Probing Leptoquark and Heavy Neutrino at LHeC

We explore leptoquark production and decay for the $\tilde{R}_{2}$ class of models at the proposed $e^{-} p$ collider LHeC, planned to operate with 150 GeV electron and 7 TeV proton beams. In addition to the coupling of leptoquark with the lepton and jet, the model also has right handed neutrinos, coupled to the leptoquark. We analyse the collider signatures of a number of final states, that can originate from leptoquark decay into the standard model particles, as well as, the final states that originate from further decay of the heavy neutrinos, produced from leptoquark. We find that the final state $\ell^{-}+\text{n-jets}\,(1\leq\text{n} \leq 2)$ has the largest discovery prospect, more than $5\sigma$ with only few fb of data to probe a leptoquark of mass 1.1 TeV, even with a generic set of cuts. The significance falls sharply with increasing leptoquark mass. However, with 100\,$\rm{fb}^{-1}$ of data, a 5$\sigma$ discovery for leptoquarks of mass upto 1.4 TeV is still achieveable. Also for the same luminosity, final state $\bar{b}\ell^{+}\tau^{-}+\text{n-jets}\,(\text{n}\geq 2)+\text{MET}$, resulting from the cascade decay of the leptoquark to an $\bar{t}$ and right handed neutrino, followed by further decays of $\bar{t}$ and the neutrino, is expected to yield a rather large number of events~($\approx 180$).

In the case of type-I seesaw [22][23][24][25], the light neutrinos acquire masses through mixings with additional Majorana RH neutrinos. To account for the tiny neutrino masses, the mass scale of these Majorana neutrinos has to be very close to the gauge coupling unification scale, in which case these massive RH neutrinos will remain inaccessible at LHC as well as, at other near future colliders. For the present and near future colliders to be able to probe the RH neutrinos, their masses have to be within the experimental reach, and the mixing with the active-neutrinos (referred as active-sterile mixing) has to be sizable. TeV scale RH neutrinos with substantially large active-sterile mixings are however possible to accommodate in type-I seesaw if cancellation exists in the light neutrino mass matrix [36]. The inverse seesaw mechanism [33][34][35] is another such scenario, where TeV scale or even smaller RH neutrino masses with sizeable active-sterile mixing can exist. In this scheme, in addition to the SM particles there are gauge singlet neutrinos, with opposite lepton numbers (+1 and -1). The light neutrino mass matrix is given in terms of the Dirac neutrino mass term, m D ∼ Y ν v (with v being the electroweak vev and Y ν , a generic Yukawa coupling), the heavy neutrino mass scale M R and, a small lepton number violating (∆L = 2) mass term µ, which ensures that M R scale remains close to TeV or less, with order one Yukawa coupling. The light neutrino mass matrix in this case is: m ν ∼ (m 2 D /M 2 R )µ. While the heavy neutrino states may lie within the kinematic reach of LHC, their production cross section falls rapidly with increasing masses and smaller active-sterile neutrino mixing. Large active-sterile mixing is possible to obtain in other seesaw secenarios as well, such as, linear seesaw [37,38], extended seesaw [39][40][41][42][43]. A substantial rise in the production cross section of the RH neutrinos is feasible in the presence of LQs. This has been explored recently for LHC in [44] for inverse seesaw, where a number of final states have been analysed in detail. Leptoquark models have also been tested recently for fitting the IceCube events [45]. For the heavy neutrino searches at LHeC in inverse seesaw model, see [46] and for the LNV signal at LHeC, see [47].
Similar studies for heavy neutrino searches also been carried out in [48,49].
In this work, we consider a particular type of scalar leptoquarkR 2 which transforms asR 2 ∈ (3, 2, 1 6 ) under the SM gauge group SU (3) c × SU (2) L × U (1) Y and for the RH neutrino, we adopt model independent framework.R 2 is a genuine LQ with fermion number F = 3B + L = 0. The colour charge of the LQ will enable in their copious production at LHC. Moreover, at e − p colliders like LHeC, they can be resonantly produced. The LHeC is a proposed e − p collider in the TeV regime after HERA, supposed to be built in the LHC tunnel [50]. LHeC will use a newly built electron beam of 60 GeV, up to possibly 150 GeV, to collide with the intense 7 TeV proton beam of the LHC. LHeC is expected to operate with 100 fb −1 integrated luminosity, and is complementary to the pp collider LHC [51]. The RH neutrino, being coupled to the LQ, can be produced from LQ decay. The decay of LQ into a lepton and a jet, and the decay of RH neutrino in different SM states give rise to a plethora of model signatures, that we study in detail. We show that among all the final states, − + n-jets(1 ≤ n ≤ 2) has the highest LQ discovery prospect, even with generic sets of cuts. Additionally, we also carry out an in-depth analysis for few other channels that arise due to the decay of a heavy neutrino. We show that with judiciously applying selection cuts the channels − + n-jets(n ≥ 3), and + τ −b + / E T + n-jets( n ≥ 2) can be made background free.
The discussion of the paper goes as follows: in Section. II, and Section. III, we review the model and the theory constraints. Following that, in Section. IV, we discuss the production and decay of LQ at LHeC. In the subsequent sections, Section. V, Section. VI and Section. VII, we present a detailed collider analysis and discuss the discovery prospects of different final states. Finally, in Section. VIII, we summarize.

II. MODEL
We consider the scalar LQR 2 charged as (3,2,1/6) under SM gauge group. In the presence of the RH neutrinos N R , the LQ has additional interaction [2,3,20], where i, j = 1, 2, 3 are flavor indices and a, b = 1, 2 are SU (2) L indices. We assume that there are three right-chiral neutrinos N j R (j = 1, 2, 3), Y ij and Z ij are the elements of arbitrary complex 3×3 Yukawa coupling matrices. Note that,R 2 comprises two LQs. One has Q = 2 3 , and the other has Q = − 1 3 . Upon expansion, the Lagrangian becomes where the superscript of LQ fields denotes electric charge of a given SU (2) L doublet component ofR 2 , U PMNS and V CKM are Pontecorvo-Maki-Nakagawa-Sakata (PMNS) and Cabibbo-Kobayashi-Maskawa (CKM) matrices. At the e − p machineR 1 3 2 can not be resonantly produced. Hence, the expected cross-section for the production ofR 1 3 2 is small. Therefore, in this work, we consider onlyR 2 3 2 and study its production. The charged current and neutral current interactions of the RH neutrinos are parametrized in a model independent way as follows, and The interaction of the heavy neutrinos with Higgs has the following form: In the above P L/R = (1 ∓ γ 5 )/2 is the left/right-chirality projection operator, and V is the mixing matrix through which light neutrinos mix with the RH neutrinos, and referred to as active-sterile mixing. We consider a diagonal basis for the charged leptons.
For the RH neutrino, coupled with LQ, we do not assume any particular model. Instead, we are interested in different frameworks of RH neutrinos, that can lead to large activesterile mixing, so that the heavy neutrinos decay inside the detector. It is widely known, that a number of different frameworks can generate large active-sterile mixing, including inverse and linear seesaw [33-35, 37, 38], extended seesaw [39][40][41][42][43], cancellation framework [36]. In the inverse seesaw, light SM neutrino masses are extremely tiny, owing to the small lepton number violating parameter of the model. The active-sterile neutrino mixing is not constrained from light neutrino masses in this model. Active-sterile mixing upto O(10 −2 ) is allowed from experimental data [52][53][54][55][56][57][58][59]. In extended seesaw, or double seesaw [60], the RH neutrino gets mass due to seesaw, and light neutrino masses are generated due to two-fold seesaw. In other frameworks, such as, cancellation, small light neutrino masses are generated due to cancellation between different RH neutrino contributions in the mass matrix [36]. The active-sterile mixing is yet unconstrained from neutrino data. In all the above mentioned frameworks, owing to the charged current and neutral current interactions as well as the interaction with the Higgs, specified above in Eqs. In the following sections, we first consider the resonant production of LQ and its decay to a lepton and jet. We next consider the production of heavy neutrinos from LQ decay, and discuss the discovery prospect of the LQ in a number of channels. As mentioned before, we consider the prompt decays of heavy neutrino for the analysis of the RH neutrino signature, that occurs due to large active-sterile neutrino mixing. We compare between the usual charged current (CC) production of heavy neutrinos vs the alternate production from LQ decay. We show that the production from LQ decay dominates over the CC production mode by order of magnitude for active-sterile mixing V 10 −2 .

III. CONSTRAINTS ON LEPTOQUARK COUPLINGS
The couplings of the LQs to fermions are constrained by low energy precision observables such as atomic parity violation, Kaon decays etc. We assume that the Yukawa coupling matrix elements, Y ij = δ ij Y ii and Z ij = δ ij Z ii , where, i, j = 1, 2, 3. Hence the LQ couples exclusively to a lepton and a quark of the same generation, although it can have non-zero couplings to fermions of more than one generation.
• LQs have been searched for and studied in the context of e + e − [62][63][64][65][66], ep [67][68][69][70][71][72], pp [73][74][75][76][77], and pp [78][79][80][81][82][83] colliders. The present tightest bounds are from the LHC [84][85][86][87][88]. LHC has studied the process pp → LQLQ → j j for LQs of first, second and third generations. Non-observation of any new physics at the LHC has ruled out LQs of masses up to 1.1 TeV at 95% C.L for the LQ decaying to ej with 100% branching ratio [85]. For second generation, the bound is even more stringent M LQ > 1.5 TeV at 95% C.L [88]. For third generation, the bound is M LQ > 900 GeV at 95% C.L [87] . At However, with non-zero Yukawa couplings, significantly large contribution to the LQ pair production may arise through a singlet t-channel diagram (see the right panel of Fig. 1). The pair production cross-section at LHC can be parametrised as [89], where the three terms correspond to the QCD pair production, an interference term and t-channel production. In Fig. 2, we show the variation of LQ pair production cross-section with the Yukawa coupling Y 11 . For small Yukawa coupling Y 11 , LQ pair production is mostly governed by QCD, as can be seen from the straight line upto Y 11 ∼ 0.5 in Fig. 2. For intermediate Yukawa couplings there exists a region with negative interference between QCD diagrams and the t-channel diagram where the total cross section decreases [89], resulting in the mild dip in the cross-section for coupling beyond 0.5, that is seen in Fig. 2

. For large
Yukawa coupling Y ii , the t-channel process dominates and significantly enhances the cross-section. The right panel of Fig. 2 shows the limit on the first-generation scalar LQ pair-production times the branching fraction to ej final state as a function of the mass. For the branching fraction LQ → ej as 100%, the bound on the pair-production of LQ becomes σ(pp → LQLQ) 3 fb, for LQ mass 1.1 TeV. Comparing the left and right panel of Fig. 2, one can see that the limit on the cross section for a LQ of mass 1.1 TeV, will be inconsistent with a Yukawa coupling larger than 1.
It is obvious from Eq. 1, that for the LQ to have 100% branching ratio in the LQ → ej decay mode, the coupling Z needs to be zero. Allowing non-zero value for coupling Z will open up new decay modes, such astN for LQ, and hence will lower the stringent bound on LQs. We however, adopt a conservative approach, and in order to be consistent with the LHC results for first generation of LQ, throughout our study, we consider M LQ ≥ 1.1 TeV. Additionally, we also keep both the couplings non-zero.
• The present bound on coupling Y from atomic parity violation are Y de < 0.34 [89]. These bounds are extracted under the assumption that only one of the two contributions is present at a given moment. These bounds allow large coupling for larger mass of LQ, and place a stringent constraint for lighter LQ.
• The most stringent bound on the diagonal couplings ofR 2 2 3 comes from LFV decay mode K L → µ − e + , as this is a tree level process. Following Refs. [89,90], the bound is given by, In order to satisfy both the APV and LFV constraints, for Y de ∼ O(0.1), the other coupling Y sµ has to be tiny. We consider Y sµ to be zero and a large value (0.3) for Y de to get large production cross-section of LQ at LHeC.
We discuss the production of a LQ, and its decay to different final states in the next section, for the benchmark points, that are in agreement with the described constraints.

IV. LEPTOQUARK PRODUCTION AND ITS DECAYS
At e − p colliders, scalar LQs can be resonantly produced through s-channel process as shown in the left panel of Fig. 3, and decay to a lepton and a jet. In addition, LQ can also be a t-channel mediator for the process e − p → l − j, that we consider in our analysis (shown in the right panel of Fig. 3).
The production cross-section of a LQ at LHeC, as well as that for both the single and pair production at LHC, are shown in Fig. 4 for varying LQ mass. Clearly, the LHeC cross-section is more than both the pair-production, as well as, the single production of LQ associated with a charged lepton at LHC. The higher LQ production cross-section as well as the lower background at LHeC will allow more precise studies for probing LQ and RH neutrinos. Once produced, the LQ can decay into a number of final states, including, a) a quark-lepton pair that gives rise to single charged lepton and a light jet, b) a light jet and a heavy neutrino, and c) a top quark accompanied with a heavy neutrino. These heavy neutrinos appearing from the decays of the LQ can again be more easily probed at LHeC through its decay products. Note that, for all these processes, the LQ can also mediate as t-channel mediator. For b) and c), there is also t-channel contribution from W gauge boson mediator, but significantly smaller for the active-sterile mixing V 10 −2 . We give numerical estimates in Section. VI. However, during computation (in Fig. 5 and for the collider analysis), we consider all the contributions together. For our computations, the LQ mass has been set to 1.1 TeV. We choose three benchmark points, with the three heavy neutrino masses and the LQ couplings, Y ii and Z ii chosen such that they are consistent with all the constraints mentioned in Sec. III, as well as with the neutrino oscillation data [46]. These parameters for the benchmark points have been specified in Table I. The production cross-section for these three processes at LHeC, with electron and proton beam energies of 150 GeV and 7 TeV respectively, are also shown in The general expression for the two body decay of a scalar LQ to i q and N i q final states are given by, In the massless limit of leptons and quarks, the branching ratios are given by, At an e − p collider LQs can be resonantly produced, followed by their decay. Hence, we can write the cross section approximately as, As can be seen, from Eq. 9, with increasing coupling Z 11 the branching ratio of σ(e − p → jN 1 ) increases, while σ(e − p → lj) decreases. This results in larger cross-section for σ(e − p → jN 1 ) for larger Z 11 . Cross section for the other channel e − p →tN 3 is also large for large value of Z 33 . The values of the cross-section in fb, for three benchmark points are given in the last column of Table. I. As can be seen, the production cross-section at LHeC is fairly large, approximately σ ∼ 221−242 fb for the chosen benchmark points. As we will show in the next sections, folded with branching ratios of heavy neutrino, top quark, the total cross-section for the different final states will be sizeable.

V. COLLIDER ANALYSIS
We implemented the model in FeynRules [91], generated the model files for MadGraph5 aMC@NLO (v2 5 5) [92] to calculate the parton level cross-section for signals and background. For the collider simulation part, we passed the MadGraph generated parton level events to PYTHIA (v6.4.28) [93], where subsequent decay, initial state radiation, final state radiation and hadronisation have been carried out. The jets are reconstructed by anti-κ t algorithim [94] implemented in Fastjet package [95] with radius parameter R = 0.4. For the analysis of signal and background events we use the following set of basic cuts, The single-lepton associated with jet is the easiest channel to probe LQ. LQ, once produced resonantly, can directly decay to a charged-lepton and a jet. Additionally, the tchannel contribution, as shown in Fig. 3 will also be present. The parton-level final state are therefore − + n-jets (n = 1). Additional jets will be present due to ISR, FSR. We demand the final state should contain − and number of jets 1 ≤ n-jets ≤ 2. The main backgrounds arise from SM process, such as, e − p → − j, − jj, that is significantly larger as compared to the signal. From Tab. II, the signal cross-section is 220 fb, while the background crosssection is 3 × 10 6 fb. We use a number of cuts on different kinematic variables to reduce the background.
In Fig. 7 we have shown the transverse momentum of the leading lepton, leading and subleading jet, as well as the invariant mass distribution of the leading jet and leading lepton, both for the signal and background. Evidently, for a very heavy LQ, a high-p T cut on leading jet or lepton, and LQ invariant mass-cut will reduce SM background drastically. In

B. Signal II
If the coupling Z ij is non-zero, the LQ can also decay to RH neutrino and a jet, as shown in the right panel of Fig. 8. The considered final state, can also arise from the tchannel W exchange diagram as shown in left panel of Fig. 8. For active-sterile mixing V ∼ 10 −2 − 10 −3 , the contribution from LQ however dominates. For example, with the BP2, the CC production cross section is ≈ 12.7 fb, while the production cross section from LQ decay is ≈ 240 fb. The subsequent decays of RH neutrino, followed by hadronic and leptonic decays of gauge bosons gives rise to a number of partonic states, that we list below.  Fig. 9. Additionally, the leading jet that is directly generated from LQ decay has a very high transverse momentum (see Fig. 9). Therefore, a large cut on the transverse momentum of the leading jet reduces the background. As can be seen from Table. III   as,   Table. IV.
We do not consider the last final states in our study because of very small cross-section and very large SM background due to large jet multiplicity.
For the final states involving τ and b, tagging can reduce the SM background significantly.
We consider the p T for the b and τ jets, as p T > 40 GeV. In this work, we adopt a minimalistic approach and consider a flat 75% efficiency for b-tagging and 60% efficiency for τ -tagging.
Similar to the previous case, the M EF F distribution is hard due to the large missing energy and large transverse momenta of final state particles. For this signal, the most dominant SM backgound comes from the processt − W + ,tZν andthν witht →bτ − ν, W + → + ν, Z → + − , and h → + − . After applying the basic cuts only, SM background drops significantly. In addition, we use missing energy / E T , leading jet p T and M EF F distribution to further reduce the SM background.   p j T , along with on LQ invariant mass, and invariant mass of RH neutrino can make the SM background negligibly small. The results are given in Table. VIII. We have also shown the cross section and number of events with 100 fb −1 integrated luminosity as a function of LQ masses in Fig. 12. As can be seen, the cross-section for the − +n-jets(1 ≤ n ≤ 2) channel is the largest, varies 10 2 − 0.42 fb for a wide range of LQ mass. With 100 fb −1 luminosity, this predicts 10 4 number of events at LHeC. The other channel with jet multiplicity (n ≥ 3) also offers a large cross-section, and large number of events O(10 2 ).

B. Non-Zero Backgound Case
For the final states, − +n-jets(1 ≤ n ≤ 2) and − +n-jets(n ≥ 3), the SM background is non zero if we do not use the invariant mass of LQ and RH neutrino. Since, the LQ and RH neutrino masses are unknown, we do not implement the mass cut, rather in this section show the cross-sections with a very generic sets of cuts. Assuming LQ mass to be more than 1 TeV, all the other cuts which we considered can be easily applied. For the above two final states we applied cuts only on p T and p j T . For final states, − + +n-jets(n ≥ 1) + / E T ,  b − τ − + + / E T andb − τ − + n-jets (n ≥ 2) + / E T we used the same cuts as in Tables. IV, V and VI respectively. We show the signal cross section and statistical significance with integrated luminosity of 100 fb −1 in Fig. 13. We also show the required luminosity to achieve 3σ and 5σ statistical significance in Fig. 14. The statistical significance has been calculated using the following expression: where, S and B denote the number of signal and background events, respectively.  ℓ − +n-jets(1 ≤n≤ 2) ℓ − +n-jets(n≥ 3) ℓ − ℓ + +n-jets(n≥ 1) Right panel: the significance with 100 fb −1 luminosity.
statistical significance is 7σ for a 1.1 TeV LQ, which decreases to 0.6σ for a 1.7 TeV LQ.
For this final state, 5σ statistical significance can be probed only for LQ mass 1.1 TeV, with integrated luminosity 100 fb −1 . Again for this case also, signal cross section is small for higher LQ mass, hence difficult to probe higher mass regime. However, it may be noted that our estimates for the cross-sections for LQs of higher masses are rather conservative, as they have been computed, assuming the coupling to be the same as that for 1.1 TeV LQ, while higher values of couplings for larger LQ masses will be permissible.

VIII. CONCLUSIONS
In this work, we study the discovery prospect ofR 2 class of LQ model at LHeC. The model contains two LQs with Q = 2 3 , and Q = − 1 3 . LQ with Q = 2 3 can be copiously produced at LHeC, due to its interaction with the electron and down type quark. We study the production and its decay to different final states, including a lepton and a jet, a jet and a RH neutrino, and RH neutrino and a top quark. The typical production cross-section for e − p → lj, jN 1 ,tN 3 are 221, 242, 222 fb for M LQ = 1.1 TeV, M N 1,3 = 150 GeV, and the couplings Y 11 = 0.3, Z 33 = 1. The produced RH neutrino further decays and give a plethora of model signatures. For the RH neutrinos, we adopt a model independent framework, and a large active-sterile mixing to ensure its decay within the detector. For the LQs, the higher production cross-section as well as the lower backgrounds at LHeC result in a much higher statistical significance for few of the signals studied.
We have analysed a number of final states, including − + n-jets (1 ≤ n ≤ 2), ± + n-jets (n ≥ 3), ± ∓ + n-jets (n ≥ 1) + / E T ,b − τ − + + / E T ,b − τ − + n-jets (n ≥ 2) + / E T , b + τ − + n-jets (n ≥ 2) + / E T . Among these, the model signature − + n-jets (1 ≤ n ≤ 2) arises due to the direct decay of LQ to a lepton and a jet. All the other final states arise due to the decay of LQ to a RH neutrino and a light jet, or to a RH neutrino and top quark, with successive decays of RH neutrino, and t quark into SM states.
We find that, among all the above mentioned final states, − + n-jets (1 ≤ n ≤ 2) has the highest discovery prospect even after giving a generic set of cuts. A LQ of mass upto 1.4 TeV in this channel can be discovered at more than 5σ C.L. with 100 fb −1 of data. The LQs will also result in the enhancement of the RH neutrino production in association with a light jet, or with top quark. If at LHeC the electron beam is polarized, the right handed neutrinolight jet production cross-section can substantially increase [46,47]. We find that among all the final states − + n-jets(n ≥ 3), andb + τ − + n-jets(n ≥ 2) + / E T are the most optimal, after implementing the selection cuts judiciously. With 100 fb −1 integrated luminosity, for LQ mass 1.1 TeV, the expected number of events for the final sates − + n-jets(n ≥ 3), and b + τ − + n-jets(n ≥ 2) + / E T are 10 4 and 180 respectively.