Signatures of R̃2 class of Leptoquarks at the upcoming ep colliders

Signatures of R̃2 class of Leptoquarks at the upcoming ep colliders Rojalin Padhan, 2, ∗ Sanjoy Mandal, 2, † Manimala Mitra, 2, ‡ and Nita Sinha 2, § Institute of Physics, Sachivalaya Marg, Bhubaneswar 751005, India Homi Bhabha National Institute, BARC Training School Complex, Anushakti Nagar, Mumbai 400094, India The Institute of Mathematical Sciences, C.I.T Campus, Taramani, Chennai 600 113, India


INTRODUCTION
LQs are hypothetical particles which can emerge from the unification of quarks and leptons in the Pati-Salam model [1]. LQs also exist in grand unified theories based on SU (5) [2] and SO(10) [3][4][5]. TeV scale LQs can also exist in extended technicolor models [6][7][8][9]. Under the Standard Model (SM) representation, there are twelve types of LQs, six of them are scalar, while the other six are vector type of LQs [10]. We consider the scalar LQ R 2 charged as (3,2,1/6) under SM gauge group. The advantage withR 2 type of scalar LQ is that in addition to the coupling with the lepton and jet, the model also has right handed (RH) neutrinos coupled to the LQ. Hence, this model provides unique signatures, that can be tested in different collider and non-collider experiments. Moreover, theR 2 also allows for matter stability [11]. LQ's can be most easily tested at ep colliders. At ep colliders such as LHeC [12,13] and FCC-eh [13,14], LQs can be resonantly produced. LHeC and FCC-eh are the proposed ep colliders, planned to operate with c.m. energies √ s = 1.3 TeV and √ s = 3.46 TeV, respectively.
LHeC (FCC-eh) will use electron and possibly positron beam of 60 GeV, to collide with the 7 TeV (50 TeV) proton beam. There are number of important phenomenological beyond standard model studies for ep colliders which have been listed in Ref. [15].
For the specific type of LQ model, that we consider in this paper, the LQ can decay to a lepton and jet, as well as, to a jet and RH neutrino. The decay of LQ into a lepton and a jet, and the decay to a jet and RH neutrino with the subsequent decays of RH neutrino in different SM states give rise to many possible final states. In this work, we have studied in detail the final state l ± j. We have considered both the scenarios of electron and positron beams. We show that with judicious application of selection cuts, final state l ± j has very good discovery prospects at LHeC and FCC-eh colliders. We find that, we can probe the LQ mass up to 1.2 TeV at more than 5σ significance with e + beam at LHeC. For FCC-eh, we can easily probe LQ mass up to 2.2 TeV (3 TeV) with e − (e + ) beams.
The paper is organized as follows: First we review the model and the existing constraints on LQ. Following this, we discuss the production of LQ at an ep collider and compare with the LHC. In the subsequent sections, we present a detailed collider analysis and discuss the discovery prospects of the final state l ± j. Finally, we conclude.

MODEL
We consider an extension of the SM with a single scalar LQ,R 2 (3, 2, 1/6) = (R 2 This is a genuine LQ (F = 3B + L = 0). The superscript ofR 2 denotes the electromagnetic (em) charge. In the presence of the RH neutrinos N R (1,1,0), the LQ has additional interaction [10,16,17], where i, j = 1, 2, 3 are flavor indices. Y and Z are the Yukawa couplings. U PMNS and V CKM represent the Pontecorvo-Maki-Nakagawa-Sakata and the Cabibbo-Kobayashi-Maskawa matrices. For simplicity, we assume that both the Yukawa couplings are diagonal, Hence in our model LQ couples to the same generation of lepton and quark. Although, most of the collider bounds on LQ mass and coupling are derived assuming only one generation is present at a time, our model can have non-zero couplings of LQ to fermions of more than one generation. In the next section, we review the existing constraints on LQ mass and couplings.

CONSTRAINTS ON LQ MASS AND COUPLINGS
Tight constraints exist onR 2 type of scalar LQ's mass and coupling from both the collider experiments as well as low energy experiments such as atomic parity violation and lepton flavour violating decays, K L → µ − e + etc. Below, we summarize present bounds on LQ mass and couplings.

Atomic Parity Violation
There are tight bounds on Yukawa couplings Y de and Y ue from atomic parity violation (APV) experiments. Following Ref. [18], the present bound on the coupling Y from atomic parity violation are Y de < 0.34 MLQ 1 TeV and Y ue < 0.36 MLQ 1 TeV . Hence for larger LQ mass the allowed values of Yukawa couplings will also be large. Note that, these bounds are derived with the assumption that only one of the Yukawa coupling is present at a time.
The tree level lepton flavour violating (LFV) process K L → µ − e + gives tight constraints on the diagonal couplings ofR 2 3 2 . Specifically this tree level process constrains the product of Yukawa couplings |Y sµ Y * de |. Following Refs. [18,19], the experimental upper bound on the decay mode K L → µ − e + results in the bound on the product of Yukawas and is given by |Y sµ Y * de | ≤ 2.1 × 10 −5 MLQ 1 TeV . Therefore, couplings of first two generations are tightly constrained.

Collider bounds
The present tightest collider bounds on LQs come from LHC [20][21][22][23][24][25]. LHC has specifically looked for the final states pp → LQ LQ → j j with the assumption that LQ decays to the final state j with 100% branching ratio. LHC constrains first [25], second [21] and third generation [20] of LQ considering to be e, µ or τ . From the non-observation of any new physics at LHC, the LQs mass up to 1.435 TeV at 95% C.L [25] has been ruled out for the first generation. In Fig. 1, we have shown the Feynman diagrams for LQ pair productions at LHC. For smaller values of Yukawa coupling Y , LQ pair production is dominated by the gluon-initiated diagrams, whereas for relatively larger Yukawa coupling Y , the t-channel quark-initiated diagram can dominate. In Fig. 2, we have shown present limit on scalar LQ pairproduction times branching fraction to ej final state as a function of LQ mass. This bound corresponds to that from CMS experiment with √ s = 13 TeV and integrated luminosity 35.9 fb −1 [25]. Black and red lines represent observed and expected limits. Note that, Ref. [25] also studies the eνjj channel. For β = 0.5, using the eνjj channel alone, LQ masses are excluded below 1.195 TeV. It also gives limit on LQ mass combining both the channels eejj and eνjj. In our analysis we have chosen the values of Yukawa couplings and LQ mass according to this combined limit presented in Fig. 9 of Ref. [25]. Note that, assuming 100% branching ratio to ej final state (which implies Z 11 = 0), bound on LQ mass is 1.435 TeV. If the branching ratio β to ej is less than 100% (which is of course possible if LQ has additional interactions such as LQ-νj, LQ-jN i ), the bound on LQ mass can be lowered. For example, to be consistent with LHC, flavour and APV constraints, for LQ mass 1 TeV, one can choose the coupling Y 11 = 0.34 and Z 11 = 1.03. We have shown this benchmark BP 1 in Table I In the next section, we discuss about the LQ production at colliders.

LQ PRODUCTION
At an ep collider, scalar LQs can be resonantly produced through s-channel process, as shown in the upper panel of Fig. 3. LQs can also mediate t-channel process ep → lj, as shown in the lower panel of Fig. 3. Single or pair production of LQs is also possible at pp colliders such as LHC. We have shown the feynman diagrams for pair production and single production of LQs at LHC in Fig. 1 and Fig. 4, respectively. We compare the production of LQ at e − p collider with that at LHC. For comparison of production cross-section of LQ at LHeC and LHC, we have shown the production cross-section of single LQ at LHeC, as well as that for both the single and pair production at LHC, in the upper panel of Fig. 5 for varying LQ mass. For this comparison, electron and proton beams are fixed at 60 GeV and 7 TeV, respectively. For LQ mass upto 1.2 TeV, the single LQ production at ep collider clearly dominates over the single and pair production of LQ at LHC. For LQ mass M LQ > 1.2 TeV, single LQ production at LHC dominates. In the lower panel of Fig. 5, we have given the comparison of single LQ production at e − p and e + p colliders. For this comparison we have fixed the electron or positron beam at 60 GeV and for proton beam we have taken 7 TeV and 50 TeV. From this plot, it is evident that LQ production at e + p collider is larger than that at the e − p collider for the chosen mass range. This occurs as the 2 3 charged LQ couples with e − −d and e + −d, respectively and a quark parton distribution function is larger than that for anti-quark.
We also compute the production cross-section for the channel lj with both polarised and unpolarised elec-  tron and positron beams to show how much the crosssections differ. There is a relative enhancement in l − j (l + j) production cross-sections at e − p (e + p) collider when the electron (positron) beam is dominantly left (right)-polarised. We have shown our results in Fig. 6 for both LHeC and FCC-eh case. We can see that the production rates improve by almost a factor of 2 over the entire range of the LQ mass in the case of polarised electron or positron beams. This enhancement occurs due to the couplings e − L −R 2 −d at e − p and e + p colliders, respectively. Hence at e − p and e + p colliders, LQ predominantly couples with left polarized electron and right polarized positron, respectively.
In Fig. 7 we present a densityplot which shows the variation of cross-section for the process e − p → l − j with the variation ofR 2 − e − d coupling (Y 11 ) and mass of the LQ (M LQ ). Here theR 2 -N R − u coupling (Z 11 ) is fixed to 1.03 and mass of the right-handed neutrino (M N1 ) is assumed to be 100 GeV. The dashed curves represent the constraints on Y 11 , for Z 11 equal to 0.42, 0.62 and 1.03, from LHC search [25]. Region below each dashed curve is allowed for respective value of Z 11 . Blue region in Y 11 − M LQ plane is disallowed due to limit on Y 11 from APV. Here we considered c.m. energy √ s = 1.3 TeV. As we can see, a 700 GeV mass LQ is still allowed, for Y 11 = 0.238 and Z 11 = 1.03. Finally, in the next section we present our analysis for the model signature ep → lj.

ANALYSIS
We consider the process e ± p → l ± j as the signal. The dominant background for this process comes from the SM processes e ± p → l ± j, l ± jj. The analysis is carried  [25], where we consider the combined limit on branching ratio from [25]. The colour bar indicates the cross-section for e − p → l − j for 60 GeV electron and 7 TeV proton beam. We consider BP1 for this plot.  [29]. Finally data analysis and plotting is done in ROOT(v6.14/04) [30]. We apply the following basic cuts P T (l) > 10 GeV, P T (j) > 20 GeV, |η(l)| < 5, |η(j)| < 5, ∆R ll > 0.  Table. I is considered.  Table. I is considered. peak which implies that the off-shell production of LQ is less than that in case of e − beam. This occurs since for e − p (e + p) collider, LQ couples to e − −d (e + − d).
The d quark carries greater fraction of proton momentum. From Fig. 9 it is evident that for FCC-eh there are no such differences between the use of e − and e + beams due to availability of enough c.m. energy for LQ mass 1 TeV. Signal (e ± p → l ± j ) and background crosssections after different selection cuts at FCC-eh for BP1.
In Table. III, we have shown the variation of signal and background cross-sections after applying different selection cuts one by one at LHeC for BP 1 . Before applying any selection cuts, the signal and background cross-sections for the channel e − p → l − j (e + p → l + j) are 4.0164 fb (39.23 fb) and 2180 fb (1440 fb) 1 , respectively. Note that due to the initial and final state radiations, additional jets will be present in the final states considered. For signal selection we demand atleast one lepton and one jet in the final state. Then we impose a cut on transverse momentum of leading jet and lepton 1 Note that, for the SM background, at the event generations level we impose two additional cuts on transverse momentum of leading lepton and jet, which are p T (l 1 ) > 200 GeV and p T (j 1 ) > 200 GeV to have better statistics as the SM background cross-section is very large.
i.e. p T (j 1 ) ≥ 400 GeV and p T (l 1 ) ≥ 400 GeV. This significantly reduces the background cross-section. Finally we select the signal events by demanding a window on lepton-jet invariant mass in between M LQ ± 100GeV. After imposing all cuts the signal cross-section for the channel e − p → l − j (e + p → l + j) is reduced to one sixteenth (one fifth) of its initial value. Similarly Table. IV represents the result for FCC-eh. We calculate the statistical significance using the expression: where s and b are number of signal and background events after all cuts, respectively.

RESULTS
We discuss the discovery prospects of LQ at LHeC and FCC-eh in the mass range 700 GeV-1200 GeV and 1 TeV-3 TeV, respectively. In the upper panel of Figs. 10 and 11, we have shown the signal cross-section after all the cuts at LHeC and FCC-eh, respectively. In the lower panel of Fig. 10, we have shown the required luminosity (L) to achieve 3σ and 5σ significance versus M LQ for LHeC. Yukawa coupling Y 11 has been fixed to its upper limit, 0.34 MLQ 1 TeV for a given M LQ according to APV constraint. Z 11 has been fixed to its lower limit for given M LQ and Y 11 , according to the upper-bound from LHC search. For M LQ ≥ 1 TeV, we use the same selection cuts as given in Table. III. For M LQ < 1 TeV, we again apply the same set of cuts except the cuts c 2 and c 3 . Here cuts c 2 and c 3 are defined as c 1 + p T ( 1 ) ≥ 300 GeV and c 2 + p T (j 1 ) ≥ 300 GeV, respectively.
Similarly, Fig. 11 shows the variation of required luminosity (L) to achieve 3σ and 5σ significance, as a function of M LQ for FCC-eh. For M LQ < 2 TeV, we use same selection cuts as given in Table. IV. For M LQ ≥ 2 TeV, we define c 2 and c 3 as c 1 + p T ( 1 ) ≥ 800 GeV and c 2 + p T (j 1 ) ≥ 800 GeV, respectively.
We find that, this final state has reasonably good discovery prospect even after giving a generic set of cuts. For LHeC with e − beam, its difficult to probe LQ due to small cross-section. From the lower panel of Fig. 10, it is evident that to probe higher LQ masses with 5σ significance, the required luminosity is very high in spite of smaller SM background, as the signal cross section is itself small for higher LQ masses. On the contary, for LHeC with e + beam, due to larger signal cross-sections, it is possible to probe the LQ mass upto 1.2 TeV at more than 5σ significance with integrated luminosity less than 100 fb −1 .
For FCC-eh, even with e − beam, we can probe LQ mass upto 2.3 TeV at more than 5σ significance with in-σ(ep) TeV. Since at LHeC c.m.energy is low to get a better signal cross-section we use the optimum values of the couplings for a given LQ mass, which are given in Table. II. tegrated luminosity less than 1000 fb −1 . For higher LQ masses, required luminosity quickly increases to probe with 5σ significance. For the case of e + beam, due to enhanced cross-section we can very easily probe the LQ mass upto 3 TeV at more than 5σ significance with integrated luminosity less than 500 fb −1 .
Finally, we would like to point out that as with the polarized electron or positron beams, lj production cross-sections are enhanced by almost a factor of two, we can probe even higher LQ masses compared to the case of unpolarized beams. We evaluate the asymmetry between the production cross-section of LQ at e + p and e − p colliders, which is defined as A ep = σ(e + p)−σ(e − p) σ(e + p)+σ(e − p) . It is found to be positive (which is evident from the upper panel of Figs Panel: Required luminosity to reach 3σ and 5σ significance versus mass of the LQ with c.m.energy 3.46 TeV. In this case we consider the same couplings for all LQ masses as given in Table. I for MLQ=1 TeV.

CONCLUSIONS
In this work we study the discovery prospect ofR 2 class of LQ model at the proposed ep colliders such as LHeC and FCC-eh. This type of LQ can be copiously produced at ep colliders, due to its interaction with the electron and down type quarks. There are many possible final states for this type of LQ model. We specifically focus on LQ production and its decay to a lepton and a jet. For this final state, we expect higher statistical significance at an ep collider compared to pp collider due to higher production cross-section as well as lower SM backgrounds. We find that at LHeC with e + beam, we can probe the LQ mass upto 1.2 TeV at more than 5σ significance. For FCC-eh, with e − beam we can probe LQ mass upto 3 TeV with 5σ significance, but the required luminosity is large. On the contrary, at FCC-eh with e + beam, we can easily probe the LQ mass upto 3 TeV at more than 5σ significance with nominal integrated luminosity. Note that at an ep collider, polarization of the electron or positron beams can result in a substantial increase in the LQ production and the resulting lepton-jet production cross-sections.