistent with SM
predictions for mW (see Sec. 10 for further details).

88.8 Massive neutrinos in weak-scale SUSYIn the minimal version of the SM and its supersymmetric extension, there are no right-handed
neutrinos, and Majorana mass terms for the left-handed neutrinos are absent. However, given the
overwhelming evidence for neutrino masses and mixing (see Sec. 14 and [250]), any viable modelof the fundamental particles must provide a mechanism for generating neutrino masses [251]. In
extended supersymmetric models, various mechanisms exist for producing massive neutrinos [252].
Although one can devise models for generating massive Dirac neutrinos [253], the most common
approaches for incorporating neutrino masses are based on L-violating supersymmetric extensions of
the MSSM, which generate massive Majorana neutrinos. Two classes of L-violating supersymmetricmodels will now be considered.
88.8.1 The supersymmetric seesaw
Neutrino masses can be incorporated into the SM by introducing SU(3)×SU(2)×U(1) singlet
right-handed neutrinos (νR ) whose mass parameters are very large, typically near the grand unification scale. In addition, one must also include a standard Yukawa couplings between the lepton
doublets, the Higgs doublet, and νR . The Higgs vacuum expectation value then induces an offdiagonal νL –νR mass on the order of the electroweak scale. Diagonalizing the neutrino mass matrix(in the three-generation model) yields three superheavy neutrino states, and three very light neutrino states that are identified with the light neutrinos observed in nature. This is the seesaw
mechanism [254].
It is straightforward to construct a supersymmetric generalization of the seesaw model of neub c = (νeR ; νR ).
trino masses [255,256] by promoting the right-handed neutrino field to a superfield N
Integrating out the heavy right-handed neutrino supermultiplet yields a new term in the superpotential [cf. Eq. (88.1)] of the form
Wseesaw =f b b b b
(HU L)(HU L) ,MR

(88.34)

where MR is the mass scale of the right-handed neutrino sector and f is a dimensionless constant.
Note that lepton number is broken by two units by Eq. (88.34), which implies that R-parity invariance is preserved. The supersymmetric analogue of the Majorana neutrino mass term in thesneutrino sector leads to sneutrino–antisneutrino mixing phenomena [256, 257]. In addition, new
Higgs-slepton interaction terms can probe the structure of the supersymmetric seesaw model [258].
b also provides an intriguing dark matter candidate [259].
The right-handed sneutrino that resides in L
The SUSY Les Houches Accord [80, 99], mentioned at the end of the introduction to Sec. 88.4,
has been extended to the supersymmetric seesaw (and other extensions of the MSSM) in Ref. [260].
88.8.2 R-parity-violating SUSY
It is possible to incorporate massive neutrinos in renormalizable supersymmetric models while
retaining the minimal particle content of the MSSM by relaxing the assumption of R-parity invariance. The most general R-parity-violating model involving the MSSM spectrum introduces
many new parameters to both the SUSY-conserving and the SUSY-breaking sectors [80,261]. Eachnew interaction term violates either B or L conservation. For example, starting from the MSSM
superpotential given in Eq. (88.1) [suitably generalized to three generations of quarks, leptons andtheir superpartners], consider the effect of adding the following new terms:
bpL
bmE
b c + (λ0 )pmn L
bpQ
bmD
b c + (λB )pmn U
b cD
bc bc
b b
WRPV =(λL )pmn LnLn
p m Dn + (µL )p Hu Lp ,31st May, 2024

(88.35)

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88. Supersymmetry, Part I (Theory)

where p, m, and n are generation indices, and gauge group indices are suppressed. Eq. (88.35)
yields new scalar-fermion Yukawa couplings consisting of all possible combinations involving two
SM fermions and one scalar superpartner.
Note that the term in Eq. (88.35) proportional to λB violates B, while the other three terms
bd
b uH
violate L. The L-violating term in Eq. (88.35) proportional to µL is the RPV analog of the µHbterm of the MSSM superpotential, in which the Y = −1 Higgs/higgsino supermultiplet Hd isbp.
replaced by the slepton/lepton supermultiplet L
Phenomenological constraints derived from data on various low-energy B- and L-violating processes can be used to establish limits on each of the coefficients (λL )pmn , (λ0L )pmn , and (λB )pmn
taken one at a time [261, 262]. If more than one coefficient is simultaneously non-zero, then thelimits are in general more complicated [263]. All possible RPV terms cannot be simultaneously
present and unsuppressed; otherwise the proton decay rate would be many orders of magnitudelarger than the present experimental bound. One way to avoid proton decay is to impose B or
L invariance (either one alone would suffice). Otherwise, one must accept the requirement thatcertain RPV coefficients must be extremely suppressed.
One particularly interesting class of RPV models is one in which B is conserved, but L isviolated. It is possible to enforce baryon number conservation (and the stability of the proton),while allowing for lepton-number-violating interactions by imposing a discrete Z3 baryon triality
symmetry on the low-energy theory [264], in place of the standard Z2 R-parity. Since the distinction
between the Higgs and matter supermultiplets is lost in RPV models where L is violated, the mixing
of sleptons and Higgs bosons, the mixing of neutrinos and neutralinos, and the mixing of charged
leptons and charginos are now possible, leading to more complicated mass matrices and mass
eigenstates than in the MSSM. The treatment of neutrino masses and mixing in this framework
can be found, e.g., in Ref. [265].
Alternatively, one can consider imposing a lepton parity such that all lepton superfields are
odd [264, 266]. In this case, only the B-violating term in Eq. (88.35) survives, and L is conserved.
Models of this type have been considered in Ref. [267]. Since L is conserved in these models, themixing of the lepton and Higgs superfields is forbidden. Moreover, neutrino masses (and mixing) are
not generated if lepton parity is an exact symmetry. However, one expects that lepton parity cannot
be exact due to quantum gravity effects. Remarkably, the standard Z2 R-parity and the Z3 baryon
triality are stable with respect to quantum gravity effects, as they can be identified as residual
discrete symmetries that arise from spontaneously broken non-anomalous gauge symmetries [264].
The symmetries employed above to either remove or suppress R-parity violating operators
were flavor independent. In contrast, there exist a number of motivated scenarios based on flavor
symmetries that can also yield the suppression as required by the experimental data (e.g., see
Ref. [268]).
The supersymmetric phenomenology of the RPV models exhibits features that are distinct fromthat of the MSSM [261]. The LSP is no longer stable, which implies that not all supersymmetricdecay chains must yield missing-energy events at colliders. A comprehensive examination of thephenomenology of the MSSM extended by a single R-parity violating coupling at the unification
scale and its implications for LHC searches has been given in Ref. [269]. As an example, the sparticle
mass bounds obtained in searches for R-parity-conserving SUSY can be considerably relaxed incertain RPV models due to the absence of large missing transverse momentum signatures [270].
This can alleviate some of the tension with naturalness (introduced in Sec. 88.7.1).
Nevertheless, the loss of the missing-energy signature is often compensated by other striking
signals (which depend on which R-parity-violating parameters are dominant). For example, supersymmetric particles in RPV models can be singly produced (in contrast to R-parity-conserving
models where supersymmetric particles must be produced in pairs). The phenomenology of pair31st May, 2024

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88. Supersymmetry, Part I (Theory)

produced supersymmetric particles is also modified in RPV models due to new decay chains notpresent in R-parity-conserving SUSY models [261].
In RPV models with lepton number violation (these include weak-scale SUSY models withbaryon triality mentioned above), both ∆L = 1 and ∆L = 2 phenomena are allowed, leading
to neutrino masses and mixing [271], neutrinoless double-beta decay [272], sneutrino-antisneutrino
mixing [273], and resonant s-channel production of sneutrinos in e+ e− collisions [274] and in charged
sleptons in pp̄ and pp collisions [275], respectively.

88.9 Extensions beyond the MSSM
Extensions of the MSSM have been proposed to solve a variety of theoretical problems [35]. One
such problem involves the µ parameter of the MSSM. Although µ is a SUSY-preserving parameter,
it must be of order the effective SUSY-breaking scale of the MSSM to yield a consistent supersymmetric phenomenology [276]. Any natural solution to the so-called µ-problem must incorporate asymmetry that enforces µ = 0 and a small symmetry-breaking parameter that generates a valueof µ that is not parametrically larger than the effective SUSY-breaking scale [277]. A number of
proposed mechanisms in the literature (e.g., see Ref. [276–278]) provide concrete examples of anatural solution to the µ-problem of the MSSM.
In extensions of the MSSM, other compelling solutions to the µ-problem are possible. For
example, one can replace µ by the vacuum expectation value of a new SU(3)×SU(2)×U(1) singlet
scalar field, as noted below Eq. 88.23. This is the NMSSM, which yields phenomena that werebriefly discussed in Sections 88.4–88.7. The NMSSM superpotential consists only of trilinear terms
whose coefficients are dimensionless. There are some advantages to extending the NMSSM further
to the ‘USSM’ [107] by adding a new broken U(1) gauge symmetry [279], under which the singlet
field is charged.
Alternatively, one can consider a generalized version of the NMSSM (called the GNMSSM in
Ref. [209]), where all possible renormalizable terms in the superpotential are allowed, which yield
new supersymmetric mass terms (analogous to the µ term of the MSSM). A discussion of theparameters of the GNMSSM can be found in Ref. [80]. Although the GNMSSM does not solve theµ-problem, it does exhibit regions of parameter space in which the degree of fine-tuning is relaxed,
as discussed in Sec. 88.7.1.
The generation of the µ-term may be connected with the solution to the strong CP problem [280].
Models of this type, which include new gauge singlet fields that are charged under the Peccei-Quinn(PQ) symmetry [281], were first proposed in Ref. [276]. The breaking of the PQ symmetry isthus intimately tied to SUSY breaking, while naturally yielding a value of µ that is of order theelectroweak symmetry breaking scale [282].
All supersymmetric models discussed so far in this review possess self-conjugate fermions—
the Majorana gluinos and neutralinos. However, it is possible to add additional chiral superfields
in the adjoint representation. The spin-1/2 components of these new superfields can pair up
with the gauginos to form Dirac gauginos [283, 284]. Such states appear in models of so-calledsupersoft SUSY breaking [285], in some generalized GMSB models [286], and in R-symmetric
SUSY models [287, 288]. Such approaches often lead to improved naturalness and/or significantly
relaxed flavor constraints. The implications of models of Dirac gauginos on the observed Higgs
boson mass and its properties are addressed in Ref. [289].
For completeness, we briefly note other MSSM extensions considered in the literature. These
include an enlarged electroweak gauge group beyond SU(2)×U(1) [290]; the addition of new Higgs
supermultiplets beyond the doublets and singlets of the MSSM/NMSSM [291]; and/or the additionof new (possibly exotic) matter supermultiplets [210,292,293] such as vector-like fermions and theirsuperpartners.
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