Lepton flavour violating signature in supersymmetric $U(1)^\prime$ seesaw models at the LHC

We consider a $U(1)'$ supersymmetric seesaw model in which a right-handed sneutrino is a thermal dark matter candidate whose relic density can be in the right range due to its coupling to relatively light $\tilde{Z'}$, the superpartner of the extra gauge boson $Z'$. Such light $\tilde Z'$ can be produced at the LHC through cascade decays of colored superparticles, in particular, stops and sbottoms, and then decay to a right-handed neutrino and a sneutrino dark matter, which leads to lepton flavor violating signals of same/opposite-sign dileptons (or multileptons) accompanied by large missing energy. Taking some benchmark points, we analyze the opposite- and same-sign dilepton signatures and the corresponding flavour difference i.e., ($2e-2\mu$). It is shown that $5\sigma$ signal significance can be reached for some benchmark points with very early data of $\sim 2$ fb$^{-1}$ integrated luminosity. In addition, $3\ell$ and $4\ell$ signatures also look promising to check the consistency in the model prediction, and it is possible to reconstruct the $\tilde{Z'}$ mass from $jj\ell$ invariant mass distribution.

Abstract: We consider a U (1) supersymmetric seesaw model in which a right-handed sneutrino is a thermal dark matter candidate whose relic density can be in the right range due to its coupling to relatively lightZ , the superpartner of the extra gauge boson Z . Such lightZ can be produced at the LHC through cascade decays of colored superparticles, in particular, stops and sbottoms, and then decay to a right-handed neutrino and a sneutrino dark matter, which leads to lepton flavor violating signals of same/opposite-sign dileptons (or multileptons) accompanied by large missing energy. Taking some benchmark points, we analyze the opposite-and same-sign dilepton signatures and the corresponding flavour difference i.e., (2e − 2µ). It is shown that 5σ signal significance can be reached for some benchmark points with very early data of ∼ 2 fb −1 integrated luminosity. In addition, 3 and 4 signatures also look promising to check the consistency in the model prediction, and it is possible to reconstruct theZ mass from jj invariant mass distribution.

Introduction
Among various reasons for requiring theories beyond the Standard Model (SM), experimental evidences for tiny neutrino masses and dark matter, and a theoretical requirement for naturalness of the electroweak scale would be key elements and related to each other. A sumpersymmetric seesaw model [1] associated with an additional gauge symmetry U (1) [2] is an example which resolves these issues in a consistent theoretical framework.
In supersymmetric theories with R-parity, the lightest supersymmetric particle (LSP) is stable and thus a neutral LSP, typically a linear combination of neutral gauginos and Higgsinos, becomes a good thermal dark matter (DM) candidate if supersymmetry (SUSY) is broken around the TeV scale [3]. The SUSY breaking can radiatively induce the U (1) breaking in addition to the usual electroweak symmetry breaking, which determines the seesaw scale also at O(TeV) [4]. The additional abelian gauge symmetry U (1) such as U (1) B−L may be a remnant of a grand unified gauge group which requires the presence of right-handed neutrinos (RHNs) for the anomaly-free condition, and thus realizes the TeV-scale seesaw mechanism. In such a framework, a right-handed sneutrino (RHNs) can be the LSP and thus another good dark matter candidate whose thermal relic density is in the right range if the U (1) gauginoZ , the superpartner of the U (1) gauge boson Z , is relatively light [5].
In this paper, we analyze LHC signatures of the extra gauginoZ which can be pairly produced mainly through third generation squark cascades and then decay to a RHN and a RHsN DM. Due to the Majorana nature of a RHN, N , it can decay to both-sign leptons, N → l ± W ∓ , leading to the lepton number violating signature of same-sign dilepton (SSD) events in addition to the usual opposite-sign dilepton (OSD) events [6]. Furthermore, Yukawa couplings of a RHN are generically flavour-dependent and thus lead to lepton flavour violating decays, e.g., Br(N → e ± W ∓ ) Br(N → µ ± W ∓ ). Taking some benchmark points, we carry out a Pythia-FastJet level collider simulation at the 14 TeV LHC for multilepton final states to study the prospect for detecting such lepton flavour violating signatures manifested by a flavour difference, e.g., 2e − 2µ, in both same-sign and oppositesign dilepton final states. Similar phenomenon should appear also in multi-lepton channels such as a tri-lepton difference 3e − 3µ. As we will see, 5σ discovery is expected in the 2e − 2µ SSD final states for an optimistic benchmark point even with ∼ 2 fb −1 integrated luminosity at the very early stage of LHC14.
This paper is organized as follows. In Section 2, we describe a generalZ phenomenology by taking a specific U (1) model. Considering all the relevant experimental constraints on SUSY parameter space, three benchmark points are set up in Section 3, and corresponding production rates and decay branching fractions of squarks are calculated in Section 4. A detailed LHC phenomenology of lepton number and flavour violating signatures are analyzed in Section 5, and we conclude in Section 6.

2.Z Phenomenology
Among various possibilities of an extra gauge symmetry U (1) and the presence of the associated right-handed neutrinos [2], we will take the U (1) χ model for our explicit analysis as in [5]. The particle content of our U (1) χ model is as follows: where SU (5) representations and U (1) charges of the SM fermions (10 F ,5 F ), Higgs bosons (5 H ,5 H ), and additional singlet fields (N, X, S 1,2 ) are shown. Here N denotes the righthanded neutrino, X is an additional singlet field fit into the 27 representation of E 6 , and we introduced more singlets S 1,2 , vector-like under U (1) , to break U (1) and generate the Majorana mass term of N [4]. Note that the right-handed neutrinos carry the largest charge under U (1) and thus the corresponding Z decays dominantly to right-handed neutrinos. Furthermore, the additional singlet field X is neutral under U (1) so that it can be used to generate a mass to the U (1) Higgsinos as will be discussed below. The gauge invariant superpotential in the seesaw sector is given by where L i and H u denote the lepton and Higgs doublet superfields, respectively. After the U (1) breaking by the vacuum expectation value S 1 , the right-handed neutrinos obtain the mass m N i = λ N i S 1 and induce the seesaw mass for the light neutrinos: In this type of models the RHN can be produced from the Z decay, Z → N N , and then can go through a flavour violating decay, N → eW . The recent mass bound of Z from LHC, pushes m Z > ∼ 2 ∼ 3 TeV depending on U (1) types [7]. This reduces drastically the production cross-section pp → Z → N N as shown in [8]. The other source of RHNs could be via the production ofZ . Once produced, it can have a decay to a RHN (N ) and a RHsN LSP (Ñ 1 ), i.e.,Z → NÑ 1 .
As shown in [8], the direct production cross-section of pp →Z Z is not encouraging even for 14 TeV at the LHC. Another supersymmetric source could be the cascade decays of the strongly interacting superpartners, i.e., squarks and gluinos. Similarly to the cascade decays of electroweak gauginos and higgsinos that can produce gauge bosons and Higgs in the minimal supersymmetric standard model [9,10], one can have extra gauginos,Z , produced via cascade production of squarks and gluinos in a supersymmetric U (1) model. From [8] (Figure 13), we can see that the right-handed down type squark mostly decays tõ Z for the U (1) χ model. In a scenario whereZ is the next LSP (NLSP), eventually total strong production will become the cross-section ofZ pair. TheZ then decays to a RHN along with a LSP. In the next sections we shall be considering the recent experimental bounds to see the prospect of these channels for further collider studies.

Experimental bounds and Benchmark points
The discovery of the Higgs boson with a mass around 125.5 GeV [11,12] has been very crucial in understanding the electroweak (EW) symmetry breaking. Although rather a large degree of fine-tuning cannot be avoided, SUSY still remains as a promising theory beyond SM stabilizing such a light Higgs boson mass. Various SUSY scenarios need large quantum corrections to have the Higgs boson mass around 125.5 GeV, which gives strong bounds to the SUSY mass spectrum contributing to the one-loop Higgs boson mass , m h . So the strongly interacting SUSY particles also get indirect bounds from the Higgs boson mass. It has been shown that for pMSSM one needs either very large stop masses or large splitting between the two mass eigen states in order to have ∼ 125 GeV Higgs. [13] . In cMSSM/mSUGRA one needs squarks masses greater than few TeV [14]. In this study we choose our parameter space to accommodate the lightest Higgs boson mass around 125.5 GeV considering the theoretical uncertainties 1 .
For the collider study we consider that recent bounds on third generation squark masses [18]. Most of the above bounds considerst 1 → tχ 0 branching fractions to be unity. So for a very light stop, one needs to go for a heavy LSP for standard decays as above. The non-standard decays, such ast 1,2 (b 1,2 ) → t(b)Z , wherẽ Z is not the LSP, will reduce the lower bounds further and reopen much lighter stop and sbottom masses. For our study we take relatively larger stop and sbottom masses given in Table 2. We also choose mg ≥ 1.4 TeV to satisfy recent gluino mass bound [19,20]. The first two generations squarks masses we have taken more than a TeV [20]. Table 1 presents the input parameters chosen for the benchmark points for the collider study. The heavy pseudo-scale boson mass m A is chosen to be 1 TeV, and thus all the heavy Higgs bosons are decoupled from the analysis.  Table 2 shows the respective SUSY particle mass spectrum generated by Suspect [15]. As we implement our vertices in CalcHEP [21] which uses Suspect for SUSY spectrum generation. One can see that for BP1 and BP3Z is NLSP, whereas for BP2 it is next to next LSP (NNLSP). In BP3 first two generations of squarks and gluino are decoupled having masses ∼ 2 TeV. In all three benchmark point we consider a right-handed sneutrinõ N 1 as the LSP with mÑ 1 = 110 GeV. We will take the corresponding right-handed neutrino mass to be m N = 100 GeV.

Production rates and decays
We calculate the production rates of the quark and gluino pairs via CalcHEP [21] at the LHC with the center of mass energy of 14 TeV. The renormalization and factorization scales are chose as mt 1 and CTEQ6L [24] is chosen as parton distribution function (PDF). One can see from Table 3 that only lighter third generation squarks have relatively large cross-sections. We do not show the cross-sections of first two generation squark pairs which are less than 10 fb.
Let us look at theZ productions rates from the third generation SUSY cascade decay. Table 4 gives the decay branching fractions of the different squarks toZ . As explained in our earlier work that for U (1) χ model right handed squark will have larger decay branching fraction toZ as compared to the left handed squarks [8]. In spite of having a larger enough branching fraction 35 ∼ 68% from right-handed squarks, the first two generations fail to contribute due to lager allowed masses. Whereas for third generations mixing between the left handed and right handed squakrs plays a role in reducing the effective branching   TheZ thus produced will decay viaZ → NÑ 1 . The right-handed neutrino then decays through lepton flavour violating eW , Zν and hν depending on m N : where in the pararenthes are shown the branching fractions for m N = 100 GeV. For this study we only focus on the mode N → e ± W ∓ which produces same number of positively and negatively charged electrons. Thus from pp → N N + X, we expect have charge symmetry considering only this lepton flavour violating decay. In BP1 and BP3 as mentioned earliert 1 andb 1 completely decays toZ which further decays to NÑ 1 , and then the right-handed neutrino prefers to decay into electron and W ± boson. The final state coming from the sbottom production and decay will have two b-jet and 4 non-b-jet at the partonic level if we demand both the W s to decay hadronically: Similarly, fort 1,2 we havet Thus, it will be our primary interest to look for the lepton flavour violating final state: with n q ≥ 4 for both same-sign or opposite-sign e or µ. In addition, 3l and 4l signatures coming from leptonic decays of W are also promising to look for the signal events. Unlike BP1 and BP3, in BP2Z is not NLSP but NNLSP (see Table 2) and the NLSP is of the Higgsino type. This results in sharing the third generation squark branching with the higgsino-like lighter neutralino (χ 0 1 ) and lighter chargino (χ ± 1 ). The effect can be seen from Table 4, which makes BP2 more challenging.

LHC Phenomenology
As discussed, relatively light third generation sqaurks can give rise to to numerous flavour violating dilepton final states in association with some b-jets and non-b-jets mainly coming from W bosons along with the missing energy. It is also important to look for the Majorana nature of the RHN which decays to both sign of electrons, i.e., N → e ± W ∓ . This suggests that determining the charge multiplicity we should expect to have similar number of lepton flavour violating events for both OSD and SSD.
When some of the W s decay leotonically these give rise to 3 and 4 signatures. In this collider study we mainly focus on the dilepton, trilepton and 4 final states. For this purpose we generated the events in CalcHEP [21] and simulated with PYTHIA [25] via the the SLHA interface [26] for the decay branching and mass spectrum.
For hadronic level simulation we have used Fastjet-3.0.3 [27] algorithm for the jet formation with the following criteria: • the calorimeter coverage is |η| < 4.5 • p jet T,min = 20 GeV and jets are ordered in p T • leptons ( = e, µ) are selected with p T ≥ 10 GeV and |η| ≤ 2.5 • no jet should match with a hard lepton in the event • ∆R lj ≥ 0.4 and ∆R ll ≥ 0.2 • Since efficient identification of the leptons is crucial for our study, we additionally require hadronic activity within a cone of ∆R = 0.3 between two isolated leptons to be ≤ 0.15p T GeV in the specified cone.
We show in Figure 1 the jet p T distributions coming from tt (left) andb 1b * 1 (right) for BP2. It is clear that the jets coming fromb 1 decay could be as hard as p T > ∼ 300 GeV, which is very unlikely in the case of tt. Figure 2 shows the jet (left) and lepton (right) multiplicity distribution forb 1b * 1 (BP2) and for the dominant background tt. We can see that though both the signal and backgrounds can have large number of jets in the final states, theb 1b  From Figure 3 (left) we can see that the leptons coming fromb 1b * 1 have a high energy tail. The contribution of this high energy tail is coming from the decay the right-handed neutrino to eW , which gets the boost from the cascade decays ofb 1 . Figure 3 (right) presents the p T distributions for the signalb 1b * 1 and for the dominant background tt. It is clear the LSP in the case ofb 1b * 1 , adds to large missing p T as compared to the neutrinos for tt. A p T cut of p T > ∼ 100 GeV will kill most of the SM backgrounds. In this article we focus on the multilepton final states with lepton number and flavour violation. In the following subsections we describe the final states with their signal and backgrounds number at LHC14.

2e and 2µ signatures and charge multiplicity
Our signal signature shown in Eq. 4.2 involves the right handed neutrino decay to eW (jj) leading to 2e + 2b + 4j+ p T in the final state. In the process of hadronization and jet formation with ISR/FSR, more number of jets are produced. We have also seen that the right-handed neutrino can decay either e + W − or e − W + , pp →b 1b * 1 should generate both same sign and opposite sign di-lepton signatures. To extract out the lepton flavour asymmetry of electron and muon we look for final states where the W ± s decay hadronically. This implies to look for a final states 2e/2µ + n j ≥6 (n b ≥2).
In Table 5 we show the number of events for 2e + n j ≥6 (n b ≥2) final states for the signal benchmark points and the SM backgrounds at 50 fb −1 of luminosity of 14 TeV LHC. We consider ttZ, ttW and ttbb as the dominant SM backgrounds which are not reducible backgrounds with ISR/FSR. We show the the contributions coming from each third generation sqaurks pair production process for both SSD and OSD, and also from the SM backgrounds. The respective signal significance are also been calculated and listed. From Table 5 we can clearly see that the signal numbers are symmetric in SSD and OSD, whereas the backgrounds prefer OSD as expected. The backgrounds come from the two opposite sign W decay or from one neutral gauge boson (Z) decay. The slightly large number of OSDs in the case of signal happens due to the decay kinematics of the righthanded neutrinos. Generally the charge symmetry is maintained when we tag two leptons coming from two different right-handed neutrinos. When N decays to e ± W ∓ , sometimes one of the electrons can not be isolated from the jets coming from the associated W boson, i.e., cannot pass the jet-lepton isolation criterion (∆R lj > 0.4 ). The original 3e events become 2e events where the second right-handed neutrino decay (N → e ± W ∓ ) can contribute to dilepton final states with the leptonic decay of the associated W boson. This would always be of opposite-sign leptons as the right-handed neutrino (N ) is charge neutral. Thus single right-handed neutrino contributing to dilepton final sates makes it opposite-sign. Relaxing the isolation criterion reduces the discrepancy.
As expected, BP1 and BP3 produce much more signal numbers than BP2 because botht 1 andb 1 fully decays toZ . In the backgrounds tt makes highest contribution due to  Table 5: Number of events in 2e + n j ≥6 (n b ≥2) final states for the benchmark points and the SM backgrounds at LHC14 with an integrated luminosity of 50 fb −1 .
its large cross-section. For opposite-sign di-electron final states BP1 and BP3 have more than 8σ significance, whereas for the same-sign di-electron they are around 21σ. BP2 fails to cross even 2σ. Table 6 presents corresponding number of di-muon events for the benchmark points and the backgrounds. As the right-handed neutrino decays only to electron flavour, the number of events for the muon final states are very low. The backgrounds numbers are similar to the electron final states in Table 5.
Next we take the difference in number of events between electrons and muons for both OSD and SSD. Table 7 shows the event numbers in the (2e − 2µ) + n j ≥5 (n b ≥2) final state for all the benchmark points and the backgrounds. For BP1 and BP3 we can have around 25σ signal significance for OSD and SSD flavour difference. It is encouraging to see that the significance of BP3 can reach to about 7σ for SSD.

3 signature
Let us now consider the 3e final state, which is possible if one of the W s from the decay of the right-handed neutrino, decays leptonically. In this case we can have final state 3e+n j ≥4 (n b ≥2) from the decay ofb 1b * 1 . Here, we also impose a missing energy cut p T ≥ 100 GeV to reduce the SM backgrounds.   Table 7: Number of events for (2e − 2µ) + n jets ≥ 6(n b ≥ 2) final states for the benchmark points and the SM backgrounds at LHC14 with an integrated luminosity of 50 fb −1 .
for a signal significance of around 9 and 8σ respectively. In case of BP2 due to the small branching fraction ofb 1 → bZ andt 1 → tZ , one needs larger luminosity to probe this 3e signal state. Table 9 presents the number of events corresponding to the 3µ final state. As expected, the signal could not contribute much for the 3µ final state. Thus, the difference in the electron and muon events are of the same order of 3e final state as can be read from   Table 9: Number of events for 3µ + n j ≥4 (n b ≥2) + p T ≥ 100 GeV final states for the benchmark points and the SM backgrounds at LHC14 with an integrated luminosity of 50 fb −1 .

4 signature
Finally we are interested where two W s from the decays of the right-handed neutrinos (see Eq. 4.2), decay leptonically. Here we do not distinguish the flavours of the charged lepton and consider both e and µ in the final state. Of course, among these 4 , two of them are electrons coming the flavour violating decays of the right-handed neutrino, i.e., N → eW . Table 11 presents the number of events for the 4 + n j ≥3 (n b ≥2) final state for the benchmark points and the SM backgrounds with an integrated luminosity of 50fb −1 at the LHC14 . BP1 and BP3 reach more than 12σ significance but BP2 fails to contribute for this final state.

Reconstruction of RHN
For the 2e or 3 final states a invariant mass of jj will give the right-handed neutrino   mass. In Figure 4(left) we demonstrate the invariant mass distribution of one electron and two jets coming from W . Here we have taken two jets satisfying |(M jj − M W )| ≤ 15 GeV. Thus M ejj reconstruct the decay of the right-handed neutrino, i.e., N → eW . To control the dominant SM background tt we have chosen n ≥ 3 + n j ≥ 4(n b ≥ 2)+ p T ≥ 100 GeV as final state. The demand of additional jets and b-jets reduce the SM backgrounds substantially. Figure 4(left) shows the total number of events coming from thet 1 ,b 1 for BP1 and the SM backgrounds at an integrated luminosity of 50 fb −1 .t 2 ,b 2 contributions are negligible. We can see that the signal peaked around ∼ 100 GeV, which is the righthanded neutrino mass, m N . Clearly it has more than 60σ signal significance a 50 fb −1 luminosity. Now if we use the flavour violating decay of the right-handed neutrino and demand that out of the three leptons two of them are electrons, then this suppresses the backgrounds much more than the signal. From Figure 4(right) we can see the corresponding invariant mass distribution of ejj for the final state of n ≥ 3(n e ≥ 2) + n j ≥ 4(n b ≥ 2)+ p T ≥ 100 GeV. It is visible that the signal stands out over the backgrounds much clearly. Thus the study of third generation squarks decays is very important which can lead to the information about right-handed neutrino mass produced in a supersymmetric cascade decay.  Figure 4: M jj distribution (left) for n ≥ 3 + n j ≥ 4(n b ≥ 2)+ p T ≥ 100 GeV and M ejj (right) n ≥ 3(n e ≥ 2) + n j ≥ 4(n b ≥ 2)+ p T ≥ 100 GeV for final state at an integrated luminosity of 50 fb −1 . The blue graph represents the total number of events and the red corresponds to the SM backgrounds only.

Conclusion
A U (1) supersymmetric seesaw model with R-parity can be motivated by simultaneous explanation of the observed neutrino masses and mixing, the existence of dark matter, and the stabilization of the Higgs boson mass assuming TeV-scale SUSY breaking scale. This can induce radiative breaking of the electroweak symmetry as well as the U (1) gauge symmetry. In this scheme, a right-handed sneutrinoÑ 1 becomes a good thermal dark matter candidate if the extra gauginoZ is relatively light. The addition of the new decay modes, reduces the experimental lower bounds of the supersymmetric particles, viz, stops and sbottoms. Considering stop and sbottom below TeV, we showed thatZ produced from third generation SUSY cascade decays can lead to significant lepton number and flavour violating signatures in final states with multi-lepton accompanied by multi-jet (+missing energy) through the decay chain ofZ → NÑ 1 → e ± W ∓Ñ 1 if allowed kinematically. These signatures are going to shed a light not only on the existence of a right-handed neutrino but also a U (1) model with a superpartner of an extra Z boson. In addition to the conventional same-sign dilepton signal, the flavour differences 2e − 2µ and 3e − 3µ, as well as the 4l final state are also promising channels to look for at the 14 TeV LHC.
In the case ofZ being NLSP, early data of LHC14 will be able to probe some optimistic benchmark points. IfZ is not the NLSP so that the third generation squark branching fraction is shared with other higgsinos and gauginos, much more data are needed to tell us about the model under consideration.
The invariant mass of jj system can successfully reconstruct the right-handed neutrino mass if it is produced in the supersymmetric cascade decays. Thus it can shed direct light to the right-handed neutrino spectrum and it's flavour violating decay directly through supersymmetric cascade decays.