Supersymmetry and the LHC Run 2

## Contents

* 1 What is supersymmetry, and why is it relevant?
* 2 Models of SUSY
* 3 Where and how we look for SUSY
* 4 Results of SUSY searches: an overview
* 5 Outlook
* 6 References

## [edit] What is supersymmetry, and why is it relevant?

The idea of exploiting potential symmetries of a physical system to impose constraints on the system's Lagrangian, and therefore on the equations of motion, has been a very fertile one. It has been used successfully in virtually all branches of physics, from classical mechanics to contemporary applications to physical modelling in all disciplines. Modern particle physics is itself based on the symmetry properties imposed by special relativity and the principle of gauge invariance, which led to the formalisation of the electroweak theory and, eventually, to the Standard Model of particle physics (SM - This book contains an excellent introduction to these topics).

Supersymmetry (SUSY) is an additional space-time symmetry of nature. In fact, it is the only way to extend space-time symmetries to internal symmetries. If SUSY is a good symmetry for a physical system, then the system is invariant for an operator that transforms all bosons into fermions and vice versa, thus restoring a symmetry between the matter and force fields of the SM. The Lagrangian of any particle physics model can be made supersymmetry-invariant, by introducing an appropriate number of bosonic and fermionic degrees of freedom: to supersymmetrise the Standard Model, one needs to introduce two scalar fields for every fermion of spin 1/2. Two physical scalar particles (indicated with $\tilde{f}_{\mathrm{1}}$ and \\(\tilde{f}_{\mathrm{2}}\\), in order of increasing mass) correspond to the two spin degrees of freedom of $f$ (spin-up and spin-down, or left- and right-chirality). In supersymmetric models, a single Higgs doublet is not sufficient to give mass to the up-type and down-type fermions without breaking SUSY, therefore at least a second Higgs doublet is introduced, leading to the prediction of five Higgs boson states. The supersymmetric partners of these Higgs boson states (the higgsinos, $\tilde{\mathbf{H}}$) mix with the supersymmetric partners of the $B$ (the bino, $\tilde{B}$) and $ \mathbf{W} $ (the wino triplet, $\tilde{\mathbf{W}}$) SM fields to give eight electroweakino states, four neutralinos $\tilde{\chi}_1^0, \dots, \tilde{\chi}_4^0$ and two pairs of charginos $\tilde{\chi}_1^{\pm},\tilde{\chi}_2^{\pm}$.

In a model for which supersymmetry is an exact symmetry of nature, all quantum numbers of the supersymmetric partners are the same as those of their standard counterparts. This implies, in particular, that the gauge couplings of SM particles and their supersymmetric partners are the same. This means also that the masses of the supersymmetric particles should be equal to those of their standard partners. This is clearly not the case (or we would have had a supersymmetric electron, or selectron with a mass of 511 keV): supersymmetry must be broken. There is no general guiding principle stating how supersymmetry is broken. In the most general case, one needs to add to the Lagrangian all terms allowed by the Poincarè group and gauge symmetry. Therefore, while the supersymmetry-conserving part of the Lagrangian is completely determined by the structure of the SM, the supersymmetry-violating part of the Lagrangian (the so-called soft-SUSY-breaking terms) introduces a large number of new parameters in the model: for example, mass terms for the scalar superpartners of the fermions are not explicitly forbidden by the $\mathrm{SU(2)}\times U(1)$ symmetry; likewise, mixing terms between the scalars are in principle allowed.

The supersymmetrisation of the Standard Model leads to more than doubling its particle content. The model that arises from minimal additions to the SM Lagrangian to make it supersymmetric is known as the Minimal Supersymmetric extension of the Standard Model, or MSSM. Of course, it is conceivable to first extend the SM to include new phenomena, and then modify the Lagrangian to make it supersymmetric, to obtain non-minimal SUSY extensions of the SM.

The arbitrariness introduced by the soft-SUSY-breaking terms, and the possibility of non-minimal SUSY extensions of the SM lead to the potential definition of an infinite number of supersymmetric models. The question is SUSY excluded? is therefore technically ill-posed: SUSY is a broken symmetry of a physical system, and, as such, cannot be excluded. Specific supersymmetric models, or even classes of models defined by a certain theoretical paradigm, can certainly be excluded.

For a comprehensive theoretical/phenomenological review of the subject, the interested reader should consult Martin (1998). An extended review of the relevant experimental results is available at Adam (2022).

## [edit] Models of SUSY

The longevity of SUSY as a means of extending the Standard Model (despite decades of negative results of SUSY searches) relies on its appeal, both from a theoretical and a phenomenological point of view. From a theoretical point, it was realised early on that making SUSY a local symmetry would tame some of the divergencies arising in previous attempts to include gravity in a quantum theory. A lot of the early theoretical development of SUSY happened thanks to the exploration of these supergravity theories.

If you ask the average high-energy physicist, they will mention three arguments to support the need of supersymmetric extensions to the Standard Model:

1. SUSY can solve the hierarchy problem. Fermionic loops induce quantum corrections to the mass of scalars that grow with the square of the energy scale involved. If the SM is extended to include a higher energy scale (possibly the GUT or Planck scale), this applies to the Higgs boson, whose natural mass becomes of the order of the higher energy scale, rather than the electroweak scale. SUSY cancels these quadratically growing fermionic corrections with equivalent corrections (opposite in sign) from the corresponding superpartners.
2. SUSY may introduce new neutral stable particles. If they interact only weakly with ordinary matter, they may be a suitable candidate for explaining the cold dark matter relic density.
3. In the MSSM, the evolution of the gauge couplings is such that their values unify at a scale not far from the Planck mass.

Figure 1: Direct pair-production cross sections (values agreed upon by the LHC SUSY Cross Section Working Group [1]

Most of the experimental effort before the LHC focused on models that could address all of the points above, starting from specific mechanisms to break SUSY. Models that belong to this family are the constrained MSSM (cMSSM, or mSUGRA), the Anomaly Mediated SUSY Breaking (AMSB), the Gauge Mediated Supersymmetry Breaking (GMSB). A more modern approach to SUSY searches has been developed by the community, with the advent of simplified models. A simplified model is one where only a few supersymmetric particles are assumed to be relevant to the experimental phenomenology through their production and decay. The estimate of the production cross sections is performed under the assumption that any supersymmetric particle other than those considered in the model contributes in a negligible way. Figure 1 shows the state-of-the-art cross sections for the pair production of different SUSY particles under these assumptions.

The striking benefit of the simplified model approach is that one can design and optimise searches that target specific topologies and kinematical domains. For example, a possible simplified model for direct stau pair-production may assume that the only relevant particles are the supersymmetric partner of the left-handed chiral component of the tau lepton, and a neutralino LSP (see Figure 2(f)). The only decay process will then be $\tilde{\tau}_1\rightarrow \tau \tilde{\chi}_1^0$, and the topology of the final state will contain two $\tau$ and invisible particles, yielding a clear experimental target.

The list of simplified models that is considered by the LHC collaborations is quite extensive and complete. Many of them are inspired by one or more of the arguments 1.-3. above.

* In the MSSM, the mass of the SM-like Higgs boson is determined at tree level by the same mass parameter that determines the mass of the higgsinos. The size of the one-loop corrections is largely determined by the mass and mixing parameters of the partners of the top quark (the stops). SUSY models that solve the hierarchy problem tend to have higgsinos with masses of maximum a few hundred GeV, and stops with masses of maximum 1-2 TeV.

* Models featuring a good potential dark matter candidate require the conservation of a multiplicative quantum number called R-parity. SM particles have R-parity of 1, while SUSY particles have R-parity of $-1$. R-parity conservation implies that SUSY particles always appear in even numbers at a production or decay vertex. In a R-parity conserving model, SUSY particles are produced in pairs. Likewise, there is always an odd number of SUSY particles in a decay of a SUSY particle: as a consequence, the lightest supersymmetric particle (LSP) is stable. If it is electrically neutral and interacts only via the weak interactions, it is a potential dark matter candidate.

* Gauge coupling unification can be obtained even in models that solve the hierarchy problem requiring a significant level of fine tuning between the model parameters. In split SUSY models, for example, the supersymmetric partners of the fermions typically have very high mass. This implies that the only particles which are energetically accessible at the LHC may be the charginos, neutralinos and gluinos, rather than the squarks and sleptons.

When interpreting results, simplified models have been used to determine exclusions on sparticle masses assuming the corresponding simplified model cross section, together with limits on the cross sections for specific sparticle masses. More extensive parameter scans have also been performed by the LHC collaborations: in this case, after simplifying the MSSM parameter space with well-justified assumptions (leading to the phenomenological MSSM, or pMSSM), the combined sensitivity of the searches designed on simplified models is assessed. These scans were produced by the collaborations in Run 1 and again using the results of Run 2.

## [edit] Where and how we look for SUSY

If SUSY is a (broken) symmetry of nature, then it could manifest itself in different ways. The plethora of new particles introduced by SUSY would imply many new Feynman diagrams to be taken into account when deriving the theoretical prediction for a given experimental test. This would have three main effects at the LHC experiments:

* SUSY particle contributions to known processes would modify the SM predictions, leading to deviations from the SM-only hypothesis in precision measurements, like production and decay rates, angular distributions, etc. A famous and compelling example (at the time of writing) is the prediction for the gyromagnetic factor of the muon: the existence of relatively a low mass $\tilde{\mu}$ could modify the SM prediction and accomodate the discrepancy between the measured and observed value Chackraborti (2006). The SUSY explanation for the muon $g-2$ anomaly is historically one of those explored with great detail, although other explanations (including that of a systematic effect in the SM theoretical prediction not fully accounted for) exist. The vast phenomenology of even the simplest SUSY extension to the SM yields a large number of precision observables potentially sensitive to SUSY: such a list includes many flavour, precision electroweak, top-quark and Higgs boson physics observables.
* The more complex Higgs sector provides a much richer Higgs-related phenomenology. The MSSM predicts the existence of five Higgs boson states, that can be directly produced, or interfere with other production processes.
* The supersymmetric partners can be directly produced (typically in pairs) in particle collisions. They would decay to stable SM particles and, in case of R-parity conserving models, to the LSP. Generally speaking, SUSY particle production would lead to new experimental signatures, not necessarily common in the SM.

Although the impact of precision measurements and Higgs-boson-related searches on the SUSY landscape is remarkable, this paper focuses on the third category above: the direct production of supersymmetric particles.

Generally speaking, there are three categories of SUSY analyses, depending on the assumptions on the structure of the SUSY model considered.

* R-parity conserving SUSY: if R-parity is conserved, then the LSP needs to be electrically neutral and insensitive to the strong force (to justify the fact that such a stable particle has not been detected so far). The experimental consequence is that the LSP will escape the LHC detectors without interacting with them. In general, SUSY particle production events will therefore have an imbalance of momentum in the plane transverse to the beam. Such missing transverse momentum ($E_{\mathrm{T}}^{\mathrm{miss}}$) is a key signature of R-parity conserving (RPC) SUSY production. The vast majority of RPC SUSY analyses heavily exploit the presence of $E_{\mathrm{T}}^{\mathrm{miss}}$ as a mean of triggering and characterising the signal while effectively rejecting the background coming from SM particle production. It is certainly conceivable to have a RPC model realised in nature, and producing only limited amounts of missing transverse momentum: examples are the so-called compressed models, where the large-mass pair-produced particle is almost degenerate in mass with the LSP, which is therefore produced with limited momentum. Other examples come from the class of models known as stealth SUSY. These models are typically targeted with dedicated search strategies: for example, compressed models can be effectively target considering topologies where the pair-produced particle system recoils against one or more jets.

* R-parity violating models: if R-parity is not conserved, SUSY particles can decay into only SM particles. R-parity violating (RPV) couplings in the Lagrangian are typically assumed to be small, to preserve consistency with existing constraints from lepton and baryon number conservation. This means that the RPV couplings become phenomenologically relevant only in the absence of competing RPC decays. The most important phenomenological consequence of the introduction of the RPV couplings is that the LSP is not stable anymore. The key RPC signature of missing transverse momentum does not apply to events of RPV SUSY models. Instead, typical analyses focus on identifying resonant signals from SUSY particle decays, or non-resonant excesses for multi-particle production. There is often a significant level of overlap between RPV SUSY signatures and other models of new phenomena. For example: if the supersymmetric partner of the top quark (the stop) can decay RPV into a $b$-quark and a $\tau$-lepton, the final state topology (two $b$-$\tau$ resonances of identical mass) would be identical to that of, e.g., a third-generation scalar leptoquark.

* Models with long-lived SUSY particles: in certain regions of the parameter space, SUSY particles can become long-lived (for example, gluinos in split SUSY may become long-lived if the mass of the squarks they decay into is very high). Depending on which SUSY particles are long-lived, their lifetime and the decay products, these signatures give rise to a number of compelling experimental challenges for the LHC experiments, designed with signatures from promptly decaying particles in mind.

## [edit] Results of SUSY searches: an overview

Looking at the values of the cross sections displayed in , one can immediately gather a feeling for how the search for SUSY has evolved with increasing integrated luminosity at the LHC. The LHC general purpose experiments (ATLAS and CMS) have first gained sensitivity to particles produced via the strong interactions (gluinos and squarks): they have been the main target of the LHC Run 1 (at a centre of mass energy of $\sqrt{s} = 8\ \mathrm{TeV}$), and, in general, the limits on their masses set with the Run 2 data (at $\sqrt{s} = 13 \ \mathrm{TeV}$) range from about $1-1.5\ \mathrm{TeV}$ (for a single generation of squarks) to about $2-2.5\ \mathrm{TeV}$ (for gluinos). On the other hand, while ATLAS and CMS managed to achieve some sensitivity to winos, selectrons and smuons already using the Run 1 data, they managed to comprehensively target the sparticles produced via the electroweak interactions (winos, binos, higgsinos and sleptons) only using Run 2 data: for staus and higgsinos, the first limits have appeared during Run 2, and they are expected to evolve quickly with Run 3 and high luminosity upgrades of the LHC.

Strong production: Gluinos and squarks are pair-produced via diagrams which are equivalent to SM QCD diagrams. The relevance of the different Feynman diagrams (and also the level of interference between different diagrams) is completely determined by the squark and gluino masses. On the other hand, (and, again, equivalently to the SM) the decay of squarks and gluinos depends not only on the mass hierarchy of the strong sector, but also on the masses and parameters of the electroweak sector. If RPC is assumed, eventually all decay chains will have to end with the production of an LSP.

Figure 2: Examples diagrams of SUSY particle production in proton-proton collisions.

To avoid making the discussion too abstract and obscure, let's consider a specific example. Let's assume a RPC SUSY model is realised in nature. The masses of the relevant particles are: gluino mass $m_{\tilde{g}} = 2$ TeV, eight-fold degenerate (the four first- and second-generation squarks, with two states corresponding to the SM quark chirality states) squark mass $m_{\tilde{q}} = 1.5$ TeV, a single bino-like stable neutralino LSP $m_{\tilde{\chi}_1^0} = 100$ GeV. In this model, production of $\tilde{g}\tilde{g}$, $\tilde{g}\tilde{q}$ and $\tilde{q}\tilde{q}$ will take place at the LHC energies. Because of the strong coupling and the availability of a lighter squark state, the gluino will decay via $\tilde{g}\rightarrow q \tilde{q}$ (see diagram a) in Figure 2). The squarks, in turn, cannot decay via strong interactions (no lighter states available for the transition), therefore they will eventually decay via an electroweak interaction to the bino state with $\tilde{q} \rightarrow q \tilde{\chi}_1^0$. The final state will always contain two $\tilde{\chi}_1^0$, yielding abundant $E_{\mathrm{T}}^{\mathrm{miss}}$ (given the mass gap with the squarks, the $\tilde{\chi}_1^0$ will have a large momentum). Depending on the production process, the final state will contain four (for $\tilde{g}\tilde{g}$), three (for $\tilde{g}\tilde{q}$) or two (for $\tilde{q}\tilde{q}$) high transverse momentum $p_{\mathrm{T}}$ jets.

It becomes clear already from this discussion that strong production is the realm of final states containing $E_{\mathrm{T}}^{\mathrm{miss}}$ and different jet multiplicities. Leptons can be present if more intermediate electroweak states exist with masses smaller than those of the pair produced particles.

Figure 3: Gluino pair-production summary plot of mass exclusion limits in the gluino-neutralino plane in ATLAS [2]. Each curve refers to a different decay mode of the gluino according to the legend.

Figure 3 shows a summary of the limits extracted by the ATLAS collaboration in simplified models of gluino production. In these simplified models, it is assumed that the only accessible SUSY particles are the gluinos themselves, and one or more electroweak states. Therefore, the only production process taking place through the strong interactions is $\tilde{g}\tilde{g}$. Different curves refer to either different assumptions on the gluino decay chain, or different analysis results. The first message of the plot is the mass scale of the exclusion of gluinos: focusing on a neutralino mass of $m_{\tilde{\chi}_1^0} = 0$, exclusions range from $m_{\tilde{g}} > 2$ TeV to $m_{\tilde{g}} > 2.4$ TeV.

It is also instructive to look more in detail at some of the models used for this plot. A zero-lepton analysis (in red in the plot), comparing the yields for selections at different jet multiplicities, large $E_{\mathrm{T}}^{\mathrm{miss}}$, and large $m_{\mathrm{eff}}$ (defined as the scalar sum of $E_{\mathrm{T}}^{\mathrm{miss}}$ and the $p_{\mathrm{T}}$ of all jets), provides the best sensitivity for a simplified model of gluino pair-production followed by the decay $\tilde{g}\rightarrow q\tilde{q}^* \rightarrow qq\tilde{\chi}_1^0$ ($\tilde{q}^*$ indicates a squark with mass much larger than that of the gluino).

Equivalent models are considered (in pink and purple), where the $\tilde{q}$ is not the superpartner of a first- or second-generation squark, but rather of a third-generation one, implicitly assuming that the third-generation squarks have significantly smaller masses than those of the first and second generation. This is an expected consequence of argument i) mentioned above: the stop mass is a key parameter in assessing the level of fine tuning between the bare Higgs boson mass and its quantum corrections, and a small fine tuning requires a stop mass within a few TeV. In these models, the SM quarks produced in the decay of the gluinos are tops and bottoms, leading to the characterising signature of jets, $b$-jets, $E_{\mathrm{T}}^{\mathrm{miss}}$ and possibly leptons.

Models involving $W$ and $Z$ bosons in the decay of the gluino stem from scenarios where more intermediate electroweak states are available: they imply longer gluino decay chains, leading to higher final state object multiplicities with, on average, lower $p_{\mathrm{T}}$ ((b)).

Most lines show a weaker limit close to the line where $m_{\tilde{g}} = m_{\tilde{\chi}_1^0}$. In this compressed regime, the neutralinos (and, in general, all final state objects) are produced with low momentum, leading typically to a lower signal acceptance of the kinematic selection. The yellow line behaves very differently from the others. It refers to a model where the neutralino is not stable, but it can rather decay to a low-mass invisible stable gravitino $\tilde{G}$ and a photon. The $E_{\mathrm{T}}^{\mathrm{miss}}$ is proportional to the mass gap between the $\tilde{\chi}_1^0$ and the $\tilde{G}$, and it is therefore maximum if $m_{\tilde{g}} = m_{\tilde{\chi}_1^0}$.

Many of the experimental approaches to gluino searches typically work for squark searches as well: similar final states (although with lower jet multiplicities) are often produced under similar assumptions. What drives the difference in sensitivity between gluino and squark pair-production is mainly the different production cross sections, which, in the case of the squarks, depend on the assumed multiplicity of squark flavours. Typical exclusion limits for a $\tilde{\chi}_1^0$ LSP with $m_{\tilde{\chi}_1^0} = 0$ range between about $m_{\tilde{q}} > 1$ TeV and $m_{\tilde{q}} > 1.8$ TeV (depending on the mass hierarchy and electroweak sector) if an eight-fold mass degeneracy is assumed, but they can be as low as a few hundred GeV if the production of a single squark is assumed.

The case of third-generation squark pair-production deserves to be singled out, both because of its connection with the hierarchy problem and naturalness. The keyword for third-generation squark searches is $b$-jets: unless flavour violation is assumed, $b$-quarks will be produced as part of the decay chain, giving a very clear experimental handle to these searches.

Figure 4: Stop limits for two- and three-body stop decay ([3])

Figure 5: Stop limits for the four-body decay ([4])

Even in its simplest possible decay mode in models with a neutralino LSP, $\tilde{t}_1 \rightarrow t^{(*)} \tilde{\chi}_1^0$ ((c)), the strategy for stop pair-production search is relatively complex: because of the large top-quark mass, and depending on the mass splitting between the $\tilde{t}_1$ and the $\tilde{\chi}_1^0$, on-shell top quarks may or may not be present in the final state. Figure 4(a) summarises the results from the CMS collaboration, assuming that the branching ratio of $\tilde{t}_1 \rightarrow t^{(*)} \tilde{\chi}_1^0$ is 100\%. Different regions are clearly visible for $\Delta m\left(\tilde{t}_1,\tilde{\chi}_1^0\right) > m_{\mathrm{top}}$ (often referred to as two-body stop decay), $m_{W} + m_{b} < \Delta m\left(\tilde{t}_1,\tilde{\chi}_1^0\right) < m_{\mathrm{top}}$ (three-body stop decay). Figure 5(b) summarises the results of a search for stop production in the compressed region $\Delta m\left(\tilde{t}_1,\tilde{\chi}_1^0\right) < m_{W} + m_{b}$ (four-body stop decay) in a single lepton final state.

Electroweak production: Charginos, neutralinos and sleptons can be produced in proton--proton collisions via the electroweak interactions. Because of this, their production cross section is significantly lower than that for gluinos and squarks. Charginos and neutralinos are the mass eigenstates that arise from the mixing of the eigenstates of the electroweak interactions (the bino, the three winos and the four higgsinos states). An additional complication in the design of suitable simplified models of chargino and neutralino production (in the following referred to as electroweakinos) is the fact that the strength of the interaction with the SM fermions and corresponding superpartners depends on the composition of the mass eigenstates in terms of the interaction eigenstates. This affects both the production cross section and the decay branching fractions of the electroweakinos. For example, a pure wino will interact with only the left handed chirality component of the SM fermions and superpartners. On top of that, because of the structure of the electroweakino mixing matrix, precise relations between the electroweakino masses exist depending on the values of the bino, wino, higgsino masses (indicated as $M_1$, $M_2$ and $\mu$ respectively) and other parameters of the electroweak sector (for example, the angle between the two Higgs field complex doublets in the MSSM). A few benchmark paradigms have been therefore assumed as a guideline to design the search analyses and to extract the electroweakino mass limits. A well-known benchmark is a RPC one where it is assumed that the LSP is a bino-like $\tilde{\chi}_1^0$, and the only other SUSY state within reach is a wino-like state, yielding a pair of charginos $\tilde{\chi}_1^{\pm}$ and a $\tilde{\chi}_2^0$ nearly degenerate in mass. The interesting production channels are therefore $\tilde{\chi}_1^+\tilde{\chi}_1^-$ and $\tilde{\chi}_1^{\pm}\tilde{\chi}_2^0$. The wino states will then transition to the LSP via $\tilde{\chi}_1^{\pm} \rightarrow W^{\pm} \tilde{\chi}_1^0$ and $\tilde{\chi}_2^0 \rightarrow Z\tilde{\chi}_1^0$ or $\tilde{\chi}_2^0 \rightarrow h\tilde{\chi}_1^0$ ($h$ representing a SM like Higgs boson). The final state is then characterised by the presence of two gauge or Higgs bosons and the $\tilde{\chi}_1^0$s ((d) and (e)). Such final states have been historically targeted by analyses looking for multiple leptons from the decay of the vector bosons and $E_{\mathrm{T}}^{\mathrm{miss}}$. The LHC Run 2 has seen the first (very successful) attempt to target these final states using final states with no leptons, using techniques of collecting the boson decay products in large-radius jets and then exploiting the jet mass and substructure to tag them as $W/Z/h$ jets.

The ATLAS collaboration summary for this benchmark scenario is shown in Figure 6. The limits shown address separately the production of $\tilde{\chi}_1^{+} \tilde{\chi}_1^{-}$ (in green) and that of $\tilde{\chi}_2^0\tilde{\chi}_1^{\pm}$. The analysis determining the sensitivity at high common $\tilde{\chi}_1^{\pm} - \tilde{\chi}_2^0$ mass is a zero lepton analysis, while multilepton analyses dominate the sensitivity in compressed regions and for difficult regions of the parameters where the gap in mass between the pair-produced particles and the LSP is similar to one of the bosons emitted in the decay.

Figure 6: Summary of the ATLAS Collaboration exclusions limits on the electroweakino masses [5] in a model where a pair of mass-degenerate winos is produced and decays to a bino-like LSP via the emission of a gauge or Higgs boson.

A second crucial benchmark considered is that of pair-production of higgsino-like states. If SUSY needs to provide a solution to the hierarchy problem, then the higgsino mass parameter cannot be too far from the electroweak scale, imposing a higgsino mass of the order of a few hundred GeV at most. It is conceivable to consider a model where the higgsino mass parameter is significantly smaller than the wino and bino mass. In such a model, four higgsino states would exist (two charginos and two neutralinos) with masses similar to each other. The exact mass separation depends mainly on the value of $M_1$ and $M_2$: it is below a GeV for $\mu \ll M_1, M_2$ and tens of GeV for differences between $\mu$ and $M_1, M_2$ of the order of few hundred GeV. Because of the small mass gap between them, particles emitted in the transition between the higgsino states typically have low $p_{\mathrm{T}}$: analyses targeting these scenarios focused on single-, di- and tri-lepton states, typically requiring that the higgsino pair system recoils against one or more jets. The summary of the results from the CMS collaboration is shown in Figure 7. Because of the small cross sections and of the challenging final states, exclusion limits are significantly weaker than those shown in Figure 6: it is only with the Run 2 data that the LHC experiments have started to have sensitivity to this type of scenarios.

Figure 7: Summary of the CMS Collaboration exclusions limits on the electroweakino masses [6] in a model where a pair of higgsino states is produced and decays to a higgsino-like LSP. The mass limit is given as a function of the $\tilde{\chi}_2^0$ (on the $x$-axis) and of its mass separation with the $\tilde{\chi}_1^0$ LSP.

In the extreme case, for very large values of $M_1$ and $M_2$, the higgsino mass separation becomes so small that the higgsino states may actually become long-lived. This scenario is phenomenologically similar to that where $M_2 \ll \mu, M_1$ (the wino-like LSP case): in this case, three nearly-degenerate states ($\tilde{\chi}_1^0$ and a pair of $\tilde{\chi}_1^{\pm}$) exist, and the chargino can be long-lived because of the small mass separation with the $\tilde{\chi}_1^0$. The long-lived higgsino and wino cases have both been targeted with dedicated disappearing track analyses, where short-tracks are required to be identified using a few silicon layers, that have no extensions in the rest of the inner detector (CMS and ATLAS disappearing track).

Slepton pair-production has been a target of the LHC analyses already in Run 1: if the pair-produced sleptons are selectrons or smuons, and focusing on simplified models where the only relevant SUSY particles are the sleptons themselves and a $\tilde{\chi}_1^0$ LSP, the final state following $\tilde{\ell}_1 \rightarrow \ell \tilde{\chi}_1^0$ will contain two non-resonant opposite-sign electrons or muons, and $E_{\mathrm{T}}^{\mathrm{miss}}$ from the LSP, giving rise to a reasonably distinctive and experimentally straightforward signature. However, if the pair-produced slepton is a stau, the most challenging identification of the $\tau$-lepton makes the analysis more difficult: this is primarily the reason why we had to wait until Run 2 to have the first constraints on the existence of $\tilde{\tau}_1$. $\tilde{\tau}_1$ pair-production is a process of particular interest from the cosmological point of view  Ellis (1998): $\tilde{\tau}_1$ co-annihilation is a process where the existence of $\tilde{\tau}_1$ of mass similar to that of a bino-like $\tilde{\chi}_1^0$ enhances the $\tilde{\chi}_1^0$ self-annihilation cross section, therefore giving a mechanism of regulation of the dark matter relic density. The LHC sensitivity is obtained by analyses identifying two hadronically decaying $\tau$-leptons plus significant missing transverse momentum. Figure 8 shows the sensitivity obtained by the CMS collaboration. Degenerate $\tilde{\tau}$ corresponding to the two $\tau$-lepton degrees of freedom are excluded up to 400 GeV for a massless LSP. The limits are significantly weaker if only $\tilde{\tau}_{L}$ are considered, and there is hardly any sensitivity to the case where only the $\tilde{\tau}_{R}$ is produced. Favourable regions of the parameter space for the $\tilde{\tau}_1$ co-annihilation are not yet excluded.

Figure 8: Summary of the CMS Collaboration exclusions limits on direct $\tilde{\tau}_1$ production [7] in a model where a pair of $\tilde{\tau}_1$ is produced and decays to a $\tau$-lepton and a bino-like LSP.

Less conventional scenarios: while the discussion so far focused mostly on R-parity conserving, prompt decays of pair-produced particles, in general the limits obtained for other models are as compelling. Figure 9 shows a summary of the ATLAS decays to gluino pair-production in RPV SUSY models. Depending which specific RPV coupling is allowed in the model, different analyses are employed. All these analyses are characterised by limited or no $E_{\mathrm{T}}^{\mathrm{miss}}$ in the final state. Many of them exploit the very large multiplicity of objects in the final state. For example, $\tilde{g}\rightarrow t\bar{t}\tilde{\chi}_1^0$ followed by the RPV decay of the neutralino can lead to up to 18 final state objects (jets, leptons, etc.). The limits obtained on the gluino mass are certainly not weaker than the RPC ones. Other RPV analyses rely on the search of resonance states, or on the presence of flavour/charge configurations strongly suppressed in the Standard Model.

Scenarios featuring long-lived particles are ubiquitous in SUSY models. SUSY particles become long-lived whenever their decay width into other particles is suppressed. The three main reasons for this to happen are, as usual, small coupling for the decay (because of, e.g., RPV), small phase space available for the decay (as in, e.g., highly compressed scenarios, like for example higgsino-like LSP with $\mu \ll M_1,M_2$, or wino-like LSP with $M_2 \ll \mu, M_1$), decay happening via mediators with very large mass (as in, e.g., split SUSY, where the gluino decay is mediated by a very high-mass squark). The signatures and corresponding experimental techniques for the reconstruction and identification of long-lived particles are subject of an extremely active area of research at the LHC. They will not be discussed further in this paper, but the interested reader should refer to the excellent paper from M. H. Genest, in this same series. Compelling limits already exist for many scenarios (like, for example, those featuring long-lived gluinos and squarks), and many more will come from future runs of the LHC, where detector upgrades are planned that will ease the reconstruction and analysis of these non-conventional final states.

Figure 9: Summary of the ATLAS Collaboration exclusions limits on gluino pair-production in models where the $\tilde{\chi}_1^0$ LSP decays via different $R$-parity violating modes [8].

## [edit] Outlook

There is hardly another area of searches for phenomena beyond those predicted by the SM where the impact of the LHC has been so dramatic. Before the beginning of the LHC data taking, SUSY was seen as a single answer to many unresolved open questions of the Standard Model. The LHC experiment research programme has first quickly excluded most of the simplest SUSY configurations, then moved to a detailed work targeting many signatures, not necessarily favoured by a theoretical prejudice. The lack of an identified SUSY signal so far is certainly a disappointing and possibly somewhat surprising outcome to many scientists.

The pre-LHC SUSY landscape was dominated by a relatively low number of frameworks arising from specific top-down approaches to the way SUSY was broken. These models became quickly disfavoured as the mass of the Higgs boson was unveiled (requiring heavy stops) and the limits on the strongly produced particles started to challenge those models' predictions. This happened largely already at the end of Run 1. The Run 2 research programme has been more agnostic. The classical paradigm of SUSY as a solution to the hierarchy problem requires higgsinos with masses of maximum few hundred GeV, stops at the TeV scale, and gluinos not too far above that. The current limits well into the few-TeV region for gluinos and TeV for stops exceed the expectations for classical naturalness definitions. For the first time, collider experiments started to extend the LEP limits on Higgsinos. Scenarios including viable dark matter candidates have been severely constrained by the LHC results. Analyses of the relevant parameter space (ATLAS pMSSM (2024)) show that a dark matter component that is predominantly a bino LSP is largely excluded, while a mixed higgsino-bino is still largely allowed, even with electroweakinos of just a few hundred GeV.

There is still information that the LHC can give us. The sensitivity to some production processes (most notably higgsino and stau pairs) has only started to emerge with the Run 2: the experimental landscape will change quickly with increasing luminosity during Run 3 and the LHC High-Luminosity (HL-LHC) phase. Also, the experimental techniques are constantly perfected by the collaborations: upgrades of the detectors and data acquisition system will allow a deeper exploration of less conventional experimental signatures, for example the long-lived ones.

And, of course, the questions connected with the hierarchy of energy scales, to the nature of dark matter, to a higher degree of unification of the fundamental interactions, are as compelling as ever: the push towards new theoretical paradigms (with or without SUSY) and the development of new creative experimental techniques is relentless. The community eagerly awaits the outcome of the future experimental endeavours (looking with interest at potential future collider efforts beyond the HL-LHC) to shed (at least some) light on these puzzles.

## [edit] References

* Thomson, Mark (2013). Modern Particle Physics, Cambridge University Press, New York. ISBN 9781107034266.

* Martin, Stephen P (1998). A Supersymmetry Primer, Adv. Ser. Direct. High Energy Phys. 18: 1-98. arXiv:hep-ph/9709356

* Adam, Wolfgang (2022). Status of searches for electroweak-scale supersymmetry after LHC Run 2, Int. J. Mod. Phys. A 37: 2130022. doi:10.1142/S0217751X21300222.arXiv:2111.10180

* Chakraborti, Manimala (2020). Improved $(g−2)_{\mu}$ Measurements and Supersymmetry, Eur. Phys. J. C 80: 984. doi:10.1140/epjc/s10052-020-08504-8.arXiv:2006.15157

* CMS Collaboration, (2020). Searches for physics beyond the standard model with the MT2 variable..., Eur. Phys. J. C 80: 3. doi:10.1140/epjc/s10052-019-7493-x.arXiv:1909.03460

* ATLAS Collaboration, (2022). Search for long-lived charginos..., Eur. Phys. J. C 82: 606. doi:10.1140/epjc/s10052-022-10489-5.arXiv:2201.02472

* Ellis, John R (1998). Neutralino-Stau Coannihilation and the Cosmological Upper Limit on the Mass of the Lightest Supersymmetric Particle, Phys. Lett. B 444: 367. doi:10.1016/S0370-2693(98)01392-6.arXiv:hep-ph/9810360

* ATLAS Collaboration, (2024). ATLAS Run 2 searches for electroweak production of supersymmetric particles interpreted within the pMSSM, CERN-EP-2024-021 : . arXiv:2402.01392