### 88.4.2 The squarks and sleptons

For a given Dirac fermion $f$, there are two superpartners, $\tilde{f}_L$ and $\tilde{f}_R$, where the $L$ and $R$ subscripts simply identify the scalar partners that are related by SUSY to the left-handed and right-handed fermions, $f_{L,R} \equiv \frac{1}{2}(1 \mp \gamma_5)f$, respectively. (There is no $\tilde{\nu}_R$ in the MSSM.) However, $\tilde{f}_L-\tilde{f}_R$ mixing is possible, in which case $\tilde{f}_L$ and $\tilde{f}_R$ are not mass eigenstates. For three generations of squarks, one must diagonalize $6 \times 6$ matrices corresponding to the basis $(\tilde{q}_{iL}, \tilde{q}_{iR})$, 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],

$$ \mathcal{M}^2 = \begin{pmatrix} M_Q^2 + m_q^2 + L_q & m_q X_q^* \\ m_q X_q & M_R^2 + m_R^2 + R_q \end{pmatrix}, \quad (88.11) $$

where

$$ X_q \equiv A_q - \mu^*(\cot \beta)^{2T_{3q}}, \quad (88.12) $$

and $T_{3q} = \frac{1}{2}[-\frac{1}{2}]$ for $q = t$ [b]. The diagonal squared-masses are governed by soft-SUSY-breaking squared-masses $M_Q^2$ and $M_R^2 \equiv M_U^2[M_D^2]$ for $q = t$ [b], the corresponding quark masses $m_t$ [$m_b$] and the electroweak correction terms:

$$ \begin{aligned} L_q &\equiv (T_{3q} - e_q \sin^2 \theta_W) m_Z^2 \cos 2\beta, \\ R_q &\equiv e_q \sin^2 \theta_W m_Z^2 \cos 2\beta, \end{aligned} \quad (88.13) $$

where $e_q = \frac{2}{3}[-\frac{1}{3}]$ for $q = t$ [b]. The off-diagonal squark squared-masses are proportional to the corresponding quark masses and depend on $\tan \beta$, the soft-SUSY-breaking $A$-parameters and the higgsino mass parameter $\mu$. 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 and second generation $\tilde{q}_L-\tilde{q}_R$ mixing is smaller than that of the third generation where mixing can be enhanced by factors of $m_t$ and $m_b \tan \beta$.

In the case of third generation $\tilde{q}_L-\tilde{q}_R$ mixing, the squark mass eigenstates (usually denoted by $\tilde{q}_1$ and $\tilde{q}_2$, with $m_{\tilde{q}_1} < m_{\tilde{q}_2}$) are determined by diagonalizing the $2 \times 2$ matrix $\mathcal{M}^2$ given by Eq. (88.11). The corresponding squared-masses and mixing angle are given by [111]:

$$ \begin{aligned} m_{\tilde{q}_{1,2}}^2 &= \frac{1}{2} \left[ \text{Tr} \mathcal{M}^2 \mp \sqrt{(\text{Tr} \mathcal{M}^2)^2 - 4 \det \mathcal{M}^2} \right], \\ \sin 2\theta_{\tilde{q}} &= \frac{2m_q |X_q|}{m_{\tilde{q}_2}^2 - m_{\tilde{q}_1}^2}. \end{aligned} \quad (88.14) $$

The one-generation results above also apply to the charged sleptons, with the obvious substitutions: $q \to \ell$ with $T_{3\ell} = -\frac{1}{2}$ and $e_\ell = -1$, and the replacement of the SUSY-breaking parameters: $M_Q^2 \to M_L^2$, $M_U^2 \to M_D^2$, and $A_q \to A_\tau$. For the neutral sleptons, $\tilde{\nu}_R$ does not exist in the MSSM, so $\tilde{\nu}_L$ is a mass eigenstate.

In the case of three generations, the SUSY-breaking scalar-squared masses $[M_Q^2, M_U^2, M_D^2, M_L^2, M_E^2]$ and the $A$-parameters $[A_U, A_D, A_E]$ are now $3 \times 3$ matrices as noted in Sec. 88.3.3. The diagonalization of the $6 \times 6$ squark mass matrices yields $\tilde{f}_{iL}-\tilde{f}_{jR}$ mixing. In practice, since the $\tilde{f}_L-\tilde{f}_R$ 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 and third 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 CP-violating 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 potentially complex parameter ($m_{12}^2$) of the

MSSM Higgs potential can be chosen real and positive by rephasing the Higgs fields, in which case $\tan \beta$ is a real positive parameter. Consequently, the physical neutral Higgs scalars are CP-eigenstates (at tree-level). The MSSM Higgs sector contains five physical spin-zero particles: a charged Higgs boson pair ($H^\pm$), two CP-even neutral Higgs bosons (denoted by $h^0$ and $H^0$ where $m_h < m_H$), and one CP-odd neutral Higgs boson ($A^0$). The discovery of a SM-like Higgs boson at the LHC with a mass of 125 GeV (see Sec. 11) strongly suggests that this state should be identified with $h^0$, 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 scalar states exist that are dominantly singlet, then they are weakly coupled to SM gauge bosons and fermions through their small mixing with the MSSM Higgs scalars. Consequently, it is possible that one (or both) of the singlet-dominated states is considerably lighter than the Higgs boson that was observed at the LHC.

### 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 two sources: (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: $\tan \beta$ [defined in Eq. (88.3)] and one Higgs mass usually taken to be $m_A$. For example, the tree-level squared mass of the charged Higgs boson is given by

$$ m_{H^\pm}^2 = m_A^2 + m_W^2, \quad (88.15) $$

where $m_A^2 = m_{H_d}^2 + m_{H_u}^2 + 2|\mu|^2$ [cf. Eq. (88.5)] and

$$ H^\pm = H_d^\pm \sin \beta + H_u^\pm \cos \beta, \quad A = \sqrt{2} (\operatorname{Im} H_d^0 \sin \beta + \operatorname{Im} H_u^0 \cos \beta). \quad (88.16) $$

The CP-even scalar mass eigenstate fields $h$ and $H$ are identified by diagonalizing the $2 \times 2$ squared-mass matrix

$$ \mathcal{M}^2 = \begin{pmatrix} m_A^2 \sin^2 \beta + m_Z^2 \cos^2 \beta & -(m_A^2 + m_Z^2) \sin \beta \cos \beta \\ -(m_A^2 + m_Z^2) \sin \beta \cos \beta & m_A^2 \cos^2 \beta + m_Z^2 \sin^2 \beta \end{pmatrix}. \quad (88.17) $$

In particular,

$$ h = -(\sqrt{2} \operatorname{Re} H_d^0 - v_d) \sin \alpha + (\sqrt{2} \operatorname{Re} H_u^0 - v_u) \cos \alpha, \quad (88.18) $$

$$ H = (\sqrt{2} \operatorname{Re} H_d^0 - v_d) \cos \alpha + (\sqrt{2} \operatorname{Re} H_u^0 - v_u) \sin \alpha, \quad (88.19) $$

with corresponding tree-level squared masses,

$$ m_{H,h}^2 = \frac{1}{2} \left( m_A^2 + m_Z^2 \pm \sqrt{(m_A^2 + m_Z^2)^2 - 4m_Z^2 m_A^2 \cos^2 2\beta} \right), \quad (88.20) $$

and mixing angle $\alpha$ given by

$$ \cos \alpha = \sqrt{\frac{m_A^2 \sin^2 \beta + m_Z^2 \cos^2 \beta - m_h^2}{m_H^2 - m_h^2}}, \quad (88.21) $$

in a convention where $|\alpha| \le \pi/2$. However, because the off-diagonal elements of $\mathcal{M}^2$ are negative, it follows that $-\pi/2 \le$
$\alpha \le 0$ [104]. In light of Eq. (88.20), the tree-level mass of the lighter CP-even Higgs boson is bounded [29,30],

$$m_h \le m_Z |\cos 2\beta| \le m_Z. \qquad (88.22)$$

This bound can be substantially modified when radiative corrections are included, as discussed in Sec. 88.5.2.

In the NMSSM, we set $\mu = 0$ in Eq. 88.1 and then add two additional terms to the superpotential,

$$W_{\text{NMSSM}} \supset \lambda \hat{H}_u \hat{H}_d \hat{S} + \frac{1}{3} \kappa \hat{S}^3, \qquad (88.23)$$

where $\hat{S}$ is a singlet Higgs superfield. In the NMSSM as defined here, all terms in $W_{\text{NMSSM}}$ are cubic in the superfields due to the presence of a discrete $\mathbb{Z}_3$ symmetry. An effective $\mu$-term is generated, $\mu_{\text{eff}} = \lambda \langle S \rangle$, where $\langle S \rangle$ is the vacuum expectation value of the scalar field component of $\hat{S}$. Moreover, due to the term proportional to $\lambda$ 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],

$$m_h^2 \le m_Z^2 \cos^2 2\beta + \frac{1}{2} \lambda^2 v^2 \sin^2 2\beta, \qquad (88.24)$$

where $v \equiv (v_u^2 + v_d^2)^{1/2} = 246$ GeV. By requiring that $\lambda$ remain finite after renormalization-group evolution up to the Planck scale, one finds that $\lambda$ is constrained to lie below about 0.7–0.8 at the electroweak scale [36] (although larger values of $\lambda$ have also been considered in Ref. [116]).

The tree-level Higgs couplings to gauge bosons and the Higgs boson self-couplings are governed by the electroweak gauge couplings and the parameter $\cos(\beta - \alpha)$. Explicitly,

$$\cos(\beta - \alpha) = \frac{m_Z^2 \sin 2\beta \cos 2\beta}{\sqrt{(m_H^2 - m_h^2)(m_H^2 - m_Z^2 \cos^2 2\beta)}}. \qquad (88.25)$$

Note that $\cos(\beta - \alpha) \to 0$ in the limit of $m_H \gg m_h, m_Z$. In this decoupling limit [117], the properties of $h$ 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 with the observed Higgs boson then $H, A$ and $H^\pm$ 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 the superpotential given in Eq. (88.1). The corresponding Higgs-fermion Yukawa couplings can be expressed in terms of the fermion masses and the separate parameters $\cos(\beta - \alpha)$ and $\tan \beta$. In particular, the Higgs sector of the MSSM is a Type-II two-Higgs doublet model [121], in which one Higgs doublet ($H_d$) couples exclusively to the right-handed down-type quark (or lepton) fields and the second Higgs doublet ($H_u$) couples exclusively to the right-handed up-type quark fields. Consequently, the diagonalization of the fermion mass matrices simultaneously diagonalizes the matrix of Yukawa couplings, resulting in flavor-diagonal tree-level couplings of the neutral Higgs bosons $h^0, H^0$ and $A^0$ to quark and lepton pairs. One can again check that in the decoupling limit where $\cos(\beta - \alpha) \to 0$, 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 supersymmetric model 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 most easily 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)] are small in comparison to the geometric mean of the two top-squark squared-masses, $M_S^2 \equiv M_{t_1} M_{t_2}$. In this case (assuming $m_A > m_Z$), the predicted upper bound for

$m_h$ is approximately given by [123]

$$m_h^2 \lesssim m_Z^2 \cos^2 2\beta + \frac{3g^2 m_t^4}{8\pi^2 m_W^2} \left[ \ln \left( \frac{M_S^2}{m_t^2} \right) + \frac{X_t^2}{M_S^2} \left( 1 - \frac{X_t^2}{12M_S^2} \right) \right], \qquad (88.26)$$

where $X_t \equiv A_t - \mu \cot \beta$ [cf. Eq. (88.12)] is proportional to the off-diagonal entry of the top-squark squared-mass matrix (where for simplicity, $A_t$ and $\mu$ are taken to be real). The Higgs mass upper limit specified by Eq. (88.26) is saturated when $\tan \beta$ is large (i.e., $\cos^2 2\beta \sim 1$) and $X_t = \sqrt{6} M_S$, which defines the so-called maximal mixing scenario.

In applying the radiatively corrected MSSM Higgs sector to the analysis of LHC Higgs data, the authors of Refs. [124, 125] suggested that a reasonable approximation would consist of re-taining the leading corrections employed in deriving Eq. (88.26), while discarding all other subleading terms. This procedure was implemented by simply replacing $M_{22}^2 \to M_{22}^2 + \Delta M_{22}^2$ in the 22 element of the CP-even Higgs squared-mass matrix given in Eq. (88.17), since $\Delta M_{22}^2$ contains the leading contributions that govern the Higgs mass radiative corrections. One can now re-diagonalize the CP-even Higgs squared-mass matrix and determine $\Delta M_{22}^2$ in terms of the parameters $m_A, \tan \beta$ and the measured Higgs mass (e.g., $m_h \simeq 125$ 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 results that 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 value of $m_h$. 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 $m_h$ shown in Fig. 88.1, as a function of $X_t$ (assumed for simplicity to be real).

In the NMSSM with $m_h \simeq 125$ GeV, the dominant radiative correction to Eq. (88.24) is the same as the one given in Eq. (88.26). However, in contrast to the MSSM, one does not need as large a boost from radiative corrections to achieve a Higgs mass of 125 GeV in certain regimes of the NMSSM parameter space (e.g., $\tan \beta \sim 2$ and $\lambda \sim 0.7$ [127]).

## 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 $L_e, L_\mu$, and $L_\tau$; (ii) unsuppressed flavor-changing neutral currents (FCNCs); and (iii) new sources of CP violation that are inconsistent with the experimental bounds.

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 [128]. This phenomenon is most easily understood in a scenario where $m_A \ll M_S$ (i.e., all five physical Higgs states are significantly lighter than the SUSY breaking scale). In this case, one can integrate out the heavy 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 (both CP-conserving and CP-violating) and Higgs-fermion interactions (beyond those of Type-II) that are consistent with electroweak gauge invariance [129].

For example, the MSSM contains new sources of CP violation [130]. Indeed, for TeV-scale sfermion and gaugino masses, some combinations of the complex phases of the gaugino-mass parameters, the A-parameters, and $\mu$ must be less than about $10^{-2}-10^{-3}$ to avoid generating electric dipole moments for the
<figure><img src="image_363.png" /><figcaption>Figure 88.1: The lighter CP-even Higgs mass in the MSSM as a function of a common SUSY mass parameter $M_S$ and of the stop mixing parameter $X_t$ (normalized to $M_S$). Figure taken from Ref. [73].</figcaption></figure>

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 of the 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 $M_{GUT}$, close to the Planck scale [7, 70, 139–141],

$$g_s(M_{GUT}) = g(M_{GUT}) = \sqrt{\frac{5}{3}} g'(M_{GUT}), \quad (88.27)$$

suggests that the high-energy structure of the theory may be considerably simpler than its low-energy realization.<fnref n="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 supersymmetric particle spectrum is then a function of a relatively small number of input parameters. This is accomplished by imposing a simple structure on the soft SUSY-breaking parameters at a common high-energy scale $M_X$ (typically chosen to be the Planck scale, $M_P$, the grand unified theory scale, $M_{GUT}$, or the messenger scale, $M_{mess}$). 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 theoretical conditions 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 such models, electroweak symmetry breaking is generated radiatively, and the resulting electroweak symmetry-breaking scale is intimately tied to the scale of low-energy SUSY breaking.

### 88.6.1 Gaugino mass relations

One prediction of many supersymmetric grand unified models is the unification of the (tree-level) gaugino mass parameters<fnref n="3" /> at some high-energy scale, $M_X$,

$$M_1(M_X) = M_2(M_X) = M_3(M_X) = m_{1/2}. \quad (88.28)$$

Due to renormalization group running, in the one-loop approximation the effective low-energy gaugino mass parameters (at the electroweak scale) are related,

$$M_3 = (g_s^2/g^2)M_2 \approx 3.5M_2, \quad M_1 = (5g'^2/3g^2)M_2 \approx 0.5M_2. \quad (88.29)$$

Eq. (88.29) can arise more generally in gauge-mediated SUSY-breaking models where the gaugino masses are generated at the messenger scale $M_{mess}$ (which typically lies significantly below the unification scale where the gauge couplings unify). In this case, the gaugino mass parameters are proportional 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, $\mu$, and $\tan\beta$. 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 $|\mu| \gg |M_1| \gtrsim m_Z$, 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 in many phenomenological studies, a truly model-independent approach would take the gaugino mass parameters $M_1, M_2$, and $M_3$ 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

<footnote n="2">²Generically, 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 $g'$. 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).</footnote>

<footnote n="3">³Non-universal gaugino mass parameters can also be a viable option in grand unified models with non-minimal gauge kinetic functions [146].</footnote>
do not satisfy Eq. (88.29). For example, the gaugino masses in AMSB models arise entirely from a model-independent contribution derived from the super-conformal anomaly [51, 149]. In this case, Eq. (88.29) is replaced (in the one-loop approximation) by:

$$M_i \approx \frac{b_i g_i^2}{16\pi^2} m_{3/2}, \qquad (88.30)$$

where $m_{3/2}$ is the gravitino mass and the $b_i$ are the coefficients of the MSSM gauge beta-functions corresponding to the corresponding U(1), SU(2), and SU(3) gauge groups, ($b_1, b_2, b_3$) = ($\frac{33}{5}, 1, -3$). Eq. (88.30) yields $M_1 \simeq 2.8 M_2$ and $M_3 \simeq -8.3 M_2$, which implies that the lightest chargino pair and neutralino comprise a nearly mass-degenerate triplet of winos, $\tilde{W}^{\pm}, \tilde{W}^0$ (cf. Table 1), over most of the MSSM parameter space. For example, if $|\mu| \gg m_Z, |M_2|$, then Eq. (88.30) implies that $M_{\tilde{\chi}^{\pm}} \simeq M_{\tilde{\chi}_1^0} \simeq M_2$ [150]. Alternatively, one can construct an AMSB model where $|\mu|, m_Z \ll M_2$, which 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 can occur 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 [48]. As a result, the soft supersymmetry-breaking parameters at the high-energy scale $M_X$ take a particularly simple form in which the scalar squared-masses and the A-parameters are flavor-diagonal and universal [46]:

$$ \begin{aligned} M_Q^2(M_X) &= M_U^2(M_X) = M_D^2(M_X) = m_0^2 \mathbf{1}, \\ M_L^2(M_X) &= M_E^2(M_X) = m_0^2 \mathbf{1}, \\ m_1^2(M_X) &= m_2^2(M_X) = m_0^2, \\ A_U(M_X) &= A_D(M_X) = A_E(M_X) = A_0 \mathbf{1}, \end{aligned} \qquad (88.31) $$

where $\mathbf{1}$ is a $3 \times 3$ 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 $M_Q^2, M_U^2$, and $M_D^2$ in Eq. (88.11). Through the renormalization group running with boundary conditions specified in Eq. (88.29) and Eq. (88.31), one can show that the low-energy values of $M_Q^2, M_U^2$, and $M_D^2$ depend primarily on $m_0^2$ and $m_{1/2}^2$. A number of useful 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 $m_0, m_{1/2}, A_0$, the Planck-scale values for $\mu$ and B-parameters (denoted by $\mu_0$ and $B_0$), and the gravitino mass $m_{3/2}$. Without additional model assumptions, $m_{3/2}$ is independent of the parameters that govern the mass spectrum of the superpartners of the SM [46]. In principle, $A_0, B_0, \mu_0$, and $m_{3/2}$ 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 values of the mSUGRA parameters, which then fixes all the parameters of the low-energy MSSM. In

particular, the two Higgs vacuum expectation values (or equivalently, $m_Z$ and $\tan\beta$) 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 $\mu_0$ and $B_0$ in terms of $m_Z$ and $\tan\beta$ [the sign of $\mu_0$, denoted sgn($\mu_0$) below, is not fixed in this process]. In this case, the MSSM spectrum and its interaction strengths are fixed by five parameters:

$$m_0, A_0, m_{1/2}, \tan\beta, \text{and sgn}(\mu_0), \qquad (88.32)$$

and an independent gravitino mass $m_{3/2}$ (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 $B_0 = A_0 - m_0$ and $m_{3/2} = m_0$ comprise the original mSUGRA proposal [44, 48, 159].

One can also relax the universality of scalar masses by decoupling the squared-masses of the 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 ($m_{H_d}^2$ and $m_{H_u}^2$) are set equal (NUHM1 [160]) or taken to be independent (NUHM2 [161]) at the high energy scale $M_X$. 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]. 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, then the gaugino mass parameters, diagonal scalar squared-mass parameters, and the SUSY-breaking trilinear scalar interaction terms (proportional to $\lambda_f A_F$) 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, $m_0^2$, is added to the AMSB expressions for the diagonal scalar squared-masses [153]. Thus, the mAMSB spectrum and its interaction strengths are determined by four parameters, $m_0^2, m_{3/2}, \tan\beta$ and sgn($\mu_0$).

The mAMSB scenario appears to be ruled out based on the observed value of the Higgs boson mass, assuming an upper limit on $M_S$ of a few TeV, since the mAMSB constraint on $A_F$ implies that the maximal mixing scenario cannot be achieved [cf. Eq. (88.26)]. Indeed, under the stated assumptions, 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 $\lambda_f A_F$ 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 gauge-mediated SUSY breaking (GMSB) because the supersymmetry breaking is communicated to the sector of MSSM fields via gauge interactions [56, 58]. In GMSB models, the mass scale of the messenger 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, $\Lambda$, that determines all low-energy 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 the resulting superpartner masses be of order 1 TeV, one must have $\Lambda \sim O(100 \text{ TeV})$. The origin of the $\mu$ 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-pa-
rameters, it is difficult to boost the contributions of the radiative corrections in Eq. (88.26) to obtain a Higgs mass as large as 125 GeV. However, this conflict can 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, the next-to-lightest supersymmetric particle (NLSP) now plays a crucial role in the phenomenology of supersymmetric particle production and decays. Note that unlike the LSP, the NLSP can be charged. In GMSB models, the most likely candidates for the NLSP are $\tilde{\chi}_1^0$ and $\tilde{\tau}_R^\pm$. The NLSP will decay into its superpartner plus a gravitino (e.g., $\tilde{\chi}_1^0 \to \gamma\tilde{G}$, $\tilde{\chi}_1^0 \to Z\tilde{G}$, $\tilde{\chi}_1^0 \to h^0\tilde{G}$ or $\tilde{\tau}_1^\pm \to \tau^\pm\tilde{G}$), with 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 co-NLSP case, all three right-handed sleptons are close enough in mass and thus can each play the role of the NLSP.

Different choices for the identity of the NLSP and its decay rate lead to a variety of distinctive supersymmetric phenomenologies [58,169]. For example, a long-lived $\tilde{\chi}_1^0$-NLSP that decays outside collider detectors leads to supersymmetric decay chains with missing energy in association with leptons and/or hadronic jets (this case is indistinguishable from the standard phenomenology of the $\tilde{\chi}_1^0$-LSP). On the other hand, if $\tilde{\chi}_1^0 \to \gamma\tilde{G}$ is the dominant decay mode, and the decay occurs inside the detector, then nearly all supersymmetric particle decay chains would produce a photon. In contrast, in the case of a $\tilde{\tau}_1^\pm$-NLSP, the $\tilde{\tau}_1^\pm$ would either be long-lived or would decay inside the detector into a $\tau$-lepton plus missing energy.

A number of attempts have been made to address the origins of the $\mu$ 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 value of the additional singlet Higgs superfield can be used to generate effective $\mu$ and $B$-parameters [172]. Such models provide an alternative GMSB framework for achieving a Higgs mass of 125 GeV, while still 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 this Section, the phenomenological MSSM (pMSSM) has been introduced [173].

The pMSSM is governed by 19 independent real supersymmetric parameters: the three gaugino mass parameters $M_1, M_2$ and $M_3$, the Higgs sector parameters $m_A$ and $\tan\beta$, the Higgsino mass parameter $\mu$, five sfermion squared-mass parameters for the degenerate first and second generations ($M_Q^2, M_U^2, M_D^2, M_L^2$ and $M_E^2$), the five corresponding sfermion squared-mass parameters for the third generation, and three third-generation $A$-parameters ($A_t, A_b$ and $A_\tau$). The first and second generation $A$-parameters are typically neglected in pMSSM studies, as their phenomenological consequences are negligible in most applications. One counterexample arises when considering the $A_\mu$ dependence of the anomalous magnetic moment of the muon, which can be as significant as other contributions due to superpartner mediated radiative corrections [174]. Since its initial proposal, the pMSSM approach has been extended to include a 20th parameter, $A_\mu$ [175]. It also has been further 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 is defined 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] are defined 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 supersymmetric particles (often two or three). Other supersymmetric particles are assumed to play no role (this may happen by virtue of them being too heavy to be produced). Experimental bounds from non-observation of a signal are usually presented in terms of the physical masses of the supersymmetric particles 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 supersymmetric particle masses.

For example, consider a search for hadronic jets plus missing transverse momentum. One can match 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 produces the missing transverse momentum, i.e. $q\tilde{q}\tilde{q} \to (q\tilde{\chi}_1^0)(q\tilde{\chi}_1^0)$. Excluded cross-sections resulting from the non-observation of a signal (which in this case could consist of some specified minimum value of 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 to present 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 other than those assumed. For example, in the $q\tilde{q}\tilde{q} \to q\tilde{\chi}_1^0q\tilde{\chi}_1^0$ 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 colored supersymmetric 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 which lighter squarks and gluinos below 1 TeV cannot be
definitely ruled out. Under the assumption of 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 and chargino 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 of supersymmetric 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 this context, the weak scale soft supersymmetry-breaking masses must be generally of the order of 1 TeV or below [183]. This requirement is most easily seen in the determination of $m_Z$ by the scalar potential minimum condition. In light of Eq. (88.5), to avoid the fine-tuning of MSSM parameters, the soft SUSY-breaking squared-masses $m_{H_d}^2$ and $m_{H_u}^2$ and the higgsino squared-mass $|\mu|^2$ should all be roughly of $\mathcal{O}(m_Z^2)$. Many authors have proposed quantitative measures of fine-tuning [183–188]. One of the simplest measures is the one advocated by Barbieri and Giudice [183] (which was also introduced previously in Ref. [184]),

$$ \Delta_i \equiv \left| \frac{\partial \ln m_Z^2}{\partial \ln p_i} \right|, \quad \Delta \equiv \max \Delta_i, \qquad (88.33) $$

where the $p_i$ are the MSSM parameters at the high-energy scale $M_X$, which are set by the fundamental SUSY-breaking dynamics. The theory is more fine-tuned as $\Delta$ becomes larger. However, different measures of fine-tuning yield quantitatively different results; in particular, calculating yield fine-tuning based on the high-scale parameters [as defined in Eq. (88.33)] yields a difference by a factor $\sim 10$ 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 $p_i$ are parameters of the model at the energy scale $M_X$ 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, $\Delta$ is sensitive to all the SUSY-breaking parameters of the model (see e.g. Ref. [191]). The computation of $\Delta$ is often based on Eq. (88.5), which is a tree-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 a prior probability distribution proportional to $1/\Delta$ [193] so that fine-tuning is balanced against the fit 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 fine-tuning resulting from a Jacobian factor when transforming to other variables.<fnref n="4" /> By comparing the results 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 the interpretation of the fine-tuning measure.

As anticipated, there is a tension between the present experimental lower limits on the masses of 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 ($m_h \simeq 125$ GeV), which is not far from the MSSM upper bound ($m_h \lesssim 135$ GeV) [which depends on the top-squark mass and mixing as noted in Sec. 88.5.2]. If $M_{\text{SUSY}}$ characterizes the scale of supersymmetric particle masses, then one would crudely expect $\Delta \sim M_{\text{SUSY}}^2/m_Z^2$. For example, if $M_{\text{SUSY}} \sim 1$ TeV then one expects a $\Delta^{-1} \sim 1\%$ fine-tuning of the MSSM parameters to achieve the observed value of $m_Z$. This separation of the electroweak symmetry-breaking and SUSY-breaking scales is an example of the little hierarchy problem [198, 199].

The fine-tuning parameter $\Delta$ can depend quite sensitively on the structure of the SUSY-breaking dynamics, such as the value of $M_X$ and relations among SUSY-breaking parameters in the fundamental high energy theory [200]. For example, in so-called focus point SUSY models [187, 201], all 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 $\Delta$ due to the observed Higgs mass of 125 GeV, one can achieve moderate values of $\Delta$ 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 constrained by the LHC data. Hence, in the absence of any relation between third generation squarks and those of the first two generations, the naturalness constraints due to present LHC data can be considerably 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 are naturally suppressed [203]. Other MSSM mass spectra that are compatible with moderate fine tuning 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 for the 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 other colored supersymmetric states.

Among the uncolored superpartners, the higgsinos are typically the most impacted by the naturalness constraints. Eq. (88.5) suggests that the masses of the two neutral higgsinos and charged higgsino pair (which are governed by $|\mu|$) should not be significantly larger than $m_Z$ to avoid an unnatural fine-tuning of the supersymmetric parameters, which would imply the existence of 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 conclusion that $\mu \sim \mathcal{O}(m_Z)$ if additional correlations among the SUSY breaking mass parameters and $\mu$ are present. 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 $|\mu| > 1$ TeV can be achieved while naturally maintaining the observed value of $m_Z$.

Finally, one can also consider extensions of the MSSM in which the degree of fine-tuning is relaxed. 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 $m_h^2$ proportional to the parameter $\lambda^2$. This means that one does not have to rely on a large contribution from radiative 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], and subsequently treated in more detail in Refs. [116, 209]. Naturalness can also be relaxed in extended su-

<footnote n="4">⁴For example, one may consider the parameters $\mu$ and $m_{12}^2$ to be more fundamental than $\tan \beta$ and $M_Z$. In this case, one would choose a flat prior probability distribution in $\mu$ and $m_{12}^2$ rather than in $\tan \beta$ and $M_Z$ [186, 194]. The Jacobian factor is then obtained from Eq. (88.4) and Eq. (88.5).</footnote>
persymmetric 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 origin of electroweak symmetry breaking, since different measures of fine-tuning noted above can lead to different assessments [189, 190]. Moreover, the maximal value of $\Delta$ that determines whether weak-scale SUSY is a fine-tuned model (should it be $\Delta \sim 10$? 100? 1000?) is ultimately subjective. 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 a future collider facility (with an energy reach beyond the LHC [214]), it would be viewed as a spectacularly 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 $\Delta$ 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 their properties, 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 of various supersymmetric particles into dark matter particles and SM particles. The resulting relic density 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 seven parameter version of the pMSSM). However, subsequent tweaks to the supersymmetric model that yield an unstable LSP, such as the introduction of R-parity violating effects, can mean that the relic 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 of supersymmetric particles can contribute to the muon anomalous magnetic moment, $a_\mu \equiv \frac{1}{2}(g-2)_\mu$, as reviewed in Ref. [216]. The SM prediction for $a_\mu$, which employs dispersion relations and low energy $e^+e^-$ scattering data to determine the hadronic corrections [217], exhibits a deviation of $5\sigma$ from the experimentally observed value [218]. The deviation of Ref. [217] from the measured value of $a_\mu$ [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 the context 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 $4\sigma$ 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 $a_\mu$ from the corresponding SM prediction still needs to be clarified.

The precision of the measured value of $a_\mu$ is not sensitive to the experimental error associated with the measured value of the fine structure constant, $\alpha$. In contrast, the comparison of the SM prediction with the experimental measurement of the anomalous

magnetic moment of the electron, $a_e$, depends critically on the value of $\alpha$. Using the experimentally determined value of $\alpha$ given in Ref. [223] yields a SM prediction for $a_e$ that is $2.4\sigma$ above its measured value [224]. However, this previous determination of $\alpha$ is in tension at the $5\sigma$ level with a more recent measurement of the fine structure constant [225]. The latter yields a SM prediction for $a_e$ that is $1.6\sigma$ below its measured value [225].

Measurements of the fine structure constant, $a_e$ and $a_\mu$ jointly constrain the pMSSM parameter space [226] due to shifts originating from supersymmetric loop effects. In particular, if the supersymmetric interpretation of the deviation in the measured value of $a_\mu$ from its SM prediction is combined with the experimental limits on the electron electric dipole moment [135], then the resulting upper bounds on MSSM CP-violating phases are even more constraining [227] than previously 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 $b \to s\gamma$ is a sensitive probe of the virtual effects of new physics beyond the SM. The experimental measurements of $B \to X_s + \gamma$ [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, the absence of any significant deviation places useful constraints on the MSSM parameter space [231].

The rare decays $B_s \to \mu^+\mu^-$ and $B_d \to \mu^+\mu^-$ are especially sensitive to supersymmetric loop effects, with some loop contributions scaling as $\tan^6\beta$ when $\tan\beta \gg 1$ [232]. At present, a combination [233] of the measurements of these rare decay modes [234] is in slight tension at the $1.6\sigma$ 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 $b \to s\mu^+\mu^-$ transitions, although the level of tension depends upon the theoretical treatment of the SM analysis. In a certain angular distribution parameter (denoted by $P'_5$) extracted from $B^0 \to K^0\mu^+\mu^-$ decays, the tension is around the $4\sigma$ level [236]. An even larger discrepancy is observed in a combination of angular distributions and rates derived from $B^\pm \to K^\pm\mu^+\mu^-$ and $B^0 \to K^0\mu^+\mu^-$ [237]. Finally, there is a $3.6\sigma$ deviation in the branching ratio of $B_s \to \phi\mu^+\mu^-$ for di-muon invariant mass squared values between $1.1 \text{ GeV}^2$ and $6.0 \text{ GeV}^2$ [238]. Finally, a recent measurement of $B^+ \to K^+\nu\bar{\nu}$ by the Belle II Collaboration [239] obtained a branching fraction that is roughly four times larger than the SM prediction [240], corresponding to $2.7\sigma$ above the SM expectation. However, it is unlikely that this result can be attributed to new contributions from the MSSM [241], as the latter are expected to be negligible in comparison with the SM.

The decays $B^\pm \to \tau^\pm\nu_\tau$ and $\bar{B} \to D(*)\tau^-\bar{\nu}_\tau$ 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 $B^\pm \to \tau^\pm\nu_\tau$ are currently consistent with SM expectations [242]. The BaBar Collaboration measured values of the rates for $\bar{B} \to D\tau^-\bar{\nu}_\tau$ and $\bar{B} \to D^*\tau^-\bar{\nu}_\tau$ [243] which exhibited a combined $3.4\sigma$ discrepancy from the SM predictions, which was also not compatible with the Type-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 $\bar{B} \to D\tau^-\bar{\nu}_\tau$ and $\bar{B} \to D^*\tau^-\bar{\nu}_\tau$ decay rates relative to the corresponding SM values has a significance 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 existence of new physics beyond the SM. Moreover, the absence of evidence for sizable deviations in other B-physics 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 $7\sigma$ above the SM prediction. Supersymmetric models have been shown to be able to accommodate the measured value by the CDF Collaboration at the $2\sigma$ level [248]. However, other measurements, including a more recent determination by ATLAS Collaboration [249], are consistent with SM predictions for $m_W$ (see Sec. 10 for further details).

## 88.8 Massive neutrinos in weak-scale SUSY

In 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 model of 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 supersymmetric models will now be considered.

### 88.8.1 The supersymmetric seesaw

Neutrino masses can be incorporated into the SM by introducing $SU(3) \times SU(2) \times U(1)$ singlet right-handed neutrinos ($\nu_R$) 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 $\nu_R$. The Higgs vacuum expectation value then induces an off-diagonal $\nu_L - \nu_R$ 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 neutrino masses [255,256] by promoting the right-handed neutrino field to a superfield $\tilde{N}^c = (\tilde{\nu}_R; \nu_R)$. Integrating out the heavy right-handed neutrino supermultiplet yields a new term in the superpotential [cf. Eq. (88.1)] of the form

$$W_{\text{seesaw}} = \frac{f}{M_R} (\hat{H}_U \hat{L}) (\hat{H}_U \hat{L}), \quad (88.34)$$

where $M_R$ 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 the sneutrino 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]. The right-handed sneutrino that resides in $\hat{L}$ also provides an intriguing dark matter candidate [259].

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]. Each new 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 and their superpartners], consider the effect of adding the following new terms:

$$W_{\text{RPV}} = (\lambda_L)_{pmn} \hat{L}_p \hat{L}_m \hat{E}_n^c + (\lambda'_L)_{pmn} \hat{L}_p \hat{Q}_m \hat{D}_n^c \\ + (\lambda_B)_{pmn} \hat{U}_p^c \hat{D}_m^c \hat{D}_n^c + (\mu_L)_p \hat{H}_U \hat{L}_p, \quad (88.35)$$

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 $\lambda_B$ violates B, while the other three terms violate L. The L-violating term in Eq. (88.35) proportional to $\mu_L$ is the RPV analog of the $\mu\hat{H}_U\hat{H}_d$ term of the MSSM superpotential, in which the $Y = -1$ Higgs/higgsino supermultiplet $\hat{H}_d$ is replaced by the slepton/lepton supermultiplet $\hat{L}_p$.

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 $(\lambda_L)_{pmn}$, $(\lambda'_L)_{pmn}$, and $(\lambda_B)_{pmn}$ taken one at a time [261, 262]. If more than one coefficient is simultaneously non-zero, then the limits 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 magnitude larger 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 that certain RPV coefficients must be extremely suppressed.

One particularly interesting class of RPV models is one in which B is conserved, but L is violated. 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 $Z_3$ baryon triality symmetry on the low-energy theory [264], in place of the standard $Z_2$ 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, the mixing 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 $Z_2$ R-parity and the $Z_3$ 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 from that of the MSSM [261]. The LSP is no longer stable, which implies that not all supersymmetric decay chains must yield missing-energy events at colliders. A comprehensive examination of the phenomenology 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 in certain 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 pair-produced supersymmetric particles is also modified in RPV models due to new decay chains not present in R-parity-conserving SUSY models [261].

In RPV models with lepton number violation (these include
weak-scale SUSY models with baryon triality mentioned above), both $\Delta L = 1$ and $\Delta L = 2$ 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 $p\bar{p}$ 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 $\mu$ parameter of the MSSM. Although $\mu$ 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 $\mu$-problem must incorporate a symmetry that enforces $\mu = 0$ and a small symmetry-breaking parameter that generates a value of $\mu$ 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 a natural solution to the $\mu$-problem of the MSSM.

In extensions of the MSSM, other compelling solutions to the $\mu$-problem are possible. For example, one can replace $\mu$ by the vacuum expectation value of a new $SU(3) \times SU(2) \times U(1)$ singlet scalar field, as noted below Eq. 88.23. This is the NMSSM, which yields phenomena that were briefly 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 $\mu$ term of the MSSM). A discussion of the parameters of the GNMSSM can be found in Ref. [80]. Although the GNMSSM does not solve the $\mu$-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 $\mu$-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 is thus intimately tied to SUSY breaking, while naturally yielding a value of $\mu$ that is of order the electroweak 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-called supersoft 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 $SU(2) \times U(1)$ [290]; the addition of new Higgs supermultiplets beyond the doublets and singlets of the MSSM/N-MSSM [291]; and/or the addition of new (possibly exotic) matter supermultiplets [210, 292, 293] such as vector-like fermions and their superpartners.

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# 89. Supersymmetry, Part II (Experiment)

Revised August 2023 by M. D’Onofrio (Liverpool U.) and F. Moortgat (CERN; Ghent U.).

## 89.1 Introduction

Supersymmetry (SUSY), a transformation relating fermions to bosons and vice versa [1–9] is one of the most compelling possible extensions of the Standard Model of particle physics (SM).

On theoretical grounds SUSY is motivated as a generalization of space-time symmetries. A low-energy realization of SUSY, i.e., SUSY at the TeV scale, is, however, not a necessary consequence. Instead, low-energy SUSY is motivated by the possible cancellation of quadratic divergences in radiative corrections to the Higgs boson mass [10–15]. Furthermore, it is intriguing that a weakly interacting, (meta)stable supersymmetric particle might make up some or all of the dark matter in the Universe [16–18]. In addition, SUSY predicts that gauge couplings, as measured experimentally at the electroweak scale, unify at an energy scale $O(10^{16})$ GeV (“GUT scale”) near the Planck scale [19–25].

In the minimal supersymmetric extension to the Standard Model, the so called MSSM [11, 26, 27], a supersymmetry transformation relates every chiral fermion and gauge boson in the SM to a supersymmetric partner with half a unit of spin difference, but otherwise with the same properties (such as mass) and quantum numbers. The MSSM Higgs sector contains two doublets, which give mass to the up-type and down-type quarks, respectively. After electroweak symmetry breaking, five Higgs bosons arise, of which two are charged. The supersymmetric partners of chiral fermions are squarks ($\tilde{q}$) and sleptons ($\tilde{l}, \tilde{\nu}$), and the “gauginos” for gauge bosons. The supersymmetric partners of the Higgs doublets are known as “higgsinos.” The weak gauginos and higgsinos mix, giving rise to charged mass eigenstates called “charginos” ($\tilde{\chi}^{\pm}$), and neutral mass eigenstates called “neutrinos” ($\tilde{\chi}^{0}$). These are often collectively referred to as electroweakinos (EWkinos). The SUSY partners of the gluons are known as “gluinos” ($\tilde{g}$). The fact that such particles are not yet observed leads to the conclusion that, if supersymmetry is realized, it is a broken symmetry. A description of SUSY in the form of an effective Lagrangian with only “soft” SUSY breaking terms and SUSY particle masses of the order of TeV maintains the cancellation of quadratic divergences of soft SUSY breaking scalar mass squared parameters.

The phenomenology of SUSY is to a large extent defined by the SUSY breaking mechanism and the SUSY breaking scale. These determine the SUSY particle masses, the mass hierarchy, the field contents of physical particles, and their decay modes. In addition, phenomenology crucially depends on whether the multiplicative quantum number of R-parity [27], $R = (-1)^{3(B-L)+2S}$, where B and L are baryon and lepton numbers and S is the spin, is conserved or violated. If R-parity is conserved, SUSY particles (sparticles), which have odd R-parity, are produced in pairs and the decays of each SUSY particle must involve an odd number of lighter SUSY particles. The lightest SUSY particle (LSP) is then stable and often assumed to be a weakly interacting massive particle (WIMP). If R-parity is violated, new terms $\lambda_{ijk}$, $\lambda'_{ijk}$ and $\lambda''_{ijk}$ appear in the superpotential, where ijk are generation indices; $\lambda$-type couplings appear between lepton superfields only, $\lambda^{\prime\prime}$-type are between quark superfields only, and $\lambda'$-type couplings connect the two. R-parity violation implies lepton and/or baryon number violation. More details of the theoretical framework of SUSY are discussed elsewhere in this volume [28].

The discovery of a Higgs boson with a mass around 125 GeV imposes constraints on SUSY models, which are discussed elsewhere [28, 29]. Low-energy data from flavor physics experiments, high-precision electroweak observables as well as astrophysical data also impose strong constraints on the allowed SUSY parameter space. Recent examples of such data include measurements of the rare B-meson decay $B_s \to \mu^+\mu^-$ [30–32], measurements of the anomalous magnetic moment of the muon [33, 34], and accurate determinations of the cosmological dark matter relic density constraint [35, 36].

Indirect constraints can be more sensitive to higher SUSY mass scales than experiments searching for direct sparticle production

at colliders, but the interpretation of these results is often strongly model dependent. In contrast, direct searches for sparticle production at collider experiments, which are the main topic of this review, are less subject to interpretation ambiguities and therefore they play a crucial role in the search for SUSY. Further discussion on indirect constraints can be found in 88.7.2.

## 89.2 Overview of the experimental search programme

The electron-positron collider LEP was operational at CERN between 1989 and 2000. In the initial phase, center-of-mass energies around the Z-peak were probed, but after 1995 the LEP experiments collected a significant amount of luminosity at higher center-of-mass energies, some $235 \text{ pb}^{-1}$ per experiment at $\sqrt{s} \ge 204$ GeV, with a maximum $\sqrt{s}$ of 209 GeV. Searches for new physics at $e^+e^-$ colliders benefit from the clean experimental environment and the fact that momentum balance can be measured not only in the plane transverse to the beam, but also in the direction along the beam (up to the beam pipe holes), defined as the longitudinal direction. Searches at LEP are dominated by the data samples taken at the highest center-of-mass energies.

The CDF and D0 experiments at the Tevatron, a proton-antiproton collider at a center-of-mass energy of up to 1.96 TeV, had an extensive search program for supersymmetric particles. CDF and D0 collected integrated luminosities between 10 and $11 \text{ fb}^{-1}$ each up to the end of collider operations in 2011.

The electron-proton collider HERA provided collisions to the H1 and ZEUS experiments between 1992 and 2007, at a center-of-mass energy up to 318 GeV. A total integrated luminosity of approximately $0.5 \text{ fb}^{-1}$ was collected by each experiment. Since at HERA baryons collide with leptons, SUSY searches typically look for R-parity violating production of single SUSY particles.

The Large Hadron Collider (LHC) at CERN started proton-proton operation at a center-of-mass energy of 7 TeV in 2010. By the end of 2011 the ATLAS and CMS experiments had collected about $5 \text{ fb}^{-1}$ of integrated luminosity each, and the LHCb experiment had collected approximately $1 \text{ fb}^{-1}$. In 2012, the LHC operated at a center-of-mass energy of 8 TeV, and ATLAS and CMS collected approximately $20 \text{ fb}^{-1}$ each, whereas LHCb collected $2 \text{ fb}^{-1}$. In 2015, the LHC started Run 2, with a center-of-mass energy of 13 TeV. At the end of Run 2 in November 2018, ATLAS and CMS had both collected approximately $140 \text{ fb}^{-1}$, and LHCb had collected almost $6 \text{ fb}^{-1}$. The Run 3 of the LHC started in July 2022 with a center-of-mass energy of 13.6 TeV. Almost $70 \text{ fb}^{-1}$ of data have been collected by ATLAS and CMS at the time of writing (August 2023), and LHCb collected around $1.3 \text{ fb}^{-1}$.

At the LHC, cross sections of QCD-mediated processes are the largest achievable at colliders, which is reflected in the higher sensitivity for SUSY particles carrying color charge, squarks and gluinos, with respect to LEP, Tevatron and HERA. In particular, proton-proton collisions at the LHC differ from proton-antiproton collisions at the Tevatron in the sense that there are no valence anti-quarks in the proton, and that gluon-initiated processes play a more dominant role. The increased center-of-mass energy of the LHC compared to the Tevatron significantly extends the kinematic reach for SUSY searches. This is reflected foremost in the sensitivity for squarks and gluinos, but also for SUSY particles produced via electroweak processes.

In this review we report results of direct searches for SUSY particles at colliders up to August 2023, mostly covering data analyses at the ATLAS and CMS experiments with reference to results from LEP, HERA and the Tevatron. For more details on LEP and Tevatron constraints, see earlier PDG reviews [37]. Analyses on the Run 3 datasets are still in progress hence the results reported for ATLAS and CMS are based on Run 1 and Run 2 data. Results are categorized depending on the targeted SUSY particles, the nature of their production and decays, and the assumption on R-parity. Brief summaries of search techniques and approaches adopted for interpretation of the results are given in Section 89.3 and 89.4, respectively. Sections 89.5-89.8 focus on results for promptly-decaying gluinos and first and second generation squarks, top and bottom squarks, electroweakinos (charginos,
neutralinos) and sleptons. Limits obtained for sparticle masses assuming R-parity violating models are also reported. Results of dedicated searches for long-lived (LL) SUSY particles are reported in Section 89.9. Finally, Section 89.10 provides examples of global reinterpretations of SUSY searches.

## 89.3 Experimental search techniques

Large background contributions from Standard Model processes pose challenges to the trigger and analysis. Such backgrounds are dominated by multijet production processes, including, particularly at the LHC, those of top quark production, as well as jet production in association with vector bosons. The proton's momentum is shared between its parton constituents, and in each collision only a fraction of the total center-of-mass energy is available in the hard parton-parton scattering. Since the parton momenta in the longitudinal direction are not known on an event-by-event basis, use of momentum conservation constraints in an analysis is restricted to the transverse plane, leading to the definition of transverse variables, such as the missing transverse momentum, and the transverse mass.

Under the assumption of R-parity conservation (RPC), the typical SUSY search signature for squarks and gluinos at hadron colliders contains high-$p_T$ jets, which are produced in the decay chains of heavy squarks and gluinos, and significant missing transverse momentum originating from the two LSPs produced at the end of the decay chains, which escape experimental detection. Electroweakino decays are often characterised by the presence of leptons, for instance from W and Z bosons in the decay chain, plus missing transverse momentum. Standard Model backgrounds with missing transverse momentum include leptonic W/Z-boson decays, associated production of jets and a Z-boson decaying into neutrinos, semi-leptonic heavy-flavor decays to neutrinos, top-quark pair production with one or both W bosons decaying leptonically, and multijet events that may be affected by instrumental effects such as jet mis-measurement.

Selection variables designed to separate the SUSY signal from the Standard Model backgrounds include transverse variables such as $H_T$, $E_T^{\text{miss}}$, and $m_{\text{eff}}$. The quantities $H_T$ and $E_T^{\text{miss}}$ refer to the measured transverse energy and the missing transverse momentum in the event, respectively. They are usually defined as the scalar sum of the transverse jet momenta or of the transverse energies of calorimeter clusters measured in the event ($H_T$), or the magnitude ($E_T^{\text{miss}}$) of the negative vector sum of transverse momenta of reconstructed objects like jets and leptons in the event ($\bar{p}_T^{\text{miss}}$). The quantity $m_{\text{eff}}$ is referred to as the effective mass of the event and is defined as $m_{\text{eff}} = H_T + E_T^{\text{miss}}$. The peak of the $m_{\text{eff}}$ distribution for SUSY signal events correlates with the SUSY mass scale, in particular with the mass difference between the primary produced SUSY particle and the LSP [38], whereas the Standard Model backgrounds dominate at low $m_{\text{eff}}$. An additional reduction of multijet backgrounds can be achieved by demanding isolated leptons or photons in the final states; in such events the lepton or photon transverse momentum may be added to $H_T$ or $m_{\text{eff}}$ for further signal-background separation. Other kinematic variables and approaches developed to increase the sensitivity to pair production of heavy sparticles with TeV-scale masses focusing on the kinematics of their decays are the $\alpha_T$ [39–43], razor [44], transverse mass ($m_{\text{T2}}$) [45], and contransverse mass ($m_{\text{CT}}$) [46] variables. The super-razor [47] and recursive jigsaw reconstruction [48] techniques are topological event reconstruction methods also used in searches for sparticles with masses in the TeV-range. In case massive, unstable SM particles such as top quarks, vector bosons or the Higgs boson arise in the decay chain, dedicated procedures are employed for their reconstruction and identification. If these are produced with a significant boost, their decay products will typically overlap and become collimated (boosted) in the direction of the parent particle. Their hadronic decay products can be reconstructed as a single large-radius jet (see studies from ATLAS [49]), and jet-substructure [50] techniques can be adopted. Machine-learning (ML) approaches are also used to identify and classify hadronic decays of highly Lorentz-boosted bosons and top quarks, see as example recent studies from CMS [51].

Most analyses still use simple combinations of selections on

kinematic variables (often referred to as *cut-and-count* analyses) and exploit multiple signal regions defined by categorizing events on the basis of several variables and their correlations. An example is provided in Fig. 89.1 from Ref. [52]: 39 signal regions are set to search for top squarks in events with one lepton, jets and missing transverse momentum. Regions depend on jet multiplicities, reconstructed top-quark candidates and thresholds applied to the most discriminant variables used for optimisation.

In recent years, shape analyses or analyses using more sophisticated techniques, e.g. machine learning, have been also developed. ML techniques often allow analyses to better capture the complexity of the events and achieve improved sensitivity with respect to counting experiments. For example, Boosted Decision Trees (BDTs) allow analyses to exploit simultaneously the discriminating power of multiple variables and that of correlations among objects in the events. Deep Neural Networks (NN) are also employed for objects reconstruction and to use directly detector-recorded energies and momenta of produced particles instead of first deriving a restricted set of physical variables. Among others, currently in use are Convolutional NN for image classification techniques, Recurrent NN, parametric NN and Generative Adversarial Networks.

Variables and approaches reported above are also used in searches for R-parity violating (RPV) SUSY models where signal events are often characterised by the presence of multiple leptons and/or jets, and little or no missing transverse momentum. However, if R-parity violating couplings are small, sparticles might be long-lived and may travel macroscopic distances before decaying. Similarly, long lifetimes may be due to small mass splittings, as in the case of pure higgsino/wino scenarios, or to heavy mediators, as in Split SUSY models [53, 54]. The identification of long-lived particles requires dedicated tools and their signatures can be very diverse. At the LHC, customized techniques (also based on NN) have been developed, for example, to reconstruct significantly displaced decay vertices or short track segments, and to identify tracks with atypical properties or unusual ionization, small and localized deposits of energy inside of the calorimeters without associated tracks, or stopped particles that decay out of time with collisions. For an overview, see Ref. [55].

## 89.4 Interpretation of search results

Since the mechanism by which SUSY is broken is unknown, a general approach to SUSY via the most general soft SUSY breaking Lagrangian adds a significant number of new free parameters. For the minimal supersymmetric standard model, MSSM, i.e., the model with the minimal particle content, these comprise 105 new real degrees of freedom. A phenomenological analysis of SUSY searches leaving all these parameters free is not feasible. For the practical interpretation of SUSY searches at colliders several approaches are taken to reduce the number of free parameters.

One approach is to assume a SUSY breaking mechanism and lower the number of free parameters through the assumption of additional constraints. Before the start of the LHC, interpretations of experimental results were often performed in constrained models of gravity mediated [56, 57], gauge-mediated [58–60], and anomaly mediated [61, 62] SUSY breaking. The most popular model was the constrained MSSM (CMSSM) [56, 63, 64], which in the literature is also referred to as minimal supergravity, or MSUGRA. These constrained SUSY models are theoretically well motivated and provide a rich spectrum of experimental signatures. However, with universality relations imposed on the soft SUSY breaking parameters, they do not cover all possible kinematic signatures and mass relations of SUSY. Furthermore, the Higgs mass measurement from LHC Run 1, together with other collider and non-collider measurements, limits substantially the allowed SUSY parameter space for these models. This indicates that very constrained models like the CMSSM are no longer good benchmark scenarios to solely characterize the results of SUSY searches at the LHC and efforts have been made to complement them with more flexible approaches.

A broader and more comprehensive subset of the MSSM can be studied via the so-called phenomenological-MSSM, or pMSSM [65–68]. It is derived from the MSSM, using experimental data to eliminate parameters that are free in principle but have
<figure><img src="image_364.png" /><figcaption>**Figure 89.1:** Example of multiple signal regions set for a typical SUSY search at the LHC. For definition of regions and acronyms specific for this search, see corresponding paper.</figcaption></figure>

<figure><img src="image_365.png" /><figcaption>**Figure 89.2:** Cross sections for pair production of gluinos and squarks as a function of their mass at the LHC for a center-of-mass energy of 13 TeV. They are calculated at approximately next-to-next-to-leading-order (NNLO) including the resummation of soft gluon emission at the next-to-next-to-leading logarithmic accuracy.</figcaption></figure>

already been highly constrained by measurements of e.g., flavor mixing and CP-violation. This effective approach reduces the number of free parameters in the MSSM to typically 19 or even less, making it a practical compromise between the full MSSM and highly constrained models such as the CMSSM.

Even less dependent on fundamental assumptions are interpretations in terms of so-called simplified models [69–72]. Such models assume a limited set of SUSY particle production and decay modes and leave open the possibility to vary masses and other parameters freely. Therefore, simplified models enable comprehensive studies of individual SUSY topologies, and are useful for optimization of the experimental searches over a wide parameter space without limitations on fundamental kinematic properties such as masses, production cross sections, and decay modes.

As a consequence, ATLAS and CMS have adopted simplified models as the primary framework to provide interpretations of their searches. In addition to using simplified models that describe prompt decays of SUSY particles, the experiments are now also

focusing more on the use of simplified models that allow for decays of long-lived SUSY particles as they can arise in different SUSY scenarios (see Section 89.9 for further discussion). Today, almost every individual search provides interpretations of their results in one or several simplified models that are characteristic of SUSY topologies probed by the analysis.

While simplified models are very convenient for the interpretation of individual SUSY production and decay topologies, care must be taken when applying these limits to more complex SUSY spectra. In particular, the branching ratio of SUSY particles into a specific final state is often assumed to be 100% and in many of the quoted limits the LSP is assumed to be massless. Therefore, simplified model limits should be seen as an approximation of the constraints that can be placed on sparticle masses in more complex SUSY spectra. Only on a case-by-case basis can it be determined whether the limit of a given simplified model represents a good approximation of the true underlying constraint that can be applied on a sparticle mass in a complex SUSY spectrum.
In the following, we will point out explicitly the assumptions that have entered the limits when quoting interpretations from simplified models.

Since none of the searches performed so far have shown significant excess above the SM background prediction, the interpretation of the presented results are exclusion limits on SUSY parameter space. Unless stated differently, all quoted exclusion limits are at the 95% confidence level. Finally, we note that many of the recent publications by ATLAS and CMS also present results in terms of model-independent limits such that their discovery potential is statistically quantified independently on a particular model.

## 89.5 Exclusion limits on gluino, first and second generation squark masses in RPC scenarios

Colored SUSY particles such as squarks and gluinos are produced via the strong interaction and have the highest cross sections at hadron colliders. Limits on squark masses of the order 100 GeV have been set by the LEP experiments [73], in the decay to quark plus neutralino, and for a mass difference between squark and quark plus neutralino of typically at least a few GeV. However, hadron collider experiments are able to set much more stringent mass limits.

The main production mechanisms at hadron colliders are squark-squark, squark-gluino and gluino-gluino production; when “squark” is used “antisquark” is also implied. Pair production usually involves both the s-channel and t-channel parton-parton interactions. However, since there is a negligible amount of bottom and top quark content in the proton, top and bottom squark production proceeds through s-channel diagrams only. Experimental analyses of squark and/or gluino production typically assume the first and second generation squarks to be approximately degenerate in mass. Cross section calculations at the center-of-mass energy of 13 TeV shown in Fig. 89.2 (from Ref. [74]) assume mass degeneracy of left- and right-handed u, d, s, c squarks. Other sparticle masses are considered decoupled, i.e. their masses are assumed to have a very high value (above 10 TeV). The LHC experiments also provide simplified model limits on individual first or second generation squarks. Top and bottom squarks have the same production cross section, identical to that of a single light squark.

In this section, we focus on results assuming R-parity conservation. Limits set by the Tevatron experiments on the gluino mass assume the framework of the CMSSM, with $\tan\beta = 5$ (CDF) or $\tan\beta = 3$ (D0). Furthermore, the common trilinear term $A_0$ is set to 0 and the higgsino mass term $\mu$ is assumed to be less than 0. The resulting lower mass limits are about 310 GeV for all squark masses, or 390 GeV for the case $m_{\tilde{g}} = m_{\tilde{g}}$ [75, 76].

ATLAS and CMS limits on the gluino mass have been established in the framework of simplified models. Assuming only gluino pair production, three main primary decay chains of the gluino have been considered by the LHC experiments for interpretations of their search results. The first decay chain, $\tilde{g} \to q\bar{q}\tilde{\chi}_1^0$, assumes gluino mediated production of first and second generation squarks (on-shell or off-shell) which leads to four light flavor quarks in the final state. Therefore, inclusive all-hadronic analyses searching for multijet plus $E_T^{\text{miss}}$ final states are utilized to put limits on this simplified model. These limits are derived as a function of the gluino and neutralino (LSP) mass. As shown in Fig. 89.3 (left) the CMS collaboration [77] excludes in this simplified model gluino masses below approximately 2.1 TeV for a light neutralino mass below about 600 GeV. In scenarios where neutralinos are not very light, the efficiency of the analyses is reduced by the fact that jets are less energetic, and there is less missing transverse momentum in the event. This leads to weaker limits when the mass difference $\Delta m = m_{\tilde{g}} - m_{\tilde{\chi}_1^0}$ is reduced. For example, for neutralino masses above about 1.2 TeV no limit on the gluino mass can be set for this decay chain. Therefore, limits on gluino masses are strongly affected by the assumption of the neutralino mass. Similar results for this simplified model have been obtained by ATLAS [78]. A second simplified model postulates a decay chain where $\tilde{g} \to q\bar{q}W\tilde{\chi}_1^0$, assuming that the intermediate (on-shell or off-shell) squark is left-handed and de-

cays to a chargino and a quark, with the chargino decaying to a W boson and the LSP. This leads to two W bosons and four light flavor quarks plus $E_T^{\text{miss}}$ in the final state. Both leptonic and hadronic decays of the W can be considered. In this scenario, the ATLAS collaboration [79, 80] excludes gluino masses below approximately 2.2 TeV for a sufficiently light neutralino and assuming the chargino mass is halfway between the gluino and neutralino mass, see Fig. 89.3 (right). Again, for neutralino masses above about 1.2 TeV, there exists no limit on the gluino mass for this decay chain. A similar reach has been obtained by the CMS collaboration [81, 82, 83, 84], where the most recent results are obtained using novel algorithms based on deep neutral networks that identify hadronically decaying W bosons.

Gluino decays are not limited to first and second generation squarks: another important decay chain of the gluino considered for interpretation in simplified models is $\tilde{g} \to b\bar{b}\tilde{\chi}_1^0$. In this case the decay is mediated via bottom squarks and thus leads to four jets originating from b-quarks and $E_T^{\text{miss}}$ in the final state. For this topology as well, inclusive all-hadronic searches provide the highest sensitivity. In this case, the use of secondary vertex reconstruction for the identification of jets originating from the b-quarks in the event provides a powerful handle to reject SM background contributions. Therefore, in addition to a multijet plus $E_T^{\text{miss}}$ signature these searches also require between 1 and 4 jets to be tagged as b-jets. As shown in Fig. 89.4 (left), for this simplified model CMS [77] excludes gluino masses below $\approx 2.3$ TeV for a sufficiently light neutralino, while for neutralino masses above $\approx 1.5$ TeV no limit on the gluino mass can be set. Comparable limits for this simplified model are provided by searches from ATLAS [83, 84], with the most recent results pushing the boundary of gluino masses to 2.44 TeV in the case of a massless neutralino.

If kinematically allowed, decays of gluinos to top squarks via $\tilde{g} \to \bar{t}t$ are also possible. This defines the next important simplified model characterizing gluino pair production, $\tilde{g} \to t\bar{t}\tilde{\chi}_1^0$, which leads to a “four tops” final state, $t\bar{t}t\tilde{\chi}_1^0\tilde{\chi}_1^0$. The topology of this decay is very rich in different experimental signatures: as many as four isolated leptons, four b-jets, several light flavor quark jets, and significant missing transverse momentum from the neutralinos in the W decay and from the two neutralinos. As shown in Fig. 89.4 (right), the ATLAS search [83] rules out gluinos with masses below $\approx 2.4$ TeV for light neutralinos in this model. For neutralino masses above $\approx 1.4$ TeV, no limit can be placed on the gluino mass. The CMS multiple b-jet search [85] and the search in full-hadronic events using identification techniques for boosted top-quarks in the final state [86] result in gluino mass limits of $\approx 2.3$ TeV.

Assuming gluinos to be heavier than squarks, squarks will predominantly decay to a quark and a neutralino or chargino, if kinematically allowed. The decay may involve the lightest neutralino (typically the LSP) or chargino, but, depending on the masses and couplings of the gauginos and on the handedness of the squarks, may involve heavier neutralinos or charginos. For pair production of first and second generation squarks, the simplest decay modes involve two jets and missing transverse momentum, with potential extra jets stemming from initial state or final state radiation (ISR/FSR) or from decay modes with longer decay chains (cascades). In cascades, isolated photons or leptons may appear from the decays of sparticles such as neutralinos or charginos.

Limits on first and second generation squark masses set by the Tevatron experiments assume the CMSSM, and amount to lower limits of about 380 GeV for all gluino masses, or 390 GeV for the case $m_{\tilde{g}} = m_{\tilde{g}}$ [75, 76]. At the LHC, limits on squark masses have been set using up to approximately $140 \text{ fb}^{-1}$ of data at 13 TeV. Interpretations in simplified models typically characterize squark pair production considering only one decay chain at the time for the squark, for instance $\tilde{q} \to q\tilde{\chi}_1^0$. It is usually assumed that the left and right-handed $\tilde{u}, \tilde{d}, \tilde{s}$ and $\tilde{c}$ squarks are degenerate in mass. Furthermore, it is assumed that the mass of the gluino is very high (above 10 TeV) and thus contributions of the corresponding t-channel diagrams to squark pair production are negligible. Therefore, the total production cross section
<figure><img src="image_366.png" /><figcaption>**Figure 89.3:** Lower mass limits, at 95% C.L., on gluino pair production for various decay chains in the framework of simplified models. Left: $\tilde{g} \to q\bar{q}\tilde{\chi}_1^0$, result of the CMS collaboration. Right: $\tilde{g} \to q\bar{q}W\tilde{\chi}_1^0$, result of the ATLAS collaboration.</figcaption></figure>

<figure><img src="image_367.png" /><figcaption>**Figure 89.4:** Lower mass limits, at 95% C.L., on gluino pair production for various decay chains in the framework of simplified models. Left: $\tilde{g} \to b\bar{b}\tilde{\chi}_1^0$. Right: $\tilde{g} \to t\bar{t}\tilde{\chi}_1^0$. Results from the CMS and ATLAS collaborations.</figcaption></figure>

for this simplified model is eight times the production cross section of an individual squark (e.g. $\tilde{u}_L$). Under these assumptions, CMS obtains a lower squark mass limit of $\approx 1.75$ TeV for light neutralinos [85], as shown in Fig. 89.5 (left). Only for neutralino masses below $\approx 800$ GeV can any squark masses be excluded. Searches for new physics in monojet channels are sensitive to very compressed scenarios, and masses up to 900 GeV are excluded for $\Delta m = m_{\tilde{q}}, m_{\tilde{\chi}_1^0} \approx 5$ GeV [87].

If the assumption of mass degenerate first and second generation squarks is dropped and only the production of a single light squark is assumed, the limits weaken significantly. For example, the CMS limit on degenerate squarks of 1750 GeV for light neutralinos drops to $\approx 1300$ GeV for pair production of a single light squark, and for neutralinos heavier than $\approx 600$ GeV no squark mass limit can be placed. It should be noted that this limit is not a result of a simple scaling of the above mentioned mass limits assuming eight-fold mass degeneracy but it also takes into account that for an eight times lower production cross section the analyses must probe kinematic regions of phase space that are closer to the

ones of SM background production.

A summary of results of the ATLAS searches also targeting more complicated decay chains, where e.g. intermediate bosons are produced as well, can be seen in Fig. 89.5 (right). Depending on the topology, the exclusion limit can reach up to $\approx 2$ TeV (this value is only reached in gaugino-mediated simplified models [88]). Recent results include a search in two-opposite-sign, same=flavor dilepton final state events [89] targeting squarks decaying through the next-to-lightest neutralino via a slepton or a Z boson and exclude mass up to $\approx 1550$ GeV for a massless neutralino. Squarks decaying via charginos and next-to-lightest neutralinos in long decay chains including W and Z bosons are constrained with searches for same-sign dilepton or three-lepton final state events [90]: in this case, masses up to $\approx 1.7$ TeV are excluded.

For single light squarks ATLAS also reports results of a dedicated search for pair production of scalar partners of charm quarks [91]. Assuming that the scalar-charm state exclusively decays into a charm quark and a neutralino, scalar-charm masses
<figure><img src="image_368.png" /><figcaption>**Figure 89.5:** Left: 95% C.L. exclusion contours in the framework of simplified models assuming a single decay chain of $\tilde{q} \to q\tilde{\chi}_1^0$, obtained by the CMS collaboration. Right: Assuming more complicated decay chains including W or Z bosons, obtained by the ATLAS collaboration.</figcaption></figure>

up to 800 GeV are excluded for neutralino masses below 260 GeV.

<table><caption>**Table 89.1:** Summary of squark mass and gluino mass limits using different interpretation approaches assuming R-parity conservation. Masses in this table are provided in GeV. Further details about the assumptions and analyses from which these limits are obtained are discussed in the corresponding sections of the text.</caption><thead><tr><th>Model</th><th>Assumption</th><th>m<sub>q̃</sub></th><th>m<sub>q̃</sub></th></tr></thead><tbody><tr><td colspan="4">Simplified models q̃q̃</td></tr><tr><td rowspan="2">q̃ → q̃q̃χ<sub>1</sub><sup>0</sup></td><td>m<sub>χ̃<sub>1</sub><sup>0</sup></sub> = 0</td><td>-</td><td>≈ 2300</td></tr><tr><td>m<sub>χ̃<sub>1</sub><sup>0</sup></sub> &gt;≈ 1200</td><td>-</td><td>no limit</td></tr><tr><td rowspan="2">q̃ → bbχ̃<sub>1</sub><sup>0</sup></td><td>m<sub>χ̃<sub>1</sub><sup>0</sup></sub> = 0</td><td>-</td><td>≈ 2440</td></tr><tr><td>m<sub>χ̃<sub>1</sub><sup>0</sup></sub> &gt;≈ 1600</td><td>-</td><td>no limit</td></tr><tr><td rowspan="2">q̃ → t<span style="text-decoration: overline">t</span>χ̃<sub>1</sub><sup>0</sup></td><td>m<sub>χ̃<sub>1</sub><sup>0</sup></sub> = 0</td><td>-</td><td>≈ 2400</td></tr><tr><td>m<sub>χ̃<sub>1</sub><sup>0</sup></sub> &gt;≈ 1400</td><td>-</td><td>no limit</td></tr><tr><td colspan="4">Simplified models q̃q̃</td></tr><tr><td rowspan="2">q̃ → q̃χ<sub>1</sub><sup>0</sup></td><td>m<sub>χ̃<sub>1</sub><sup>0</sup></sub> = 0</td><td>≈ 1900</td><td>-</td></tr><tr><td>m<sub>χ̃<sub>1</sub><sup>0</sup></sub> &gt;≈ 800</td><td>no limit</td><td>-</td></tr><tr><td rowspan="2">ũ<sub>L</sub> → q̃χ<sub>1</sub><sup>0</sup></td><td>m<sub>χ̃<sub>1</sub><sup>0</sup></sub> = 0</td><td>≈ 1300</td><td>-</td></tr><tr><td>m<sub>χ̃<sub>1</sub><sup>0</sup></sub> &gt;≈ 600</td><td>no limit</td><td>-</td></tr></tbody></table>

A summary of the most important squark and gluino mass limits for different interpretation approaches assuming R-parity conservation is shown in Table 89.1. For gluino masses rather similar limits of about 2.3–2.4 TeV are obtained from different model assumptions, indicating that the LHC is indeed probing direct gluino production at the TeV scale and beyond. However, for neutralino masses above approximately 1.2 to 1.6 TeV, even in the best case scenarios, ATLAS and CMS searches do not place any limits on the gluino mass. Limits on direct squark production, on the other hand, depend strongly on the chosen model. For neutralino masses above ≈ 600 GeV no limits on any direct squark pair production scenario are placed by the LHC.

## 89.6 Exclusion limits on bottom and top squarks masses in RPC scenarios

Besides placing stringent limits on first and second generation squark masses, the LHC experiments also search for the production of third generation squarks. SUSY at the TeV-scale is often motivated by naturalness arguments, most notably as a solution to cancel quadratic divergences in radiative corrections to the Higgs boson mass. In this context, the most relevant terms for SUSY phenomenology arise from the interplay between the masses of the third generation squarks and the Yukawa coupling of the top quark to the Higgs boson. This motivates a potential constraint on the masses of the top squarks and the left-handed bottom squark. Due to the large top quark mass, significant mixing between $t̄_L$ and $t̄_R$ is expected, leading to a lighter mass state $t̄_1$ and a heavier mass state $t̄_2$. In the MSSM, the lightest top squark ($t̄_1$) can be the lightest squark.

Bottom squarks are expected to decay predominantly to $b\tilde{\chi}_1^0$ giving rise to the characteristic multi b-jet and $E_T^{miss}$ signature. Direct production of bottom squark pairs has been searched for at the Tevatron and at the LHC. Limits from the Tevatron are $m_{t̄_b} > 247$ GeV for a massless neutralino [92, 93]. ATLAS has set a lower limit of $m_{t̄_b} > 1250$ GeV for massless neutralinos in this model [94] exploiting the two b-jets and missing transverse momentum analysis. For $m_{\tilde{\chi}_1^0} \approx 800$ GeV or higher no limit can be placed on direct bottom squark pair production in this simplified model. Limits from CMS are comparable [85]. Reinterpretations of the monojet analysis results can be used to constrain very compressed scenarios, with exclusion of masses up to 550 GeV in case of $\Delta m = m_{t̄_b} - m_{\tilde{\chi}_1^0} \approx 5$ GeV [87]. Further bottom squark decay modes have also been searched for by ATLAS [95–97] and CMS [77, 98, 99], for instance targeting more complex decay chains that include Higgs bosons in the cascade.

The top squark decay modes depend on the SUSY mass spectrum, and on the $t̄_L-t̄_R$ mixture of the top squark mass eigenstate. If kinematically allowed, the two-body decays $t̄ \to t\tilde{\chi}_1^0$ (which requires $m_{t̄} - m_{\tilde{\chi}_1^0} > m_t$) and $t̄ \to b\tilde{\chi}_\pm^0$ (which requires $m_{t̄} - m_{\tilde{\chi}_\pm} > m_b$) are expected to dominate. If not, the top squark decay may proceed either via the two-body decay $t̄ \to c\tilde{\chi}_1^0$ or through $t̄ \to b\tilde{f}'\tilde{\chi}_1^0$ (where $f'$ and $f$' denote a fermion-antifermion pair with appropriate quantum numbers). For $m_{t̄} - m_{\tilde{\chi}_1^0} > m_b$ the latter decay chain represents a four-body decay with a W boson,
<figure><img src="image_369.png" /><figcaption>**Figure 89.6:** A summary of the 95% C.L. exclusion contours in the stop-neutralino mass plane for various possible decay chains, including two-, three- and four-body decays, as obtained in dedicated analyses by ATLAS.</figcaption></figure>

charged H$^{±}$, slepton $\tilde{\ell}$, or light flavor squark $\tilde{q}$, exchange. If the exchanged W boson and/or sleptons are kinematically allowed to be on-shell, the three-body decays $\tilde{t} \to Wb\tilde{\chi}_1^0$ and/or $\tilde{t} \to b\nu\tilde{\ell}(\bar{\nu}\ell)$ will become dominant. For further discussion on top squark decays see for example Ref. [100].

Limits from LEP on the $\tilde{t}_1$ mass are $m_{\tilde{t}} > 96$ GeV in the charm plus neutralino final state, and > 93 GeV in the lepton, b-quark and sneutrino final state [73].

The Tevatron experiments have performed a number of searches for top squarks, often assuming direct pair production. In the $b\ell\bar{\nu}$ decay channel, and assuming a 100% branching fraction, limits are set as $m_{\tilde{t}} > 210$ GeV for $m_{\tilde{\nu}} < 110$ GeV and $m_{\tilde{t}} - m_{\tilde{\nu}} > 30$ GeV, or $m_{\tilde{t}} > 235$ GeV for $m_{\tilde{\nu}} < 50$ GeV [101, 102]. In the $\tilde{t} \to c\tilde{\chi}_1^0$ decay mode, a top squark with a mass below 180 GeV is excluded for a neutralino lighter than 95 GeV [103, 104]. In both analyses, no limits on the top squark can be set for heavy sneutrinos or neutralinos. In the $\tilde{t} \to b\tilde{\chi}_1^{\pm}$ decay channel, searches for a relatively light top squark have been performed in the dilepton final state [105, 106]. The CDF experiment sets limits in the $\tilde{t} - \tilde{\chi}_1^0$ mass plane for various branching fractions of the chargino decay to leptons and for two values of $m_{\tilde{\chi}_1^{\pm}}$. For $m_{\tilde{\chi}_1^{\pm}} = 105.8$ GeV and $m_{\tilde{\chi}_1^0} = 47.6$ GeV, top squarks between 128 and 135 GeV are excluded for W-like leptonic branching fractions of the chargino.

The LHC experiments have improved these limits substantially and developed dedicated searches for all potential top-squark decay mode. This is well depicted in Fig. 89.6, where a summary of the limits on the top squark mass are reported for the single decay chain of $\tilde{t} \to t\tilde{\chi}_1^0$ as well as for the three-body and four-body decay modes. Interpretations are carried out in simplified models hence care must be taken when interpreting these limits in the context of more complete SUSY models.

The most important searches for the top squark decay topology are dedicated searches requiring zero or one isolated lepton, modest $E_T^{\text{miss}}$, and four or more jets out of which at least one jet must be reconstructed as a b-jet [52, 77, 85, 86, 107–109]. For example, using an all-hadronic analysis, CMS excludes top squarks with masses below about 1300 GeV in this model for light neutralinos, while for $m_{\tilde{\chi}_1^0} > 700$ GeV no limits can be provided [86]. Equivalent constraints are set by the ATLAS all-hadronic search [107].

Analyses targeting final state events characterised by the presence of one lepton (electron or muon) have similar albeit lower sensitivity in the $\tilde{t} \to t\tilde{\chi}_1^0$ decay mode, see for instance the recent ATLAS result [109] using NN methods.

Single and di-lepton analyses are most sensitive to models where the top squark decay exclusively proceeds via the chargino mediated decay chain $\tilde{t} \to b\tilde{\chi}_1^{\pm}$, $\tilde{\chi}_1^{\pm} \to W^{\pm(*)}\tilde{\chi}_1^0$. In this case, top squark mass exclusion limits vary strongly with the assumptions made on the $\tilde{t} - \tilde{\chi}_1^{\pm} - \tilde{\chi}_1^0$ mass hierarchy. For example, for $m_{\tilde{\chi}_1^{\pm}} = (m_{\tilde{t}} + m_{\tilde{\chi}_1^0})/2$, a stop mass below $\approx 1150$ GeV for a light $\tilde{\chi}_1^0$ is excluded by CMS, while no limit can be placed for $m_{\tilde{\chi}_1^0} > 550$ GeV [52]. These limits, however, can weaken significantly when other assumptions about the mass hierarchy or the decay of the charginos are imposed [52, 110–112].

Other analyses with zero, one or two leptons target this kinematic region [113–119], also providing sensitivity in the case of alternative decay modes of the top squark, i.e. including tau sleptons [112, 120–122] or Higgs and Z bosons in the decay chain [123].

If the decays $\tilde{t} \to t\tilde{\chi}_1^0$ and $\tilde{t} \to b\tilde{\chi}_1^{\pm}$, $\tilde{\chi}_1^{\pm} \to W^{\pm(*)}\tilde{\chi}_1^0$ are kinematically forbidden, the decay chains $\tilde{t} \to Wb\tilde{\chi}_1^0$ and $\tilde{t} \to c\tilde{\chi}_1^0$ can become important. The one-lepton ATLAS search [108] provides for the kinematic region $m_{\tilde{t}} - m_{\tilde{\chi}_1^{\pm}} > m_b + m_W$ lower limits on the top squark mass of $\approx 700$ GeV for a neutralino lighter than $\approx 570$ GeV, as also shown in Fig. 89.6. For the kinematic region in which even the production of real W bosons is not allowed, ATLAS and CMS improve the Tevatron limit on $\tilde{t} \to c\tilde{\chi}_1^0$ substantially. Based on a monojet analysis [87] ATLAS excludes top squark masses below $m_{\tilde{\chi}_1^0} \approx 550$ GeV along the kinematic boundary for the $\tilde{t} \to c\tilde{\chi}_1^0$ decay. The ATLAS monojet analysis also places similar boundaries for the other decay chain relevant in this phase region, $\tilde{t} \to b\tilde{f}\tilde{f}'\tilde{\chi}_1^0$. Other analyses, such as the search in events with one leptons and jets, are also sensitive to this decay mode: the ATLAS one-lepton analysis excludes up to $\approx 650$ GeV for $\Delta m = m_{\tilde{t}}, m_{\tilde{\chi}_1^0} \approx 50$ GeV [108]. The CMS collaboration uses the hadronic searches [86, 114, 116] to place constraints on stop decay into charm-quark and excludes $m_{\tilde{t}} \approx 630$ GeV for $\Delta m = m_{\tilde{t}}, m_{\tilde{\chi}_1^0} \approx 50$ GeV [86]. Constraints are also set in case of
$\tilde{t} \to b f \bar{f}' \tilde{\chi}_1^0$, with CMS excluding masses up to 480 and 700 GeV for $\Delta m = m_{\tilde{t}}, m_{\tilde{\chi}_1^0} \approx 10$ and 80 GeV, respectively [124].

<table><caption>**Table 89.2:** Summary of bottom and top squark mass limits using different interpretation approaches assuming R-parity conservation. Masses in this table are provided in GeV. Further details about the assumptions and analyses from which these limits are obtained are discussed in the corresponding sections of the text.</caption><thead><tr><th>Model</th><th>Assumption</th><th>$m_{\tilde{q}}$</th></tr></thead><tbody><tr><td>$\tilde{b} \to b \tilde{\chi}_1^0$</td><td>$m_{\tilde{\chi}_1^0} = 0$<br>$m_{\tilde{\chi}_1^0} \gtrsim 700$</td><td>$\approx 1250$<br>no limit</td></tr><tr><td>$\tilde{t} \to t \tilde{\chi}_1^0$</td><td>$m_{\tilde{\chi}_1^0} = 0$<br>$m_{\tilde{\chi}_1^0} \gtrsim 600$</td><td>$\approx 1300$<br>no limit</td></tr><tr><td>$\tilde{t} \to b \tilde{\chi}_1^\pm$<br>($m_{\tilde{\chi}_1^\pm} = (m_{\tilde{t}} - m_{\tilde{\chi}_1^0})/2$)</td><td>$m_{\tilde{\chi}_1^0} = 0$<br>$m_{\tilde{\chi}_1^0} \gtrsim 550$</td><td>$\approx 1150$<br>no limit</td></tr><tr><td>$\tilde{t} \to W b \tilde{\chi}_1^0$<br>($m_W < m_{\tilde{t}} - m_{\tilde{\chi}_1^0} < m_t$)</td><td>$m_{\tilde{\chi}_1^0} \lesssim 570$</td><td>$\approx 700$</td></tr><tr><td>$\tilde{t} \to c \tilde{\chi}_1^0$</td><td>$m_{\tilde{t}} - m_{\tilde{\chi}_1^0} \approx 50$<br>$m_{\tilde{t}} \approx m_{\tilde{\chi}_1^0}$</td><td>$\approx 630$<br>$\approx 550$</td></tr><tr><td>$\tilde{t} \to b f f' \tilde{\chi}_1^0$<br>($m_{\tilde{t}} - m_{\tilde{\chi}_1^0} < m_W$)</td><td>$m_{\tilde{t}} - m_{\tilde{\chi}_1^0} \approx 50$<br>$m_{\tilde{t}} \approx m_{\tilde{\chi}_1^0}$</td><td>$\approx 650$<br>$\approx 550$</td></tr></tbody></table>

In general, the variety of top squark decay chains in the phase space region where $\tilde{t} \to t \tilde{\chi}_1^0$ is kinematically forbidden represents a challenge for the experimental search program but more data and refined analyses in Run 2 have further improved the sensitivity in this difficult but important region of SUSY parameter space, and more is expected for future Runs of the LHC. Precision SM measurements can also provide important insights to such challenging regions. For instance, analyses of $t\bar{t}$ spin correlations can be used to set constraints on top squark masses close to the top-quark mass. ATLAS [125] excludes the mass region between 150 and 230 GeV for kinematically allowed values of the neutralino mass. CMS [126] results cover a similar mass range.

## 89.7 Exclusion limits on the masses of charginos, neutralinos and sleptons in RPC scenarios

Charginos, neutralinos and sleptons are produced through electroweak interactions and their cross sections depend on the assumptions made for mass and mixing parameters.

Charginos and neutralinos carry no color charge and the mixing of the charged wino and higgsino states (for charginos), and the neutral bino, wino and higgsino states (for neutralinos), is determined by a limited number of parameters. For charginos these are the wino mass parameter $M_2$, the higgsino mass parameter $\mu$, and $\tan \beta$, and for neutralinos these are the same parameters plus the bino mass parameter $M_1$. If one of the parameters $M_1, M_2$ or $\mu$ is substantially smaller than the others, the chargino/neutralino composition would be dominated by specific states, which are referred to as bino-like ($M_1 \ll M_2, \mu$), wino-like ($M_2 \ll M_1, \mu$), or higgsino-like ($\mu \ll M_1, M_2$). If gaugino mass unification at the GUT scale is assumed, a relation between $M_1$ and $M_2$ at the electroweak scale follows: $M_1 = 5/3 \tan^2 \theta_W M_2 \approx 0.5 M_2$, with $\theta_W$ the weak mixing angle. The largest production rates at hadron machines are obtained when the LSP is bino-like and the lightest chargino and next-to-lightest neutralino are wino-like, forming an approximately mass degenerate SU(2) triplet. If the higgsino mass is much smaller than the gaugino masses, the two lightest neutralinos and the lightest chargino form an approximately mass degenerate Dirac SU(2) doublet, production rates are lower and the mass spectrum is compressed.

In models with slepton and gaugino mass unification at the GUT scale, the superpartner of the right-handed lepton, $\tilde{\ell}_R$, is expected to be lighter than the left-handed one, $\tilde{\ell}_L$. Cross sections depend on the L-R mixing which in turn depends on the mass of the fermion and other parameters. For tau sleptons there may

be considerable mixing between the L and R states, leading to a significant mass difference between the lighter $\tilde{\tau}_1$ and the heavier $\tilde{\tau}_2$.

Fig. 89.7 [127,128] (see also Ref. [74] for further details) shows typical production cross sections at hadron colliders for electroweakinos under various assumptions and considering a center-of-mass energy of 13 TeV. For masses of several hundreds of GeV, they are at least two orders of magnitude smaller than for colored SUSY particles. Thanks to the large data samples collected at the LHC, the sensitivity of LEP and Tevatron searches for direct chargino/neutralino or slepton production has been surpassed in several regions of SUSY parameter space.

<figure><img src="image_370.png" /><figcaption>**Figure 89.7:** Cross sections for pair production of charginos, neutralinos and sleptons calculated for 13 TeV centre-of-mass energy at NLO plus next-to-leading-log (NLL) precision. Wino-like $\tilde{\chi}_1^\pm \tilde{\chi}_2^0$ and $\tilde{\chi}_1^+ \tilde{\chi}_1^-$ productions are calculated in a limit of mass-degenerate $\tilde{\chi}_1^\pm$ and $\tilde{\chi}_2^0$, light bino $\tilde{\chi}_1^0$, and with all the other sparticles assumed to be heavy and decoupled. Higgsino-like cross sections assume mass-degenerate $\tilde{\chi}_1^\pm, \tilde{\chi}_2^0$ and $\tilde{\chi}_1^0$. For slepton pair cross sections, left(right)-handed only or 50% mixed states are considered.</figcaption></figure>

### 89.7.1 Exclusion limits on chargino masses

The lightest chargino $\tilde{\chi}_1^\pm$ is searched for either considering pair production, or through associated production of $\tilde{\chi}_1^\pm$ and the next-to-lightest neutralino, $\tilde{\chi}_2^0$, if they are assumed to be degenerate in mass. If kinematically allowed, two body decay modes such as $\tilde{\chi}^\pm \to \tilde{f} \tilde{f}'$ (including $\ell \bar{\nu}$ and $\tilde{\ell} \nu$) are dominant. If not, three-body decays $\tilde{\chi}^\pm \to \tilde{f} \tilde{f}' \tilde{\chi}_1^0$, mediated through virtual W bosons or sfermions, become dominant. If sfermions are heavy, the W mediation dominates, and $\tilde{f} \tilde{f}'$ are distributed with branching fractions similar to W decay products (barring phase space effects for small mass gaps between $\tilde{\chi}^\pm$ and $\tilde{\chi}^0$). If, on the other hand, sleptons are light enough to play a significant role in the decay, leptonic final states will be enhanced.

At LEP, $\tilde{\chi}_1^+ \tilde{\chi}_1^-$ is the dominant production process. Charginos have been searched for in fully-hadronic, semi-leptonic and fully leptonic decay modes [129, 130]. A general lower limit on the lightest chargino mass of 103.5 GeV is derived, except in corners of phase space with low electron sneutrino mass, where destructive interference in chargino production, or two-body decay modes, play a role. The limit is also affected if the mass difference between $\tilde{\chi}_1^\pm$ and $\tilde{\chi}_1^0$ is small; dedicated searches for such scenarios set a lower limit of 92 GeV.

At the Tevatron, charginos have been searched for via associated production of $\tilde{\chi}_1^\pm \tilde{\chi}_2^0$ [131, 132]. Decay modes involving multilepton final states provide the best discrimination against the large multijet background. Analyses have looked for at least three charged isolated leptons, for two leptons with missing transverse momentum, or for two leptons with the same charge. Depending
on the $(\tilde{\chi}_1^\pm - \tilde{\chi}_1^0)$ and/or $(\tilde{\chi}_2^0 - \tilde{\chi}_1^0)$ mass differences, leptons may have soft transverse momentum.

At the LHC, the search strategy is similar to that at the Tevatron but it also exploits the large datasets collected to target missing momentum and fully-hadronic final states and consider $\tilde{\chi}_1^+\tilde{\chi}_1^-$ production only i.e. releasing the assumption of $\tilde{\chi}_1^\pm, \tilde{\chi}_2^0$ mass degeneracy. Furthermore, compressed SUSY spectra are searched for considering low-momentum leptons recoiling against an ISR jet.

Chargino pair production is searched for in the dilepton plus missing momentum final state. In a simplified model interpretation of the results where only the $\tilde{\chi}_1^+\tilde{\chi}_1^-$ process is considered and assuming mediation of the chargino decay by light sleptons ($\tilde{e}$ and $\tilde{\mu}$), ATLAS [133] and CMS [134] set limits on the chargino mass up to 1 TeV for massless LSPs, but no limits on the chargino mass can be set for $\tilde{\chi}_1^0$ heavier than 480 GeV. Limits are fairly robust against variation of the slepton mass, unless the mass gap between sparticles becomes small enough that the charged lepton has a too low-momentum to be reconstructed. For decays mediated through $\tilde{\tau}$ or $\tilde{\nu}_\tau$, limits of 630 GeV are set by ATLAS [135] for LSPs not heavier than 200 GeV. The CMS experiment provides similar limits [136].

In case slepton masses are higher than the mass of the chargino, searches are performed for charginos decaying via a W boson. ATLAS sets limits on this kind of simplified models [133] for pure-wino charginos: masses below 420 GeV are excluded for massless LSPs, but no limits are set for LSPs heavier than 120 GeV. In the challenging case where the difference between chargino and LSP masses is close to that of the W boson, dedicated searches have been carried out by ATLAS [137], excluding chargino masses up to 140 GeV. Searches have also been carried out in the single lepton final state. ATLAS excludes chargino masses between 260 and 520 GeV for a massless LSP [138].

Further extension of the sensitivity in the high-mass region is achieved by ATLAS [139] and CMS [140] exploiting the large branching ratio of fully-hadronic final states and the identification of boosted W bosons through large-radius jets and jet sub-structure information. Considering only wino-like chargino pair production in case of a massless LSP, ATLAS excludes a region between 630 GeV and 760 GeV, while CMS excludes between 290 GeV and 670 GeV. No limits are set for LSP masses above 200 GeV.

Several final states characterised by the presence of missing transverse momentum are studied to set limits on the chargino mass through $\tilde{\chi}_1^\pm\tilde{\chi}_2^0$ associated production. Usually, wino-like $\tilde{\chi}_1^\pm$ and $\tilde{\chi}_2^0$ and bino-like $\tilde{\chi}_1^0$ are assumed with $m_{\tilde{\chi}_1^\pm} = m_{\tilde{\chi}_2^0}$, leaving $m_{\tilde{\chi}_1^\pm}$ and $m_{\tilde{\chi}_1^0}$ free. Again, the branching fraction of leptonic final states is determined by the slepton masses. If the decay is predominantly mediated by a light $\tilde{\ell}_L$, i.e. $\tilde{\ell}_R$ is assumed to be heavy, the three charged-lepton flavors will be produced in equal amounts, and a multi-lepton signature is favored. It is assumed that $\tilde{\ell}_L$ and sneutrino masses are equal, and diagrams with sneutrinos are included. In this scenario, ATLAS [141] and CMS [142] exclude chargino masses below 1450 GeV for massless LSPs; no limits are set for LSP masses above 1 TeV. If the decay is dominated by a light $\tilde{\ell}_R$, the chargino cannot be a pure wino but needs to have a large higgsino component, preferring the decays to tau leptons. Limits are set in various scenarios. If, like for $\tilde{\ell}_L$, a flavor-democratic scenario is assumed ($\tau$-enriched scenario), CMS sets limits of 1150 GeV on the chargino mass for massless LSPs, but under the assumption that both $\tilde{\chi}_1^\pm$ and $\tilde{\chi}_2^0$ decay leads to tau leptons in the final state ($\tau$-dominated scenario), the chargino mass limit deteriorates to 970 GeV for massless LSPs [142]. ATLAS assumes a simplified model in which staus are significantly lighter than the other sleptons and search for a similar multi-tau final state, setting a lower limit on the chargino mass of 760 GeV in this model [135].

If sleptons are heavy, the chargino is assumed to decay to a W boson plus LSP, and the $\tilde{\chi}_2^0$ into Z plus LSP (WZ searches) or H plus LSP (WH searches). Searches for the WZ channel exploit events with either two same-sign leptons or three leptons, a single

lepton, or with no leptons (all-hadronic). If $\tilde{\chi}_2^0$ decays through a Higgs boson with $m_H = 125$ GeV, search strategies depend on the SM-like Higgs decay modes and branching fractions. Higgs bosons in the final state are identified by either two jets originating from bottom quarks ($h \to b\bar{b}$), two photons ($h \to \gamma\gamma$), or leptons from the WW, ZZ or $\tau\tau$ decay modes. Interpretations are also provided in scenarios where the $\tilde{\chi}_2^0$ does not decay exclusively into a Z or H boson and the LSP.

ATLAS has searched for $\tilde{\chi}_1^\pm\tilde{\chi}_2^0$ production in events with three leptons and missing momentum and wino-like chargino masses below 640 GeV are excluded for massless LSPs [143]. Results have also been reported for searches including same-sign lepton signal regions [144] and single lepton searches [138]. Limits are reduced in case of higgsino-like scenarios, with exclusion down to 210 GeV. CMS [142] excludes chargino masses below 650 GeV for massless LSPs using events with two leptons of the same sign, or with three or more leptons targeting final states with W and Z bosons. No constraints are set for $\tilde{\chi}_1^0$ masses above 300 GeV.

Additional constraints on the chargino mass are placed exploiting all-hadronic analyses. A wino-like (higgsino-like) chargino with mass up to 1060 (900) GeV is excluded by ATLAS [139] when the LSP mass is below 400 (240) GeV and the mass splitting is larger than 400 (450) GeV. CMS [140] excludes wino-like charginos with mass up to 970 GeV for massless LSP.

Searches exploiting the presence of b-jets are most sensitive in the high-mass chargino region if $\tilde{\chi}_2^0$ decays through a Higgs boson, while analyses targeting photon pair or leptonic decays provide the best sensitivity in the region of low masses. CMS [145] sets lower limits on the chargino mass up to 820 GeV for massless LSPs, but vanish for LSP masses above 350 GeV using events with at least one lepton and missing transverse momentum in the final state. Similar sensitivity is achieved by the ATLAS analyses [146–148].

In both the wino region (a characteristic of anomaly-mediated SUSY breaking models) and the higgsino region of the MSSM, the mass splitting between $\tilde{\chi}_1^\pm$ and $\tilde{\chi}_1^0$ is small. The chargino decay products are very soft and may escape detection. These compressed spectra are hard to detect, and have triggered dedicated search strategies. ATLAS [149] and CMS [150] have performed searches for charginos and neutralinos in a compressed mass spectrum using initial state radiation and two or three low-momentum (soft) leptons. For wino-like charginos, assuming degenerate $\tilde{\chi}_1^\pm$ and $\tilde{\chi}_2^0$, exclusion contours in the chargino-mass versus $\Delta m(\tilde{\chi}_1^\pm - \tilde{\chi}_1^0)$ plane are derived. As an example, such charginos are excluded by CMS (ATLAS) below 280 (240) GeV for $\Delta m(\tilde{\chi}_1^\pm - \tilde{\chi}_1^0) = 10$ GeV. Considering the higgsino model, the masses probed by CMS reach up to 215 GeV for a mass difference of 7.5 GeV and 150 GeV in the highly compressed region with a mass difference of 3 GeV. CMS has also searched for chargino-pair production through vector-boson-fusion [151], also targeting compressed mass spectra. Assuming degenerate $\tilde{\chi}_1^\pm$ and $\tilde{\chi}_2^0$, charginos with a mass below 112 GeV are excluded for $\Delta m(\tilde{\chi}_1^\pm - \tilde{\chi}_1^0) = 1$ GeV. CMS has published further searches for such compressed spectra with a soft tau lepton [152].

### 89.7.2 Exclusion limits on neutralino masses

In a considerable part of the MSSM parameter space, and in particular when demanding that the LSP carries no electric or color charge, the lightest neutralino $\tilde{\chi}_1^0$ is the LSP. If R-parity is conserved, such a $\tilde{\chi}_1^0$ is stable. Since it is weakly interacting, it will typically escape detectors unseen. Limits on the invisible width of the Z boson apply to neutralinos with a mass below 45.5 GeV, but depend on the Z-neutralino coupling. Such a coupling could be small or even absent; in such a scenario there is no general lower limit on the mass of the lightest neutralino [153]. In models with gaugino mass unification and sfermion mass unification at the GUT scale, a lower limit on the neutralino mass is derived from limits from direct searches, notably for charginos and sleptons, and amounts to 47 GeV [154]. Assuming a constrained model like the CMSSM, this limit increases to 50 GeV at LEP; however the strong constraints now set by the LHC increase such CMSSM-derived $\tilde{\chi}_1^0$ mass limits to well above a few hundred GeV (the latest reinterpretation only uses Run 1 data and indicates 200 GeV [155–
157]).

In gauge-mediated SUSY breaking models (GMSB), the LSP is typically a gravitino, and the phenomenology is determined by the nature of the next-to-lightest supersymmetric particle (NLSP). A NLSP neutralino will decay to a gravitino and a SM particle whose nature is determined by the neutralino composition. Final states with two high $p_T$ photons and missing momentum are searched for, and interpreted in gauge mediation models with bino-like neutralinos [158–164].

Assuming the production of at least two neutralinos per event, neutralinos with large non-bino components can also be searched for by their decay in final states with missing momentum plus any two bosons out of the collection $\gamma, Z, H$. A number of searches at the LHC have tried to cover the rich phenomenology predicted by the general gauge mediation model (GGM) where $\tilde{\chi}_1^0$ might decay with various BR in the $Z$ and $H$ decay modes [98, 147, 161, 165–175]. As an example, strong constraints for decay modes involving $Z$ and $H$ bosons arise from full-hadronic searches on higgsino-like scenarios: ATLAS [139] excludes higgsino production with decays into a massless gravitino LSP in a mass range between 450 (500) GeV and 940 (850) GeV assuming BR = 1 (0.5) for the $Z$ plus gravitino decay mode. Assuming only $H$ plus gravitino decays, CMS [176] excludes a higgsino mass range between 175 GeV and 1025 GeV. Assuming a mix of $\gamma, Z, H$ decays, a CMS search [164] in the photonic final state excludes higgsinos up to 1.05 TeV.

Heavier neutralinos, in particular $\tilde{\chi}_2^0$, have been searched for in their decays to the lightest neutralino plus a $\gamma$, a $Z$ boson or a Higgs boson. Limits on electroweak production of $\tilde{\chi}_2^0$ plus $\tilde{\chi}_1^\pm$ from all-hadronic, same-sign dilepton and trilepton analyses have been discussed in the section on charginos; the assumption of equal mass of $\tilde{\chi}_2^0$ and $\tilde{\chi}_1^\pm$ make the limits on chargino masses apply to $\tilde{\chi}_2^0$ as well. Multilepton analyses have also been used to set limits on $\tilde{\chi}_2^0\tilde{\chi}_3^0$ production; assuming equal mass and decay through light sleptons, limits are set up to 680 GeV for massless LSPs [177]. Again, compressed spectra with small mass differences between the heavier neutralinos and the LSP form the most challenging region.

In $\tilde{\chi}_2^0$ decays to $\tilde{\chi}_1^0$ and a lepton pair, the lepton pair invariant mass distribution may show a structure that can be used to measure the $\tilde{\chi}_2^0 - \tilde{\chi}_1^0$ mass difference in case of a signal [38]. This structure, however, can also be used in the search strategy itself, as demonstrated by ATLAS [178, 179] and CMS [98, 180].

Fig. 89.8 summarizes some of the most recent results from ATLAS and CMS for chargino pair or chargino and next-to-lightest neutralino pair productions comparing results obtained with various assumptions. The limits on weak gauginos in simplified models are also summarized in Table 89.3. Only results for promptly decaying charginos and neutralinos are included: limits from searches focusing on long-lived particle scenarios are reported in Section 89.9. A combination of several searches for the electroweak production of wins, bins, higgsinos, and sleptons has also been recently been presented by CMS [181].

Interpretations of the search results outside simplified models, such as in the phenomenological MSSM [182–187], show that the simplified model limits can translate into accurate constraints but must also be interpreted with care. Electroweak gauginos in models that are compatible with the relic density of dark matter in the Universe, for example, have particularly tuned mixing parameters and mass spectra, which are not always captured by the simplified models used.

### 89.7.3 Exclusion limits on slepton masses

The most model-independent searches for selectrons, smuons and staus originate from the LEP experiments [188]. Smuon production only takes place via s-channel $\gamma^*/Z$ exchange. Search results are often quoted for $\tilde{\mu}_R$, since it is typically lighter than $\tilde{\mu}_L$ and has a weaker coupling to the $Z$ boson; limits are therefore conservative. Decays are expected to be dominated by $\tilde{\mu}_R \to \mu\tilde{\chi}_1^0$, leading to two non-back-to-back muons and missing momentum. Slepton mass limits are calculated in the MSSM under the assumption of gaugino mass unification at the GUT scale, and depend on the mass difference between the smuon and $\tilde{\chi}_1^0$. A $\tilde{\mu}_R$

<table><caption>**Table 89.3:** Summary of weak gaugino mass limits in simplified models, assuming R-parity conservation. Masses in the table are provided in GeV. Further details about assumptions and corresponding analyses are discussed in the text. Details on constraints for compressed scenarios from soft-lepton searches are not represented in the table.</caption><thead><tr><th>Assumption</th><th>m<sub>χ</sub></th></tr></thead><tbody><tr><td>χ<sub>1</sub><sup>±</sup>, all Δm(χ<sub>1</sub><sup>±</sup>, χ<sub>1</sub><sup>0</sup>)</td><td>&gt; 92</td></tr><tr><td>χ<sub>1</sub><sup>±</sup> Δm &gt; 5, m<sub>ν</sub> &gt; 300</td><td>&gt; 103.5</td></tr><tr><td>χ<sub>1</sub><sup>±</sup>, m<sub>(ℓ,ν)</sub> = (m<sub>χ<sub>1</sub><sup>±</sup></sub> + m<sub>χ<sub>1</sub><sup>0</sup></sub>)/2</td><td></td></tr><tr><td>m<sub>χ<sub>1</sub><sup>0</sup></sub> ≈ 0</td><td>&gt; 1000</td></tr><tr><td>χ<sub>1</sub><sup>±</sup>, m<sub>χ<sub>1</sub><sup>0</sup></sub> &gt; 480</td><td>no LHC limit</td></tr><tr><td>χ<sub>1</sub><sup>±</sup>, m<sub>ℓ</sub> &gt; m<sub>χ<sub>1</sub><sup>±</sup></sub></td><td></td></tr><tr><td>m<sub>χ<sub>1</sub><sup>0</sup></sub> ≈ 0</td><td>&gt; 420 and 290 – 760</td></tr><tr><td>χ<sub>1</sub><sup>±</sup>, m<sub>χ<sub>1</sub><sup>0</sup></sub> &gt; 120</td><td>no LHC limit</td></tr><tr><td>m<sub>χ<sub>1</sub><sup>±</sup></sub> = m<sub>χ<sub>2</sub><sup>0</sup></sub>, m<sub>ℓ</sub> = (m<sub>χ<sub>1</sub><sup>±</sup></sub> + m<sub>χ<sub>1</sub><sup>0</sup></sub>)/2</td><td></td></tr><tr><td>m<sub>χ<sub>1</sub><sup>0</sup></sub> ≈ 0</td><td>&gt; 1450</td></tr><tr><td>m<sub>χ<sub>1</sub><sup>0</sup></sub> &gt; 1000</td><td>no LHC limit</td></tr><tr><td>m<sub>χ<sub>1</sub><sup>±</sup></sub> = m<sub>χ<sub>2</sub><sup>0</sup></sub>, m<sub>ℓ</sub><sub>R</sub> = (m<sub>χ<sub>1</sub><sup>±</sup></sub> + m<sub>χ<sub>1</sub><sup>0</sup></sub>)/2</td><td>flavor-democratic</td></tr><tr><td>m<sub>χ<sub>1</sub><sup>0</sup></sub> ≈ 0</td><td>&gt; 1150</td></tr><tr><td>m<sub>χ<sub>1</sub><sup>0</sup></sub> &gt; 700</td><td>no LHC limit</td></tr><tr><td>m<sub>χ<sub>1</sub><sup>±</sup></sub> = m<sub>χ<sub>2</sub><sup>0</sup></sub>, m<sub>τ</sub> = (m<sub>χ<sub>1</sub><sup>±</sup></sub> + m<sub>χ<sub>1</sub><sup>0</sup></sub>)/2</td><td>τ-dominated</td></tr><tr><td>m<sub>χ<sub>1</sub><sup>0</sup></sub> ≈ 0</td><td>&gt; 970</td></tr><tr><td>m<sub>χ<sub>1</sub><sup>0</sup></sub> &gt; 450</td><td>no LHC limit</td></tr><tr><td>m<sub>χ<sub>1</sub><sup>±</sup></sub> = m<sub>χ<sub>2</sub><sup>0</sup></sub>, m<sub>ℓ</sub> &gt; m<sub>χ<sub>1</sub><sup>±</sup></sub>, BF(WZ) = 1</td><td></td></tr><tr><td>m<sub>χ<sub>1</sub><sup>0</sup></sub> ≈ 0</td><td>&gt; 960</td></tr><tr><td>m<sub>χ<sub>1</sub><sup>0</sup></sub> &gt; 300</td><td>no LHC limit</td></tr><tr><td>m<sub>χ<sub>1</sub><sup>±</sup></sub> = m<sub>χ<sub>2</sub><sup>0</sup></sub>, m<sub>ℓ</sub> &gt; m<sub>χ<sub>1</sub><sup>±</sup></sub>, BF(WH) = 1</td><td></td></tr><tr><td>m<sub>χ<sub>1</sub><sup>0</sup></sub> ≈ 0, wino-like χ<sub>1</sub><sup>±</sup></td><td>&gt; 1060</td></tr><tr><td>m<sub>χ<sub>1</sub><sup>0</sup></sub> ≈ 0, higgsino-like χ<sub>1</sub><sup>±</sup></td><td>&gt; 900</td></tr><tr><td>m<sub>χ<sub>1</sub><sup>0</sup></sub> &gt; 400(240)</td><td>no LHC limit wino(higgsino)-like</td></tr></tbody></table>

with a mass below 94 GeV is excluded for $m_{\tilde{\mu}_R} - m_{\tilde{\chi}_1^0} > 10$ GeV. The selectron case is similar to the smuon case, except that an additional production mechanism is provided by t-channel neutralino exchange. The $\tilde{e}_R$ lower mass limit is 100 GeV for $m_{\tilde{\chi}_1^0} < 85$ GeV. Due to the t-channel neutralino exchange, $\tilde{e}_R\tilde{e}_L$ pair production was possible at LEP, and a lower limit of 73 GeV was set on the selectron mass regardless of the neutralino mass by scanning over MSSM parameter space [189]. The potentially large mixing between $\tilde{\tau}_L$ and $\tilde{\tau}_R$ not only makes the $\tilde{\tau}_1$ light, but can also make its coupling to the $Z$ boson small. LEP lower limits on the $\tilde{\tau}$ mass range between 87 and 93 GeV depending on the $\tilde{\chi}_1^0$ mass, for $m_{\tilde{\tau}} - m_{\tilde{\chi}_1^0} > 7$ GeV [188].

At the LHC, pair production of sleptons is not only heavily suppressed with respect to pair production of colored SUSY particles but the cross section is also almost two orders of magnitude smaller than the one of pair production of charginos and neutralinos. With the full data sets of Run 1 and Run 2, however, ATLAS and CMS have surpassed the sensitivity of the LEP analyses under certain assumptions.

ATLAS and CMS have searched for direct production of selectron pairs and smuon pairs at the LHC, with each slepton decaying to its corresponding SM partner lepton and the $\tilde{\chi}_1^0$ LSP. In simplified models, ATLAS [133] and CMS [180] set lower mass limits on sleptons of 700 GeV for degenerate $\tilde{\ell}_L$ and $\tilde{\ell}_R$, for a massless $\tilde{\chi}_1^0$ and assuming equal selectron and smuon masses, as shown in Fig. 89.9. The limits deteriorate with increasing $\tilde{\chi}_1^0$ mass due to decreasing missing momentum and lepton momentum. As a consequence, no limits are set for $\tilde{\chi}_1^0$ masses above 400 GeV. Limits are also derived without the assumption of slepton mass degeneracy [133,190]. A dedicated search for sleptons with small mass difference between $\tilde{\ell}$ and $\tilde{\chi}_1^0$ is performed by ATLAS [149] demanding
<figure><img src="image_371.png" /><figcaption>**Figure 89.8:** LHC exclusion limits on chargino and neutralino masses in a number of simplified models. Left: CMS limits on chargino and neutralino masses for pair production of charginos, pair production of heavier neutralinos, or pair production of chargino and neutralino, under a variety of assumptions including light sleptons mediating the decays. Right: ATLAS limits on chargino and neutralino masses for pair production of chargino and neutralino, under the assumption of decoupled sleptons, and chargino/neutralino decay through on- or off-shell W and Z or H.</figcaption></figure>

<figure><img src="image_372.png" /><figcaption>**Figure 89.9:** LHC exclusion limits on slepton (selectron, smuon and stau) masses, assuming equal masses of selectrons and smuons, degeneracy of $\tilde{\ell}_L$ and $\tilde{\ell}_R$, and a 100% branching fraction for $\tilde{\ell} \to \ell\tilde{\chi}_1^0$.</figcaption></figure>

the presence of ISR jets. Although no sensitivity is achieved for intermediate mass differences between the slepton and the LSP, slepton masses up to 260 GeV are excluded for slepton-LSP mass differences of 10-20 GeV. ATLAS [137] followed up with a dedicated search in the challenging regime where the mass difference between the slepton and the LSP is close to the W mass. Sleptons with masses up to 150 GeV are excluded for a mass-splitting between the sleptons and the LSP of 50 GeV.

ATLAS and CMS have also searched for $\tilde{\tau}$-pair production. In simplified models, ATLAS excludes $\tilde{\tau}$ masses between 120 and 390 GeV assuming light $\tilde{\chi}_1^0$, combining the production of degenerate left- and right-handed $\tilde{\tau}$s [191]. The CMS analysis [192] covers lower masses and closes the mass gap with LEP. In a search for hadronic final states, CMS [193] excludes purely left-handed $\tilde{\tau}$s with masses between 115 and 340 GeV for massless LSP. A recent ATLAS search [194] for stau pair production in the fully hadronic final state excludes, for a massless LSP, degenerate left- and right-handed $\tilde{\tau}$s with masses up to 480 GeV, purely left-handed $\tilde{\tau}$s with masses up to 410 GeV and purely right-handed $\tilde{\tau}$s with masses up

to 330 GeV.

In gauge-mediated SUSY breaking models, sleptons can be (co-)NLSPs, i.e., the next-to-lightest SUSY particles and almost degenerate in mass, decaying to a lepton and a gravitino. This decay can either be prompt, or the slepton can have a non-zero lifetime. Combining several analyses, lower mass limits on $\tilde{\mu}_R$ of 96.3 GeV and on $\tilde{e}_R$ of 66 GeV are set for all slepton lifetimes at LEP [195]. In a considerable part of parameter space in these models, the $\tilde{\tau}$ is the NLSP. The LEP experiments have set lower limits on the mass of such a $\tilde{\tau}$ between 87 and 97 GeV, depending on the $\tilde{\tau}$ lifetime. ATLAS and CMS have searched for final states with $\tau$s, jets and missing transverse momentum, and have interpreted the results in GMSB models setting limits on the model parameters [196, 197]. CMS has interpreted a multilepton analysis in terms of limits on gauge mediation models with slepton NLSP [198]. CDF has put limits on gauge mediation models at high $\tan\beta$ and slepton NLSP using an analysis searching for like-charge light leptons and taus [199].

The invisible width of the Z boson puts a lower limit on the
<figure><img src="image_373.png" /><figcaption>**Figure 89.10:** Summary of 95% C.L. limits on sparticle masses in RPV models, for $\lambda_{121}$ (left), and for $\lambda_{313}$ (right). The vertical red line represents the direct production mass bound when the sparticle under consideration is the LSP.</figcaption></figure>

sneutrino mass of about 45 GeV. Tighter limits are derived from other searches, notably for gauginos and sleptons, under the assumption of gaugino and sfermion mass universality at the GUT scale, and amount to approximately 94 GeV in the MSSM [200].

The limits on sleptons in simplified models are summarized in Table 89.4.

<table><caption>**Table 89.4:** Summary of slepton mass limits from LEP and LHC, assuming R-parity conservation and 100% branching fraction for $\tilde{\ell} \to \ell \chi_1^0$. Masses in this table are provided in GeV.</caption><thead><tr><th>Assumption</th><th>m<sub>˜&ell;</sub></th></tr></thead><tbody><tr><td>m&#x0304;<sub>R</sub>, Δm(m&#x0304;<sub>R</sub>, χ<sub>1</sub><sup>0</sup>) > 10</td><td>> 94</td></tr><tr><td>e&#x0304;<sub>R</sub>, Δm(e&#x0304;<sub>R</sub>, χ<sub>1</sub><sup>0</sup>) > 10</td><td>> 94</td></tr><tr><td>e&#x0304;<sub>R</sub>, any Δm</td><td>> 73</td></tr><tr><td>τ&#x0304;<sub>R</sub>, Δm(τ&#x0304;<sub>R</sub>, χ<sub>1</sub><sup>0</sup>) > 7</td><td>> 87</td></tr><tr><td>ν&#x0304;<sub>e</sub>, Δm(ν&#x0304;<sub>e</sub>, χ<sub>1</sub><sup>0</sup>) > 10</td><td>> 94</td></tr><tr><td>m<sub>e&#x0304;<sub>L,R</sub></sub> = m&#x0304;<sub>L,R</sub>, m&#x0304;<sub>1</sub><sup>0</sup> &approx; 0</td><td>> 700</td></tr><tr><td>m&#x0304;<sub>1</sub><sup>0</sup> &gt;&approx; 400</td><td>no LHC limit</td></tr><tr><td>m&#x0304;<sub>L</sub> = m&#x0304;<sub>R</sub>, m&#x0304;<sub>1</sub><sup>0</sup> &approx; 0</td><td>> 480</td></tr><tr><td>m&#x0304;<sub>1</sub><sup>0</sup> &gt;&approx; 200</td><td>no LHC limit</td></tr><tr><td>m&#x0304;<sub>L</sub>, m&#x0304;<sub>1</sub><sup>0</sup> &approx; 0</td><td>> 410</td></tr><tr><td>m&#x0304;<sub>1</sub><sup>0</sup> &gt;&approx; 150</td><td>no LHC limit</td></tr><tr><td>m&#x0304;<sub>R</sub>, m&#x0304;<sub>1</sub><sup>0</sup> &approx; 0</td><td>> 330</td></tr><tr><td>m&#x0304;<sub>1</sub><sup>0</sup> &gt;&approx; 100</td><td>no LHC limit</td></tr></tbody></table>

## 89.8 Exclusion limits on promptly decaying sparticle masses in RPV scenarios

The large variety of possible configurations for non-zero $\lambda$, $\lambda'$ and $\lambda''$ couplings in RPV SUSY scenarios leads to a huge number of potential signatures to be investigated in the strong and electroweak sectors. The lightest supersymmetric particle can be any particle, although most of the models considered at the LHC assume that to be the lightest neutralino, and can decay promptly or displaced. Cascade decays of sparticles down to the LSP can also be prompt or displaced depending on the strength of the couplings involved, although typically such cascades are dominated by gauge couplings which are prompt. In this section,

we review dedicated searches for scenarios with small non-zero RPV couplings but still large enough to allow prompt decays of the sparticles involved (see Section 89.9 for examples of dedicated signatures searches for non-prompt decays).

Prompt RPV gluino and squark decays are searched for in final states generally characterised by the presence of multiple leptons or jets and moderate or no missing transverse momentum.

Searches in multilepton final states [90, 166, 201–205] set lower gluino mass limits between 1 and 2.5 TeV for decays mediated by $\lambda$ and $\lambda'$ couplings, with strong dependence on the couplings involved, the neutralino mass and the lepton flavor. Multijet final states have been used to search for fully hadronic gluino decays involving $\lambda''$ couplings, by CDF [206], ATLAS [96, 207–210] and CMS [211–213]. Lower gluino mass limits range between 600 and 2000 GeV depending on neutralino mass and flavor content of the final state. ATLAS [214] also searched in events characterised by high jet multiplicity, at least one isolated light lepton and either zero or at least three b-tagged jets, reaching as high as 2.4 TeV in gluino mass. If gluinos decay via top squark into *tbd* final states through a non-zero $\lambda''_{313}$ coupling, a search for same-sign/three-lepton events can be effective. ATLAS [90] sets a lower limit of 1.65 TeV for a top squark with a mass below 1.45 TeV.

RPV production of single squarks via a $\lambda'$-type coupling has been studied at HERA. In such models, a lower limit on the squark mass of the order of 275 GeV has been set for electromagnetic-strength-like couplings $\lambda' = 0.3$ [215]. At the LHC, prompt [202, 205, 216] R-parity violating squark decays have been searched for, with mass limits very model-dependent.

Dedicated searches for RPV top squarks (at production and/or in decays) have been carried out in the past decades. Production of single top squarks has been searched for at LEP, HERA, and the Tevatron. For example, an analysis from the ZEUS collaboration [217] makes an interpretation of its search result assuming top squarks to be produced via a $\lambda'$ coupling and decay either to $b\tilde{\chi}_1^{\pm}$ or R-parity-violating to a lepton and a jet. Limits are set on $\lambda'_{131}$ as a function of the top squark mass in an MSSM framework with gaugino mass unification at the GUT scale. CMS and ATLAS have performed several searches for top squarks using a variety of multilepton and/or multijet final states and Run 2 data. The $\lambda'$-mediated top squark decay $\tilde{\ell} \to b\ell$ has been studied by ATLAS for prompt decays [218], and by ATLAS and CMS for non-prompt decays [219–221], setting limits up to 1.4–1.6 TeV in simplified models for this mode. CMS also searched for the $\lambda'$-mediated