Boosting vector leptoquark searches with boosted tops

At the LHC, a TeV-scale leptoquark (LQ) that decays dominantly to a top quark ($t$) and a light charged lepton ($\ell=e,\mu$) would form a resonance system of boosted-$t$ $+$ high-$p_{\rm T}$-$\ell$. We consider all possible vector LQ models within the Buchm\"{u}ller-R\"{u}ckl-Wyler classifications with the desired decay. We propose simple phenomenological Lagrangians that are suitable for bottom-up/experimental studies and, at the same time, can cover the relevant parameter spaces of these models. In this simplified framework, we study the pair and single production channels of vector LQs at the LHC. Interestingly, we find that, like the pair production, the cross sections of some single production processes also depend on the parameter $\kappa$ that appears in the gluon-vector LQ coupling. We adopt a search strategy of selecting events with at least one boosted hadronic top quark and exactly two high-$p_{\rm T}$ leptons of the same flavor and opposite sign. This combines events from the pair and single production processes and, therefore, can enhance the discovery potential than that of the pair-production-only searches. For $5\sigma$ discovery we find that vector LQs can be probed up to 2.55 TeV for $100\%$ branching ratio in the $t\ell $ decay mode and $\mathcal{O}(1)$ new couplings at the 14 TeV LHC with 3 ab$^{-1}$ of integrated luminosity.


I. INTRODUCTION
In the recent past, several experimental collaborations have reported some hints of lepton flavor universality violation in the heavy meson decays. Collectively, these point toward the existence of some physics beyond the Standard Model (SM) as the SM gauge interactions are flavor-blind. Intriguingly, these seem to be quite tenacious and have created a lot of excitement in the particle physics community. Initially, the BaBar collaboration found two significant anomalies in the flavor changing charged current decays of the B-meson via the b → cτ ν transition. They reported the anomalies in terms of excesses in the R D ( * ) observables defined as the ratios of branching ratios (BRs) to reduce some systematic and hadronic uncertainties [1,2]. Since then, the excesses have survived the later measurements by the LHCb [3][4][5] and Belle [6][7][8][9] collaborations. The statistical average of these two observables obtained in the RD − RD * plane by the HFLAV group puts the anomalies away from the corresponding SM predictions [10][11][12][13] by a combined significance of ∼ 3.1σ. The LHCb collaboration has also observed downward deviations of about 2.5σ [14][15][16][17][18] from the SM predictions [19,20] in the flavor changing neutral current transition b → sµµ measured in terms of the R K ( * ) observables. Similarly, an excess of about ∼ 2σ is found in another observable R J/ψ [21]. In addition, a long-standing discrepancy of about ∼ 3.5σ exists in the muon anomalous magnetic moment measurement [22].
It is known that TeV-scale leptoquarks (LQs) are good candidates to address the flavor anomalies. Moreover, their phenomenology has been explored in various other contexts also . LQs are color-triplet bosons [either scalars (sLQs) or vectors (vLQs)] predicted by many beyond the SM theories [76][77][78][79][80]. They have fractional electric charges and carry both lepton and baryon numbers. In general, a LQ can couple to a quark and a lepton of either the same or different generations. The flavor anomalies suggest that LQs couple more strongly to the third generation fermions than the other two. Cross-generational couplings of LQs could generate flavor changing neutral currents -the ones involving the first and second generations are tightly constrained. However, bounds are relatively weaker when a fermion of third generation is involved.
Among the current LHC programs, the search for LQs is one of the important ones. Usually, the LHC searches are done for LQs that couple to quarks and leptons of the same generation and are labeled accordingly. For example, pair production of a scalar LQ that decays to a top quark and tau lepton (or a bottom quark and a tau lepton or neutrino), i.e, a third generation LQ, has been extensively analyzed by both the AT-LAS [81,82] and the CMS [83,84] collaborations. Altogether, the current bound on the third generation LQ is roughly about a TeV (this, of course, depends on various assumptions and we refer the reader to the actual papers for details). However, the flavor-motivated LQ models with sizable cross-generational couplings would have exotic signatures and require different search strategies. Of late, the nonstandard decay modes of LQs have started to gain attention; the CMS collaboration has published their first results on the pair production searches of LQs in the ttµµ channel [85]. Based on the 13 TeV data, they performed a prospect study for the pair production of sLQs in the ttµµ channel at the high-luminosity LHC (HL-LHC) [86].
Earlier in Ref [87], we investigated the HL-LHC prospects of sLQs that couple dominantly to the top quark in some detail. There, we focused on charge 1/3 and 5/3 sLQs that decay to a top quark and a charged lepton. Even though we considered only third generation quarks, interestingly, we found that in some scenarios single productions can improve their prospects significantly. 1 In this paper, we present a similar follow-up study for vLQs. Here too, we concentrate on a specific subset of possible vLQs that dominantly couple with a top quark and can decay to a top quark and a light charged lepton (e or µ) with a substantial BR. Since, an analysis of the pair production of vLQs that decay to a top quark and a neutrino at the LHC is available from Ref. [88], in this paper we do not analyze this channel again. Instead, we present a set of simplified models that covers all the possibilities of a vLQ decaying to a top quark and any lepton. These are suitable for experimental analysis. We also demonstrate how they are related to the known models of vLQs [89,90].
Our main motivation for considering this specific type of vLQs is to investigate their collider discovery/exclusion potential by making use of the boosted top signature coming from the LQ decay. They form an exotic resonance system with a boosted top and a high-pT lepton and provide a novel way to search for these models at the LHC. Various flavor anomalies suggest that cross-generational Yukawa-type LQ couplings with top and lepton might be large. A large coupling makes various single production channels important, especially in the high mass region. In our analysis, we adopt the same search strategy as the one we proposed for the sLQs [87]. We identify our signal by selecting at least one boosted hadronic top and exactly two high-pT leptons and demand the highest-pT top (if there are more than one tops) and one of the selected lepton to reconstruct a heavy system, i.e., the LQ. As we have demonstrated before [87,[91][92][93], such selection strategy combines pair and single production events and increase the LHC reach. Although, pair production is suitable for probing the low mass region, single production takes over when the LQ becomes heavy. Compared to the sLQs, the pair production cross sections for vLQs are relatively bigger and hence, the current mass limits obtained for the pair productions are generally higher for the vLQs than for the sLQs. In case for vLQs, the importance of single productions becomes visible for relatively higher mass compared to sLQs. We shall see that the discovery prospects of the vLQs at the HL-LHC is significantly improved if the new couplings controlling single productions are of order unity.
Before we proceed further, we note that since this is a followup paper of Ref. [87], we shall frequently refer to that paper and omit some details that are common while ensuring that our presentation is self-contained. The rest of the paper is organized as follows. In section II, we describe the vLQ models and introduce simplified models suitable for experimental analysis. In section III, we discuss the LHC phenomenology and illustrate our search strategy and then present our estimations in section IV. Finally, we summarize and conclude in section V.

II. VECTOR LEPTOQUARK MODELS
To conserve electromagnetic charge, vLQs that decay to a toplepton pair would have either electric charge equal to ±1/3 or ±5/3 (if the lepton is a charged one) or 2/3 (if lepton is a neutrino). This means that amongst the vLQs listed in Refs. [89,90], the weak singlets U1 andŨ1, doublets V2 and V2 and the triplet U3 would qualify for our study. Below, we display the relevant terms in the interaction Lagrangians following the notation of Ref. [90]. To avoid proton decay constraints, we ignore the diquark operators.
where uR and R are an SM right-handed up-type quark and a charged lepton respectively and i, j = {1, 2, 3} are the generation indices. The color indices are suppressed. For our purpose, we consider only those terms that would connect vLQ to a third generation quark and ignore the rest, The necessary interaction terms for the charge 2/3 U1 can be written as, where QL, LL and dR are the SM left-handed quark doublet, lepton doublet and a down-type right-handed quark, respectively. The i = 3 terms can be written explicitly as, V2 = (3,2,5/6): For V2, the Lagrangian is as follows, The superscript C denotes charge conjugation. Expanding the Lagrangian we get, where U and V represent the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) neutrino mixing matrix and the Cabibbo-Kobayashi-Maskawa (CKM) quark mixing matrix, respectively. We assume U to be unity, as the LHC is blind to the flavor of the neutrinos. Similarly, since the small off-diagonal terms of the CKM matrix play negligible role at the LHC, we assume a diagonal CKM matrix for simplicity. Hence, the terms relevant for our analysis are, Ṽ 2 = (3,2,−1/6): ForṼ2, the Lagrangian becomes, Expanding it we get, The terms with the third generation quarks are U3 = (3,3,2/3): The necessary interaction terms for the triplet U3 are,  in the RLCSS/OS scenarios where 0 ≤ β < 100% (for β = 100%, these two scenarios become the same as the RC scenario). The exceptional scenarios are marked by an asterisk. Here, λ is a generic free coupling parameter. For simplicity, we have chosen only this one coupling to control all the non-zero new couplings in every benchmark. This essentially means choosing β to be 50% in the exceptional scenarios also.
where τ k denotes the Pauli matrices. This can be expanded as The terms for the third generation quarks can be written explicitly as,

A. Simplified model and benchmark scenarios
The above models can be simplified into the following phenomenological Lagrangians, where we have suppressed the lepton generation index. We denote a generic charge ±n/3 vLQ by χn. Here, ηL and ηR = 1−ηL are the charged lepton chirality fractions [87,92]. In Eq. (15), we have introduced a sign term, R = ±1 to incorporate a possible relative sign between the left-handed and the right-handed terms [see Eq. (7)]. We shall consider only real couplings in our analysis for simplicity. As we did for the sLQs [87], here also we identify some benchmark scenarios with the simplified models (see Table I).
Each scenario corresponds to one of the realizable models described above [see Eqs. (1) -(13)]. Here, we have ignored any possible mixing among the vLQs. The BR for a χ to decay to a top-quark, β is fixed in any model [Eqs. (1) -(13)], except for two cases that we shall describe shortly. For simplicity, we choose only one free coupling λ parametrizing the non-zero new couplings in every benchmark scenario (see the fourth and seventh columns of Table. I. By doing this, we are essentially choosing β to be 50% in the free β scenarios also.) • In the Left Coupling (LC) scenario, a χ can directly couple with left-handed leptons. We set λ equal to Λ (for χ1) orΛ (for χ5) orΛν (for χ2) and put all other couplings to zero. For χ1 and χ5 we set ηL = 1. Here , χ1 represents aṼ In this scenario, a χ1 or χ5 decay to t pairs and a χ2 decays to tν pairs all the time.
• If a χ2 is of U1 or U 2/3 3 type, it can also decay to a b pair. Hence, it is possible that a χ2 couples with left-handed leptons but the BR for the χ2 → tν (β) is 50%. Such possibilities are captured in the Left Couplings with the Same Signs (LCSS) or the Left Couplings with Opposite Signs (LCOS) scenarios. The difference between these two comes from the different relative signs between the χ2b and χ2tν couplings. In LCSS, the χ2 behaves as a U1 withΛ =Λν = x LL 1 3j , whereas in LCOS it behaves as a U 2/3 3 It is important to note that in the U1 case, it is possible to have β < 50% if we consider a non-zero x RR 1 3j . This can be seen from Eq. (4). Unlike the sLQ case [87], the LCSS and LCOS scenarios for the vLQ yield the same single production cross section as there is no interference among the contributing diagrams.
• The Right Coupling (RC) scenario, where a LQ couples only to right-handed charged leptons, is exclusive to χ1 and χ5. Like the LC scenario, here we have β = 100%.
In this case a χ1 behaves as a V 1/3 2 with Λ = x LR 2 3j and a χ5 behaves as aŨ1 withΛ = x RR 1 3j .
FIG. 1. Representative Feynman diagrams for the LQ production at the LHC. Diagrams (a) and (b) show pair production processes and (c) and (d) show single production processes.
• Unlike the sLQ φ1 (see Ref. [87]), the χ1 type vLQs (if it is V 1/3 2 ) can decay to both t and bν pairs, provided Λ and Λν both are nonzero. We design two scenarios, namely, Right (lepton) Left (neutrino) Couplings with the Same Signs (RLCSS) where χ1 ≡ V . In these two scenarios, β can be anything between zero and 100% as both involve two independent couplings [x LR 2 3j and x RL 2 3j , see Eq. (7)]. We, however, consider only β = 50% for these benchmarks. We introduce these two benchmarks for completeness, though, for our purpose, these two are equivalent. As there is no interference contribution sensitive to this sign flip, all the production processes would have the same cross sections in both the scenarios.
Before we move on, we note that the kinetic terms for a vector leptoquark contains a free parameter, usually denoted as κ [90], where χµν stands for the field strength tensor of χ. This parameter κ can change pair and interestingly, some single production cross sections through the modification of the χχg vertex. 2 We take two benchmark cases with κ = 0 and κ = 1 in our analysis.

III. LHC PHENOMENOLOGY & SEARCH STRATEGY
We keep our computational setup the same as before [87]. We use FeynRules [94]

A. Production at the LHC
The vLQs would be produced resonantly at the LHC through the pair and the single production channels. The dominant pair production diagrams are free of the new couplings and depend only on the universal strong coupling (there are diagrams with t-channel lepton exchange that involve new couplings, but their contribution to the total pair production cross section is small [92]), hence the process is mostly model-independent up to a choice of κ. The pair production would lead to the following final states, Here, as we did in case of the sLQs [87], we ignore those channels with no top quark and consider only symmetric channels i.e., both the vLQs decay to the same final state. Constraining ourselves to such channels will restrict the possible SM backgrounds and make our signal easier to detect. It is generally believed that the symmetric modes have good prospect [103]. 3 With this considerations, we are now left with only (t )(t ) (for χ1 or χ5) and (tν)(tν) (for χ2) channels. With similar consideration for the single production processes, where a LQ is produced in association with a lepton and either a jet or a top-quark, the possible final states are given as,  Table I). Here, stands for either an electron or a muon and the j in the single production processes includes all the light jets as well as b-jets. Their cross sections are generated with a cut on the transverse momentum of the jet, p j T > 20 GeV.
In Fig. 1, we show some representative Feynman diagrams of the pair and the single productions of vLQs. In Fig. 2 We see that in the LC scenario with κ = 0, the single production cross section σ(pp → χ1 j) overtakes the pair production cross section at about 1.8 TeV while σ(pp → χ1t ) always remains smaller. For κ = 1, the pair production cross section increases moving the crossover point with σ(pp → χ1 j) to about 2.6 TeV. Interestingly, we find that σ(pp → χ1t ) depends on the choice of κ despite being a single production process as it contains κ-dependent χ1χ1g vertex. In the RC scenario, σ(pp → χ1 j) is reduced by almost two orders of magnitude than that in the LC scenario. This happens because in the RC scenario, a χ1 couples to a right-handed top that comes from another left-handed top generated in the charged current interaction through a chirality flip. If the Λν coupling alone is turned on, the cross section for pp → χ1 j process is negligible (see in Figs. 2(a) and 2(b)). (Note, however, a nonzero Λν can still affect the BRs. For example, we can consider RLCSS and RLCOS scenarios where the BR for the χ1 → t and χ1 → bν modes are 50% each.) Now, because of the small contribution from the Λν dependent diagrams and the fact that there is no interference in both the RLCSS and RLCOS scenarios, the pp → χ1 j process would have the same cross section as in the RC scenario. For χ2, pair production pp → χ2χ2 always dominates over single production pp → χ2tν up to 3 TeV mass with λ = 1 coupling for both κ = 0 and κ = 1. In this case, we obtain a tt plus large / E T signature which is analyzed in Ref. [88]. The χ5 vLQ is similar to the χ1 and yield similar signatures at the colliders.
The distinctive feature of our signal is the presence of boosted top quarks and high-pT charged leptons. In symmetric modes, we have at least one top quark in the final state for single productions while the pair production give rise to two top quarks. In both the cases, we have two high-pT charged leptons. Therefore, as already indicated in the Introduction, we combine events from both pair and single productions by demanding at least one top-jet (a hadronically decaying top quark forming a fatjet) and exactly two high-pT same-flavoropposite-signs (SFOS) leptons in the final state to enhance the signal sensitivity. Note that the same final state can arise from the pair and the single productions. For example, the t t state can come from both pp → χ1,5χ1,5 and pp → χ1,5t processes [see Figs. 1(c) & 1(d)]. This can lead to double counting the contribution of some diagrams while generating signal events. One can bypass it by ensuring both χ and χ † are not on-shell simultaneously in any single production event [92].

B. Backgrounds and selection
Since the topology of the vLQ signal is identical to that of the sLQ signal [i.e., at least one (hadronic) top fatjet and exactly two high-pT SFOS leptons], our background analysis essentially remains the same as before [87]. We therefore point the reader to the earlier paper for a detailed discussion on the possible SM background processes. Here, we present the gist of our discussions there. The dominant SM background processes for our desired signal can arise from processes having two leptons and significant cross sections at the LHC. The top- like fatjet can appear either from an actual top quark decaying hadronically or from a bunch of QCD/non-QCD jets mimicking its signature. We find that pp → Z + jets and pp → tt + jets processes contribute majorly to the background. The single top, diboson and ttV (V = W, Z) production processes are subdominant. There are SM processes with large cross sections e.g. pp → W + jets → ν + jets can in principle act as backgrounds because of a jet faking as a lepton. However, we found that these processes actually contribute negligibly, thanks to a very small misidentification efficiency.
In Table II, we list the relevant SM processes and their higher-order cross sections. We consider these backgrounds after adjusting with appropriate K-factors to include higherorder effects. Although the bare cross sections (i.e. without any cut) of some background processes are seemingly huge, we control them by applying strong selection cuts. These cuts are designed in a way such that they would drastically reduce the background without harming the signal much since our signal possesses specific kinematic features very different from the backgrounds. However, some backgrounds are so big at the beginning (e.g., Z + jets), in order to save computation time and have better statistics, we apply the following strong cuts at the generation level. Here i denotes the i th pT-ordered lepton (e/µ). After generating events with the above generation-level cuts, we apply the following final selection criteria sequentially on the signal and the background events at the analysis level.  We use the combined pair and single productions for the signals in the LC and RC scenarios. We also show the pair production significance for 50% and 100% BR in the χ → t decay mode. We have considered λ = 1 while computing the signals.

IV. DISCOVERY POTENTIAL
We use the following formula to estimate the signal significance Z. The pair-production-only regions for 50% and 100% BRs in the χ → t decay mode are shown with shades of green. Since the pair production is insensitive to λ, a small coupling is sufficient to attain 5σ significance within the green regions.
where the number of signal and background events surviving the final selection cuts, as listed in the previous section, are denoted by NS and NB, respectively. In Fig. 3, we show expected significance as functions of vLQ masses. As discussed earlier, the choice of κ affects the pair and as well as some single productions. In Figs. 3(a) and 3(b), we present Z for χ1 with κ = 0 and κ = 1, respectively. Similarly, Figs. 3(c) and 3(d) are for χ5. These curves are obtained for the 14 TeV LHC with 3 ab −1 of integrated luminosity. We have used λ = 1 to estimate the significance for the combined signal (i.e. the pair and single production events together). We note the following points: • The LC100 (RC100) curves for χ1 and the LC (RC) curves for χ5 represent the significances in the LC (RC) scenario where the BR of the χ1 → t decay is 100%.
• For χ1, the LC50 and RC50 curves represent the cases where the BR of χ1 → t decay mode is 50%. Although, they are not realized in the LC and RC scenarios, such a situation is possible if there are other decay modes of χ1 (which play no role in our analysis beyond modifying the BR). Hence, we show these plots to give some estimates of how the significance would vary with the BR.
• For comparison, we also show the expected significance obtained with only the pair production events for the 50% and 100% BR cases. For instance, for 100% BR in the χ1 → t mode, the HL-LHC (3 ab −1 ) discovery mass reach (i.e., Z = 5σ) with only pair production is about 2.05 (2.35) TeV for κ = 0 (κ = 1).
• When the LC coupling is unity, the discovery reach goes up to 2.35 (2.50) TeV once the single productions are included. However, in case of the RC scenario, the improvement is minor. This happens because σ (pp → χ1 j) is larger in the LC scenario than that in the RC scenario.
• Unlike the scalar case, there is no interference among the different signal diagrams and hence, the signal significance in the RLCSS or RLCOS benchmarks are the same as that in the RC scenario.  respectively. There is a suppression in the RC channel because of similar reason as for χ1, a χ5 LQ also couples to a right chiral top.
In Table III we collect all the numbers for Z = 2σ, 3σ and 5σ.
Since, we can parameterize the combined signal cross section for any Mχ as the combined signal cross section increases with λ for any fixed Mχ. By recasting the figures shown in 3 which are for λ = 1, we can obtain the reach in the λ-Mχ plane, as we show in Figs. 4 and 5. We show the 5σ discovery curves in Fig. 4 while the 2σ exclusion curves are displayed in Fig. 5. These plots show the lowest value of λ required to observe the vLQ signal for a varying Mχ with 5σ confidence level for discovery. For the exclusion plots, all points above the curves can be excluded with 95% confidence level at the HL-LHC.

V. SUMMARY AND CONCLUSIONS
Usually, in the direct LQ searches, it is assumed that LQs only couple to quarks and leptons of the same generation. Col-lider signatures of TeV scale LQs with large cross-generational couplings, motivated by the persistent flavor anomalies, are completely different than what is considered in the usual LQ searches at the LHC. It is then important to explore these possibilities in detail. In a previous paper [87], we investigated the HL-LHC prospects of all scalar LQ models within the Buchmüller-Rückl-Wyler classifications [89] that would produce boosted-t + high-pT-signatures at the LHC. In this follow-up paper, we investigate the case for the vector LQs with the same signature. The vLQs that decay to a top-quark can have three possible electric charges, ±1/3, ±2/3 and ±5/3. Among these, our primary focus is on the charge ±1/3, ±5/3 vLQs that can decay to a top quark and an electron or a muon as a unique top-lepton resonance system would appear from the decays of these LQs.
In this paper, we have introduced some simple phenomenological Lagrangians suitable for bottom-up/experimental studies. These simple models can cover the relevant parameter spaces of the full models described in listed in Refs. [89,90]. In this simplified framework, we study the pair and single production channels of vector LQs at the LHC. Pair production of the vLQs produce final states with two boosted top quarks and two high-pT leptons and determine the LHC discovery reach in the low mass region. Whereas, the single productions produce final state with at least one boosted top quark and two high-pT leptons. We observe two interesting points about the single productions: 1) despite considering vLQ couplings with only third generation quarks, we see that the single production cross sections are not necessarily very small, provided, of course, the new couplings controlling them are not negligible, 2) like the pair production, some single production processes can also depend on the parameter κ that appears in the gluonvector LQ coupling. In some scenarios, for order one new coupling(s), the single productions would control the LHC reach in the high mass region.
We have adopted a search strategy of selecting events with at least one boosted hadronic top quark and exactly two high-pT leptons of same flavor and opposite sign. This combines events from the pair and single productions and, therefore, enhance the discovery reach by about 300 GeV from the usual pair pro-duction searches at the LHC . Our results show that charge 1/3 and 5/3 vector LQs can be probed up to 2.35 (2.50) TeV for 100% branching ratio in the t decay mode for κ = 0 (κ = 1) and order one new couplings at the 14 TeV LHC with 3 ab −1 of integrated luminosity with 5σ significance. Alternately, in absence their discoveries, they can be excluded up to 2.65 (2.75) TeV at 95% confidence limit. Since the single production cross sections scale as λ 2 , we also show how the discovery/exclusion reach would vary with λ within its perturbative domain.