TeV-scale leptoquark searches at the LHC and their E$_6$SSM Interpretation

We perform a model-independent search for leptoquarks (LQs) at the Large Hadron Collider through their pair-production and subsequent decay into $t\bar t \tau\tau$ intermediate states. We show that, assuming full luminosity of the Run 2, a fully hadronic signal emerging from this intermediate state can surpass in sensitivity the established searches relying on leptons in the final state. Our conclusion is supported by a thorough Monte-Carlo analysis, and we advocate the deployment of our proposed search channel in the proper experimental setting of the Run 3. Furthermore, in order to highlight the full scope of this approach for constraining LQ theories, we interpret our results in the context of the string-inspired Exceptional Supersymmetric Standard Model, which naturally predicts the $S_1-$type scalar LQ states that we analyse here.

Naturally, they were extensively searched for at the ep colliders. In fact, at HERA, DESY, an excess was found in 1997 [14] which would have been compatible with the so-called first-generation LQs (i.e., those coupling to first-generation leptons and quarks). However, further searches at HERA (and also at Tevatron, where most of the partonic processes were initiated by quarks inside the (anti-)proton) failed to confirm the anomaly. As a result, mass limits near 300 GeV [15,16] were set on first-generation LQs. Even weaker limits were placed on second-generation LQs. These (relatively) weak limits were owing to the fact that the partonic energy accessible at HERA and Tevatron was limited to the sub-TeV range. However, as colour triplets, LQs also interact with gluons. At the Large Hadron Collider (LHC), since the amount of gluons inside the proton can well exceed that of (both valence and sea) quarks, the availability of much higher centre-of-mass energy may enable the pair-production of third-generation LQs.
The ATLAS and CMS experiments have both, therefore, analysed the datasets from the LHC Runs 1 and 2 to put exclusion limits on the mass of the third-generation LQs, based on their ttτ τ , tbτ ν, and tbµν decay channels, in the TeV range [17][18][19][20]. In this paper, we revisit the ttτ τ search channel for LQs of the S 1 type (i.e., LQs with charge -1/3 and spin zero) having couplings solely with the third generation fermions. However, unlike the existing searches for this channel, which have relied on leptons in the final state, we investigate the scope of the fully hadronic final state. Following a sophisticated detector-level analysis tensioning the signal to the most relevant backgrounds, we show that such a complementary channel can offer sensitivities to this specific type of LQ comparable to, if not better than, what has so far been achieved in their probes exploiting leptons. We then interpret the results of our analysis in the Exceptional Supersymmetric Standard Model (E 6 SSM) [21,22] (for a recent review, see Ref. [23]), as a theoretically well-motivated prototypical framework that naturally accommodates the S 1 -type LQs.
The plan of the paper is as follows. In the next section, we will describe our phenomenological setup in a model-independent way, which will then be subjected to a Monte-Carlo analysis in the following section. In Sec. 4 we will discuss the E 6 SSM and its possible scenarios that are amenable to experimental investigation by ATLAS and CMS, before concluding in Sec. 5.

Third-generation Scalar LQs
We consider a simple extension of the SM by adding one scalar S 1 -type LQ, which is charged as (3, 1, −1/3) under its gauge group, SU (3) C × SU (2) L × U (1) Y . We limit our discussion to LQs with the same quantum numbers as the coloured triplet of scalars, which, along with the Higgs doublet, constitutes the fundamental representation of SU (5) [24,25]. LQs with such quantum numbers have been proposed as a solution to the a µ and B-flavour anomalies (see, e.g., [26][27][28]).
We assume that our LQ, denoted by D henceforth, couples only to the third-generation fermions, with the corresponding Lagrangian being where Q L = (t, b) T L , and L = (ν τ , τ ) T L . This implies that the left-handed t quark couples to the left-handed τ lepton, while the right-handed t couples to the right-handed τ . The mass of the t quark at the one-loop level can thus cause the τ chirality to flip. Based on the above interaction terms, the D can decay into tτ and bν τ pairs. However, we assume λ λ, so that the decay to tτ is significantly enhanced. This LQ can be produced in pairs at the LHC through the processes shown in Fig. 1.
Given that the D couples only to the third-generation fermions, the diagram (e) in Figure 1 is negligible (unless the corresponding λ and λ couplings are extremely large, which is never the case here). The cross section for the D pair production is therefore almost completely determined by QCD, and can be considered to be model-independent. 1 We can write this in the form where BR stands for branching ratio. In our setup, we have BR(D → tτ − ) = BR(D →tτ + ) 1, and the leading order (LO) partonic cross sections σ(gg → DD) and σ(qq → DD) are given by [29] σ(gg where β = (1 − 4M 2 D /ŝ) 1/2 , and √ŝ is the partonic centre-of-mass energy. In Tab. 1 we show the LO and next-to-LO (NLO) pair-production cross sections, computed with MadGraph5 v3.2.0 [30] and Prospino v2.1 [31], using the NNPDF 3.1 parton distribution functions (PDFs) at LO [32]. Note that our cross section values are somewhat smaller than those in [19], mainly due to the different PDF sets used, but they agree well with those in Ref. [33], which also uses NNPDF 3.1.  Our simplified extension of the SM with a S 1 -type LQ can be considered an effective low-energy limit of some GUT framework. In the simplest GUTs, however, our D would have non-zero quarklepton as well as quark-quark couplings, which can lead to rapid proton decay. To ensure the model's phenomenological viability, all the couplings of the D to quark pairs can be set to zero, implying the conservation of baryon and lepton numbers.
Furthermore, GUT models have a large separation of scales, and to stabilise the ensuing hierarchy, supersymmetry is often preferred. If we supersymmetrise our model, we need to introduce two chiral superfields, one of which has λ-type and the other λ -type superpotential couplings. The 'exotic' particle spectrum then consists of two scalars and one fermion. The scalar states may have non-zero mixing, and one of the resulting physical states decays dominantly to tτ , while the other has nearly equal BRs in the tτ and bν decay channels. The decay of their superpartner fermion LQ leads to third-generation fermions and missing transverse momentum. Such a low-energy effective model can be embedded in a supersymmetric (SUSY) GUT framework such as the E 6 SSM, which is free from the severe constraints from proton stability noted above.
3 The fully hadronic ttτ τ final state at the LHC Run 2 The S 1 -type LQ under consideration here is subject to stringent limits from the ATLAS and CMS collaborations, most recent ones of which are given in Table 2. These limits correspond to the BR(tτ ) at the 95% confidence level (CL), and are obtained either directly when tτ is the exclusive decay channel of the LQ, or translated from the measurements in the bν decay channel, when it is additionally open, as 1 − BR(bν). For the tτ decays of the pair-produced scalar LQs, these searches have so far employed final states with e and µ leptons as well as hadrons from the τ decays [19,34]. For collider analyses of some other possible LQ decay channels, see [35,36] and references therein. In this study, we propose to complement the ATLAS and CMS probes of the ttτ τ intermediate state with the fully hadronic topology emerging when the decays of both the τ leptons and both the t quarks result in only jets in the final state. The analysis should select events with two τ h (i.e., τ decaying hadronically) and at least six jets, with two of them being b-tagged. The fully hadronic final state has two advantages compared with the semi-leptonic final state. First, it allows one to fully reconstruct the D mass. Second, it increases the number of the signal events. We will show that the level of the background can be reduced to that of the expected signal using τ -and b-tagging techniques, and by exploiting the large mass of the τ h pair.
To investigate the scope of the fully hadronic final state, we performed a signal-to-background analysis for the pair-production of LQs with a mass of 1 TeV each. The signal and the dominant ttjj background cross sections were calculated with Madgraph aMC@NLO at the LO, with selections ∆R(jj) > 0.4, p j T > 20 GeV, and |η j | < 2.4 for the jet transverse momentum and pseudorapidity, respectively, at the parton level. τ leptons in the signal sample were forced to decay hadronically, τ ± → π ± π 0 ν τ . We used the efficiencies and fake rates for τ -and b-tagging, and the τ τ mass resolution for the Z and Higgs bosons from the CMS publications (cited in the following), implying • p τ 1(2) = 0.7 (0.5) efficiency of τ -tagging for real τ , with p τ h T > 100 GeV (< 100 GeV), and p τ −fake = 3 × 10 −3 mistagging rate for the jets (using the DeepTau algorithm [37]); • p b = 0.8 efficiency of b-tagging for real b-jets, and p b−fake = 10 −2 mistagging rate for non-bjets (using the DeepJet algorithm [38]); • 15% di-τ mass resolution for the Z and Higgs bosons, i.e., 14 GeV and 19 GeV, respectively (using the SVFit algorithm [39]).  Fig. 3 the distribution of the p T of the parton jets, i.e., the two partons in addition to tt, and in the right panel the di-jet effective mass, m jj , reconstructed from these parton jets.
We exploit the difference between the m τ τ and m jj distributions to suppress the dominant ttjj background while preserving a high efficiency for the signal. The event selection strategy we propose is the following. Events must contain eight jets with two of them b-tagged. Six jets in the event, including the two b-tagged jets, must be associated with the two t quarks. The remaining two jets have to be τ -tagged, and their effective mass should be greater than 200 GeV. In our estimate of the efficiency of the di-jet mass selection for the ttjj background we assume correct association of the jets with the t quarks for 100% of the events. It was shown, however, in [40] that the fraction of the ttjj events with correct j-to-t association is 60%. It could thus potentially make the suppression factor of the di-jet mass selection used in our estimates weaker.
For the estimation of the efficiency of the di-jet mass selection we used the generator-level τ h for the signal, and parton jets with p T > 40 GeV and |η| < 2.4 for the ttjj background. This corresponds to the CMS di-τ high-level trigger selections used for the Run 2 [41]. Table 3 shows the cut-flow for the signal and background cross sections before and after the selections, along with the selection efficiencies. The selections considered here are the double-b-and double-τ -tagging, and the constraint on the di-τ mass.
For the signal cross section we have used two values. The cross section corresponding to the upper limit obtained in the ATLAS analysis [19] at m D =1 TeV, and (in parenthesis) the cross section of the pp → DD process in the E 6 SSM at the 13 TeV LHC. The latter includes the NLO corrections calculated with Prospino [31]. For the background, we have used the CMS measurements of the inclusive ttjj and ttbb cross sections [40, 42], the inclusive cross sections for the ttZ(→ τ τ ) [43] and ttW (→ jj) [44] processes, and the Z(→ τ τ ) + 6j [45] and W (→ jj) + 6j [46] cross sections, as functions of the jet multiplicity. The cross section for the Higgs boson production, in association with two t quarks, followed by its τ τ decay was taken from Ref. [47].
For two background processes, ttW and W + 6j, we did not evaluate the efficiency of the di-τ mass selection, and therefore give the upper limit on the corresponding cross sections after the selections. Note that the di-τ mass selection cuts eliminates the ttZ and Z + 6j backgrounds due to more than 5σ difference between the di-jet threshold and m Z . It likewise eliminates the ttH background, since the jj threshold is also about 4σ above the m H .
As one can see from Table 3, the selection cuts used eliminate all the backgrounds except the dominant ttjj. Further suppression of this background can be done by exploiting the possibility of the full reconstruction of the D mass as the effective mass of the tτ pair. The association of the two τ 's and the two t quarks to the DD pair can be done as follows. Out of the two τ h objects, we take the one which gives the tτ mass, m tτ 1 , closest to the D mass (1 TeV here). At this step, the t quark out of the two is taken randomly. The other τ h and the other t quark then give the mass of the second LQ, m tτ 2 . Fig. 4 shows the distribution of the m tτ 1 (left) and m tτ 2 (right) for the ttjj events after the selections p j T > 40 GeV, |η j | < 2.4, and m jj > 200 GeV.   One can see that selecting a large m tτ 2 can sufficiently suppress the ttjj background. The detector mass resolution of the t quark in a fully hadronic decay mode is 14 GeV (8% of the m t ) at the CMS [48]. We assume conservatively a detector resolution of 10% for the tτ mass, so that σ exp D = 100 GeV for m D = 1 TeV, and the natural width of D to be negligible compared to it. The efficiency of the selection m tτ 2 > m D − 2σ exp D = 800 GeV is 0.06 for the ttjj background, and 0.95 for the signal. It leads to cross sections of 0.14 (1.65) fb for the signal and 0.04 fb for the ttjj background after all the selections imposed so far. It corresponds to 20 (231) expected signal events and 6 background events for an integrated luminosity of 140 fb −1 , resulting in a signal significance, 2( √ S + B − √ B) [49], of 5.2 (> 5). Moreover, the expected upper limit on the number of the signal events is 6.2 corresponding to the cross section of 0.16 fb, which is better than the 0.5 fb limit at m S 1 = 1 TeV from the ATLAS analysis [19].
Two factors related to the detector reconstruction effects can potentially weaken the performance of our analysed channel. First, as mentioned already, the association of the jets and b-jets to the t quarks is not expected to be 100% correct. And second, the jet energy smearing can reduce the impact of the m jj > 200 GeV selection cut. Such effects can be estimated with a full detector simulation and reconstruction. We, however, do not expect our reported results to change drastically, and emphasise that the fully hadronic ttτ τ final state can nicely complement the current Run 2 analyses employing final states with leptons.

Interpretation in the E 6 SSM
As an application of our analysis to a top-down framework, we look at how it constrains the parameter space of the E 6 SSM, which is one of the best-motivated E 6 -inspired SUSY models. The simple extension of the SM discussed in section 2 may be embedded in this model, with the sparticle mass scale lying in the few TeV range.
In the E 6 SSM, the E 6 group is broken down to the SM gauge group plus an extra U (1) N symmetry around the GUT scale M X , where The U (1) ψ and U (1) χ symmetries are associated with the subgroups It is assumed that in the E 6 SSM the matter parity Z M 2 = (−1) 3(B−L) is also preserved.
In this SUSY model the anomalies are automatically cancelled if the low-energy particle spectrum includes three complete 27-dimensional representations of E 6 (27 i with i = 1, 2, 3). Each 27-plet contains one generation of ordinary matter, a SM singlet field Φ i carrying non-zero U (1) N charge, up-and down-type Higgs doublets H u i and H d i , as well as exotic scalar quarks (squarks) D i and D i with electric charge ±1/3. In order to avoid rapid proton decay via these exotic states, one must impose some additional symmetry. The simplest options are a Z L 2 symmetry, which implies that all supermultiplets except the lepton ones are even, or a Z B 2 symmetry, under which exotic quark and lepton supermultiples are odd, whereas the others remain even [21,22]. In the first case the D i and D i are diquarks (Model I). In the second scenario the exotic squarks carry baryon (B D = 1/3 and B D = −1/3) and lepton (L D = 1 and L D = −1) numbers simultaneously so that they are LQs (Model II).
The exotic states in the E 6 SSM may also give rise to flavour changing processes.
and the effective µ-term. In Eq. (6) g 1 andQ Φ = 5 √ 40 are the U (1) N gauge coupling and the U (1) N charge of the superfield Φ, respectively. The LHC constraints require the extra U (1) N gauge boson to be heavier than 4.5 TeV [50,51]. To satisfy this constraint ϕ should be larger than 12 TeV. The conservation of the Z M 2 symmetry and R-parity in the E 6 SSM ensures that the lightest SUSY particle (LSP) is stable. In this case the LSP and the next-to-LSP (NLSP) are mostly composed of the fermion components of the superfields Φ α . Using the approach discussed in [52][53][54], it was shown that the masses of the LSP and NLSP (H 0 1 andH 0 2 ) are smaller than 60-65 GeV [55]. The couplings of these fermions to the SM particles tend to be rather small. Nevertheless if the LSP had a mass close to m Z /2, it could account for some of the observed cold dark matter (DM) relic density [55]. In these scenarios, the lightest Higgs boson decays mainly into eitherH 0 1 or H 0 2 , while all other BRs are highly suppressed. Such scenarios have been already excluded by the LHC experiments. When the LSP and NLSP are considerably lighter than m Z , the annihilation cross section ofH 0 1H 0 1 → SM particles becomes too small, resulting in a DM density which is substantially larger than its measured value.
The exotic matter in the E 6 SSM may result in distinctive LHC signatures [21,22,59,62,91,[99][100][101][102], and can give rise to non-standard Higgs decays [55,57,86,[103][104][105][106][107][108][109]. In the E 6 SSM with approximate Z H 2 symmetry, one can also impose an exact Z S 2 symmetry, which implies that only components of the Φ α superfields are odd [63]. In this case the LSP and NLSP become massless and decouple. The presence of such massless fermions does not affect the Big Bang Nucleosynthesis if the Z boson is rather heavy [63]. In this variant of the model, one of the lightest R-parity odd state is stable and may account for some of the observed DM matter relic abundance. Neglecting non-renormalisable interactions and all the Z H 2 symmetry-violating couplings, the superpotential of this model can be written as where N c i and L j are supermultiplets of the right-handed neutrinos and left-handed leptons. In the case of Model II, the Z H 2 symmetry-violating terms, that permit the lightest exotic quark or squark to decay, can be presented in the following form Here e c i , u c k , d c i , and Q i are supermultiplets of the right-handed charged leptons, the right-handed uptype quarks, the right-handed down-type quarks, and the left-handed quark doublets, respectively.
The VEV of the Φ induces the masses of the exotic fermions, which in the leading approximation are given by where µ D i are the exotic quark masses, while µH α are the masses of the states composed of the fermionic components of the H d α and H u α . These masses are determined by the values of the Yukawa couplings κ i and λ α .
The breakdown of the gauge symmetry in the E 6 SSM may give rise to a substantial mixing between the scalar components of the supermultiplets D i and D i . Since we choose the field basis such that the Yukawa couplings of the D i and D i to Φ are flavour diagonal, which leads to a mixing only between the exotic squarks from the same family. As a consequence, the calculation of the exotic squark masses reduces to the diagonalisation of three 2 × 2 matrices These contributions are negative because the U (1) N charge of the Φ and the U (1) N charges of the D i and D i are opposite. On the other hand, the U (1) N D-term contributions to the masses of the ordinary squarks and sleptons are positive. As a result, in some parts of the E 6 SSM parameter space the exotic squarks tend to be substantially lighter than the superpartners of the SM fermions. The magnitude of mixing in the exotic squark sector is governed by the mixing parameters X D i as well as the masses µ D i . If µ D i are sufficiently large, the mixing effects can be so substantial that the corresponding lightest exotic squarks may be among the lightest SUSY particles in the E 6 SSM. In the Model II, the lightest scalar LQ, which we denote by D in order to identify it with the S 1 -type LQ in our low-energy SM extension discussed earlier, is a linear superposition of the scalar components of the D 1 and D 1 supermultiplets. Its mass is given by To simplify our analysis of the D in the E 6 SSM, we set A κ 1 = 0. Because of this the mixing between the scalar components of the D 1 and D 1 is almost always small unless M 2 11 (1) ≈ M 2 22 (1). In this case, D is almost entirely the scalar component of either D 1 or D 1 , and m 2 D is defined by the smallest diagonal element of the mass matrix in Eq. (10), i.e., either by M 2 11 (1) or M 2 22 (1). Moreover, we focus on the part of the E 6 SSM parameter space where m D is smaller than the mass of the lightest fermion LQ. Therefore, the decay modes of the D are entirely determined by the Yukawa interactions in Eq. (8).
We examine the BRs of the decays of the D assuming that Z H 2 is mainly broken by the operators involving quarks and leptons of the third generation. In other words, we ignore all the  [19] and [20], respectively. The red points correspond to some particular configurations of the E 6 SSM parameter space, explained in the text.
In Fig. 5, we show the expected exclusion contour (blue) from our signal-to-background analysis in the {m S 1 , BR(D → tτ )} plane, together with those from two recent ATLAS results (black and green). Also plotted are a number of E 6 SSM parameter space points, obtained for different values of m 2 D 1 , m 2 D 1 , and κ 1 . When the D is predominantly the scalar component of the D 1 , it couples to the doublets and thus has BR(D → tτ ) close to 50%. When it is mostly the D 1 it couples to singlets and decays solely to tτ if the right-handed neutrino is heavier than the D. Hence a lot of the points have the BR(D → tτ ) close to either 50% or 100%. In the intermediate cases M 2 11 (1) ≈ M 2 22 (1), and there can be sizeable mixing between the scalar components of the D 1 and D 1 .
We see in the figure that, when the D is D 1 -like, our proposed analysis of the fully hadronic final state resulting from its pair-production can exclude its mass up to about 1340 GeV. The exclusion bound is even stronger, close to 1580 GeV, if the D is predominantly the D 1 instead.

Summary and Conclusions
LQs, being coloured, can be produced efficiently at the LHC, and are thus among the potential signatures of BSM physics. We have studied the possibility of searching third-generation LQs decaying to tτ in the fully hadronic channel. This analysis would complement the existing searches relying on the semi-leptonic channel, and has the advantage of larger statistics, as well as the possibility of reconstructing the mass of the LQ.
In the fully hadronic channel, we expect larger backgrounds from QCD processes. But we have shown that, with suitable event selection, the noise from the ttjj background gets under control, while the other backgrounds are negligible. We have found that this channel has a large estimated sensitivity for the Run 2 data, leading to an expected exclusion limit of up to 1580 GeV (assuming BR(S 1 → tτ ) = 1) on the LQ mass, thereby besting the ATLAS result based on the semi-leptonic channel by about 150 GeV.
As an example of a model with such a signature, we have discussed a variant of the E 6 SSM in which the fundamental representation contains scalar (and fermion) LQs. The D-term contributions to the scalar LQ masses are negative, unlike those to other sfermions, and they can hence be among the lightest BSM states of this SUSY scenario. If their couplings to the fermions of the first two generations are vanishing, one of the two lightest scalar LQs can decay dominantly to tτ , and the other to tτ and bν, with nearly equal BR in each of these channels. We have then demonstrated how the fully hadronic channel is expected to improve the sensitivity of the LHC searches to the pair-production of the lightest LQ in this model. [41] CMS collaboration, Searches for additional Higgs bosons and for vector leptoquarks in τ τ final states in proton-proton collisions at √ s = 13 TeV, 2208.02717.