For the remainder of this paper we are interested in the sensitivity of the LHC with respect to the \(\Lambda _{\not R_p} \)–CMSSM, i.e. to a complete supersymmetric model. In particular, we are interested in how the presence of R-parity-violating operators affects the well-known results for the R-parity-conserving CMSSM [20, 22, 172, 173]. For this purpose we shall use the program CheckMATE [63,64,65]. As we saw in Sects. 3 to 5, the LHC experiments mainly set bounds on simplified supersymmetric R-parity-violating models. They set little or no bounds on the complete \(\Lambda _{\not R_p}\)–CMSSM model. We here use CheckMATE to recast ATLAS and CMS searches and thus set bounds on the various \(\Lambda _{\not R_p}\)–CMSSM models.
Method
The program CheckMATE automatically determines if a given parameter point of a particular model beyond the Standard Model (BSM) is excluded or not by performing the following chain of tasks. First, the Monte Carlo generator MadGraph [174] is used to simulate proton–proton collisions. The resulting parton level events are showered and hadronized using Pythia 8 [175]. The fast detector simulation Delphes [176] applies efficiency functions to determine the experimentally accessible final-state configuration, including the determination of the jet spectrum using FastJet [177, 178]. Afterwards, various implemented analyses from ATLAS and CMS designed to identify different potentially discriminating final-state topologies are used.
Events which pass well-defined sets of constraints are binned in signal regions for which the corresponding prediction for the Standard Model and the number of experimentally observed events are known. By comparing the predictions of the Standard Model and the user’s BSM model of interest to the experimental result using the CL\(_{\text {S}}\) prescription [179], CheckMATE concludes if the input parameter combination is excluded or not at the 95% confidence level. For more information we refer to Refs. [63,64,65].
Model setup
For the proper description of the RPV Feynman rules in MadGraph , we take the model implementation from SARAH, which we already used in Sect. 2, and export it via the UFO format [180]. For a given set of \(\Lambda _{\not R_p}\)–CMSSM parameters, we make use of the respective SPheno libraries created from the same SARAH model used to determine the low-energy particle spectrum, the mixing matrices and the decay tables. We calculate the SUSY and Higgs masses including RPV-specific two-loop corrections [181,182,183] which are particularly important for light stops [55]. As discussed in Sect. 2.4, four-body decays of the stau can be dominant and lead to important experimentally accessible final states. In regions where this occurs, see for instance Sect. 6.2.4, the four-body stau decays have been determined using MadGraph.
Monte Carlo simulation
In \(\Lambda _{\not R_p}\)–CMSSM parameter regions where the entire SUSY spectrum is kinematically accessible at the LHC, i.e. with masses at or below \(\mathcal {O}(\)1 TeV), there exist a plethora of possible final-state configurations. To maintain computational tractability in our study, we applied the following list of simplifying assumptions:
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We include only two-body supersymmetric final-state production: \(p p \rightarrow A B+X_{\mathrm {soft}}\), and require both supersymmetric particles, A and B, to be produced on-shell. Note that in RPC supersymmetry, additional hard QCD radiation, i.e. \(pp \rightarrow A B j\), is important in parameter regions with highly degenerate spectra due to the resulting kinematic boost of the decay products; see e.g. Ref. [184]. However, due to the instability of the LSP in the \(\Lambda _{\not R_p}\)–CMSSM, this additional boost is not needed and therefore this final state is not expected to contribute sizably to the final constraining event numbers.
-
We do not consider final-state combinations which are strongly suppressed by the relevant parton density distributions and/or which only exist in RPV supersymmetry. Most importantly, this excludes flavor-off-diagonal combinations of “sea”-squarks (we clarify the meaning below) or squark–slepton combinations.
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We include in our simulations production processes, which can only proceed via the electroweak interactions, i.e. the production of sleptons, electroweak gauginos and the mixed production of electroweak gauginos and squarks or gluinos. However, in the case of electroweak gauginos we only include the production of the two lightest neutralinos and the lightest chargino, i.e. the dominantly bino and wino states in a CMSSM setup. We expect no sizable contributions from the ignored Higgsinos, since these are typically significantly heavier and therefore have negligible production rates in comparison with the lighter winos. Similarly, we do not include flavor-off-diagonal slepton combinations and mixed electroweak gaugino–squark production with “sea”-squarks.
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With the above considerations, the resulting set of production channels that we consider are listed below: Strong processes:
-
\(\tilde{g} \tilde{g}\)
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\(\tilde{g} \tilde{q}_{V}^{(*)}\)
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\(\tilde{q}_{V}^{(*)} q_{V}^{(*)}\) (all combinations)
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\(\tilde{q}_{S}\tilde{q}_{S}^{*}\) (only flavor-diagonal)
Electroweak processes:
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\(\tilde{\chi } \tilde{\chi }\) [all (non-Higgsino) combinations]
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\((\tilde{\ell }_{L}, \tilde{\ell }_{R}, \tilde{\nu }_{\ell , L}) (\tilde{\ell }_{L}^* , \tilde{\ell }_{R}^* , \tilde{\nu }_{\ell , L}^*)\) (only flavor-diagonal)
Mixed processes:
Here, \(\tilde{q}_V\) refers to the superpartners of the light quarks: \(\tilde{u}_{L,R}, \tilde{d}_{L,R}\) and \(\tilde{s}_{L,R}\), while \(\tilde{q}_S\) refers to the remaining squarks \(\tilde{c}_{L,R}, \tilde{b}_{1,2}\) and \(\tilde{t}_{1,2}\). Furthermore, \(\tilde{\chi }\) subsumes the two lightest neutralinos \(\tilde{\chi }_{1, 2} ^0\) and the lightest chargino \(\tilde{\chi }_1^\pm \).
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Decays of supersymmetric particles are performed within Pythia 8 , using the information from the decay table determined in SPheno. This ignores any potential spin-dependent information, which could be relevant when performing the proper matrix-element calculation. However, as we do not assume spin-effects to be important here, we take the computationally faster approach of using decay tables.
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To take into account the sizable contributions from higher-order QCD effects in the production cross section, we multiply the leading-order production cross sections taken from MadGraph with the next-to-leading-logarithm K-factors determined by NLLFast [185,186,187,188,189,190,191]. This tool interpolates gluino- and squark-mass-dependent higher-order cross sections for all “strong Processes” listed above. NLLFast assumes degenerate first- and second-generation squark sector where we use the median of the squark masses for the calculation. Note that this degeneracy is present in models with small \(\Lambda _{\not R_p}\) but not necessarily if \(\Lambda _{\not R_p}\) is large.Footnote 8 Stops and sbottoms are always treated separately and obtain individual K-factors. The consideration of higher-order effects for the remaining processes is computationally far more involved as tools like PROSPINO [192] need to perform the full NLO calculation. These effects are, however, expected to be significantly smaller compared to the strong production processes and thus we neglect them here.
Incorporated analyses
CheckMATE provides a large set of implemented ATLAS and CMS results from both the \(\sqrt{s} = 8\) and the 13 TeV runs of the LHC. These analyses target a large variety of possible final states which typically appear in theories beyond the Standard Model. The vast majority, however, are designed to target RPC supersymmetry. This implies cuts which require significant amounts of missing transverse momentum and/or highly energetic final-state objects, namely leptons or jets. As many of the most prominent decay chains in the \(\Lambda _{\not R_p}\)–CMSSM indeed correspond to these signatures, it is therefore interesting to determine the relative exclusion power of these tailored analyses in comparison to the RPC–CMSSM.
In this study we consider all \(\sqrt{s} = 8\) TeV LHC analyses implemented in CheckMATE 2.0.1. For a full, detailed list we refer to the documentation in Refs. [63,64,65] and the tool’s website.Footnote 9 We discuss the target final states of the relevant analyses in more detail below, alongside our results. Some final states have been reanalyzed and the corresponding bounds have been updated with new LHC results taken at \(\sqrt{s} = 13\) TeV center-of-mass energy. For our purpose of comparing the relative exclusion power when going from an RPC to an RPV scenario, using the \(\sqrt{s} = 8\) TeV analysis set has the advantage of covering a much larger variety of final states.
To set the limit, CheckMATE tests all signal regions in all selected analyses, determines the one signal region with the largest expected sensitivity and checks if the corresponding observed result of that signal region is excluded at 95% C.L. or not. In the following, the analysis which contains this limit-setting signal region is referred to as the “most sensitive analysis” for a given \(\Lambda _{\not R_p}\)–CMSSM parameter point. Due to the lack of information about correlations in systematic uncertainties between different signal regions, CheckMATE is currently incapable of combining information from different signal regions.Footnote 10
The list of analyses we employ are shown in Fig. 10. In bold is the name under which the analysis is listed in CheckMATE. Underneath in small italics type we briefly denote the physical signature. Here \(\ell \) refers to a charged lepton, \(\not \!\!E_T\) refers to missing transverse energy. j refers to a jet in the final state, b specifically a b-jet. The references for the analyses are given in the CheckMATE documentation. The boxes of the analyses carry different colors and hatchings. This is employed in the later exclusion plots, to show which analysis within CheckMATE is the most sensitive.
Scanned parameter regions
Even though one of the appealing features of the CMSSM is the small number of free parameters compared to other supersymmetric theories, we still need to fix certain degrees of freedom in order to be able to show results in an understandable 2-dimensional parameter plane. As the masses of the supersymmetric particles will be one of the most important variables when it comes to the observability of a model realization at the LHC, we show results in the \(M_0\)–\( M_{1/2}\) plane. To be specific we scan over the parameter range
$$\begin{aligned} M_0\in [0,\,3000]\,{\mathrm{GeV}},\quad M_{1/2}\in [200,\,1000]\,{\mathrm{GeV}}. \end{aligned}$$
(50)
For better comparison, we choose the remaining model parameters as in the RPC–CMSSM ATLAS analysis in Ref. [172] where \(\tan \beta \) is fixed to a relatively large value of 30 while \(A_0\) is set via the standard formula \(A_0 = -2 M_0\), which ensures a realistically large value of the lightest neutral CP-even Higgs boson mass. The sign of the \(\mu \) parameter is fixed to be positive, to avoid further tension with \((g-2)_\mu \) [194]. Allowing any number of RPV operators to have non-vanishing values would yield an unmanageable set of possible scenarios to study. We therefore restrict ourselves to cases where only one of the many operators has a non-zero value at the unification scale. In order to directly compare our results to the RPC–CMSSM, we use a small RPV coupling at the GUT scale, \(\Lambda _{\not R_p}|_{\mathrm{GUT}}=0.01\). This essentially mimics the RPC mass spectrum, but allows for the prompt decay of the LSP. We comment on possible effects of increasing the RPV coupling in Sect. 7.
Results
In this section we show the results of the scans. Throughout the analysis of these results, the principal questions which we seek to answer are the following:
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1.
To what extent do the existing ATLAS and CMS analyses, which largely focus on RPC supersymmetry, exclude the parameter space of the \(\Lambda _{\not R_p}\)–CMSSM?
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2.
Does breaking R-parity weaken the bounds of the RPC–CMSSM due to a gap in the coverage of possible final states?
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(a)
If the answer is yes, how could these gaps be closed?
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(b)
Alternatively, if the answer is no, in the cases where the bounds become stronger, which of the effects mentioned in previous sections lead to this result?
When presenting our results, we show in the figures for each parameter point, which CheckMATE analysis is the most sensitive. We do this by using the color code of Fig. 10.
R-parity conserving CMSSM
We start with a short discussion of the RPC–CMSSM in Fig. 11. On the left we denote in the \(M_0\)–\(M_{1/2}\) plane by the thick, solid black line the 95%-CL exclusion range we obtained using CheckMATE for this model. Thus below the curve, in red is the excluded parameter area. Above the curve, in green is the allowed area. The remaining CMSSM parameters have been set to \(\tan \beta =30\), \(\mu > 0\) and \(A_0 =-2 M_0\). In light gray we present supersymmetric mass iso-curves for the LSP (dot-dashed), the first two generation squarks (dotted), the gluino (solid), and the lightest stop (dashed). The white region on the far left at low \(M_0\) results in a \(\tilde{\tau }\)-LSP, which is not viable phenomenologically if R-parity is conserved: as there is no possible decay channel for the stau, these regions result in stable charged particles which e.g. spoil big bang nucleosynthesis [195]. This is why here and in the following, we do not show any RPC results in the stau-LSP region. We do include them in the RPV cases.
In the right plot, we show in the \(M_0\)–\(M_{1/2}\) plane, which analysis implemented in CheckMATE is most sensitive at a given parameter point or region. We use the color and hash code of Fig. 10. Thus for example the point (\(M_0=1000\,\)GeV, \(M_{1/2}=500\,\)GeV) is excluded by the analysis denoted atlas_1405_7875 [129] in CheckMATE.
The most sensitive analyses target either a 0-lepton multi-jet (atlas_1405_7875), or \(a \ge 3b\)-jet final state (atlas_conf_2013_061, Ref. [196]), both requiring a significant amount of missing transverse momentum. The former final state is especially sensitive when light gluinos decay into jets via on-shell squarks and therefore – as can be seen in our results – covers the low \(M_{0}\) region where the squarks are relatively light. On the contrary, the latter targets stop and gluino pair production, the dominant modes for large \(M_0\), which can produce four b-jets due to the resulting top quarks in the final decay chain.
In this particular set of plots, see Fig. 11, we also show the nominal ATLAS exclusion limit taken from Ref. [172] as an additional, black dotted line. The discrepancies arise due to the ATLAS result including a statistical combination of the orthogonal sets of 0 and \(1\ell \) signal regions which CheckMATE cannot perform. Apart from this combination, the detailed, analysis-dependent results given in Ref. [172] match our determination of the respective most sensitive analysis in this model.
\(LL\bar{E}\), \(\Lambda _{\not R_p}\)-CMSSM
We now consider the case of a small non-zero \(LL\bar{E}\) operator as discussed in Sects. 3.2 and 4. We determine excluded parameter regions of the corresponding \(\Lambda _{\not R_p}\)–CMSSM and compare with the LHC exclusion line obtained in the RPC case. Since we consider a small RPV coupling, the particle spectrum remains virtually unchanged with respect to the RPC–CMSSM. However, as emphasized before, such an operator leads to the decay of the LSP. Therefore, a neutralino LSP will decay into two leptons and one neutrino for a generic \(\lambda _{ijk}\) coupling, cf. Eq. (18) and Table 3. In parameter regions where the lightest stau is the LSP its possible decay modes are: (i) directly into \(\ell _i \nu _j\), if either i, j or k equals 3, (ii) via the RGE-generated \(\lambda _{i33}\) coupling if the non-zero RPV coupling at \(M_X\) is of form \(\lambda _{ijj}\) and \(\{i,j\}\ne 3\) (see Sect. 2.4) or (iii) via a four-body decay, cf. see Sect. 4, and Table 9 for the corresponding LHC signatures. However, the four-body decay does not happen here due to the large \(\tan \beta \) value employed and the consequential dominance of the two-body decay through the RGE-generated operators.
The generic LHC searches for RPC supersymmetry look for missing energy in combination with jets and/or leptons. They should thus also perform well for the \(LL\bar{E}\) models, due to the many extra leptons from the RPV decay. Although the amount of missing energy is in general not as pronounced as for an RPC model, the energy carried away by the neutrino in the final decay can still be sizable enough to produce a striking signature; see Refs. [197, 198].
In the top row of Fig. 12 we show the CheckMATE exclusion in the \(M_0\)–\(M_{1/2}\) plane for the \(\Lambda _{\not R_p}\)–CMSSM with \(\lambda _{122}\ne 0\). The remaining CMSSM parameters and the light gray iso-mass curves are as in Fig. 11. The excluded region is shown below the thick solid black curve in red. The allowed region is shown above this curve in green. The dark red/green colored regions correspond to a \(\tilde{\tau }\)-LSP, which must be considered in the RPV case. The light red/green colored regions correspond to a \(\tilde{\chi }^0_1\)-LSP, as in the RPC case. The RPC–CMSSM exclusion line of Fig. 11 (thick solid black curve there) is shown here as a thick dashed black curve for comparison. It does not extend into the \(\tilde{\tau }\)-LSP region, as that is not viable in the RPC–CMSSM.
The plot on the upper right in Fig. 12 shows in thick solid black the same CheckMATE exclusion from the upper-left plot, as well as the RPC exclusion from Fig. 11 as a solid dashed line. It furthermore shows which LHC analysis implemented in CheckMATE is most sensitive at a given parameter region using the same color code as in Fig. 11. When comparing with Fig. 10, we see that most of the parameter range is most sensitively covered by atlas_conf_2013_036, Ref. [56].
As a result of \(\lambda _{122}\ne 0\), the neutralino decays will lead to four more charged first- or second-generation leptons compared to the RPC case, in regions where the neutralino is the LSP, cf. Table 3. Consequently, analyses looking for four or more leptons, Ref. [56], are very sensitive to this scenario and yield a stronger limit than in the RPC case. Thus the solid black curve in the upper-left plot is more restrictive than the dashed black curve. The search in Ref. [56] contains separate signal regions designed for both RPC and RPV signatures, respectively. It is interesting that, although their signal regions designed for the RPV signatures are the ones with the best exclusion power, the RPC signal region performs almost equally well.
When looking more closely at the CheckMATE output we see that it is a specific search region in Ref. [56], which is most sensitive to the \(LL\bar{E}\) case we are considering here, namely the electroweak pair production of neutralinos and charginos. This production channel is not very promising for RPC models in the CMSSM, as the largest electroweak cross section is usually obtained by the production of a charged and a neutral wino. Within the CMSSM boundary conditions, both would decay to the bino by emitting a W and a Higgs/Z-boson, respectively. This comparably small electroweak signal rate is usually not enough for these final states to be detected over the background. Thus the RPC–CMSSM is most stringently constrained by gluino pair production. In contrast, in the \(\Lambda _{\not R_p}\)–CMSSM the neutralino decays via \(\lambda _{122}\) lead to a clean signal with many charged leptons.
Specifically, for \(M_{0}\gtrsim 500\,\hbox {GeV}\), corresponding to regions with a neutralino LSP, we can exclude values of \(M_{1/2} \lesssim 950\,\hbox {GeV}\), which feature a bino of \(m_{\tilde{\chi }^0_1}\simeq 400\,\hbox {GeV}\) and winos of \(m_{\tilde{\chi }^0_2}\simeq m_{\tilde{\chi }^\pm _1}\simeq 800\,\hbox {GeV}\), as well as gluinos over 2.1 TeV. Thus within the \(\Lambda _{\not R_p}\)-CMSSM, the indirect constraint on the gluino mass (via the universal gaugino mass) is much stricter than the RPC gluino search can reach.
These \(\Lambda _{\not R_p}\)-CMSSM bounds obtained using CheckMATE can, to some degree, be compared to the results from Ref. [110], where bounds on the pair production of winos decaying via \(LL\bar{E}\) are set. This analysis excludes wino masses up to 900 GeV also for \(\lambda _{122}\ne 0\), when assuming that the neutral wino is the LSP. In the case at hand we have a lighter bino to which the wino will decay. This change in kinematics (for instance, the final-state leptons will be less energetic) with respect to the simplified model analysis in Ref. [110] explains the small differences observed in the bounds on the wino mass.
In the upper-left plot of Fig. 12, the low \(M_{0}\) region features a stau LSP and we observe that the exclusion power close to the LSP-boundary drops significantly. This occurs due to the produced neutralinos (charginos) decaying into \(\tau \tilde{\tau }_1\) \((\nu \tilde{\tau }_1)\) and \(\tilde{\tau }_1\) decaying via the RGE-induced \(\lambda _{133}\) coupling. As a result the stau has equal branching ratios into both \(e\nu \) and \(\tau \nu \) final states. Compared to the expected signatures of neutralino-LSP regions explained above, several final-state first- and second-generation leptons are now replaced by \(\tau \) leptons. The expected event rates therefore drop by powers of the leptonic tau branching ratio and hence significantly affect the resulting bound from the same analysis.
As \(M_0\) approaches 0, the mass of all sleptons and sneutrinos further decreases. This slowly opens further decays of the neutralino (chargino) into other \(\ell _i \tilde{\ell }_i\) and \(\nu _i \tilde{\nu }_i\) (\( \nu _i \tilde{\ell }_i\) and \(\ell _i \tilde{\nu }_i\)) combinations, which for \(i \ne 3\) lead to the same decay signatures via \(\lambda _{122}\) as discussed before for the neutralino LSP region. Hence, the exclusion line approaches the earlier, stricter bound for \(M_0 \rightarrow 0\).
Next, we consider the bottom row of Fig. 12 with a non-zero \(\lambda _{123}\) coupling. The labeling and the included curves are to be understood as for the upper two plots. In the lower-right plot we see that again atlas_conf_2013_036, Ref. [56], is most sensitive over most of the parameter region, however only if \(M_0\) is large and the squarks are decoupled. For small \(M_0\), analyses which look for jets and like-sign charged leptons, for example Ref. [133] (denoted atlas_1404_2500 in Fig. 10), become more sensitive. In fact, these two analyses are similarly sensitive in this parameter region but the second gets contributions from light squark decays and thus starts dominating the exclusion line for small \(M_0\).Footnote 11 To the right in the lower-left plot, the neutralino-LSP decay now always involves tau leptons, cf. Eq. (18). The resulting overall bound, the thick solid black curve, is weaker than in the \(\lambda _ {122}\) case in the neutralino-LSP region, as the multilepton signal is diluted by these taus. However, it is still stricter than the RPC case, the thick black dashed curve. We furthermore observe in the lower-left plot that the considered analyses in the stau LSP region, i.e. for small \(M_0\), are more sensitive, than in the \(\lambda _{122}\) case. The search is more sensitive, as the stau now decays via \(\lambda _{123}\) into a neutrino and a first- or second-generation charged lepton, rather than a tau final state as in the \(\lambda _{122}\) case, cf. the second line in Eq. (31).
For completeness we in turn show the results for non-zero values of \(\lambda _{131}\) and \(\lambda _{133}\) in Fig. 13, respectively. The notation is as in Fig. 12. Again, the differences with respect to Fig. 12 can be explained by, respectively, considering the number of charged first- and second-generation charged leptons versus the number of tau leptons in the relevant final states. For the \(\lambda _{131}\) coupling, the neutralino will decay with almost equal branching ratios into both \(e e\nu \) and \(e\tau \nu \), which is why the excluded region is larger than for \(\lambda _{123}\) but smaller than for \(\lambda _{122}\). For the case of \(\lambda _{133}\), the neutralino decays either into \(\tau \tau \nu \) or \(e \tau \nu \), which is why the LHC sensitivity is lower compared to all previous cases. Remarkably, however, it is still more sensitive than the RPC case for most values of \(M_0\). Lastly, we note there are only minor differences between the exclusion lines if we were to exchange RPV couplings to (s)electrons by couplings to (s)muons which is due to the comparable identification efficiency between electrons and muons at both ATLAS and CMS. All other \(LL\bar{E}\) couplings are obtained by exchanging flavor indices \(1\leftrightarrow 2\) and the respective bounds can therefore be inferred from the scenarios shown above.
For all of the \(LL\bar{E}\) cases, we see that the \(4\ell \)+\({\mathrm {MET}}\) search of atlas_conf_2013_036, Ref. [56] and the jets\(+\)SS\(\ell \) search of atlas_1404_2500, Ref. [133], are the most sensitive in CheckMATE over most of the parameter range. The most important message from looking at the different \(LL\bar{E}\) operators and comparing to the RPC case is, however, that within the CMSSM, as a complete supersymmetric model, the LHC is actually more sensitive to scenarios in which R-parity is violated via an \(LL\bar{E}\) operator than if R-parity is conserved. This statement holds even if the signal regions designed for RPV in Ref. [56] are disregarded.
\(LQ\bar{D}\), \(\Lambda _{\not R_p}\)-CMSSM
We now turn to the discussion of the \(LQ\bar{D}\) operator. In general, when compared to the previous \(LL\bar{E}\) case, it is clear that the LHC sensitivity is reduced, as we have to replace either a neutrino and a charged lepton or even two charged leptons by two quarks in the final RPV decay of a neutralino LSP; see Eq. (24) and Sect. 3.3. Stau LSPs, in turn, will mostly decay either into a pair of quarks or via a four-body decay into a tau, a charged lepton or neutrino, and two quarks; see Eq. (31) and Sect. 4.
Furthermore, we generally observe that the electroweak gaugino-pair production is no longer relevant in the case of a decay via \(LQ\bar{D}\). Due to the hadronic decay products of the neutralino- or stau-LSP the efficiency in the electroweak gaugino case is no longer significantly higher than in the strong production case. The latter then wins due to the significantly higher production cross section.
Let us discuss the results for the individual couplings. The first row in Fig. 14 shows the case of a non-zero \(\lambda '_{222}\) operator. As just mentioned, the overall exclusion sensitivity is significantly lower than in the \(LL\bar{E}\) case, and is comparable to the RPC–CMSSM case, shown here as the thick black dashed line. In a small region around \(M_0=750\,\)GeV, the RPC is even stricter. In the neutralino-LSP region with high \(M_0\), we find that analyses which look for jets and like-sign charged leptons, for example Ref. [133] (denoted atlas_1404_2500 in Fig. 10), are most sensitive. See the right-hand plot. In this region the first- and second-generation squarks are relatively heavy. Therefore gluino and stop pair production are the most dominant production modes. These produce final states with many b-jets, lower quark generation jets, and leptons and hence populate the 3b signal region of a “2 same-sign \(\ell \) or \(3 \ell \,\)” analysis (atlas_1404_2500), for which the Standard Model background is nearly zero. Here, the high final-state multiplicity induced by the \(LQ\bar{D}\) decay results in a slightly increased sensitivity when compared to the RPC case.
In regions with lower \(M_0\) where gluino–squark associated production and squark pair production become relevant, generic squark–gluino searches like atlas_conf_2013_062, Ref. [199], which look for jets, leptons and missing transverse momentum dominate. For these, the increased final-state multiplicity via the additional \(LQ\bar{D}\)-induced decays results in a worse bound than for the RPC case. This is due to the signal regions setting strong cuts on the required momentum of the final-state objects and the missing transverse momentum of an event. These are necessary to sufficiently reduce the Standard Model background contribution, especially from multiboson production, which also produces final states with high jet and lepton multiplicity and some missing transverse momentum. Since the expected missing transverse momentum of the event is significantly larger in RPC models for which the LSP does not decay, breaking R-parity weakens the bounds in these regions, cf. Refs. [197, 198].
Within the stau LSP region, the wedge at low \(M_0\), the stau will undergo two-body decays due to the RGE-generated \(\lambda _{233}\) operator for large \(\tan \beta \); see also Fig. 7. Therefore, this scenario mimics the results from the stau LSP region in the case where \(\lambda _{233}\) is already present at \(M_X\); see our discussion of the phenomenologically almost identical \(\lambda _{133}\) operator, in Sect. 6.2.2.
We continue with the discussion of \(\lambda '_{113}\), with the only phenomenologically relevant difference that \(\bar{D}_2\) is replaced by \(\bar{D}_3\). Hence, in the neutralino LSP case, the only phenomenological difference is that two b-jets replace two normal jets. (We found that the sensitivity in the \(\lambda '_{222}\) and \(\lambda '_{112}\) cases are almost identical, since the experimental efficiencies for muons and electrons are similar.) Due to the good b-jet tagging efficiency, this clearly improves the distinguishability with respect to the Standard Model background and results in an increase in sensitivity. This effect is most prominent for large values of \(M_0\). The same analysis as in the previous \(\lambda '_{222}\) case, see atlas_1404_2500, Ref. [133], provides the most stringent bounds as it contains special signal regions which tag additional b-jets. For smaller values of \(M_0\) barely any change in sensitivity is visible in comparison to before.
In the \(\tilde{\tau }\)-LSP region of the \(\lambda '_{113}\) case, the stau will almost always undergo a four-body decay, thereby decaying into both \(\tau e b j\) and \(\tau \nu b j\) at approximately equal rates. The increase in sensitivity with respect to the neutralino-LSP region comes from the additional tau leptons in the final state.
We continue with the cases \(\lambda '_{131}\) and \(\lambda '_{133}\) in the lower two rows in Fig. 14 and focus on the neutralino LSP region first. Here, the top quark in the decay products does not improve the sensitivity when compared to the \(\lambda '_{222}\) case. When comparing to the \(\lambda '_{113}\) case, we see the sensitivity also goes down. On the one side we no longer have the bottom quark jet in every decay and on the other hand the operator \(\lambda '_{13i}\) in principle allows for neutralino decays into both \(t+\ell +j_i\) and \(b + \nu +j_i\). However, the mass of the LSP is so low in the relevant parameter range that the decay into the top quark is kinematically suppressed. Hence, most of the neutralinos will decay via the neutrino mode and as such do not produce the final-state leptons which are required for the aforementioned “2 same-sign \(\ell \) or \(3 \ell \,\)” analysis to be sensitive. Instead, the most relevant analysis in the large-\(M_0\) region turns out to be a search looking for events with more than seven jets plus missing energy; see atlas_1308_1841, Ref. [128]. The high jet multiplicity for \(\lambda '_{131,133}\) arises from hadronically decaying tops, produced from the standard \(\tilde{g} \rightarrow t \tilde{t}, \tilde{t} \rightarrow t\tilde{\chi }^0_1\) decay chains in this parameter region, as well as jets from the final neutralino decay \(\tilde{\chi }^0_1\rightarrow jj\not \!\!E_T\).
Comparing \(\lambda '_{133}\) to \(\lambda '_{131}\), the two additional b-jets in the final state result in a slightly improved exclusion power via the multi-b analysis in atlas_conf_2013_061, Ref. [196].
In the \(\tilde{\tau }\)-LSP region, the case \(\lambda '_{133}\) is analogous to the \(\lambda '_{222}\) case in that the RGE-generated \(\lambda _ {133}\) operator determines the \(\tilde{\tau }\) decay, leading to similar bounds. In the \(\lambda '_{131}\) case, the situation is similar to \(\lambda '_{113}\ne 0\) in that the four-body decay dominates. However, as the final state including the top quark, \(\tilde{\tau } \rightarrow \tau e t j\), is kinematically suppressed, the stau almost exclusively decays into \(\tau \nu b j\). This scenario therefore exhibits the worst LHC measurement prospects of all the \(\lambda '_{aij} \ne 0\), \(a=1,2\), \(\tilde{\tau }\)-LSP scenarios.
Turning to the \(\lambda '_{311}\) scenario shown in the top row of Fig. 15, all differences with respect to the former \(\lambda ^\prime _{222}\) case can be explained by the exchange of muons by taus in the final state, which reduces the overall final state identification efficiency. As a result, the searches looking for leptons lose sensitivity and, similar to the above \(\lambda ^\prime _{i3i}\) cases, the best constraints are instead provided by the high jet multiplicity analysis described in atlas_1308_1841, Ref. [128]. Whilst in the \(\lambda ^\prime _{222}\) scenario the lower \(M_0\) region was most constrained by the squark–gluino searches in atlas_conf_2013_062, Ref. [199], here this region is again covered by the high multiplicity jet analyses. A closer look at the event rates, however, reveals that these two analyses are almost equally sensitive and hence the resulting bounds are nearly the same.
In the \(\lambda '_{311}\) scenario, the LHC sensitivity does not change significantly when traversing from the neutralino- into the stau-LSP region since the stau itself decays directly into light quark jets. Hence, only the kinematics change when the LSP cross-over occurs, while the final-state signatures stay the same. This is why we see exactly the same behavior for the other \(i=3\) cases \(\lambda '_{313}\), \(\lambda '_{331}\) and \(\lambda '_{333}\ne 0\), the former of which we show in the second row of Fig. 15. Consequently, the additional b-tagging in these scenarios does not noticeably improve the exclusion power of atlas_1308_1841, Ref. [128].
Here we have considered all distinct types of non-zero \(LQ\bar{D}\) operators, which in principle have differing LHC phenomenology. In the region where the neutralino is the LSP and \(M_0\lesssim 1.2~\)TeV, the corresponding LHC bounds that we obtain using CheckMATE are slightly weaker compared to the R-parity-conserving CMSSM. This corresponds to the region where squark pair production dominates and the additional decay of the neutralino LSP reduces the \(\not \!\!E_T\). In the parameter region where gluino and stop pair production dominates, i.e. for large \(M_0\), we instead find most \(LQ \bar{D}\) scenarios are as constrained as the RPC–CMSSM because of the equally good performance of the multi-jet searches preferred by RPV and the multi-b searches sensitive to RPC. The special cases \(L_i Q_j \bar{D}_3\) with \(i, j \in \{1, 2\}\) are significantly more constrained in this region of the R-parity-violating CMSSM, due to the additional extra leptons and b-jets in the final state. In all cases, regions with stau LSP are well covered by either multilepton or combined lepton+jet searches and yield comparable bounds as in parameter regions with a neutralino LSP.
\(\bar{U} \bar{D} \bar{D}\)
Here we discuss the \(\bar{U} \bar{D} \bar{D}\) operator for which one typically expects the weakest LHC bounds as there is no striking missing energy signal nor any additional leptons; see for example Refs. [169, 200, 201].
In Fig. 16, we show the results in analogy with the previous subsections. We first consider the case of \(\lambda ''_{121}\), the top row, where the neutralino LSP decays into three light jets. Therefore multi-jet searches should yield the most stringent limits for such scenarios. Indeed, as can be seen in the top right plot of Fig. 16, the analysis in atlas_1308_1841, Ref. [128] provides the best exclusion power for the entire neutralino-LSP region. Interestingly, the bounds on the parameter space which we obtain are almost as strong as the bounds on the RPC scenario, the thick black dashed line, cf. Fig. 11. This can be regarded as an impressive success for the experimental groups, since multi-jet analyses belong to the most challenging signatures at a hadron collider.
In the large \(M_0\) region where the exclusion lines from RPC and RPV are very similar, gluino pair production has the highest cross section. The gluinos then decay down to a top quark and a stop which itself decays to a top and a neutralino. The dominant \(\tilde{g} \rightarrow t \tilde{t}\) decay occurs because of the large stop mixing in this region which significantly reduces the \(\tilde{t}_1\) mass with respect to the other squark masses. The neutralino then eventually undergoes a three-body decay into three light jets. As a result, the signal region looking for \(\ge 10\) jets and missing energy (denoted “10j50” in Ref. [128]) provides the best constraints. This is somewhat surprising as the analysis vetoes against isolated leptons while requiring missing energy. Naively, one would have expected searches for b-jets, missing energy and leptons to dominate. However, we find that only the next-best analysis looks for that, Ref. [202], with the best applicable signal region “SR-1\(\ell \)-6j-C” looking for one lepton, more than six jets and missing energy. Furthermore, note that also in Ref. [128], a RPV interpretation has been performed, assuming gluino pair production, which decay into \(\tilde{t} \bar{t}\), with \(\tilde{t}\rightarrow bs\), obtaining bounds of \(m_{\tilde{g}}\gtrsim 1\,\)TeV. Translating the bounds we obtain for the CMSSM-like scenario to gluino mass bounds, we obtain even stricter mass limits, which is due to the higher jet multiplicity in the final state from including the intermediate neutralino in the decay chain.
In the lower \(M_0\) regions where the exclusion in RPC parameter space is stricter, squark–gluino associated production is dominant. Therefore, while the RPC–CMSSM provides a large missing energy signal and is therefore probed by analyses like atlas_1405_7875, Ref. [129], the RPV counterpart is still best covered by the 10-jet signal region of [128].
We once more want to emphasize that the bounds we obtain on the parameter space rely on the boundary conditions which we impose at the high scale and the corresponding (s)particle spectrum. In particular, as seen in the large \(M_0\) region, the presence of top quarks in the decay is of major importance. In Ref. [203], a re-interpretation of LHC results in a natural SUSY context and \(\lambda _{212}'' \ne 0\) has been performed. In their scenario the stop and gluino masses are varied independently and a Higgsino LSP decaying into three light jets is assumed. The results show that in this case, gluino and stop masses are generically less constrained when compared to the RPC analog.
In the stau LSP region, at low \(M_0\), the decay into one tau lepton and three light jets dominates for \(\lambda ''_{121}\) and into one tau, one b-jet and two light jets for \(\lambda ''_{113}\). Interestingly, most of this area is best covered by the 4-lepton analysis atlas_conf_2013_036 [56], which requires, in the case of a \(\tau \)-tag, at least three additional light leptons. This means that the other three \(\tau \)-leptons can only be identified through their leptonic decay, reducing the overall acceptance by a factor \([{\mathrm{BR}}(\tau \rightarrow \ell \nu \nu )]^3 \simeq 0.044\). In addition, we find that wino pair production is important in this part of parameter space and that the charged wino state decays into \(e/\mu +\tilde{\nu }\) in up to 30% of the cases, further contributing a charged lepton in the final state.
Turning to the cases where third-generation quarks are among the LSP decays, namely the bottom three rows in Fig. 16, we see that the bounds in the low \(M_0\) region are similar to the \(\lambda ''_{121}\) case, and that again the multi-jet search is most sensitive. In the high \(M_0\) region, where gluino and stop pair production becomes relevant, searches for same-sign leptons, atlas_1404_2500 [133], become effective for \(\lambda ''_{3ij}\). This is due to leptonically decaying top quarks in the final state and has already been analyzed in detail in Ref. [204]. Note that in comparison to the \(LQ\bar{D}\) operator \(\lambda '_{i3j}\), where the top in the final state was phase-space suppressed and as such the alternative neutrino decay mode was favored, there is no other comparable decay mode for the neutralino in the case of \(\bar{U} \bar{D} \bar{D}\). As such, it will always decay into a top quark whenever kinematically accessible, which is the case for \(M_{1/2}\gtrsim 400~\)GeV. Therefore, both gluino and stop pair production can lead to same-sign leptons from the leptonic top decay modes, rendering this scenario slightly more constrained than the \(\lambda ''_{aij}\), \(a=1,2\), cases. Interestingly, we find that in addition electroweak gaugino production is even more important for the limit setting in the large-\(M_0\) area than stop pair production.
For the \(\lambda ''_{323}\) case, the additional possibility of tagging more b-jets further improves sensitivity, such that the large-\(M_0\) region is considerably more constrained than the RPC analog.
For \(\lambda ''_{312}\), in regions with the stau being the LSP, the kinematical suppression of final states with top quarks results in the most abundant decay chains going via off-shell charginos into a neutrino, a bottom quark and two light-flavor jets, cf. Eq. (34). Therefore, searches for missing energy and several jets, e.g. atlas_1308_1841 and atlas_1405_7875, Refs. [128, 129], provide a good coverage. At very low \(M_0\), where the mass difference \(m_{\tilde{\chi }^0_1} - m_{\tilde{\tau }_1}\) is largest, even the search for same-sign leptons, atlas_1404_2500 [133], which is sensitive to the leptonically decaying taus from \(\tilde{\chi }^0 \rightarrow \tilde{\tau } \tau \), becomes effective enough to exclude the area below \(M_{1/2}\lesssim 730~ \)GeV. However, the sensitivity of this analysis to the scenario at hand quickly drops off with decreasing \(m_{\tilde{\chi }^0_1}-m_ {\tilde{\tau }_1}\), as can be seen in the \(\lambda ''_{312}\) case of Fig. 16. For \(\lambda ''_{323} \ne 0\), which features at least four b-jets in the final state, the search for large missing transverse momentum and at least three b-jets, atlas_conf_2013_061 [196], is furthermore able to exclude the rest of the \(\tilde{\tau }\)-LSP parameter space below around \(M_{1/2}\simeq 760~\)GeV.
Summarizing, \(\bar{U} \bar{D} \bar{D}\) couplings within the CMSSM are almost as well covered by LHC analyses as the RPC counterpart. Similarly to the \(LQ\bar{D}\) case, regions with low \(M_0\) are harder to detect at the LHC than the RPC scenario. For large \(M_0\) the searches for many jets and missing energy are very sensitive, leading to bounds as strong as in the RPC–CMSSM, while in the case of a \(\lambda ''_{3i3}\) coupling the bounds are even stricter. We stress again that these results are, in particular in the large \(M_0\) region, specific to the CMSSM boundary conditions. For instance, if the stops were heavier than the gluinos, the bounds which one could set on the corresponding scenario would be considerably weaker [203]. In the stau LSP region, searches for several leptons provide the best constraints whereas for \(\lambda ''_{323}\), multi-b-jet analyses are even more sensitive.
Finally, a comment is in order. Much of the considered parameter space can be excluded or detected in the near future due to the decay products of intermediate top quarks in the final state. This is a consequence of the CMSSM boundary conditions where the stops often appear in either the production or decay channels. At the LHC, there are two methods to identify top quarks. The first method involves reconstructing the individual decay products of the top quark. The second, referred to as top-tagging, involves reconstructing the top-quark decay products inside a single fat-jet. This is possible by analysing the jet substructure if the top is boosted enough, and tagging is in principle already possible if \(p_{T,t} \gtrsim 150\,\)GeV [205]. While top quarks can be produced directly from squark or gluino decays in \(LQ\bar{D}\) and \(\bar{U} \bar{D} \bar{D}\) scenarios, this does not happen in the considered scenarios because of the small couplings and the lighter neutralinos to which each colored sparticle will decay first. Moreover, the lightest neutralino decays to a top final states in \(\lambda ''_{3ij}\) scenarios, but boosted tops would require much heavier neutralinos than what is accessible in the near future (in a CMSSM context). Hence the only possibility for boosted tops is through the stop decays into \(\tilde{\chi }^0_1 t\), which happens for \(\tilde{t}_R\) mainly (while \(\tilde{t}_L\) would decay to the wino first). A naive estimate shows that \(p_{T,t}\) of \(\mathcal {O}(150\,{\mathrm{GeV}})\) is possible in all the remaining parameter space which has not yet been excluded in the figures above. Requiring, however, a significant boost of \(\mathcal {O}(400\,{\mathrm{GeV}})\) for which the tagging efficiency is greatly improved [206], then this occurs only in the upper part of the \(M_0\)–\(M_{1/2}\) plane, as this requires a significant mass splitting between the stop, the top and the neutralino. The associated region is not yet accessible with stop pair production, in particular not at the 8 TeV LHC, since it requires stop masses beyond a TeV, featuring a production cross section of sub-fb. This region will, however, be accessible with more accumulated data at the 13 TeV LHC.