Estimated constraints on t-channel Leptoquark exchange from LHC contact interaction searches

The t-channel exchange of a first generation leptoquark could contribute to the cross-section for q q-bar to e+e-. The leptoquark is off-shell, so this process can be sensitive to leptoquarks beyond the mass reach of pair production searches at the LHC (currently m_{LQ}>830 GeV). We attempt to analytically translate ATLAS bounds on $ (\bar{q} \gamma ^\mu q) (\bar{e} \gamma _\mu e) $ contact interactions to the various scalar leptoquarks, and obtain a bound on their quark-lepton coupling of order $\lambda^2 \leq (m_{LQ}/2$ TeV)$^2$. The greatest difficulty in this translation is that the leptoquarks do not induce the contact interaction studied by ATLAS, so the interference with the Standard Model is different. If bounds were quoted on the functional dependance of the cross-section on s-hat, rather than on particular contact interaction models, this difficulty in applying experimental bounds to theoretical models could be circumvented.


Introduction
The LHC has sensitivity to new particles from beyond its kinematic reach, which could materialise as an excess or deficit of events at high energy. Such modifications of the high energy tail of distributions are commonly parametrised by four-fermion "contact interactions", with coefficient ± 4π Λ 2 . Experimental results are quoted as lower bounds on Λ, for a selection of contact interactions. The question that interests us, is whether such bounds provide useful constraints on the New Physics which could affect the tails of distributions.
Concretely, we will consider the partonic process qq → e + e − . A first generation leptoquark (for reviews, see [1]) exchanged in the t-channel (see figure 1) could mediate this process. This process could be sensitive to heavier leptoquarks than could be pair-produced via strong interactions at the LHC (the current bound on pair-produced first generation leptoquarks is m LQ > ∼ 830 GeV [2]). However, this process occurs via the leptoquark-quark-lepton coupling, so will only be observable for O(1) couplings. From the ATLAS bound [3] on (qγ µ P L q)(eγ µ P L e) contact interactions, we attempt analytically to estimate bounds on the mass and quark-lepton coupling of the leptoquark. Two issues will arise. First, in section 3, we treat the leptoquark exchange as a contact interaction. However, none of the seven possible leptoquarks interfere with the SM in the same way as the ATLAS operator. We attempt to circumvent this problem by assuming the bound comes from the interference term, and making simple approximations to the Parton Distribution Functions (pdfs). The second hurdle is the leptoquark propagator ∼ 1/(m 2 LQ +t), which is taken into account in section 4. As expected, for m LQ < ∼ 2 √ŝ , the propagator reduces the cross-section, and therefore weakens the bounds. This effect seems less significant and easier to estimate that the consequences of interference. Section 5 concludes with a summary of the bounds we obtain on first generation leptoquarks, and a discussion of the difficulties of translating contact interaction bounds to any realistic model without doing a full analysis. Were it possible for experimentalists to set bounds on the functional form of the cross-section, it could be easier for theorists to translate such limits to their favourite models.
Combined constraints on leptoquarks, from pair production and single leptoquark exchange in s and t channels, has been studied at HERA (see e.g. [4]). Constraints from LHC contact interaction searches on hypothetical new particles exchanged in the t-channel have previously been calculated for Z ′ s [5]. Single leptoquark production via t-channel diagrams has also recently been studied in [6].

The ATLAS analysis, Leptoquarks and Kinematics
This section provides a brief overview of the experimental analysis we use [3], leptoquarks, and our notation. ATLAS searched [3] for contact interactions of the form where η = +/ − 1 corresponds to destructive/constructive interference with Z/γ exchange. The 95 % confidance level (CL) bounds obtained with 5 fb −1 of data at √ s = 7 TeV, are The analysis presents the number of events expected and observed as a function of the invariant mass-squared of the e + e − pair in bins of width given in the left colomn of .085 + .035±.04 0.87 ±.14 0 e + e − = (p e + p e ) 2 ). The following colomns are from the ATLAS paper [3]: the expected number of events due to the SM, due to the SM plus a contact interaction with Λ = 12 TeV and constructive interference with the SM, and finally the data.
We consider scalar leptoquarks, with renormalisable B and L conserving interactions, which generate an interaction between first generation quarks and electrons. In the notation of Buchmuller,Ruckl and Wyler [7], the leptoquark interactions with quarks and leptons can be added to the SM Lagrangian as: where τ 2 is a Pauli matrix, so iτ 2 provides the antisymmetric SU(2) contraction. These leptoquarks can contribute to qq → e + e − (where q ∈ {u, d}) via t-channel exchange. The diagrams, and interfering SM processes are given in figure 1. It is convenient to write the matrix element M(qq X → e + e − Y ) as a spinor contraction T XY multiplied by a propagator P qX eY . For instance, for S o with coupling λ R , the spinor contraction after a Fiertz transformation can be written T RR = (uγ µ P R u)(eγ µ P R e). For XX ∈ {LL, RR}, or XY ∈ {LR, RL} where the bar indicates an average over incident colour and spin, the momenta are as in figure 1, and The contact interaction analysis of ATLAS is atŝ > (400GeV) 2 , so we neglect m Z , and propagate the massless B and W 0 . Then the propagators of figure 1 give where F is the leptoquark fermion number which is 2 for doublet leptoquarks and zero otherwise,τ =t for F = 0 leptoquarks andû for F = 2, and the (×2) applies only in the case of triplet leptoquark exchange coupled to d quarks.
Recall that the hypercharge and SU(2) quantum numbers of SM fermions are For each leptoquark, the Fierz-rearrangedqqe + e − vertices, with the appropriate propagators representing Z/γ and leptoquark exchange, are given in colomn two of table 2. To obtain the contact interaction mediated by a leptoquark, τ → 0 in this table and the SM part of the propagator should be dropped. In most cases, the coefficient of the contact interaction is λ 2 /(2m 2 LQ ). ATLAS bins its data inŝ = M 2 e + e − (see eqn 3), so it is convenient to express the total cross-section for pp → e + e − as where f q is the parton distribution function (pdf) for the quark q in the proton, √ s e −η+ are the fraction of the proton's momentum carried by the parton, there is a 2 because the valence quark could be in either incident proton, and the cross-section is separated into the gauge boson mediated part plus a New Physics part. The integration limits on η + = (η e + ηē)/2 andt should be determined from the experimental cuts on the rapidities η e + , η e − of the e + e − , however, for simplicity, we integrate over the whole phase space.

Leptoquark exchange as a contact interaction
In this section, we approximate leptoquark exchange as a contact interaction, and try to constrain the leptoquarkmediated contact interactions from the ATLAS analysis. The challenge will be to deal with the different flavours and chiralities of the leptoquark-induced operators, which will affect the number of New Physics events, and the distribution in M 2 e + e − .To see this, the total cross-section of eqn (8) can be written where the NP part is divided into interference-with-the-SM, and with itself. C(ŝ) is dimensionless, fixed by the electroweak interactions of the quarks and electrons, and includes an integral over the pdfs. The ǫs are also dimensionless, and satisfy − 3/8 ≤ ǫ int / √ ǫ N P ≤ 3/8. The magnitude of ǫ N P will depend on which flavours of quark couple to a given leptoquark, and the magnitude and sign of ǫ int will depend on the flavour and chirality of the participating fermions, because these control the interference with the γ/Z.
To analytically compare the ǫs induced by leptoquarks, to those arising from the ATLAS contact interaction, we suppose a simplistic weighting of parton distribution functions (pdfs) in the proton, such that This approximation will allow to estimate ǫ int and ǫ N P from the partonic matrix elements (given in the second colomn of table 2).
To make such estimates, we first schematically write the partonic Z/γ-exchange cross section, multiplied by pdfs, as: where inside the square brackets is the pdf-weighted "propagator" |P| 2 of eqn (7), mutiplied byŝ 2 . With the approximations s 2 W = 1/4, g ′ = g/2, and eqn (10): dη dη where the pdf integral over η + is a constant that we do not need to know. (We also neglected experimental cuts in integrating overt, which is hopefully an acceptable approximation because thet dependence of |T | 2 is common to the SM and NP, and there is not dependance of the propagators in the contact approximation.) The ATLAS analysis [3] follows pythia [8] in summing over u and d flavours, and restricts to doublet ("left-handed") quarks. Using again the approximation (10), and identifying 4π/Λ 2 = λ 2 /2m 2 , the cross-section in the presence of the ATLAS contact interaction, with constructive interference with the SM, can be approximated as so ǫ int = 1/6 and ǫ N P = 1 for the ATLAS contact interaction.
interaction Table 2: Fierz-transformed two-electron-two quark matrix elements induced by the leptoquark, γ and Z exchange diagrams of figure 1, in the limit m Z → 0. (possible quark-neutrino interactions are not included).τ can bet orû. The third and fourth colomns estimate the coefficients in eqn (9), using the approximation of eqn (10). The second last colomn is the bound on λ 2 , for m LQ = 2 TeV, assuming the ATLAS limits on contact interactions can be translated to leptoquarks using eqn (13). The last colomn is an estimate of the confidence level (see eq. 16) of that bound, obtained with the cross-section of eqn (9).
The contact interactions induced by the various leptoquarks differ from the one studied by ATLAS, as can be seen from the second colomn of table 2. The values of ǫ int and ǫ N P can be estimated, as above, and are given in table 2. One sees that the leptoquarks with constructive interference (positive ǫ int ) have a smaller ǫ int / √ ǫ N P ratio than the ATLAS operator. So one could hope to constrain these leptoquarks by simply rescaling the ATLAS bound. We conservatively rescale the bound as where Λ cons = 11.7 TeV and ǫ int = 1/6 on the left side. For the leptoquark S 2 with coupling λ L , this excludes above the red diagonal line of the left figure 2.
Analytic estimates suggest that eqn (13) is conservative: if the bound arises from the interference term, then one expects ǫ AT LAS int 4π . However, if the bound came from the |N P | 2 term, then one might expect , which would give a stronger bound on the leptoquark couplings. For the ATLAS contact interaction, and leptoquark-induced contact interactions with ǫ int / √ ǫ N P ≃ 1/6, the N P 2 term becomes larger than the interferences at √ŝ > ∼ 900 GeV, so dominates the last bin with data (= 1200 TeV < √ŝ < 1800 TeV). For leptoquarks that have destructive interference with the SM, the ratio ǫ int / √ ǫ N P departs significantly from the ratio of the ATLAS operator. Nonetheless we again take the translation rule of eqn (13) with Λ des ≥ 9.3 TeV. On the right in figure 2 is plotted the exclusion for S o with coupling λ R . In the second last colomn of figure 2, eq. (13) is used to translate the ATLAS bound to all the singlet and doublet leptoquarks. The next subsection contains some simple statistics to support eqn (13).
A bound was estimated for the triplet leptoquark S 1 , for which the interference almost vanishes, as where 1/Λ 2 on the left side is the average of the ATLAS constructive and destructive bounds 1 2 ( 1 11.7 2 + 1 9.3 2 ), and ǫ N P = 4/3 on the right side. For the doublet S 2 , whose interference with the SM is very suppressed, the estimated bound from eqn (13) is to weak to be interesting. If instead, the interference term is neglected, a bound of λ 2 ≥ 1.64 for m LQ = 2 TeV can be estimated from eqn (14), (this is given in parentheses in the table).  (see table 2). Left plot for S 2 with coupling λ L , right plot for S o with coupling λ R . The blue region to the left with horizontal hashes is excluded by CMS searches [2] for pairs of first generation leptoquarks. The contact interaction approximation should apply to the right of the dashed line; if the leptoquark propagator is taken into account, only the region above the stars is excluded (see section 4).

Comparing partonic cross-sections to ATLAS data
None of the leptoquarks induce the contact interaction constrained by ATLAS, so it is not clear how to translate the ATLAS bound to leptoquarks. In particular, the shape inŝ of the differential cross-section, eqn (9), will depend on the different values of the ǫs. This will change the number of New Physics events in the ATLAS bins, and affect the overall deviation from the SM. The aim of this subsection is to confirm that eqn (13) is conservative, using simple statistics and partonic cross-sections.
We focus on the last three ATLAS bins inŝ (see table 1), and suppose that the pdfs are decreasing fast enough with increasingŝ, that in each bin b, the number of signal NP events plus background SM events can be estimated as is the ratio of the Z/γ + N P cross-section of eqn (9) to the Z/γ cross-section (this neglects experimental acceptances), taken at the left side of eachŝ bin, calculated for m LQ = 2 TeV with the value of λ 2 give in the second last colomn of table 2, and using the correct ǫ int and ǫ N P for each leptoquark. The n(Z/γ) and n(SM ) are respectively the number of Z/γ events and total number of SM events, expected by ATLAS (see table 1).
Then we estimate a "confidance level" CL for the exclusion as follows. Consider first the case of constructive interference, where the SM+NP cross-section is larger than the SM cross-section. Then assuming Poisson statistics for the last three bins, the probability of counting less than or equal the observed number N events, when expecting ν, is The "confidence level" then is estimated as In the case of a leptoquark-mediated contact interaction with destructive interference, the expected number of SM+NP events in some bins is less than the expectation for the SM alone. For those bins, P b is taken as the probability of observing more than or equal to the observed number N . These "confidence levels" are listed in the last colomn of table 2, for the bounds quoted in the second last colomn (obtained from eqn (13) and (14)). Our CL estimates are higher for the leptoquark limits than for the ATLAS contact interaction, which reassures us that our bounds are conservative. However, the variation in the CLs indicates that eqn (13) is not a reliable approximation. To obtain a consistent confidance level for various values of ǫ int / √ ǫ N P would require a more sophisticated study.

Including the leptoquark propagator
The aim of this section is to estimate the consequences of including the leptoquark propagator 1/(m 2 LQ −τ ) (wherê τ ∈ {t,û}, and recallt,û < 0), which only reduces to a contact interaction in the |τ | < ∼ŝ ≪ m 2 LQ limit. It is interesting to explore the m 2 LQ < ∼ŝ range, because the lower bound on the mass of pair-produced first generation leptoquarks is 830 GeV [2], whereas the highest bin in the ATLAS analysis is √ŝ > 1800 GeV. The effect of the massive LQ propagator can be seen by comparing the partonic cross-sections for LQ exchange versus a contact interaction. With the approximation of eqn(10), dσ/dŝ for the ATLAS contact interaction is given in eqn (12). In the same approximation, with the same values of ǫ int = 1/6 and ǫ N P = 1, dσ/dŝ for leptoquark exchange is In figure 3 are plotted the differential cross-sections for leptoquarks of masses 1, 2 and 3 TeV, and the contact interaction they induce (the leptoquark couplings are adjusted such that all three give the same contact interaction). It is clear that for m 2 LQ > 4ŝ, the contact interaction approximation reproduces leptoquark exchange to within 20%. This is represented in figure 2 as the dotted vertical lines, to the right of which the contact interaction approximation is justified.
To constrain a leptoquark of m LQ < ∼ 3.6 TeV using the ATLAS contact interaction analysis, we should account for the differences in cross-section shape, as a function ofŝ. For the leptoquarks represented in figure 2, we estimate the value of λ which can be excluded for masses of 1,2 and 3 TeV, by requiring that the estimated confidence level of the leptoquark exclusion, exceed the CL for the ATLAS contact interaction. For instance, for S 2 with coupling λ L , the bounds in the contact interaction (CI) approximation, and including the propagator are The bounds obtained with the propagator are plotted as stars in figure 2. As expected, the effect of the leptoquark mass, for m 2 LQ < ∼ŝ is to weaken the bound.  12) and (17). From the lowest curve upwards, the New Physics is Z/γ exchange plus a t-channel leptoquark with m LQ = 1 (solid),2 (dotted) and 3 (dash-dotted) TeV, and finally Z/γ exchange plus a contact interaction with coefficient λ 2 /(2m 2 ) = 1/8 TeV −2 (blue). The leptoquark couplings are chosen to reproduce the contact interaction in theŝ → 0 limit. On the left, destructive interference, constructive on the right.

Discussion
A signal for a contact interaction of coefficient λ 2 /m 2 is a plateau at the high energy end of a decreasing distribution, possibly preceded by a valley in the the case of destructive interference with the SM. For qq → e + e − , the exchange of Z/γ is the principle SM contribution, responsable for the cross-section decreasing as 1/ŝ. So one expects sensitivity to where M 2 e + e − ,max is the e + e − invariant mass-squared of the highest bin. The contact interaction approximation is expected to be valid for where the strong-coupling upper limit on λ is approximately a unitarity bound, and where we impose theŝ ≪ m 2 LQ condition asŝ ≤ m 2 LQ /4 (from fig 3). So contact interaction searches at colliders are sensitive to a triangular area in λ, m LQ parameter space (above the diagonal line of figure 2). This illustrates that the "contact interaction approximation", where the two parameters {λ, m LQ }, are replaced by a single parameter λ 2 /m 2 LQ , is not well-satisfied at colliders.
In addition, there is another effect to parametrise. Many of the channels in which contact interactions are searched for (for instance qq → qq, or qq → e + e − as studied here) can be mediated by the SM. In the present case, both the SM and leptoquark cross-sections have the same angular dependence (because the four-fermion interactions are of the form (qγ µ P X q)(eγ µ P Y e)) so the cross-section for leptoquark plus Z/γ exchange is of the form given in eqn (9): where the sign and magnitude of the interference term (encoded in ǫ int ) depend on the flavours and chiralities of the contact interaction. If the interference term is large and negative, there will be a valley before the plateau, if it is positive, there will be excess events before reaching the plateau... and if it is negative and small, the plateau can merely be delayed. In general, experimental bounds are quoted on a few discrete choices of the ratio ǫ int /ǫ N P . The aim of this paper was twofold. First, to extract bounds on leptoquarks from the LHC searches for contact interactions in the process pp → e + e − + X. Leptoquarks interacting with electrons and first generation quarks could contribute to qq → e + e − via t-channel exchange. In table 2 are quoted the resulting limits, obtained via some hopefully conservative analytic arguments. The limits have varying confidance levels, indicating the difficulty of translating current bounds to leptoquarks. The limits in table 2 neglect the first effect discussed above, of the the leptoquark propagator. For small leptoquark masses (∼ TeV), the effect of the propagator is to weaken the bound on λ 2 by less than a factor 2 (see eqn (18), and figure 3).
The second aim of this paper was to explore whether experimental bounds on a selection of contact interactions can be readily translated to New Physics scenarios. The answer is no. There are two issues which arise: 1. the flavour and chirality of the experimentally bounded interactions is unlikely to correspond to most models (as in the case here, none of the leptoquarks induce the contact interaction studied by ATLAS). A conservative and fastidious solution would be for the experimental collaborations to set bounds on a complete set of contact interactions, which do not interfere among themselves, but do induce every possible helicity amplitude. An alternative (which we will explore in a subsequent publication), would be for the experimental collaborations to set bounds on ǫ N P and ǫ int , that is, perform a two parameter fit to the coefficients of the interference and |N P | 2 terms in the (partonic) cross-section. Then for a given model, one merely needs to estimate these parameters, which can be done from partonic cross-sections with naive approximations to the pdfs.
Notice that table 2 and eqn (18) suggest that this effect is more important than the following one: bounds vary more with changes in the coefficient of the interference term, than when the propagator in taken into account.
2. The four-momentum of the new particle mediating the "contact interaction" can only be neglected if the new particle is very heavy and strongly coupled. If the propagator ∼ 1/(t − m 2 ) of a new particle exchanged in the t-channel is retained, the contact interaction is suppressed, becauset ∼ −ŝ. In the case of leptoquarks, we estimated the this weakens the bounds on λ 2 by at most 50% (see eqn (18)).
In summary, we estimated that contact interaction searches in pp → e + e − at the LHC can exclude first generation leptoquarks with couplings λ 2 > ∼ m 2 LQ /(2 TeV) 2 (see table 2 for specific bounds). Two difficulties arose in translating the experimental bounds to all the possible leptoquarks. The most significant problem is that the sign and size of the interference with the SM varies from one leptoquark to another, and significantly affects the shape of cross-section and therefore the bounds. Secondly, the leptoquarks are rarely heavy enough to justify neglecting their four-momentum in the propagator, which, when included, can weaken the bound on λ 2 by < 50%. To address the first problem, it could be interesting if the experimental collaborations set simultaneous bounds on the contact interaction coefficient 4π/Λ 2 , and on the size and magnitude of the interference with the SM.