# Sampling the \(\mu \nu \)SSM for displaced decays of the tau left sneutrino LSP at the LHC

## Abstract

Within the framework of the \(\mu \nu \mathrm{SSM}\), a displaced dilepton signal is expected at the LHC from the decay of a tau left sneutrino as the lightest supersymmetric particle (LSP) with a mass in the range 45–100 GeV. We compare the predictions of this scenario with the ATLAS search for long-lived particles using displaced lepton pairs in *pp* collisions, considering an optimization of the trigger requirements by means of a high level trigger that exploits tracker information. The analysis is carried out in the general case of three families of right-handed neutrino superfields, where all the neutrinos get contributions to their masses at tree level. To analyze the parameter space, we sample the \(\mu \nu \)SSM for a tau left sneutrino LSP with proper decay length \(c\tau > 0.1 \, \hbox {mm}\) using a likelihood data-driven method, and paying special attention to reproduce the current experimental data on neutrino and Higgs physics, as well as flavor observables. The sneutrino is special in the \(\mu \nu \mathrm{SSM}\) since its couplings have to be chosen so that the neutrino oscillation data are reproduced. We find that important regions of the parameter space can be probed at the LHC run 3.

## 1 Introduction

The search for low-energy supersymmetry (SUSY) is one of the main goals of the LHC. This search has been focused mainly on signals with missing transverse energy (MET) inspired in *R*-parity conserving (RPC) models, such as the minimal supersymmetric standard model (MSSM) [1, 2, 3, 4]. There, significant bounds on sparticle masses have been obtained [5], especially for strongly interacting sparticles whose masses must be above about 1 TeV [6, 7]. Less stringent bounds of about 100 GeV have been obtained for weakly interacting sparticles, and even the bino-like neutralino is basically not constrained due to its small pair production cross section. Qualitatively similar results have also been obtained in the analysis of simplified *R*-parity violating (RPV) scenarios with trilinear lepton- or baryon-number violating terms [8], assuming a single channel available for the decay of the LSP into leptons. However, this assumption is not possible in other RPV scenarios, such as the ‘\(\mu \) from \(\nu \)’ supersymmetric standard model (\(\mu \nu \mathrm{SSM}\)) [9], where the several decay branching ratios (BRs) of the LSP significantly decrease the signal. This implies that the extrapolation of the usual bounds on sparticle masses to the \(\mu \nu \mathrm{SSM}\) is not applicable.

The most recent analyses of signals at the LHC for LSP candidates in the \(\mu \nu \mathrm{SSM}\) have been dedicated to the left sneutrino [10, 11], and to the bino-like neutralino [12].^{1} In the latter case, it was shown that no points of the parameter space of the \(\mu \nu \mathrm{SSM}\) were excluded when the left sneutrino is the next-to-LSP (NLSP) and hence a suitable source of binos. In the region of bino (sneutrino) masses 110–120 (120–140) GeV it was found a tri-lepton signal compatible with the local excess reported by ATLAS [19]. If this excess were due to a statistical fluctuation,^{2} the prospects for the bounds on the parameter space of the sneutrino-bino mass in the \(\mu \nu \mathrm{SSM}\) were discussed for the 13-TeV search with an integrated luminosity of 100 and \(300 \, \hbox {fb}^{-1}\).

Concerning the left sneutrino LSP, in Ref. [10] the prospects for detection of signals with di-photon plus leptons or MET from neutrinos, and multi-leptons, from the pair production of left sneutrinos/sleptons and their prompt decays (\(c\tau \lesssim 0.1 \, \hbox {mm}\)), were analyzed. A significant evidence is expected only in the mass range of about 100–300 GeV. The mass range of 45–100 GeV (with the lower limit imposed not to disturb the decay width of the *Z*) was covered in Ref. [11] for the tau left sneutrino (\({\widetilde{\nu }}_{\tau }\)) LSP. First, it was checked that no constraint on the \({\widetilde{\nu }}_{\tau }\) mass is obtained from previous searches. In particular, since the sneutrino has several relevant decay modes, the LEP lower bound on its mass mass of about 90 GeV [21, 22, 23, 24, 25, 26] obtained under the assumption of BR one to leptons, via trilinear RPV couplings, is not applicable. Similar conclusions were obtained from LEP mono-photon search (gamma+MET) [27], and LHC mono-photon and mono-jet (jet+MET) searches [28, 29]. Concerning LEP searches for staus [21, 22, 23, 24, 25, 26], in the \(\mu \nu \mathrm{SSM}\) the left stau does not decay directly but through an off-shell *W* and a \({\widetilde{\nu }}_{\tau }\), and therefore searches for its direct decay are not relevant in this model. Although the sneutrino mass can in principle be constrained using searches for final states as those of the \(\mu \nu \mathrm{SSM}\) from the production of a pair of \({\widetilde{\nu }}_{\tau }\) from staus, it was also checked in Ref. [11] that this is not the case. Then, the displaced-vertex decays of the \({\widetilde{\nu }}_{\tau }\) LSP producing signals with di-lepton pairs was studied. Using the present data set of the ATLAS 8-TeV dilepton search [30], the conclusion was that one can constrain the sneutrino in some regions of the parameter space of the \(\mu \nu \mathrm{SSM}\), especially when the Yukawa couplings and mass scale of neutrinos are rather small. In order to improve the sensitivity of this search, it was proposed an optimization of the trigger requirements exploited in ATLAS based on a high level trigger that utilizes the tracker information.

The above analyses were carried out in the simplest case of the \(\mu \nu \mathrm{SSM}\) with one right-handed neutrino superfield. Thus only one of the light neutrinos gets a nonvanishing tree-level contribution to its mass, whereas the other two masses rely on loop corrections. Basically, the only experimental constraint imposed in those works was that the heavier neutrino mass should be in the range \(m_\nu \sim [0.05, 0.23] \, \hbox {eV}\), i.e. below the upper bound on the sum of neutrino masses \(\sim 0.23 \, \hbox {eV}\) [31], and above the square root of the mass-squared difference \(\Delta m_{\mathrm{atm}}^2\sim 2.42\times 10^{-3}\mathrm {eV}^2\) [32]. In addition, the simplified assumption that all neutrino Yukawas have the same value was also applied. Although these analyses were useful to get a first idea of the accelerator constraints on the left sneutrino LSP, the lack of experimental bounds on the masses of the superpartners in the \(\mu \nu \mathrm{SSM}\) makes it peremptory a detailed study reproducing the whole neutrino physics. This is the aim of this work. We will reconsider the analysis of Ref. [11], but in the context of the \(\mu \nu \)SSM with three families of right-handed neutrino superfields where all the neutrinos get contributions to their masses at tree level, and different values of the neutrino Yukawas are necessary to reproduce neutrino physics. In particular, we will study the constraints on the parameter space by sampling the model to get the \({\widetilde{\nu }}_{\tau }\) LSP in the range of masses \(45-100 \, \hbox {GeV}\), with a decay length of the order of the millimeter. We will pay special attention to reproduce the experimental neutrino masses and mixing angles [33, 34, 35, 36]. The different values of the neutrino Yukawas will imply that certain regions of the parameter space are excluded by the LEP analysis, unlike the result of Ref. [11]. In addition, we will impose on the resulting parameters to be in agreement with Higgs data and other observables.

The paper is organized as follows. In Sect. 2, we will briefly review the \(\mu \nu \mathrm{SSM}\) and its relevant parameters for our analysis of the neutrino/sneutrino sector, emphasizing the special role of the sneutrino in this scenario since its couplings have to be chosen so that the neutrino oscillation data are reproduced. In Sect. 3, we will introduce the phenomenology of the \({\widetilde{\nu }}_{\tau }\) LSP, studying its pair production channels at the LHC, as well as the signals. These consist of two dileptons or a dilepton plus MET from the sneutrino decays. Then, we will consider the existing dilepton displaced-vertex searches, and discuss its feasibility and significance on \({\widetilde{\nu }}_{\tau }\) searches. In Sect. 4, we will discuss the strategy that we employed to perform scans searching for points of the parameter space of our scenario compatible with current experimental data on neutrino and Higgs physics, as well as flavor observables. The results of these scans will be presented in Sect. 5, and applied to show the current reach of the LHC search on the parameter space of the \({\widetilde{\nu }}_{\tau }\) LSP based on the ATLAS 8-TeV result [30], and the prospects for the 13-TeV searches. Finally, our conclusions are left for Sect. 6.

## 2 The \(\mu \nu \)SSM

*R*-parity charges consistently to the right-handed neutrinos (\(\nu _{iR}\)), thus producing explicit RPV (harmless for proton decay). Note nevertheless, that in the limit \(Y_{{\nu }_{ij}} \rightarrow 0\), \({{\hat{\nu }}}^c\) can be identified in the superpotential as a pure singlet superfield without lepton number, similar to the next-to-MSSM (NMSSM) [40], and therefore

*R*parity is restored. Thus, the neutrino Yukawa couplings \(Y_{\nu _{ij}}\) are the parameters which control the amount of RPV in the \(\mu \nu \mathrm{SSM}\), and as a consequence this violation is small. After the electroweak symmetry breaking (EWSB) induced by the soft SUSY-breaking terms of the order of the TeV, and with the choice of CP conservation, the neutral Higgses (\(H_{u,d}\)) and right (\({{\widetilde{\nu }}}_{iR}\)) and left (\({\widetilde{\nu }}_i\)) sneutrinos develop the following vacuum expectation values (VEVs):

The new couplings and sneutrino VEVs in the \(\mu \nu \mathrm{SSM}\) induce new mixing of states. The associated mass matrices were studied in detail in Refs. [10, 14, 37]. Summarizing, there are eight neutral scalars and seven neutral pseudoscalars (Higgses-sneutrinos), eight charged scalars (charged Higgses-sleptons), five charged fermions (charged leptons-charginos), and ten neutral fermions (neutrinos-neutralinos). In the following, we will concentrate in briefly reviewing the neutrino and left sneutrino mass eigenstates, which are the relevant ones for our analysis.

As we can understand from these equations, neutrino physics in the \(\mu \nu \mathrm{SSM}\) is closely related to the parameters and VEVs of the model, since the values chosen for them must reproduce current data on neutrino masses and mixing angles.

### 2.1 Neutrino/sneutrino physics

Since reproducing neutrino data is an important asset of the \(\mu \nu \mathrm{SSM}\), as explained above, we will try to establish here qualitatively what regions of the parameter space are the best in order to be able to obtain correct neutrino masses and mixing angles. In particular, we will determine natural hierarchies among neutrino Yukawas, and among left sneutrino VEVs.

In addition, left sneutrinos are special in the \(\mu \nu \mathrm{SSM}\) with respect to other SUSY models. This is because, as discussed in Eq. (7), their masses are determined by the minimization equations with respect to \(v_i\). Thus, they depend not only on left sneutrino VEVs but also on neutrino Yukawas, and as a consequence neutrino physics is very relevant. In particular, if we work with Eq. (11) assuming the simplest situation that all the \(A_{{\nu }_i}\) are naturally of the order of the TeV, neutrino physics determines sneutrino masses through the prefactor \({Y_{{\nu }_i}v_u}/{v_i}\). Considering the normal ordering (NO) for the neutrino mass spectrum, which is nowadays favored by the analyses of neutrino data [33, 34, 35, 36], representative solutions for neutrino/sneutrino physics using diagonal neutrino Yukawas in this scenario are summarized below. Note that these solutions take advantage of the dominance of the gaugino seesaw for some of the three neutrino families.

(1) \(M<0\), with \(Y_{\nu _1}<Y_{\nu _2}, Y_{\nu _3}\), and \(v_1 > v_2, v_3\).

As explained in Refs. [38, 41], a negative value for *M* is useful in order to reproduce neutrino data with \(Y_{\nu _1}\) the smallest Yukawa and \(v_1\) the largest VEV. Essentially, this is because a small tuning in Eq. (3) between the gaugino seesaw and the \(\nu _{R}\)-Higgsino seesaw is necessary in order to obtain the correct mass of the first family. Here the contribution of the gaugino seesaw is always the largest one. On the contrary, for the other two neutrino families, the contribution of the \(\nu _{R}\)-Higgsino seesaw is the most important one and that of the gaugino seesaw is less relevant for the tuning. Following the above discussion about the prefactor of Eq. (11), these hierarchies of Yukawas and VEVs determine that \(m_{{\widetilde{\nu }}_{1}}\) is the smallest of all the sneutrino masses.

(2) \(M>0\), with \(Y_{\nu _3}< Y_{\nu _1} < Y_{\nu _2}\), and \(v_1<v_2\sim v_3\).

In this case, it is easy to find solutions with the gaugino seesaw as the dominant one for the third family. Then, \(v_3\) determines the corresponding neutrino mass and \(Y_{\nu _3}\) can be small. On the other hand, the NO for neutrinos determines that the first family dominates the lightest mass eigenstate implying that \(Y_{\nu _{1}}< Y_{\nu _{2}}\) and \(v_1 < v_2,v_3\), with both \(\nu _{R}\)-Higgsino and gaugino seesaws contributing significantly to the masses of the first and second family. Taking also into account that the composition of these two families in the second mass eigenstate is similar, we expect \(v_2 \sim v_3\). Now for this solution we will have \(m_{{\widetilde{\nu }}_{3}}\) as the smallest of all the sneutrino masses.

(3) \(M>0\), with \(Y_{\nu _2}< Y_{\nu _1} < Y_{\nu _3}\), and \(v_1<v_2\sim v_3\).

These solutions can be deduced from the previous ones in (2) interchanging the values of the third family, \(Y_{\nu _3}\) and \(v_3\), with the corresponding ones of the second family, \(Y_{\nu _2}\) and \(v_2\). A small adjust in the parameters will lead again to a point in the parameter space satisfying neutrino data. This is clear from the fact that \(\theta _{13}\) and \(\theta _{12}\) are not going to be significantly altered, whilst \(\theta _{23}\) may require a small tuning in the parameters. If the gaugino seesaw dominates for the second family, \(v_2\) determines the corresponding neutrino mass and \(Y_{\nu _2}\) can be small. Then, \(m_{{\widetilde{\nu }}_{2}}\) will be the smallest of all sneutrino masses.

We will see in the next subsection that solutions of type (2) are the ones interesting for our analysis.

Let us finally point out that when off-diagonal neutrino Yukawas are allowed, it is not possible to arrive to a general conclusion regarding the hierarchy in sneutrinos masses, specially when the gaugino seesaw is sub-dominant. This is because one can play with the hierarchies among \(v_i\) with enough freedom in the neutrino Yukawas in order to reproduce the experimental results. Therefore, there is no a priori knowledge of the hierarchies in the sneutrino masses, and carrying out an analysis case by case turns out to be necessary.

### 2.2 \({\widetilde{\nu }}_{\tau }\) LSP

In the \(\mu \nu \mathrm{SSM}\), because of RPV any SUSY particle can be a candidate for the LSP. Nevertheless, the case of the \({\widetilde{\nu }}_{\tau }\) LSP turns out to be particularly interesting because of the large value of the tau Yukawa coupling, which can give rise to significant BRs for decays to^{3} \(\tau \tau \) and \(\tau \ell \), once the sneutrinos are dominantly pair-produced via a Drell-Yan process mediated by a virtual *W*, *Z* or \(\gamma \), as we will discuss in the next section.

There is enough freedom in the parameter space of the \(\mu \nu \mathrm{SSM}\) in order to get light left sneutrinos. Assuming as discussed above that the \(A_{{\nu }_i}\) are naturally of the order of the TeV, values of the prefactor of Eq. (11) \({Y_{{\nu }_i}v_u}/{v_i}\) in the range of about 0.01–1, i.e. \(Y_{{\nu }_i}\sim 10^{-8}-10^{-6}\), will give rise to left sneutrino masses in the range of about 100–1000 GeV. Thus, with the hierarchy of neutrino Yukawas \(Y_{{\nu }_{3}}\sim 10^{-8}-10^{-7}<Y_{{\nu }_{1,2}}\sim 10^{-6}\), we can obtain a \({\widetilde{\nu }}_{\tau }\) LSP with a mass around 100 GeV whereas the masses of \({\widetilde{\nu }}_{e,\mu }\) are of the order of the TeV. Clearly, we are in the case of solutions for neutrino physics of type (2) discussed in Sect. 2.1. Actually this type of hierarchy, with significant values for \(Y_{{\nu }_{1,2}}\), increases the dilepton BRs of the \({\widetilde{\nu }}_{\tau }\) LSP producing signals that can be probed at the LHC, as the analysis of the next sections will show.

It is worth noticing here that in this scenario the left stau can be naturally the NLSP, since it is only a little heavier than the \({\widetilde{\nu }}_{\tau }\) because they are in the same *SU*(2) doublet, with the mass splitting mainly due to the usual small D-term contribution, \(-m_W^2 \cos 2\beta \). As we will see in the next section, this has implications for the production of the left sneutrino LSP at the LHC, because the direct production of sleptons and their decays is a significant source of sneutrinos.

## 3 Searching for \({\widetilde{\nu }}_{\tau }\) LSP at the LHC

*pp*collisions at \(\sqrt{s}= 8~\hbox {TeV}\), as well as the prospects for the 13-TeV searches.

The direct production of \({\widetilde{\nu }}_{\tau }\) occurs via a *Z* channel giving rise to a pair of scalar and pseudoscalar left sneutrinos, as shown in Fig. 1a. Note that they are co-LSPs since they have essentially degenerate masses, as explained in the previous section. On the other hand, since the left stau is typically the NLSP its direct production and decay is another important source of the \({\widetilde{\nu }}_{\tau }\) LSP. In particular, pair production can be obtained through a \(\gamma \) or *Z* decaying into two staus, as shown in Fig. 1b, with the latter having a dominant RPC prompt decay into a (scalar or pseudoscalar) sneutrino plus an off-shell *W* producing a soft meson or a pair of a charged lepton and a neutrino. Note that although RPV decays of the stau are possible, e.g. stau into a tau plus a neutrino, they are extremely supressed compared to the RPC one. Numerically, the stau has partial decay widths through RPV diagrams \(\sim 10^{-14} - 10^{-13} \, \hbox {GeV}\), while the ones corresponding to the RPC three-body decays are \(\sim 10^{-7} \, \hbox {GeV}\). Therefore, its proper decay length is \(\sim 10^{-9} \, \hbox {m}\), with the BRs corresponding to the RPV decays \(< 10^{-6}\). Sneutrinos can also be pair produced through a *W* decaying into a stau and a (scalar or pseudoscalar) sneutrino as shown in Fig. 1c, with the stau decaying as before.

*i*and

*l*neutrino physical and flavor indices, respectively. The relevant diagrams for \({\widetilde{\nu }}_{\tau }\) searches that include this decay mode are the same as in Fig. 1, but substituting one of the \(\tau \,\ell /\tau \) vertices by a two-neutrino vertex.

Let us remark that other decay channels of the \({\widetilde{\nu }}_{\tau }\) can be present and have been taken into account in our numerical computation, but they turn out to be negligible for the sneutrino masses that we are interested in this work.

One muon with \(p_{\mathrm{T}}>50~\hbox {GeV}\) and \(|\eta |<1.07\), one electron with \(p_{\mathrm{T}}>120~\hbox {GeV}\) or two electrons with \(p_\mathrm{T}>40~\hbox {GeV}\) each,

One pair \(e^+e^-\), \(\mu ^+\mu ^-\) or \(e^\pm \mu ^\mp \) with \(p_\mathrm{T} >10~\hbox {GeV}\) and \(0.02<|\eta |<2.5\) for each particle.

^{4}

At least one muon with \(p_{\mathrm{T}}>24~\hbox {GeV}\).

One pair \(\mu ^+\mu ^-\) or \(e^\pm \mu ^\mp \) with \(p_{\mathrm{T}} >10~\hbox {GeV}\) and \(0.02<|\eta |<2.5\) for each particle.

At least one electron or muon with \(p_{\mathrm{T}}>26~\hbox {GeV}\).

One pair \(\mu ^+\mu ^-\), \(e^+e^-\), or \(e^\pm \mu ^\mp \) with \(p_{\mathrm{T}} >10~\hbox {GeV}\) and \(0.02<|\eta |<2.5\) for each particle.

*ee*channels, can be found in Tables III–IX of Ref. [11]. As pointed out also in that work, this possible improvement is not only for the ATLAS analysis but also for the CMS one [42].

Let us finally remark that in our analysis below, we scan the parameter space of the model and therefore \(m_{{{\widetilde{\nu }}}_\tau }\) can be regarded as a continuous variable, unlike Ref. [11] where the sneutrino masses used were 50, 60, 80 and 100 GeV. For the selection efficiency we used a polynomial fitting from the discrete values of \(\epsilon _{\mathrm{sel}}\) given in Ref. [11] for each production mode, whereas for the vertex-level efficiency, the fitting function is of the form \(e^{P[\log (c\tau )]}\), where *P*[*x*] is a polynomial in the variable *x*.

## 4 Strategy for the scanning

In this section we describe the methodology that we employed to search for points of our parameter space that are compatible with the current experimental data on neutrino and Higgs physics, as well as ensuring that the \({\widetilde{\nu }}_{\tau }\) is the LSP with a mass in the range of 45–100 GeV. In addition, we demanded the compatibility with some flavor observables. To this end, we performed scans on the parameter space of the model, with the input parameters optimally chosen.

### 4.1 Sampling the \(\mu \nu \)SSM

For the sampling of the \(\mu \nu \mathrm{SSM}\), we used a likelihood data-driven method employing the Multinest [46] algorithm as optimizer. The goal is to find regions of the parameter space of the \(\mu \nu \mathrm{SSM}\) that are compatible with a given experimental data.

To compute the spectrum and the observables we used SARAH [47] to generate a SPheno [48, 49] version for the model. We condition that each point is required not to have tachyonic eigenstates. For the points that pass this constraint, we compute the likelihood associated to each experimental data set and for each sample all the likelihoods are collected in the joint likelihood \({\mathcal {L}}_{\text {tot}}\) (see Eq. (18) above).

### 4.2 Likelihoods

*x*, \(\sigma _T^2= \sigma ^2 +\tau ^2\) with \(\sigma \) and \(\tau \) being respectively the experimental and theoretical uncertainties on the observable

*x*.

*p*takes \(+1\) when \(x_0\) represents the lower limit and \(-1\) in the case of upper limit, while erfc is the complementary error function.

The last class of likelihood function we used is a step function in such a way that the likelihood is one/zero if the constraint is satisfied/non-satisfied.

It is important to mention that in this work unless explicitly mentioned, the theoretical uncertainties \(\tau \) are unknown and therefore are taken to be zero. Subsequently, we present each constraint used in this work together with the corresponding type of likelihood function.

**Tau left sneutrino mass**

In order to concentrate the sampling in the area in which the mass of the tau left sneutrino \(m_{{\widetilde{\nu }}_{\tau }} \in (45 , 100) \, \hbox {GeV}\), we constructed a likelihood function \({\mathcal {L}}_{{{\widetilde{\nu }}}_\tau }\) which is a Gaussian (see Eq. (19)) with mean value \(\mu _{m_{{\widetilde{\nu }}_{\tau }}} = 70 \, \hbox {GeV}\) and width \(\sigma _{m_{{\widetilde{\nu }}_{\tau }}} = 10 \, \hbox {GeV}\), and included it in the combined likelihood.

**Neutrino observables**

^{5}where \(\delta m^2=m_2^2-m_1^2\) and \(\Delta m^2=m_3^2-(m_2^2+ m_1^2)/2\). For each of the observables listed in the neutrino sector, the likelihood function is a Gaussian (see Eq. (19)) centered at the mean value \(\mu _{exp}\) and with width \(\sigma _{exp}\). Concerning the cosmological upper bound on the sum of the masses of the light active neutrinos given by \(\sum m_{\nu _i} < 0.12 \, \hbox {eV}\) [50], even though we did not include it directly in the total likelihood, we imposed it on the viable points obtained.

Neutrino data used in the sampling of the \(\mu \nu \)SSM

Parameters | \(\sin ^2{\theta _{12}}\) | \(\sin ^2 {\theta _{13}}\) | \(\sin ^2 {\theta _{23}}\) | \( \delta m^2\, /\, 10^{-5} \, ({\hbox {eV}}^2)\) | \(\Delta m^2\, /\, 10^{-3} \, ({\hbox {eV}}^2)\) |
---|---|---|---|---|---|

\(\mu _{exp}\) | 0.297 | 0.0215 | 0.425 | 7.37 | 2.525 |

\(\sigma _{exp}\) | 0.017 | 0.0007 | 0.021 | 0.17 | 0.042 |

**Higgs observables**

Before the discovery of the SM-like Higgs boson, the negative searches of Higgs signals at the Tevatron, LEP and LHC, were transformed into exclusions limits that must be used to constrain any model. Its discovery at the LHC added crucial constraints that must be taken into account in those exclusion limits. We have considered all these constraints in the analysis of the \(\mu \nu \mathrm{SSM}\), where the Higgs sector is extended with respect to the MSSM as discussed in Sect. 2. For constraining the predictions in that sector of the model, we interfaced HiggsBounds v5.3.2 [51, 52] with MultiNest. First, several theoretical predictions in the Higgs sector (using a \(\pm 3 \, \hbox {GeV}\) theoretical uncertainty on the SM-like Higgs boson) are provided to determine which process has the highest exclusion power, according to the list of expected limits from LEP and Tevatron. Once the process with the highest statistical sensitivity is identified, the predicted production cross section of scalars and pseudoscalars multiplied by the BRs are compared with the limits set by these experiments. Then, whether the corresponding point of the parameter under consideration is allowed or not at 95% confidence level is indicated. In constructing the likelihood from HiggsBounds constraints, the likelihood function is taken to be a step function. Namely, it is set to one for points for which Higgs physics is realized, and zero otherwise. Finally, in order to address whether a given Higgs scalar of the \(\mu \nu \mathrm{SSM}\) is in agreement with the signal observed by ATLAS and CMS, we interfaced HiggsSignals v2.2.3 [53, 54] with MultiNest. A \(\chi ^2\) measure is used to quantitatively determine the compatibility of the \(\mu \nu \mathrm{SSM}\) prediction with the measured signal strength and mass. The experimental data used are those of the LHC with some complements from Tevatron. The details of the likelihood evaluation can be found in Refs. [53, 54].

**B decays**

\(b \rightarrow s \gamma \) is a flavour changing neutral current (FCNC) process, and hence it is forbidden at tree level in the SM. However, its occurs at leading order through loop diagrams. Thus, the effects of new physics (in the loops) on the rate of this process can be constrained by precision measurements. In the combined likelihood, we used the average value of \((3.55 \pm 0.24) \times 10^{-4}\) provided in Ref. [55]. Notice that the likelihood function is also a Gaussian (see Eq. (19)). Similarly to the previous process, \(B_s \rightarrow \mu ^+\mu ^-\) and \(B_d \rightarrow \mu ^+\mu ^-\) are also forbidden at tree level in the SM but occur radiatively. In the likelihood for these observables (19), we used the combined results of LHCb and CMS [56], \( \text {BR} (B_s \rightarrow \mu ^+ \mu ^-) = (2.9 \pm 0.7) \times 10^{-9}\) and \( \text {BR} (B_d \rightarrow \mu ^+ \mu ^-) = (3.6 \pm 1.6) \times 10^{-10}\). Concerning the theoretical uncertainties for each of these observables we take \(\tau = 10 \%\) of the corresponding best fit value. We denote by \({\mathcal {L}}_{\text {B physics}}\) the likelihood from \(b \rightarrow s \gamma \), \(B_s \rightarrow \mu ^+\mu ^-\) and \(B_d \rightarrow \mu ^+\mu ^-\).

\(\mu \rightarrow e \gamma \) and \(\mu \rightarrow e e e\)

We also included in the joint likelihood the constraint from \(\hbox {BR}(\mu \rightarrow e\gamma ) < 5.7\times 10^{-13}\) and \(\hbox {BR} (\mu \rightarrow eee) < 1.0 \times 10^{-12}\). For each of these observables we defined the likelihood as a step function. As explained before, if a point is in agreement with the data, the likelihood \({\mathcal {L}}_{\mu \text { decay}}\) is set to 1 otherwise to 0.

Let us point out here that we did not try to explain the interesting but not conclusive \(3.5\sigma \) discrepancy between the measurement of the anomalous magnetic moment of the muon and the SM prediction, \(\Delta a_{\mu }= a_{\mu }^{\text {exp}}- a_{\mu }^{\text {SM}}= (26.8 \pm 6.3\pm 4.3) \times 10^{-10} \) [5]. Since we decouple the rest of the SUSY spectrum with respect to the tau left sneutrino mass, we do not expect a large SUSY contribution over the SM value. We checked for the points fulfilling all constrains discussed in Sect. 5, that the extra contribution \(a_{\mu }^{\text {SUSY}}\) is within the SM uncertainty.

**Chargino mass bound**

In RPC SUSY, the lower bound on the lightest chargino mass of about 94 GeV depends on the spectrum of the model [5, 57]. Although in the \(\mu \nu \mathrm{SSM}\) there is RPV and therefore this constraint does not apply automatically, to compute \({\mathcal {L}}_{m_{{{\widetilde{\chi }}}^\pm }}\) we have chosen a conservative limit of \(m_{{{\widetilde{\chi }}}^\pm _1} > 92 \, \hbox {GeV}\) with the theoretical uncertainty \(\tau = 5 \%\) of the chargino mass.

### 4.3 Input parameters

In order to efficiently scan for the \({\widetilde{\nu }}_{\tau }\) LSP in the \(\mu \nu \)SSM with a mass in the range \(45-100 \, \hbox {GeV}\), it is important to identify first the parameters to be used, and optimize their number and their ranges of values. This is what we carry out here, where we discuss the most relevant parameters for obtaining correct neutrino and Higgs physics, providing at the same time the \({\widetilde{\nu }}_{\tau }\) as the LSP with the mass in the desired range.

The relevant parameters in the neutrino sector of the \(\mu \nu \mathrm{SSM}\) are \(\lambda , \kappa , v_R, v_i, Y_{\nu _i}, \tan \beta \) and *M* (see Eq. (6)). Since \(\lambda , \kappa \) and \(v_R\) are crucial for Higgs physics, we will fix first them to appropriate values. The parameter \(\tan \beta \) is also important for both, Higgs and neutrino physics, thus we will consider a narrow range of possible values to ensure good Higgs physics. Concerning *M*, which is a kind of average of bino and wino soft masses (see Eq. (5)), inspired by GUTs we will assume \(M_2 = 2M_1\), and scan over \(M_2\). On the other hand, sneutrino masses introduce in addition the parameters \(T_{{\nu }_i}\) (see Eq. (7)). In particular, \(T_{{\nu }_3}\) is the most relevant one for our discussion of the \({\widetilde{\nu }}_{\tau }\) LSP, and we will scan it in an appropriate range of small values. Since the left sneutrinos of the first two generations must be heavier, we will fix \(T_{{\nu }_{1,2}}\) to a larger value.

Summarizing, we will perform scans over the 9 parameters \(Y_{\nu _i}, v_i, T_{{\nu }_3}, \tan \beta , M_2\), as shown in Table 2, using log priors (in logarithmic scale) for all of them, except for \(\tan \beta \) which is taken to be a flat prior (in linear scale). The ranges of \(v_i\) and \(Y_{\nu _i}\) are natural in the context of the electroweak-scale seesaw of the \(\mu \nu \mathrm{SSM}\). The range of \(T_{\nu _{3}}\) is also natural if we follow the usual assumption based on the supergravity framework discussed in Eq. (11) that the trilinear parameters are proportional to the corresponding Yukawa couplings, i.e. in this case \(T_{\nu _{3}}= A_{\nu _{3}} Y_{\nu _3}\) implying \(-A_{\nu _{3}}\in \) (\(1, 10^{4}\)) GeV. Concerning \(M_2\), its range of values is taken such that a bino at the bottom of the neutralino spectrum leaves room to accommodate a \({\widetilde{\nu }}_{\tau }\) LSP with a mass below 100 GeV. Scans 1 (\(S_1\)) and 2 (\(S_2\)) correspond to different values of \(\tan \beta \), and other benchmark parameters as shown in Table 3.

Range of low-energy values of the input parameters that are varied in the two scans, where \(Y_{\nu _{i}}\), \(v_i\), \(T_{\nu _{3}}\) and \(M_2\) are \(\log \) priors while \(\tan \beta \) is a flat prior. The VEVs \(v_i\), and the soft parameters \(T_{\nu _{3}}\) and \(M_2\), are given in GeV

Scan 1 (\(S_1\)) | Scan 2 (\(S_2\)) |
---|---|

\(\tan \beta \in (10, 16)\) | \(\tan \beta \in (1, 4)\) |

\(Y_{\nu _{i}} \in (10^{-8} , 10^{-6})\) | |

\(v_i \in (10^{-6} , 10^{-3})\) | |

\(-T_{\nu _{3}} \in (10^{-6} , 10^{-4})\) | |

\(M_2 \in (150 , 2000)\) |

Low-energy values of the input parameters that are fixed in the two scans. The VEV \(v_R\) and the soft trilinear parameters, soft gluino masses and soft scalar masses are given in GeV

Parameter | Scan 1 (\(S_1\)) | Scan 2 (\(S_2\)) |
---|---|---|

\(\lambda \) | 0.102 | 0.42 |

\(\kappa \) | 0.4 | 0.46 |

\(v_R\) | 1750 | 421 |

\(T_{\lambda }\) | 340 | 350 |

\(-T_{\kappa }\) | 390 | 108 |

\(-T_{u_{3}}\) | 4140 | 1030 |

\(m_{{\widetilde{Q}}_{3L}}\) | 2950 | 1972 |

\(m_{{\widetilde{u}}_{3R}}\) | 1140 | 1972 |

\(M_3\) | 2700 | |

\(m_{{\widetilde{Q}}_{1,2L}}, m_{{\widetilde{u}}_{1,2R}}, m_{{\widetilde{d}}_{1,2,3R}}, m_{{\widetilde{e}}_{1,2,3R}}\) | 1000 | |

\(T_{u_{1,2}}\) | 0 | |

\(T_{d_{1,2}}\), \(T_{d_{3}}\) | 0, 100 | |

\(T_{e_{1,2}}\), \(T_{e_{3}}\) | 0, 40 | |

\(-T_{\nu _{1,2}}\) | \(10^{-3}\) |

For scan \(S_2\), where we choose a large value for \(\lambda \), we are in a similar situation as in the NMSSM, and a small value of \(\tan \beta \), and moderate values of \(|T_{u_{3}}|\) and soft stop masses, are sufficient to reproduce the correct SM-like Higgs mass. Now, a moderate value of \(v_R\) is sufficient to obtain a large enough value of \(\mu \). In particular, we choose \(v_R=421 \, \hbox {GeV}\) giving rise to \(\mu \approx 375 \, \hbox {GeV}\). This value of \(v_R\) implies that \(|T_{\kappa }|\) cannot be as large as for scan \(S_1\) because then a too large value of \(\kappa \) would be needed to avoid tachyonic scalar right sneutrinos. Thus we choose \(T_{\kappa }=-108 \, \hbox {GeV}\), and \(\kappa =0.42\), which produces scalar and pseudoscalar sneutrinos lighter than in scan \(S_1\) but still heavier than \({\widetilde{\nu }}_{\tau }\) LSP and left stau NLSP. In particular, their masses are in the ranges 225–256 \(\, \hbox {GeV}\) and 345–355 \(\hbox {GeV}\), respectively.

The values of the parameters shown below \(m_{{\widetilde{u}}_{3R}}\) in Table 3, concerning gluino, and squark and slepton masses, and quark and lepton trilinear parameters, are not specially relevant for our analysis, and we choose for each of them the same values for both scans. Finally, compared to the values of \(T_{\nu _{3}}\), the values chosen for \(T_{\nu _{1,2}}\) are natural within our framework \(T_{\nu _{1,2}}= A_{\nu _{1,2}} Y_{\nu _{1,2}}\), since larger values of the Yukawa couplings are required for similar values of \(A_{\nu _{i}}\). In the same way, the values of \(T_{d_3}\) and \(T_{e_3}\) have been chosen taking into account the corresponding Yukawa couplings

## 5 Results

By using the methods described in the previous sections, we evaluate now the current and potential limits on the parameter space of our scenario from the displaced-vertex searches with the 8-TeV ATLAS result [30], and discuss the prospects for the 13-TeV searches.

To find regions consistent with experimental observations we have performed about 72 million of spectrum evaluations in total and the total amount of computer required for this was approximately 380 CPU years.

To carry this analysis out, we follow several steps. First, we select points from the scan that lie within \(\pm 3\sigma \) of all neutrino physics observables, namely the mixing angles and mass squared differences. Second, we put \(\pm 3\sigma \) cuts from \(b \rightarrow s \gamma \), \(B_s \rightarrow \mu ^+\mu ^-\) and \(B_d \rightarrow \mu ^+\mu ^-\). The points that pass these cuts are required to satisfy also the upper limits of \(\mu \rightarrow e \gamma \) and \(\mu \rightarrow eee\). The third step in the selection of our points is to ensure a tau left sneutrino LSP with \(m_{{{\widetilde{\nu }}}_\tau } \in (45, {100}) \, \hbox {GeV}\), and the left stau as the NLSP. In the fourth step we impose that Higgs physics is realized. As already mentioned, we use HiggsBounds and HiggsSignals taking into account the constraints from the latest 13-TeV results. In particular, we require that the p-value reported by HiggsSignals be larger than 5 %. It is worth noticing here that, with the help of Vevacious [58], we have also checked that the EWSB vacua corresponding to the previous allowed points are stable.

### 5.1 Constraints from neutrino/sneutrino physics

As discussed in detail in Sect. 2, reproducing neutrino physics is an important asset of the \(\mu \nu \mathrm{SSM}\). It is therefore important to analyze first the constraints imposed by this requirement on the relevant parameter space of the model when the \({\widetilde{\nu }}_{\tau }\) is the LSP.

Let us finally note that \(m_{{{\widetilde{\nu }}}_{\tau }}\) is always larger than about 61 GeV, which corresponds to half of the mass of the SM-like Higgs (remember that we allow a \(\pm \,3 \, \hbox {GeV}\) theoretical uncertainty on its mass). For smaller masses, the latter would dominantly decay into sneutrino pairs, leading to an inconsistency with Higgs data.^{6}

In Fig. 3, we show \(v_3\) vs. \(Y_{\nu _3}\) for scan \(S_1\) (left) and scan \(S_2\) (right), with the colours indicating now different values of *M*. There we can see that the greater \(v_3\) is, the greater *M* becomes. In addition, for a fixed value of \(v_3\), *M* is quite independent of the variation in \(Y_{\nu _3}\). This confirms that, as explained in solution (2) of Sect. 2.1, the gaugino seesaw is the dominant one for the third neutrino family. From the figure, we can see that the range of *M* reproducing the correct neutrino physics is 346–2223 GeV for scan \(S_1\) and 248–2100 \(\hbox {GeV}\) for \(S_2\), corresponding to \(M_2\) in the range 236–1515 \(\hbox {GeV}\) and 169–1431 \(\hbox {GeV}\), respectively. Note that for a fixed value of \(v_3\), when \(Y_{\nu _3}\) is sufficiently large the \({\widetilde{\nu }}_{\tau }\) becomes heavier than 100 GeV, and these points are not shown in the figure. As can also be seen, \(Y_{\nu _3}\) acquires larger values in scan \(S_2\) than in \(S_1\), in agreement with the discussion of Fig. 2.

### 5.2 Constraints from accelerator searches

Once the neutrino (and sneutrino) physics has determined the relevant regions of the parameter space of the \({\widetilde{\nu }}_{\tau }\) LSP in the \(\mu \nu \mathrm{SSM}\), we are ready to analyze the reach of the LHC search.

Given that for each scan the largest neutrino Yukawa is \(Y_{\nu _{2}}\), the most important contribution to the dilepton BRs comes from the channel \({\widetilde{\nu }}_{\tau }\rightarrow \tau \mu \). We also expect that the BR(\({{\widetilde{\nu }}}_\tau \rightarrow \mu \mu \)) is larger for scan \(S_1\) than for \(S_2\), as can be checked in Fig. 5 (left plots),^{7} where BR(\({{\widetilde{\nu }}}^{{\mathcal {R}}}_{\tau } \rightarrow \mu \mu \)) is plotted vs. *M*, for the points fulfilling all constraints from neutrino/sneutrino physics (although not shown here, a similar figure is obtained in the case of the pseudoscalar \({{\widetilde{\nu }}}^{{\mathcal {I}}}_{\tau }\)). The main reason is the smaller (larger) value of \(\lambda \) (\(\tan \beta \)) for scan \(S_1\) with respect to \(S_2\), which are crucial parameters in Eq. (13) for the partial decay width. Although \(\tan \beta \) does not appear explicitly in that equation, note that \(Y_{\tau }= ({\sqrt{2} m_{\tau }}/{v})\sqrt{\tan ^2\beta + 1}\). In addition, as shown in Fig. 4, the value of \(Y_{\nu _{2}}\) is larger for scan \(S_1\) than for \(S_2\), contributing therefore to larger BRs. We can also observe in both plots of Fig. 5 for the BRs that they increase with larger values of *M*. This can be understood from Eq. (14) showing that larger values of *M* decrease the decay width to neutrinos. In Fig. 5 (right plots), we show the proper decay length of the \({{\widetilde{\nu }}}^{{\mathcal {R}}}_{\tau }\) vs. *M*. Clearly, this is larger for scan \(S_2\) than for \(S_1\) because the BRs into charged leptons are smaller in the former case, as discussed before. Let us finally remember that the lower and upper bounds on *M* in the figure, have their origin in the analysis of the previous section reproducing neutrino (sneutrino) physics.

It is apparent that in scan \(S_2\) for *M* larger than about 1000 GeV, the points that we find fulfilling all constraints are not uniformly distributed. This happens essentially because the value of \(v_R\) is smaller than in scan \(S_1\) modifying the relevant contribution of the \(\nu _R\)-Higgsino seesaw for the first two families, in such a way that is more difficult to reproduce neutrino physics unless more accurate values of the neutrino Yukawas are input in the computation. As obtained in Sect. 5.1, and can be seen in Fig. 4, the allowed values of \(Y_{\nu _3}\) are larger for \(S_2\). This makes more complicated to obtain the correct mixing, producing a tuning in the parameters. To obtain these more accurate values, we would have had to run Multinest a much longer time making the task very computer resources demanding. This is not really necessary since it is not going to affect the shape of the figure, and therefore neither the conclusions obtained. In addition, let us point out that we could have also modified the values of the parameters used for scan \(S_2\) reproducing more easily neutrino physics, e.g. increasing \(v_R\) and modifying accordingly the other parameters to keep the good Higgs physics.

Concerning scan \(S_2\), we can see in Fig. 5 that the BRs into charged leptons are about two orders of magnitude smaller than for \(S_1\), and therefore following the above discussion we have checked that no points are excluded by LEP results in this case. Note that although these BRs are smaller, still a significant number of points with signal events above 3 can be obtained when *M* increases because of the larger value of the decay length, which gives rise to a larger vertex-level efficiency.

Figure 5 also shows that the sensitivity of the dilepton displaced-vertex searches to \({\widetilde{\nu }}_\tau \) is limited by their small efficiency for \(c \tau \lesssim 1~\hbox {mm}\), especially for the \(S_1\) case. It is, however, worth noticing that we may even probe such a short lifetime region by optimizing the search strategy for the sub-millimeter displaced vertices, as discussed in Refs. [62, 63]. Our result highly motivates a dedicated work for such an optimization, which we defer to another occasion.

As discussed in Sect. 3, the value of \(Y_{\nu _{1}}\) is rather large in our scenario, and therefore we expect a sizable branching fraction for the \({{\widetilde{\nu }}}_\tau \rightarrow ee\) channel. In fact, the ratio of the branching fractions for the \({\widetilde{\nu }}_\tau \rightarrow ee\) and \({{\widetilde{\nu }}}_\tau \rightarrow \mu \mu \) channels has important implications for our scenario since it reflects the information from the neutrino data via the neutrino Yukawa couplings (see Fig. 4). To see this, we plot it against the parameter *M* in Fig. 6. It is found that for the \(S_1\) case, the ratios \(R_{\mu /e} \equiv \mathrm{BR}({\widetilde{\nu }}^{{{\mathcal {R}}}}_\tau \rightarrow \mu \mu )/\mathrm{BR}({\widetilde{\nu }}^{{{\mathcal {R}}}}_\tau \rightarrow ee)\) are in the range \(3 \lesssim R_{\mu /e} \lesssim 5\), while for the \(S_2\) case they are more widely distributed: \(1 \lesssim R_{\mu /e} \lesssim 4.6\). This different behaviour can be understood if we realize that for scan \(S_1\) the second term of \(\text {BR}({\widetilde{\nu }}_{\tau }\rightarrow \mu \mu )\) in Eq. (17) is negligible with respect to the first one, and the same for the corresponding terms of \(\text {BR}({\widetilde{\nu }}_{\tau }\rightarrow ee)\). Thus, with the approximation in Eq. (13) one gets \(R_{\mu /e}\approx (Y_{\nu _{\mu }}/Y_{\nu _{e}})^2\), which using the results for the neutrino Yukawas in Fig. 4 gives rise to the above range around 3.5. However, for scan \(S_2\) the term of \(\text {BR}({\widetilde{\nu }}_{\tau }\rightarrow ee)\) proportional to \(\text {BR}({\widetilde{\nu }}_{\tau }\rightarrow \tau \tau )\) is not negligible with respect to the one proportional to \(\text {BR}({\widetilde{\nu }}_{\tau }\rightarrow \tau e)\), which is much smaller than in scan \(S_1\), due to the contribution of the first term in Eq. (12). This implies that the ratio \(R_{\mu /e}\) in scan \(S_2\) can be smaller than in \(S_1\), as can be seen in the figure. Now, if we particularly focus on the parameter points that can be probed at the 13-TeV LHC, the \(S_2\) case predicts \(R_{\mu /e} \lesssim 3.6\), and thus we can in principle distinguish this case from the \(S_1\) case by measuring this ratio in the future LHC experiments such as the high-luminosity LHC.

Finally, we show in Fig. 7\(m_{{\widetilde{\nu }}_{\tau }}\) vs. *M*. For scan \(S_1\) (left plot), tau left sneutrino masses in the range 74–91 \(\hbox {GeV}\) can be probed, corresponding to a gaugino mass parameter *M* in the range 532–1801 \(\hbox {GeV}\), i.e. \(M_2\in \) (363–1228) \(\hbox {GeV}\). Clearly, red points appear in these regions because smaller sneutrino masses produce larger decay lengths. Since decay lengths are larger for scan \(S_2\), the range of sneutrino masses that can be probed is also larger than for \(S_1\). In particular, we can see in the right plot that the range of sneutrino masses is 63–95 \(\hbox {GeV}\). In this scenario, *M* is in the range 625–2100 \(\hbox {GeV}\), correspoding to \(M_2\in \) (427–1431)\( \, \hbox {GeV}\). Let us finally mention that points with sneutrino masses slightly larger than 100 GeV, and with \(c\tau > 0.1 \, \hbox {mm}\), exist, but since they are not constrained by the number of signals events and therefore cannot be probed at the LHC run 3, we do not show them in the figures. In any cases, if we actually detect the \({\widetilde{\nu }}_\tau \) signal and measure its mass^{8} and lifetime in future experiments, we can considerably narrow down the allowed parameter region, which plays an important role in testing the \(\mu \nu \mathrm{SSM}\).

## 6 Conclusions

In the framework of the \(\mu \nu \mathrm{SSM}\), where there is RPV and the several decay BRs of the LSP significantly decrease the signals, there is a lack of experimental bounds on the masses of the sparticles. To fill this gap in SUSY searches, it is then crucial to analyze the recent experimental results that can lead to limits on sparticle masses in this model, and the prospects for the searches with a higher energy and luminosity.

With this purpose, we recast the result of the ATLAS 8-TeV displaced dilepton search from long-lived particles [30], to obtain the potential limits on the parameter space of the tau left sneutrino LSP in the \(\mu \nu \mathrm{SSM}\) with a mass in the range \(45-100 \, \hbox {GeV}\). A crucial point of the analysis, which differentiates the \(\mu \nu \mathrm{SSM}\) from other SUSY models is that neutrino masses and mixing angles are predicted by the generalized electroweak scale seesaw of the \(\mu \nu \mathrm{SSM}\) once the parameters of the model are fixed. This is obtained at tree level when three generations of right-handed neutrinos are considered. Therefore, the sneutrino couplings have to be chosen so that the neutrino oscillation data are reproduced, which has important implications for the sneutrino decay properties.

The sneutrino LSP is produced via the *Z*-boson mediated Drell-Yan process or through the *W*- and \(\gamma /Z\)-mediated process accompanied with the production and decay of the left stau NLSP. Due to the RPV term present in the \(\mu \nu \)SSM, the left sneutrino LSP becomes metastable and eventually decays into the SM leptons. Because of the large value of the tau Yukawa coupling, a significant fraction of the sneutrino LSP decays into a pair of tau leptons or a tau lepton and a light charged lepton, while the rest decays into a pair of neutrinos. A tau sneutrino LSP implies in our scenario that the tau neutrino Yukawa is the smallest coupling, driving neutrino physics to dictate that the muon neutrino Yukawa is the largest of the neutrino Yukawas. As a consequence, the most important contribution to the dilepton BRs comes from the channel \({\widetilde{\nu }}_{\tau }\rightarrow \tau \mu \). It is found then that the decay distance of the left sneutrino tends to be as large as \( > rsim 1~\hbox {mm}\), which thus can be a good target of displaced vertex searches. The strategy that we employed to search for these points was to perform scans of the parameter space of our scenario imposing compatibility with current experimental data on neutrino and Higgs physics, as well as flavor observables.

The final result of our analysis for the 8-TeV case is that no points of the parameter space of the \(\mu \nu \mathrm{SSM}\) can be probed. This is also true even considering the optimization of the trigger requirements proposed in Ref. [11]. Nevertheless, important regions can be probed at the LHC run 3 with the trigger optimization, as summarized in Fig. 7. We in particular emphasize that a trigger optimization for muons has more significant impact on the search ability than that for electrons because of the larger muon neutrino Yukawa coupling in our scenario. Our observation, therefore, suggests that optimizing only the muon trigger already has great benefit. In addition, searching for “sub-millimeter” dilepton displaced vertices is also promising. We thus highly motivate both the ATLAS and CMS collaborations to take account of these options seriously.

If the metastable \({{\widetilde{\nu }}}_\tau \) signature is actually found in the future LHC experiments, we may also measure the mass, lifetime, and decay branching fractions of \({{\widetilde{\nu }}}_\tau \) through the detailed analysis of this signature. We can then include these physical observables into our scan procedure as well in order to further narrow down the allowed parameter space. For instance, we can distinguish the \(S_1\) and \(S_2\) cases by measuring the ratio \(\mathrm{BR}({\widetilde{\nu }}^{{{\mathcal {R}}}}_\tau \rightarrow \mu \mu )/\mathrm{BR}({\widetilde{\nu }}^{{{\mathcal {R}}}}_\tau \rightarrow ee)\) as shown in Fig. 6. We can also restrict the parameter *M* through the measurements of the mass and decay length of \({{\widetilde{\nu }}}_\tau \), which allows us to infer the gaugino mass scale and thus gives important implications for future high energy colliders.

## Footnotes

- 1.
The phenomenology of a neutralino LSP was analyzed in the past in Refs. [13, 14, 15, 16]. In the recent works [17, 18], in addition to perform the complete one-loop renormalization of the neutral scalar sector of the \(\mu \nu \mathrm{SSM}\), interesting scenarios with right sneutrinos lighter than the standard model-like Higgs boson were studied.

- 2.
The recent emulated recursive jigsaw reconstruction [20] confirmed the \(3\sigma \) excess with 36 \(\hbox {fb}^{-1}\), but sees only a small \(1.27 \sigma \) excess of data with respect to predictions with full \(139 \, \hbox {fb}^{-1}\).

- 3.
In what follows, the symbol \(\ell \) will be used for an electron or a muon, \(\ell =e,\mu \), and charge conjugation of fermions is to be understood where appropriate.

- 4.
We could have required a lower threshold for the electron trigger as well, but we do not consider this optimization since we are unable to estimate the increase in the number of background events caused by the relaxation in the trigger requirement [11].

- 5.
While we were doing the scan, we updated neutrino observables from a new neutrino global fit analysis [36].

- 6.
In this scenario the SM-like Higgs decays into pairs of scalar/pseudoscalar tau left sneutrinos via gauge interactions, mostly from D-terms \(\sim \frac{1}{4}(g^2+g'^2){{\widetilde{\nu }}}_i {{\widetilde{\nu }}}_i^* H^0_uH^{0*}_u\), since its largest component is \(H^0_u\).

- 7.
Notice that the partial decay widths into neutrinos for the \(S_1\) and \(S_2\) cases are similar in size for a given value of

*M*, as can be seen from Eq. (14) and Fig. 4. Therefore, a larger partial decay width of the \({{\widetilde{\nu }}}_\tau \rightarrow \mu \tau \) channel for scan \(S_1\) implies a larger value of BR(\({{\widetilde{\nu }}}^{{\mathcal {R}}}_{\tau } \rightarrow \mu \mu \)), compared with that for scan \(S_2\). - 8.
As discussed in Ref. [11], we can in principle measure the mass of \({{\widetilde{\nu }}}_\tau \) by using hadronically decaying tau leptons.

## Notes

### Acknowledgements

We would like to thank J. Moreno for his collaboration during the early stages of this work, specially concerning the computing tasks carried out at CESGA. The work of EK, IL and CM was supported in part by the Spanish Agencia Estatal de Investigación through the Grants FPA2015-65929-P (MINECO/FEDER, UE), PGC2018-095161-B-I00 and IFT Centro de Excelencia Severo Ochoa SEV-2016-0597. The work of EK was funded by Fundación La Caixa under ‘La Caixa-Severo Ochoa’ international predoctoral grant. The work of IL was supported in part by IBS under the project code, IBS-R018-D1 The work of DL was supported by the Argentinian CONICET, and also acknowledges the support of the Spanish Grant FPA2015-65929-P (MINECO/FEDER, UE). The work of NN was supported in part by the Grant-in-Aid for Young Scientists B (No.17K14270) and Innovative Areas (No. 18H05542). NN would like to thank the IFT UAM-CSIC for the hospitality of the members of the institute during the Program “Opportunities at future high energy colliders,” where this work was finished. RR acknowledges partial funding/support from the Elusives ITN (Marie Sklodowska-Curie Grant agreement No. 674896), the “SOM Sabor y origen de la Materia” (FPA 2017-85985-P) and the Spanish MINECO Centro de Excelencia Severo Ochoa del IFIC program under Grant SEV-2014-0398. EK, IL, CM, DL and RR also acknowledge the support of the Spanish Red Consolider MultiDark FPA2017-90566-REDC.

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