The Leptoquark Hunter's Guide: Pair Production

Leptoquarks occur in many new physics scenarios and could be the next big discovery at the LHC. The purpose of this paper is to point out that a model-independent search strategy covering all possible leptoquarks is possible and has not yet been fully exploited. To be systematic we organize the possible leptoquark final states according to a leptoquark matrix with entries corresponding to nine experimentally distinguishable leptoquark decays: any of {light-jet, b-jet, top} with any of {neutrino, $e/\mu$, $\tau$}. The 9 possibilities can be explored in a largely model-independent fashion with pair-production of leptoquarks at the LHC. We review the status of experimental searches for the 9 components of the leptoquark matrix, pointing out which 3 have not been adequately covered. We plead that experimenters publish bounds on leptoquark cross sections as functions of mass for as wide a range of leptoquark masses as possible. Such bounds are essential for reliable recasts to general leptoquark models. To demonstrate the utility of the leptoquark matrix approach we collect and summarize searches with the same final states as leptoquark pair production and use them to derive bounds on a complete set of Minimal Leptoquark models which span all possible flavor and gauge representations for scalar and vector leptoquarks.


I. INTRODUCTION AND OVERVIEW
A leptoquark (LQ) is a particle with a coupling which allows it to decay to a quark (or anti-quark) and a lepton, Fig. 1. It carries color and electric charge, and possibly also weak charges. Because of their color, leptoquarks can be pair-produced as LQ anti-LQ pairs with large QCD cross sections. Searches based on this production mode with subsequent decay, pp → φφ → (lq)(lq), are the subject of this paper (for single LQ production see [1]). Here φ and φ are the leptoquark and its antiparticle, q can be any Standard Model (SM) quark or antiquark, l is any lepton, andq andl are the corresponding antiparticles. We use parenthesis to indicate which final state particles reconstruct resonances. Our goal is to provide a simple organizing principle which makes it straightforward to systematically search for all possible leptoquarks. The idea is that we identify a minimum set of independent final states which must be searched for. These final states can be arranged into a 3 × 3 matrix which we call the "leptoquark final state matrix", or simply "LQ matrix", Fig. 1. For each final state, what is needed from experiment is an upper bound on the cross section times branching fraction as a function of LQ mass. Lower bounds on the mass of an arbitrary leptoquark can then be obtained by calculating the theoretical cross section times branching fraction into the final states in the LQ matrix and comparing to the experimental cross section bounds. This paper consists of two complementary parts. In Part 1, we introduce the LQ matrix and show that bounds on the cross sections into each of the 9 final states of the matrix are both efficient and sufficient for searching for all possible LQs. In Part 2 we demonstrate the utility of this approach. We collect the best currently available cross section bounds from LHC searches organized by which element of the LQ matrix they cover and use them to put bounds on a complete set of "Minimal Leptoquark" (MLQ) models. The MLQs include scalar and vector LQs with all possible flavor, charge, and isospin quantum numbers.
We begin with a few words on the theoretical motivation for leptoquarks. LQs are predicted in many physics beyond the SM scenarios. Vector LQs may be related to unification, squarks automatically become scalar LQs in supersymmetry (SUSY) with R-parity violation (RPV), and LQs arise in models in which quarks and leptons are composites of the same underlying dynamics. Recently, additional motivation for LQs arrived from a number of anomalies observed in B meson decays [2,3] which could be explained in models with leptoquarks of TeV scale masses. Leptoquarks with couplings to muons might also be relevant to the persistent anomaly in the muon anomalous magnetic moment [4]. On the other hand, there are no convincing theoretical arguments for preferring a particular pattern of leptoquark flavors or quantum numbers, and the experimental hints, however intriguing, may still change or disappear. This motivates a systematic approach to search for a minimal but sufficient set of final states which can discover LQs with arbitrary decays to quarks and leptons.
What are the possible LQ final states? By definition, a leptoquark can decay to any of the six SM quarks with any of the three charged leptons or three neutrinos. The quarks and leptons can be particle or antiparticle and leftor right-handed (we allow for right-handed neutrinos), corresponding to hundreds of different possibilities. However, many of these are covered by the same searches. For example, left-and right-handed final state particles are generally not distinguishable. Light quarks and antiquarks u, d, s, c,ū,d,s,c from leptoquark decays all show up as light-jets. 1 Heavy quarks, t and b can be distinguished. For leptons, we have three distinct charged leptons but neutrino flavor is not observable in leptoquark decays. Thus we arrive at a matrix of nine distinct final states from the decay of a single LQ, three "flavors" of quarks j, b, t times three "flavors" of leptons 2 ν, (= e/µ), τ , see Fig. 1b.
To obtain the final states of pair production we must consider the decay of the LQ and its antiparticle. For a LQ with a unique coupling to a quark-lepton bilinear ql, the decay of the LQ anti-LQ pair is also unique to the "symmetric" final state (lq)(lq). Enumerating all possible such symmetric final states is trivial, they are determined by adding the corresponding antiparticles to the final states of a single LQ. Thus we can use the same 3 × 3 LQ matrix to classify the distinct symmetric final states.
However, when LQs have multiple decay channels then LQ pair production also produces "asymmetric" final states which are not covered by the classification in terms of a single 3 × 3 matrix. We now argue that the symmetric final states are all we need. Consider for simplicity the case where a LQ couples to two quark-lepton pairs, lq and l q . We obtain the pair production final states: symmetric : (lq)(lq) + (l q )(l q ) asymmetric : (lq)(l q ) + (l q )(lq) The two symmetric ones are contained in the LQ matrix classification, the asymmetric ones are not. But here is the point: all leptoquarks which produce asymmetric final states necessarily also produce symmetric final states. Therefore the symmetric final states are sufficient to search for all possible leptoquarks. Moreover, one might expect the couplings of leptoquarks to different quark-lepton pairs to not all be the same. Then the branching fraction to the quark-lepton pair with the largest coupling lq will dominate the decays, and the branching fraction to the symmetric state (lq)(lq) will be largest. To sum up, symmetric final states from pair production of LQs can be represented by the LQ matrix. They are sufficient for searching for all possible leptoquarks. When LQ couplings to quark-lepton pairs are at least somewhat hierarchical, a symmetric final state will usually also yield the most sensitive search.
An interesting special case arises when a symmetry relates couplings of the leptoquark to different quark-lepton pairs. In fact, isospin, SU (2) weak , is such a symmetry. Consider for concreteness an SU (2) weak singlet scalar leptoquark which couples to the SU (2) weak doublets q L and l L , both of the first generation. SU (2) weak enforces that the coupling of the LQ is to the combination dν − ue − so that pair production gives rise to the four possible final states (νd)(νd) + (e − u)(e +ū ) + (νd)(e +ū ) + (e − u)(νd), each with 25% branching fraction. One could search for this leptoquark with the symmetric LQ matrix final states (e − j)(e + j) and jj / E T or with non-matrix final states (e + j)(j / E T ) and (e − j)(j / E T ). However even in this case where no single symmetric final state dominates the "easiest" and likely most sensitive search is the symmetric search for (e − j)(e + j). We will discuss many more examples with multiplicities of final states from SU (2) weak in Sec. III.
So far, we have argued that in order to search for all possible LQs, it is sufficient for experiments to cover 9 symmetric final states which are in one-to-one correspondence with elements of the LQ matrix. What information do we need from each of these 9 searches in order to be able to put bounds on arbitrary leptoquarks?
In order to maximize sensitivity, LHC leptoquark searches adjust their cuts as a function of the mass of LQs that are being searched for. In addition, signal efficiencies depend on the LQ mass. Therefore, the natural result of a leptoquark search is a "Brazil plot" which shows the experimentally observed upper bound on the allowed signal cross section times branching fraction as a function of LQ mass (along with the range of expected bounds). This plot of the upper bound on σ × Br for each final state in the LQ matrix is precisely the result which is needed from experiment. With the 9 plots from each of the LQ matrix searches anyone can then easily determine a lower bound on the mass of their favorite leptoquark by comparing their theoretical σ × Br as a function of mass to the bound. This is true for LQs of arbitrary spin, charge, or couplings.
This last claim requires further examination. In order to determine the bound on σ × Br experimenters need to divide the observed bound on the signal cross section for events in a signal region by the efficiency of signal events to pass the signal region cuts. In general, this signal efficiency can depend on many properties of the signal under consideration, and one may worry that model-independent bounds on σ × Br cannot be obtained. Fortunately, the situation for symmetric LQ searches is nice. To fairly good precision (better than 10 %) the signal efficiency only depends on the mass of the leptoquark. The basic reason for this model independence is easy to understand: current LQ bounds already require masses well above 500 GeV and center of mass energies in excess of 1 TeV. Thus leptoquarks are produced with only moderate boosts. Furthermore they two-body-decay directly to SM particles, yielding highly energetic widely separated SM particles with approximately isotropic distributions. Thus signal efficiencies for standard kinematic cuts are high and only very weakly dependent on any details of the LQ model other than the LQ mass. Two concerns which could have invalidated this argument are dependence on LQ spin and width, we discuss them in turn.
Spin: Leptoquarks can be spin 0 or 1, i.e., scalars or vectors. In either case the cross section is dominated by gluongluon interactions. Scalars are produced in a p-wave, implying that the cross section is more suppressed near the threshold, and typical scalar LQ events have more energy than typical vector events. Thus we expect the efficiencies of p T cuts to be slightly higher for scalars. This is a small effect because even LQs decaying at rest would give their decay products sufficient momentum to pass basic p T cuts. More important are angular distributions resulting from LQ spin. Because of the large LQ mass, decay products are well-isolated and approximately back-to-back. However the pseudo-rapidity, η, distribution of vector LQs is more forward than for the scalars, causing the efficiency of especially lepton η cuts to be lower in the vector case. But this effect is also small. To demonstrate this we show the ratio of the two signal efficiencies, S / V , as a function of mass for the analysis cuts used in two typical ATLAS [5] and CMS [6] analyses in Fig. 2 (details are shown in Appendix B). This analysis is for LQs decaying to muons and light quarks and with signal simulated at the parton level. One sees that for LQ masses in the range of current bounds (800 GeV -2 TeV) the difference between scalar and vector efficiencies is always smaller than 10%.
Ratios of signal efficiencies S / V for pair-produced scalar and vector LQs with various masses. The LQ decays are to muons and jets. The panels show the relative efficiencies of a series of selection cuts from recent ATLAS [5] and CMS [6] analyses. Details of the cuts and plots of phase space distributions are in Appendix B.
We conclude that searches with cuts optimized for scalar LQ detection are also very sensitive to vectors and the differences in efficiencies are small enough that we can ignore them. An analysis aiming for higher precision could account for the small difference in efficiencies between vectors and scalars by determining separate cross section bounds for the two cases.
Width: For leptoquarks with very large couplings ( 1) to quarks and leptons the width becomes large and a significant number of events may have one or both LQs off-shell. In that case signal efficiencies can become much more model-dependent. However large couplings to quarks and leptons of the first and second generation for TeV scale leptoquarks are ruled from single leptoquark production, e + e − collisions, and precision electroweak constraints. Large couplings to third generation particles remain a possibility which could introduce some model dependence. We ignore this special possibility and assume that LQ widths are smaller than 10% so that on-shell LQ production dominates.
To summarize our results so far, we have argued that in order to search for arbitrary LQs it is sufficient to consider 9 final states corresponding to the LQ matrix. Experiments should provide the upper bound on the cross section times branching fraction as a function of LQ mass for each of the different final states (Brazil plot). The signal efficiencies needed to produce the cross section bounds are largely model independent and can be determined from Monte Carlo (MC) for any convenient leptoquark implementation (scalar or vector with arbitrary couplings). Armed with cross section bounds for each LQ matrix final state one can determine lower mass bounds on arbitrary LQs by computing the cross section times branching fractions and comparing to the bounds.
The leading Feynman diagrams contributing to the LQ pair-production cross section are shown in Fig. 3. Gluon- initiated diagrams dominate for all but the largest LQ masses. The diagram proportional to the square of the leptoquark-lepton-quark coupling λ is model dependent but it is never important for pair production. This is because once the coupling is large enough to make a difference for pair production it also contributes to single LQ production at the same order in λ and single production will yield a stronger limit [1,7]. Therefore we will ignore this diagram in the following. LQ cross sections times branching fractions into the dominant final states are large because LQs carry color and are strongly produced. Choosing for example a leptoquark mass of 1 TeV, hundreds of scalar leptoquarks or thousands of vector leptoquarks (would) have been produced with the ∼ 36 fb −1 data of 13 TeV accumulated in 2016 at the LHC, see   [8] for scalar S and vector V leptoquarks (K-factors for pair-production cross sections at hadron colliders are given in [8][9][10][11][12]). For details of the vector model see Appendix C. The cream band is explained in Sec. II.
The vector case has a subtlety related to unitarity of theories with massive vectors. A consistent theory must have additional states beyond the LQ with masses not too far above the LQ mass. For computing the vector cross section in the figure we used a minimal model implementation described in in Appendix C. The model has a massive gluon partner, g , which contributes to LQ pair production. Varying the mass of the gluon partner across its theoretically allowed range we obtain the two limiting cross sections labeled m g /m V = 0, √ 2 in Fig. 4. We now show as an example how to obtain lower limits on LQ masses using the LQ matrix final state (µ + j)(µ − j), which is obtained from pair-production and decay of a LQ decaying to muons and jets. A nice analysis for this final state has been presented by CMS [6]. CMS provided a 95% confidence level (CL) upper bound on the cross section times branching fraction to this final state which we represent with the solid black curve in the left panel of Fig 10 -2 Bounds on scalar and vector leptoquarks which decay to (µj)(µj) final states. The observed 95% CL upper limit on the production cross section times branching fraction (black solid line) is taken from a search for leptoquarks in this final state at CMS [6]. The search stopped at 1.4 TeV. In order to obtain a plausible estimate for the mass limit on vector LQs we extended the cross section limit beyond 1.4 TeV by assuming that it becomes mass-independent (black dashed line). The theoretical cross sections for different MLQs which can decay to (µj)(µj) final states are shown as colored lines and labeled with corresponding cross sections. The intersects (vertical dotted lines) determine the lower limits on the MLQ masses. (right plot) Bounds on scalar and vector MLQs which decay into (µb)(µb) final states. Since only a 95% CL on the stop/leptoquark mass (black dot) is provided in [13] we cannot derive rigorous mass bounds on other LQs. To obtain an estimate, we assume that the experimental upper limit on the cross section is flat across the LQ mass range and agrees with the value for the scalar fiducial cross section, σS, at the stop/leptoquark mass limit (black dashed horizontal line). Cross sections for other MLQs that can decay into (µb)(µb) final states are shown as colored lines and labeled with their cross sections. The intersects (vertical dotted lines) determine our estimates for lower limits on the MLQ masses. models which we define in the next Section. Four of the models have scalar LQs with pair production cross sections times branching fraction ranging from 0.25σ S to 2σ S . Here σ S is the "fiducial" scalar cross section shown in Fig. 4. The factors 0.25, 1, 1.25, 2 correspond to multiplicity factors and branching fractions which will also be explained in Sec. II. We just use them as examples to make the point that there are many different LQ models which can contribute to this final state. We can easily obtain lower bounds on the different MLQs' masses by using the plot of the experimental cross section bound. For example, the red leptoquark labeled σ S is excluded up to 1.17 TeV. This leptoquark was the designated target of the CMS search and the limit 1.17 TeV can also be found in [6]. However none of the other MLQs were considered explicitly by CMS, and had the collaboration decided to only publish the mass bound a simple determination of mass bounds for all LQs would have been impossible.
Unfortunately, while CMS did provide the very useful plot of the 95% CL cross section limit they did not extend the mass reach of their plot far enough in the leptoquark mass to allow a reliable extrapolation to vector leptoquarks which have much larger cross sections. In absence of a cross section limit provided by experimenters we are forced to guess. In this case it seems reasonable to guess that the cross section limit flattens out for large LQ masses because signal efficiencies from very heavy LQs approach saturation. Having made this guess (the dotted black line extension of the experimental limit) we can then put lower bounds on the masses of the four vector MLQ models. However, we emphasize that these mass limits are estimates based on our guess for how the sensitivity of the CMS search might continue to higher masses.
The situation depicted in the right panel of Fig. 5 is less satisfactory. This plot corresponds to the LQ matrix search for (µ − b)(µ +b ). ATLAS performed this search as a SUSY search for RPV stop decays and obtained an impressive lower mass bound of 1.4 TeV. This RPV squark is identical in production and decay to a particular scalar leptoquark, and therefore this mass bound can be directly applied to the scalar leptoquark model. Given this bound and our knowledge of the theoretical cross section times branching fraction for this leptoquark we can infer that ATLAS found a 95% CL upper bound of 4.5 × 10 −4 pb on the signal cross section for a LQ with mass 1.4 TeV (indicated with the black dot in Fig. 5). Unfortunately, that is all the information we have. From this one data point it is not possible to infer bounds on the masses of any other LQ models with differing cross sections or branching fractions. The best we can do is make the assumption that perhaps the 95% CL limit on the cross section is approximately mass independent j b t ν [14][15][16][17][18][19][20] [13,49], (τ b) [50][51][52][53][54][55], and (τ t) [56].
(as it appears to be for the CMS search in the left panel for larger LQ masses). Using our guess for the cross section bound (horizontal dashed line) we can now provide estimates for the mass bounds of other MLQ models by reading off the intersects of the cross section predictions with the extrapolated cross section bound.
In the remainder of the paper we present results for LHC searches for all 9 of the LQ matrix elements. We found fully satisfactory cross section limit plots in six cases, somewhat less satisfactory mass limits for specific leptoquark models in one case and no dedicated leptoquark (or equivalent RPV squark searches) in two cases. In all cases, we perform a recast of the bounds to obtain mass limits in our MLQ models. For the cases where experimenters provided full 95% CL bounds on the cross section our MLQ model limits can be regarded with confidence. In the other cases we regard them as estimates for what the limits could have been.
We close this Section with the LQ matrix with elements filled with references to experiments which performed searches corresponding to the LQ matrix final states. Green indicates that at least one experiment published the 95% CL upper bound on the pair production cross section as a function of LQ mass.

II. MINIMAL LEPTOQUARK MODELS
Here we define a set of MLQ models. The idea behind those models is simple. Each model has one leptoquark in a single SU (3) color ×SU (2) weak ×U (1) Y representation coupling to only one SM lepton-quark fermion bilinear. Therefore to define the MLQ models we just need to identify all possible lepton-quark bilinears and their transformation properties under the SM gauge group. For each MLQ model we then obtain the lower bound on the mass of the MLQ by comparing the predicted cross section times branching fraction into LQ final states with the upper bound from experiment. To do so we need to determine the cross section times branching fraction into the possible symmetric finals states for each MLQ. It will be convenient to parameterize the answer as µ × σ S or µ × σ V for scalar or vector MLQs. Here σ S and σ V are the fiducial cross sections shown in Fig. 4 and µ is a number defined as µ may be greater than 1 when multiple LQs are produced and less when branching fractions into the final state are nontrivial.
To define the MLQ models, we identify the lepton-quark bilinears in the SM that we can couple to LQs. Ignoring the three generations for the moment, these are given by arbitrary quark or antiquark fields multiplying arbitrary lepton fields. Enumerating all the possibilities is easiest when all SM fermions are interpolated by left chirality fields. 4 One generation of fermion fields is Q, U c , D c , L, E c , N c . Note that we have included possible singlet right-handed neutrinos. Scalar S and vector V leptoquarks couple to quark-lepton bilinears as respectively. Hereσ µ is vector of 2 × 2 matrices which contracts the vector index of the LQ with the fermion spinors. This matrix appears in all vector LQ couplings to fermions. To avoid notational clutter we suppress it in subsequent formulas. Q and L can be any of the quark and lepton fields, i.e., Q ∈ {Q, U c , D c } and L ∈ {L, E c , N c }. For both scalars and vectors there are 3 × 3 different possibilities (before considering generation number). Adding in the three generations, each of the above described scalar or vector LQs we can couple to one of 3 generations of quarks and 3 generations of leptons for an additional multiplicity of 9. Thus in total there are 3(Q, U c , D c ) × 3(L, E c , N c ) × 3(quark gen.) × 3(lepton gen.) × 2(S or V ) = 162 different MLQ models. Note that we are defining the MLQ models in the mass eigenstate basis for the quarks and charged leptons. In doing so, we ignore the small mismatch between mass eigenstate bases for the doublet up-and down-type quark fields, this is equivalent to ignoring leptoquark decays which are proportional to λ times small CKM matrix elements. The basis for neutrinos is irrelevant because the different neutrino flavors are indistinguishable final states.
For example, we can have a scalar leptoquark coupling as λS(Q 1 E c 1 ) where the subscripts are generation indices. This is a scalar which is an SU (2) weak doublet. One of its components couples to up quarks and e + and the other couples to down quarks and e + . Thus this MLQ model actually contains two mass-degenerate scalar leptoquarks which means that the total cross section for leptoquark production is twice as large as σ S . Each of the two components of S couple to only one fermion bilinear. Therefore both decay with 100% branching fraction, one decays to (e + u), the other to (e + d). In terms of experimentally distinguishable final states both LQs decay to (e + j). Thus the total cross section times branching fraction to the final state (e + j)(e − j) in this model is two times the scalar fiducial cross section in Fig. 6, and the µ-factor for this model is µ = 2. We show the pair production and decay µ-factors of all MLQs in Table I. Consider as another example the leptoquark coupling to U c † 3 E c 1 . This is a vector leptoquark with a unique coupling and decay to top quarks and e + . It is an SU (2) weak singlet and therefore has no multiplicity. The cross section times branching to the final state (e −t )(e + t) is then equal to the fiducial cross section σ V , and the µ-factor is 1. Note that the final state of this leptoquark is distinguishable from the final state of the scalar leptoquark coupling to U c 3 E c 1 which decays to (e +t ) giving the pair production final state (e − t)(e +t ) which has the same final particles but different resonance pairs. Its µ-factor is also 1.
To get MLQs with non-trivial branching fractions we must consider scalar leptoquarks coupling to QL and vectors coupling to Q † L. In each case, there are two possibilities, the leptoquark can be isospin singlet or triplet.
Turning first to the singlet case, the scalar couples to αβ Q α L β while δ αβ Q † α L β is the vector coupling. 5 Consider as an example the singlet LQ coupling to third generation quarks and first generation leptons. It couples to the fermion bilinear (te − − bν). Thus there are 4 distinct final states where the 1/4 indicates that each final state has branching fraction 25%. This is an example in which isospin predicts symmetric final states (e − t)(e +t ) and bb / E T as well as asymmetric final states (e − t)b / E T , (e +t )b / E T . Again, we focus only on the symmetric case because it does not have neutrinos and is usually more sensitive than the asymmetric one even though that has multiplicity of 2. In the literature this case is sometimes called β = 1/2 because the LQ has two different decays with branching fractions Br ≡ β = 1/2.
An isospin singlet vector coupling to third generation quarks and first generation leptons couples to (tν +be − ) and also gives rise to 25% branching fractions In both the scalar and the vector cases the branching fractions to any single final state are reduced to 25%. There is no multiplicity factor from the cross section, thus both have µ = 0.25.
An isospin triplet MLQ model contains three LQs which each couple to a different combination of fermions. For example, choosing the MLQ with couplings to third generation quarks and first generation leptons, the couplings are 5 Here the = 0 1 −1 0 and δ = 1 0 0 1 .
to tν, (te − + bν)/ √ 2, be − for the scalar case and t e − , (tν −be − )/ √ 2,bν for the vector case. Since each of the three LQs can be pair produced with the fiducial cross section one obtains the final states Here the most promising search channels are the final states (e − b)(e +b ) and (e −t )(e + t) which occur with µ-factors 1.
Finally, for isospin triplet MLQs coupling to quarks of the first or second generation the final state quarks produce indistinguishable light-jets. Thus here the final states are yielding µ-factors of 1.25 for both of the symmetric final states (e − j)(e + j) and jj / E T . We summarize all possible µ-factors in Table I ) µ-factors for production and decay σ(pp → SS → (lq)(lq)) in different scalar MLQ models. Rows correspond to different final states whereas columns correspond to different MLQ models. For example, the first row of the first column corresponds to pair production of a scalar LQ which decays to (νj) via a coupling in which Q and L are contracted into an isospin triplet. For any final state (ql)(ql), the µ-factor is the factor which multiplies the fiducial cross section σS in To place bounds on all MLQ models with the same final state, it is important for the experimental collaborations to search for sufficiently high and low LQ masses. The lower masses are required to set bounds on MLQs with small cross sections whereas the highest masses are relevant for MLQs with large cross sections. The range in cross sections of MLQ models is bounded by 0.25 × σ S and 2 × σ V . We illustrate this range as the cream band in Fig. 4. Other models can in principle have even larger or smaller µ factors because of larger LQ multiplicities or smaller branching fractions to the final state under consideration. Thus a search mass window that is as large as possible is desirable.

III. LEPTOQUARK SEARCHES ORGANIZED BY THE LQ MATRIX
LQs may be classified by their decay products and organized into the LQ matrix. The symmetric final states from pair-production where the LQ and its antiparticle decay in the same way give the strongest bounds. We therefore focus on the symmetric final states which can also be organized into the LQ matrix. The current status of searches for each final state is summarized in Fig. 6. The color of each entry of the matrix indicates whether a cross section bound as a function of mass has been made available; the numbers indicate relevant references. 6 In the following subsections we review LQ searches organized by entries of the LQ matrix.
A. νj, νb and νt Searches for pair-produced scalar leptoquarks decaying into (νj)(νj), (νb)(νb), and (νt)(νt) final states are identical to those for pair-produced squarksq 1 [14][15][16][17][18][19][20] (q 1 =ũ 1 ,d 1 ,s 1 , orc 1 ),b 1 [16][17][18][21][22][23][24][25][26], andt 1 [16-18, 22, 23, 25, 27-37], namely the pair-produced particles share the same spin, charge, decay final states and kinematics, if the neutralinõ χ 0 1 is taken to be massless. An example is shown in Fig. 7 with the leptoquark production and decay diagram on the left and the corresponding squark diagram on the right. Thus constraints on squarks can be directly applied to S νq and vise visa. Such a translation has already been performed by ATLAS [22]. We obtain the strongest 95% CL level limits by applying squark bounds from CMS Run 2 analyses, m Sνj ≥ 1.05 TeV [16,17], m S νb ≥ 1.175 TeV [17], and m Sνt ≥ 1.07 TeV [17]. The recast of the bounds to other MLQs is performed as follows. We digitize the temperature plot for the observed 95% CL upper limit on the light squark pair production cross section from searches at CMS [17], setting mχ0 1 = 0. The digitized cross section limit is shown as the black solid curve in Fig. 8. The colored lines in the same plot show the theoretical cross sections of different MLQs that can decay into (νj)(νj) final states. The curves are labeled with the cross sections given in terms of the µ-factor times the relevant fiducial cross section, σ S or σ V , for scalar or vector LQs respectively. The intersects (dotted black lines) of the cross section curves with the experimental bound correspond to the lower mass bounds obtained in the different models. We obtain the bounds 630 GeV, 1 TeV, 1.1 TeV, and 1.2 TeV for scalar leptoquarks with µ-factors of 0.25, 1, 1.25, and 2, respectively, and 1.4 TeV and 1.7 TeV for vector leptoquarks with µ-factors of 0.25 and 1, respectively. The CMS search stopped at 1.7 TeV and therefore we cannot obtain reliable experimental bounds on heavier leptoquark candidates. To get an estimate, we extrapolated the experimental bound on the cross sections as independent of LQ mass for masses larger than 1.7 TeV (shown as the black dotted line). This allows us to recast bounds on vector leptoquarks of 1.75 TeV and 1.8 TeV for µ-factors of 1.25 and 2 respectively. Referring to the row labeled S νj and V νj in Table I we can then associate the mass bounds corresponding to different µ-factors to the different MLQ models. These bounds are summarized in Table II, Table III,  and Table IV Fig. 8. The bounds apply to LQs coupling to the first or second generation quarks and neutrinos of any generation. (lower ) 95% CL lower limit on the mass of vector MLQs Vνj which decay into (νj)(νj) final states. For bounds greater than 1.7 TeV, we have assumed that the experimental upper limit on the cross section is flat as the mass increases. Since these bounds are based on our assumption they cannot be considered solid and we marked them with parentheses. Note that [17] obtain a mass bound of 1050 GeV while our plot Fig. 8 shows an intersect at ∼ 1 TeV, this small discrepancy may be a reflection of the inaccuracy of our digitization of the experimental cross section bound from the provided temperature plot or a difference in assumed K-factor.  ≥ 830 GeV Obs. 95% CL upper limit  In addition ATLAS and CMS also looked for asymmetric decays into (ej)(νj) or (µj)(νj) final states. These searches require leptoquark decays to both (ej) and (νj) which is true for the SU (2) weak singlet or triplet coupling to QL or Q † L. The latest 95% CL upper limit on the leptoquark mass from those channels is m Sνj ≥ 850 GeV [47]. This is comparable to the bound from searches for the symmetric final state (ej)(ej) in the case of a singlet leptoquark coupling to [Q 1,2 L 1,2,3 singlet], but significantly weaker than the bound from the symmetric search for the triplet leptoquark coupling to [Q 1,2 L 1,2,3 triplet], see Table II.
Also note that many searches for (νj)(νj) final states and light squark searches did not veto c-jets. Therefore the limits also apply to φ νc . One might hope to improve the sensitivity of these searches to φ νc with c-tagging. However, c-tagging efficiencies are currently too small for this to be effective. For example, Ref. [19] (8 TeV, 20 fb −1 ) conducted ac 1 search with efficiencies: c-tag 19%, b-misstagged-as-c 13%, and j-misstagged-as-c 0.5%. They obtained the bound mc 1(Sνc) ≥ 540 GeV which is weaker than the bound from the flavor inclusive analysis with the same data set [15], mc 1(Sνc) ≥ 580 GeV. Efficiencies of c-tagging did not improve significantly in Run 2 [58].
SUSY with RPV allows a superpotential couplingLQD c for the superfields. This superpotential coupling gives rise to a Yukawa couplingd c † QL in which thed c squark couples like a leptoquark. The final-state signature and kinematic distributions of pair-produced squarks which decay via this Yukawa coupling are identical to production and decay of scalar leptoquarks (e.g., see Fig. 12). Such a one-to-one correspondence to RPV squarks exists for all the leptoquarks coupling to [LQ] or [LD c ]. ATLAS have conducted this type of RPV stop search with Run 1 [49] and Run 2 [13] data. The process searched for is pp →t 1t1 wheret 1 decays into the final states (eb) or (µb) with at least one b-tag required in [13]. No excess was found for the stop masses searched. We take the strongest bounds from [13] (13 TeV, 36 fb −1 ) at the special points where the decays are 100% to (eb) or 100% to (µb). These can be directly reinterpreted as bounds on a scalar leptoquark decaying to (µb) or (eb). S eb and S µb are excluded up to 1.5 TeV and 1.4 TeV with 95% CL, respectively. Assuming that the bounds on the cross section times branching fraction to the signal regions do not depend on the LQ mass for even heavier LQs we can recast these bounds to vector MLQs (see the right panel in Fig. 5 for the cross section bound plot for (µb) and Fig. 13 for (eb)). We obtain the estimated bounds shown in parenthesis in Table VI. 10 -3 13. Bounds on scalar and vector MLQs which decay into (eb)(eb) final states. The 95% CL on the stop/leptoquark mass (black dot) is provided in [13]. The remainder of the analysis is analogous to the right plot of Fig. 5.
TABLE VI. (left) 95% CL lower limit on the masses of scalar LQs S eb and S µb . Both occur only with µ-factors equal to one so that no recasts are necessary. The limits are adopted from RPV stop searches from ATLAS [13] (see Fig. 13 and Fig. 5).
(right) Estimated 95% CL recast lower limits on the masses of vector LQs V eb and V µb .
where subscript h or l of t indicates the top quark decays hadronically or leptonically, respectively. The first two cases of Eq. Here we focus on the semi-leptonic case (Eq. (8b) , see Fig. 14). CMS [60] presented a 13 TeV stop search with 35.9 fb −1 of data with 50 distinct signal regions. The final states can have multiple b-tags, triple leptons, and large missing energy. We generated signals with MadGraph5 aMC@NLO 2.5.4 [61] at leading order parton level. We considered leptoquark masses ranging from 500 GeV to 1.6 TeV with 100 GeV steps, and applied the NLO K-factor from [8]. The signal can be triggered by two sets of dilepton triggers. The first set requires the loose isolation (I mini, < 0.4) of two light leptons, with transverse momentum p T satisfying p T 1 , p Tµ2(e2) > 23, 8(12) GeV. The second set has no isolation requirement. It requires leptons with p T 1,2 > 8 GeV and H 30 T ≥ 300 GeV. The jets are clustered with the anti-k T algorithm with R = 0.4. The reconstructed jets are required to have p T ≥ 30 GeV and pseudorapidity |η| < 2.4. Reconstructed leptons are required to be well-isolated. They need to satisfy I mini,µ(e) < 0.16 (0.12) together with p ratio Tµ(e) > 0.69 (0.76) or p rel Tµ(e) > 6.0 (7.2) GeV. Besides isolation, the analysis requires the lepton pseudo-rapidity |η µ(e) | < 2.4 (2.5) and the p T of the leading three leptons p T 1 , p Tµ2(e2) , p T 3 ≥ 25, 10 (15) After the trigger and jet and lepton reconstructions, we pass the events through the baseline selection and divide them into the 25 off-Z signal regions (SR's) and super signal regions (SSR's). b-tagging is performed on jets with p T ≥ 25 GeV and |η| < 2.4. The b-tagging, c miss-tagging, and light-jet miss-tagging rates are (70%, 10%, 1%). We then compare the number of remaining signal events with the number of the observed events and expected background given in [60]. Signal regions SR5, SR6, SR9, SR10, and SR16a are most sensitive to leptoquarks. Cuts for these regions, together with baseline cuts, are summarized in Table VII.
Using the central value of the background estimation in Table VII and assuming Poisson statistics, the resulting 95% CL lower limit on the mass of S µt is 800 GeV from SR5 and SR9 and 700 GeV from SR6, SR10, and SR16a. The resulting constraint on the mass of S et is 900 GeV from SR16a, 800 GeV from SR10 and 700 GeV from SR5, SR6, SR9. A common feature of the most sensitive search regions is that they allow relatively low hadronic activity. Note that our recast, using a simple cut and count search, is not optimized. Better sensitivity could be reached with an analysis that reconstructs at least one of the resonances of the signal. In Table VIII [60]. We also list the observed and expected number of events for each SR. See text for more details.  [60].
In analogy with the (et)(et) or (µt)(µt) cases, the τ 's in (τ j)(τ j) final states can be either hadronic or leptonic. The branching fractions for φ τ jφτ j decays are given by We are not aware of any dedicated searches for these final states. However, three types of existing searches can be relevant: (1) leptoquarks which decay to (τ b)(τ b) final states are intensively studied by ATLAS and CMS (see Sec. III F). An analysis without imposing the b-tags would cover the (τ j)(τ j) channel; (2) searches for right-handed W R and right-handed neutrinos N using the process pp → W R → τ N → τ (τ qq) has been conducted by [54,55]. The final particles of this process are identical to φ τ jφτ j . To apply this search one would have to adjust the signal kinematics and re-analyze the data; (3) our LQ final state can be also be covered by SUSY or Exotics searches which look for large missing energy with high p T -jets and zero, one, or two opposite-sign leptons. Given the similarity of this final state with the well-covered (τ b)(τ b) final state and given that SM backgrounds are small at large invariant masses we expect that a designated search will obtain bounds on leptoquark masses which are comparable to those for the (τ b)(τ b) case. We have not been able to obtain reliable bounds for any of the MLQs in this final state and therefore left the corresponding entries in Table IX blank. F. τ b LQ decays into the (τ b)(τ b) final state have been well-covered by LHC searches for LQs coupling to third generation quarks and leptons with Run 1 and Run 2 data [50][51][52][53][54][55]. The latest 95% upper limit on the only scalar MLQ with this final state is 850 GeV [55]. We recast the bound for the vector MLQs in Table X.  TABLE IX. MLQ models which decay into the final state (τ j)(τ j). We have not been able to find searches which can reliably and easily be recast to obtain mass limits for these models.  15. Bounds on scalar and vector MLQs which decay into (τ b)(τ b) final states. The observed 95% CL upper limit on the production cross section times branching fraction (black solid line) is taken from searches for LQs in this final state at CMS [55] that stopped at 1.5 TeV. The remainder of the analysis is analogous to Fig. 8. TABLE X. 95% CL lower limits on the mass of MLQs which decay into (τ b)(τ b) final states for the scalar leptoquarks (left) and the vector leptoquarks (right).

Obs. 95% CL upper limit
has the most complicated final states of the nine types of LQs in the LQ matrix because each of the four decay particles, τ + , τ − , t ort, can decay leptonically or hadronically. The multiplicity of possible final states, neglecting charge differences, is 10.
Further complications can arise from t l → τν τb decays where τ can decay either hadronically or leptonically.
Only one of the many possible final states has been covered. Ref. [56] searched for pp → φ τ tφτ t → (t h τ − h )(t l τ + l ) with t l decays to e or µ at Run 1 (8 TeV, 20 fb −1 ). The mass limit is m Sτt ≥ 685 GeV. Our recast bounds for other MLQs are shown in Table XI. Given that several other final states for φ τ t pair-production appear promising, we hope that searches in additional channels will also be performed.  16. Bounds on scalar and vector MLQs which decay into (τ t)(τ t) final states. The observed 95% CL upper limit on the production cross section times branching fraction (black solid line) is taken from LQ searches in this final state at CMS [56] which stopped at 800 GeV. The remainder of the analysis is analogous to Fig. 8.

IV. DISCUSSION AND SUMMARY
In this paper, we have advocated that LQ pair production searches be organized by the leptoquark matrix. This naturally leads to 9 distinct final states to be searched for. In order to make recasts to arbitrary leptoquark models possible it would be desirable if the experimental collaborations publish bounds on the cross section times branching fraction as a function of LQ mass for each of these final states. We have found strong bounds from existing searches for leptoquarks or equivalent searches for RPV squarks in 6 of the 9 cases. For ( b)( b) we have only been able to find a lower mass bound on a particular squark model which only allowed us to obtain recast estimates for other models in this channel. For ( t)( t) we were able to obtain bounds by recasting a general purpose cut-and-count SUSY search. These bounds could certainly be improved with a more dedicated search. Finally for (τ j)(τ j) were not able to find a search which allowed us to obtain reliable strong bounds.
We also note that even in the cases where cross section bounds as a function of mass have been published it would be useful if the search range could be extended as far as possible in LQ mass. This is important because the theoretical cross section times branching fraction for LQs to particular final states can vary by orders of magnitude. At a minimum, the mass range covered should include the range required to set bounds on the MLQ states which can have cross sections times branching fractions ranging from 0.25 × σ S to 2 × σ V .
We have focused on pair production of LQs with prompt decays. For leptoquarks with large couplings single (and off-shell) production can become important and potentially yield stronger lower bounds on LQ masses. This is especially true when the large coupling is to first generation quarks with their large parton distribution functions. We hope to extend the "Leptoquark Hunter's Guide" to single and off-shell production in a future publication [1]. The LQ matrix is also convenient for organizing the possible single LQ production final states.
Another generalization that we have not covered here is the possibility of displaced vertices from LQ decays with long life times or even LQs that are stable on detector time scales. In the former case exotic new searches using displaced vertices or kinks in tracks would be interesting, in the latter case the LQs would leave strongly ionizing tracks similar to R-hadrons in SUSY. For LQs with TeV scale masses, moderate boosts, and decay width Γ = mλ 2 /(16π) the vertex displacement is d ∼ 10 −15 /λ 2 cm. LQs with couplings larger than λ ∼ 10 −6.5 have prompt decays with vertex displacements smaller than typical LHC detector impact parameter resolution ∼ 100 µm. Couplings smaller than 10 −9 lead to "stable" LQs with decays outside the detector.
Is there a theoretically preferred range for the size of couplings? LQs are unlike any particles we have observed so far and therefore we can only make guesses. If they are vectors one should perhaps expect couplings similar to SM gauge couplings, 0.1 -1. If they are scalars one might expect couplings similar to SM Yukawa couplings, 10 −6 − 1. For LQs related to the recent hints for new physics from B meson decays the interesting range of couplings is 10 −2 ≤ λ ≤ 1. All these would imply prompt decays but smaller couplings are certainly possible. Before closing we would like to address a common misconception about leptoquark "generations". It is often said that stringent bounds on flavor violation require the couplings of leptoquarks to be limited to first, second, or third generation leptoquarks. This is statement is both incorrect and uses misleading language. What is correct is that large couplings to more than one generation of quarks lead to large quark flavor violation (e.g. K −K mixing), and couplings to more than one generation of leptons lead to lepton flavor violation (e.g. µ → eγ). However the generation number of the quark and the lepton do not have to be the same to avoid flavor violation. For example, a leptoquark coupling only to right-handed strange quarks and electrons (in the mass eigenstate basis for the fermions) exactly preserves all relevant U (1) fermion number symmetries (strangeness and electron-number). Thus flavor constraints do allow the more general form for leptoquark couplings and decays that fill out the full LQ matrix. We discourage the use of the misleading terms "first, second, third generation LQs" and instead suggest the use of more accurate language in which both the quark and lepton generation are specified.  We also list the leptoquark nomenclature for the MLQ models used in this paper (1 st column), their SM gauge quantum numbers (2 nd column), the names used in Ref. [62] (PDG, 3 rd column), and in Ref. [63] (BRW, 4 th column). Note that in our MLQ models two leptoquarks with the same gauge quantum numbers but different flavor quantum numbers are considered distinct and therefore have different names. In the PDG and BRW notation, LQs with the same gauge quantum numbers are given the same name. That is why the map between our models and PDG and BRW is not one-to-one.

Appendix B: Comparison between vector and scalar LQ searches
The kinematics of pair-produced vector LQs is very similar to pair-produced scalar LQs. We generated the processes pp → S µj S µj → (jµ + )(jµ − ) and pp → V µj V µj → (jµ + )(jµ − ) at leading order parton level using MadGraph5 aMC@NLO 2.5.4 which retains polarization effects in production and decays. We then passed the events through the selection cuts for ATLAS' [5] and CMS' [6] analyses. The ratios of the relative efficiencies for various leptoquark masses are shown in Fig. 2. One sees that the efficiencies for scalar leptoquarks S and vector leptoquarks V are very similar. To better understand the origin of this similarity, we show in Fig. 17 and Fig. 18 the scalar and vector distributions for different leptoquark masses, and in the following we detail the variables used: pTµ 1 , pTµ 2 transverse momenta of the two leading leptons, mµµ di-muon invariant mass, ST scalar sum of the pT of the two leading jets and two leading muons, m min LQ lower of the two balanced leptoquark-pair masses. The balanced leptoquark-pair is defined as the combination of leptons with quarks that yields a smaller mass differences between the two reconstructed leptoquarks, m max LQ higher of the two balanced leptoquark-pair masses.
Note that vector LQ production is slightly more forward than scalar LQ production. This results in softer p T and larger η for V V compared to SS. Consequently, the relative efficiency for V V decay products to pass through the p T and η cuts are lower than those of SS decay products. This can be seen from the first two panels of Fig. 2. The difference is more pronounced for light leptoquarks where also overall efficiencies are smaller, but for m φ ≥ 500 GeV the difference in efficiencies is always below 10%.

Appendix C: A vector leptoquark model
Vector leptoquarks may either arise as strongly coupled states within a spectrum of composite resonances from a strongly coupled theory or they may be fundamental, i.e., gauge bosons which obtain masses from spontaneous symmetry breaking. In fact, even composite spin-1 particles must have a spontaneously broken gauge invariance if the compositeness scale is parametrically larger than the mass of the spin-1 particles [64]. In this Appendix we present a renormalizable model in which the vector leptoquarks are massive gauge bosons of a spontaneously broken gauge theory.
Leptoquarks carry color and hypercharge, and therefore the gauge generators associated with the leptoquarks must not commute with color and hypercharge. The smallest gauge group which contains color and hypercharge generators and vector leptoquarks is SU (4) × U (1) X ⊃ SU (3) × U (1) Y where the U (1) X is necessary to obtain the correct hypercharges for the SM fermions. The spectrum of gauge bosons of this minimal model contains the SM gauge bosons in addition to a complex color triplet leptoquark V , and a massive singlet, a Z . This Z is a phenomenological problem. The problem is that the Z cannot be much heavier than the leptoquark because their masses have a common origin. The Z couples strongly to SM fermions and therefore has a large production cross section as an s-channel resonance. Resonance searches for the Z are much more sensitive probes of this model than leptoquark pair production. In other words, regions of parameter space with potentially discoverable leptoquarks are already ruled out by the non-observation of Z signals at the LHC. This problem with "light" Z resonances is generic to the most simple vector leptoquark models and makes viable models with TeV scale leptoquark masses non-trivial to construct. In the following we offer a generic structure for leptoquark models which solves this problem. The trick is familiar from "partial compositeness" models in which heavy gauge bosons arise as composite states from a "strong sector". They obtain small mixing with weakly coupled gauge bosons under which the SM fermions are charged. In this way the heavy "composite" gauge bosons couplings to SM fermions are suppressed by the small mixing angle and direct  [5] and CMS [6] searches for pair-produced Sµj.

Trigger
At least one muon with |η| < 2.5, pT > 26 GeV At least one muon with and isolated within ∆R = 10 GeV/pT pT > 45 GeV, |η| < 2.1 or pT > 50 GeV without isolation Muon pT, η pTµ 1,2 > 40 GeV, pTµ 1,2 > 50 GeV, |ηµ 1,2 | < 1.01 or 1.10 < |ηµ 1,2 | < 2.5 |ηµ 1,2 | < 2.1 Jet pT, η pTj 1,2 > 50 GeV, pTj 1,2 > 50 GeV, |ηj 1,2 | < 2. . The minimal choice is SU (4) but larger groups can also be used. For example, a particularly interesting choice is SU (5) in which case all the SM gauge bosons (including SU (2) weak ) can mix with the heavy gauge bosons. Such a non-minimal structure is also necessary to obtain vector leptoquarks which carry SU (2) weak quantum numbers. For definiteness we describe a particular SU (4) model in which the vector leptoquark V µ couples to the SM fermion bilinear Q †σ µ L. Here Q ≡ Q i and L ≡ L j can be chosen to be from any generation i, j = 1, 2, 3 to give 9 different possible leptoquark couplings. For ease of notation we suppress generation indices in the following. The fields and representations are shown in Table XV.  U (1) SU (4) The gauge group spontaneously breaks at the TeV scale when a vev for the bifundamental field Ω develops. It is useful to think of the SU (3) × U (1) gauge groups as a subgroup of a large global SU (4) symmetry. Then the reducible field Ω transforms as a 4 ⊗4 representation under this SU (4) times the gauged SU (4), and the Ω vacuum expectation value (vev) breaks the two SU (4) symmetries to the diagonal. This leaves an SU (3) × U (1)| diagonal = SU (3) color × U (1) Y gauge group unbroken. The vev can be written as a diagonal square matrix The Higgs mechanism generates masses for a full SU (4)'s worth of gauge bosons. The massive gauge bosons are a color octet of massive "gluons" g , a Z , and the complex color triplet leptoquark V . We are interested in the limit of large g 4 , there the vector masses become approximately m g = g 2 4 + g 2 3 f 3 g4 g3 g 4 f 3 , (C2) m Z = 1 2 g 2 4 + 2g 2 1 /3 f 2 3 + 3f 2 1 g4 g1 g 4 2 f 2 3 + 3f 2 1 , (C3) These formulas show that by varying f 3 and f 4 one can obtain arbitrary leptoquark masses with m g /m V ∈ [0, √ 2]. The gauge bosons which remain massless after this symmetry breaking are the gluons with gauge coupling g s = g 3 g 4 / g 2 4 + g 2 3 g 3 and the hypercharge boson with coupling g Y = g 1 g 4 / g 2 4 + 2g 2 1 /3 g 1 . For pair production of the vector leptoquark we need to know the couplings of the leptoquark to gluons. These are given in part by the covariant derivatives in the kinetic term of V . But there is another renormalizable coupling between gluons and the leptoquark that is allowed by gauge invariance where G a µν is the gluon field strength and T a is a Gell-Mann color matrix. The parameter κ s is not fixed by gauge invariance of the low energy theory, but it can be derived by expanding out the gauge kinetic terms of the unbroken SU (3) × SU (4) gauge theory in terms of the massive and massless vectors. One finds that κ s = 0. The vanishing of κ s can also be derived without looking at the full theory by considering unitarity of leptoquark pair production. Only for κ s = 0 does the pair production amplitude gg → VV have good UV behavior.
In addition, there is a contribution to qq → VV from a diagram with the heavy gluon g as an s-channel resonance. The masses of the vectors are such that this resonance is off-shell, but it is required to unitarize the qq → VV pair production amplitude. Therefore we need to determine the couplings of the heavy vectors to fermions. G and Z are the orthogonal linear combinations to the gluons and hypercharge. The G coupling to colored fermions is and the Z couplings are proportional to hypercharges times g 2 1 g 2 1 + 3g 2 4 /2 We see that the light fermions decouple from the heavy gauge bosons in the limit g 4 g 3 , g 1 , making the model safe from precision electroweak constraints and g or Z resonance searches at the LHC.
The s-channel diagram for pair production of leptoquarks with intermediate g coupling to leptoquarks is not suppressed in the large g 4 limit. This is because the coupling of the heavy gluon to the leptoquarks is proportional to g 4 . Then the s-channel is proportional to couplings g 2 s /g 4 × g 4 = g 2 s just like the QCD diagrams. We included this contribution in Fig. 4 and varied the mass of g within its allowed limits. In the end, this contribution turns out to not be very important because the large gg parton luminosity enhances the gluon initiated diagrams over the quark initiated ones.
Finally, we consider leptoquark decay through its coupling to light fermions. In the model described so described so far the coupling vanishes because leptoquark is an SU (4) gauge boson and the fermions are not charged under SU (4). There is also no mixing of the leptoquark with any of the light gauge bosons. However, it does couple to the heavy fermions Ψ. Thus we introduce mixing between Ψ and Q and L of the SM to generate a coupling of V to the light fermions. The relevant mass terms are L mix = λ Ω (Q, L)ΩΨ c + M ΨΨ c + h.c. (C8) where -for simplicity -we have imposed a global SU (4) symmetry on the couplings of Q and L. In the limit λ Ω Ω M the light fermions are still approximately Q and L but with a small sin(θ Ψ ) = λ Ω Ω / M 2 + (λ Ω Ω ) 2 λ Ω Ω /M admixture of Ψ. Therefore the effective coupling of the vector leptoquark to light fermions becomes λ = g 4 (sin θ Ψ ) 2 . (C9)