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PHYSICAL REVIEW D 101, 115017 (2020)

Scalar leptoquark pair production at

† ‡ Christoph Borschensky ,1,* Benjamin Fuks ,2,3, Anna Kulesza,4, and Daniel Schwartländer4,§ 1Institute for Theoretical Physics, University of Tübingen, Auf der Morgenstelle 14, 72076 Tübingen, Germany 2Sorbonne Universit´e, CNRS, Laboratoire de Physique Th´eorique et Hautes Énergies, LPTHE, F-75005 Paris, France 3Institut Universitaire de France, 103 boulevard Saint-Michel, 75005 Paris, France 4Institute for Theoretical Physics, WWU Münster, D-48149 Münster, Germany

(Received 4 March 2020; accepted 2 June 2020; published 16 June 2020)

We revisit scalar leptoquark pair production at hadron colliders. Apart from QCD contributions, we include the t-channel exchange diagrams relevant in the light of the recent B-flavor anomalies. We evaluate all contributions at next-to-leading order in QCD and resum, in the threshold regime, soft- radiation at next-to-next-to-leading-logarithmic accuracy. All corrections are found equally relevant. Our predictions consist of the most precise leptoquark cross section calculations available to date and are necessary for the best exploitation of leptoquark LHC searches.

DOI: 10.1103/PhysRevD.101.115017

I. INTRODUCTION accuracy in the strong coupling αs [11–15], sometimes also supplemented by logarithmic threshold corrections [9,10], Many extensions of the (SM) predict the are used. Thus the predictions include contributions at existence of scalar leptoquarks [1–8], i.e., scalar Oðα2Þ and Oðα3Þ, or possibly of higher order in α , but are coupling to a and a lepton simultaneously. Evidence s s s independent of y [34,35]. Bearing in mind the B-anomalies for their existence is consequently vastly searched for at the and ðg − 2Þ motivation, the limits may thus be incorrectly LHC. However, none of the recent ATLAS [9,10] and CMS μ estimated. [11–15] analyses find any hint for these leptoquarks, so that In this paper, we perform for the first time a full NLO-QCD their mass is now constrained to be larger than 1–1.5 TeV. cross section calculation for scalar leptoquark pair produc- Recently, scalar leptoquarks have gained significant interest tion at hadron colliders, in which we include both the QCD as they may provide an explanation [16–22] for the B- – and t-channel contributions. Hadronic production of heavy anomalies [23 30] and address [31] the discrepancy between systems, which is the case considered here, inevitably probes theoretical predictions [32] and experimental measurements partonic center-of-mass energies close to the production [33] of the anomalous magnetic moment of the threshold given by twice the leptoquark mass m . In this ðg − 2Þ LQ μ. In this context, favored scenarios generally feature limit, radiative corrections are dominated by soft-gluon large lepton-quark-leptoquark Yukawa couplings y. emissions, manifesting themselves as large logarithmic terms The most stringent bounds originating from LHC direct that must be consistently resummed to all orders [36–39].We searches for leptoquark pair production and decay are report here threshold-resummed results at next-to-next-to- extracted by assuming that leptoquarks are solely produced leading-logarithmic (NNLL) accuracy and showcase pre- via strong interactions. In other words, non-QCD diagrams 2 dictions obtained by matching them to our new NLO results. involving lepton t-channel exchanges of Oðy Þ are In the following, we first present the considered theoretical neglected. In the associated limit setting procedure, signal framework and provide brief technical computational details. cross sections evaluated at next-to-leading-order (NLO) We then show an illustrative selection of results that under- lines how all considered corrections affect the results in *[email protected] comparable and significant ways. Our predictions, which are † [email protected] the most precise to date, are hence required to derive limits ‡ [email protected] consistently, in particular when assessing the influence of the § [email protected] leptoquark Yukawa couplings. Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. II. THEORETICAL FRAMEWORK Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, We focus on a simplified model in which the SM is and DOI. Funded by SCOAP3. supplemented by several species of scalar leptoquarks

2470-0010=2020=101(11)=115017(7) 115017-1 Published by the American Physical Society CHRISTOPH BORSCHENSKY et al. PHYS. REV. D 101, 115017 (2020)

˜ ˜ S1, S1, R2, R2 and S3. Inspired by standard naming III. SCALAR LEPTOQUARK PAIR ð3 1Þ PRODUCTION AT THE LHC conventions [40,41], these leptoquarks lie in the ; −1=3, ð3 1Þ ð3 2Þ ð3 2Þ ð3 3Þ ; −4=3, ; 7=6, ; 1=6 and ; −1=3 representations We present selected predictions for scalar leptoquark pair of the SM gauge group respectively, and we target their production at the 13 TeV LHC for the three most commonly Yukawa interactions involving exactly one lepton and discussed types of scalar leptoquarks in the context of the ð2Þ quark. The latter are collected in the Lagrangian: flavor anomalies: the SU L singlet state S1 (denoted by ð−1=3Þ S1 due to its of −1=3),doublet state R2 and L ¼ yRR ¯ c l † þ yLLð ¯ c Þ † þ y˜RR ¯ c l ˜ † the triplet state S3. More specifically, in the last two cases, we int 1 uR RS1 1 QL ·LL S1 1 dR RS1 consider the pair production of the R2 mass eigenstate of LR † RL RL ¯ ˜ ð5 3Þ þ y e¯ Q R þ y u¯ ðL · R2Þþy˜ d ðL ·R2Þ = 2 R L 2 2 R L 2 R L electric charge of 5=3 (denoted by R2 ) andthe one of the S3 þ yLLð ¯ c σ Þð kÞ† þ ð Þ mass eigenstate of electric charge of −4=3 (denoted by 3 QL · kLL S3 H:c: 1 ð−4=3Þ S3 ). In all our calculations, we treat the leptoquark mass m as a free parameter and assume the Cabibbo-Kobayashi- In this expression, all flavor indices are suppressed for LQ σ Maskawa matrix (CKM) matrix to be diagonal. While the clarity, k stands for the Pauli matrices and the dot for the determination of a scenario compatiblewith flavor constraints invariant product of two fields lying in the (anti)funda- and Z-pole observables is desirable [56],thisgoesbeyondthe ð2Þ mental representation of SU . The QL and LL spinors scope of this study. We consider instead benchmarks moti- denote the SM weak doublets of left-handed and vated by Ref. [21]. The values of the Yukawa couplings found l , and uR, dR and R are the corresponding weak in this study were obtained in a fit to low-energy observables singlets. Moreover, the y=y˜ couplings are 3 × 3 matrices in and did not involve constraints from direct searches for the flavor space, the first index of any element yij=y˜ij leptoquarks at the LHC. Given that the description of lepton referring to the quark generation and the second one to the flavor university-violating observables involves both lepto- lepton generation in the gauge basis. quark couplings and masses, optimally one should aim at a The calculations reported in this work concern scalar global fit based on direct and indirect constraints, in which case the calculations presented in this work will play a crucial leptoquark pair production and include fixed order con- tributions at leading order (LO) and NLO in QCD. In role. For S1S1 production, we adopt a minimal flavor ansatz ðyLLÞ ¼ −0 15 contrast with previous work [34,35,42,43], we not only for the leptoquark Yukawa couplings, 1 22 . and LL LL 2 3 ðy Þ ¼ 3 with all other y elements set to 0. For R2R consider the QCD components at OðαsÞ and Oðαs Þ,but 1 32 1 2 also include the t-channel lepton exchange contributions at production, we similarly consider as the only nonvanishing RL Oð 4Þ Oð 4α Þ Oð 2α Þ Oð 2α2Þ coupling ðy2 Þ22 ¼ 1.5, a value still allowed by direct y and y s as well as the y s and y s interference of the t-channel diagrams with the QCD ones. exclusion bounds [21], while for S3S3 production, we adopt ðyLLÞ ¼ −ðyLLÞ The full NLO-accurate predictions are collectively coined 3 22 3 32, keeping the actual coupling value free “NLO w=t-channel” in the following, in contrast to the pure and setting all other couplings to 0. QCD ones that we refer to as the “NLO-QCD” predictions. Our results are obtained by convoluting the partonic The NLO w=t-channel cross sections are then additively results with two different sets of parton distribution functions (PDFs), NNPDF3.1 [57] and CT18 [58]. Unless stated matched with the resummed NNLL soft-gluon contribu- otherwise, NLO sets are employed for NLO-QCD and tions, resulting in cross section predictions at NLO þ NLO w=t-channel predictions, while NNLO sets are used w=t-channel NNLL accuracy. Threshold resummation is for NLO þ NNLL calculations. We set the renormalization performed in Mellin space (see e.g., [39]) and involves (μR) and factorization (μF) scales equal to a common value one-loop matching coefficients [44]. μ ¼ μR ¼ μF. The central scale choice μ ¼ μ0 is fixed to To ensure the correctness of the results, we perform the μ ¼ μ 0 mLQ, and scale uncertainties are estimated by varying calculations in two independent ways: We first implement by a factor of 2 up and down. the above model into FeynRules [45], which we jointly use In Fig. 1, we present cross section predictions for ð−1 3Þ ð1 3Þ ð5 3Þ ð−5 3Þ with NLOCT [46] and FeynArts [47] to renormalize the S = S = (left column) and R = R = (right column) Oðα Þ 1 1 2 2 bare Lagrangian of Eq. (1) at s . We then generate a production, both for the NNPDF3.1 (upper row) and CT18 UFO model file [48] that we use to evaluate fixed-order (lower row) parton densities. We estimate the relative LO and NLO predictions within the MG5_aMC framework importance of the various corrections studied in this work [49]. The latter are cross-checked with results obtained with respect to NLO-QCD predictions, and assess the size within the POWHEG-BOX framework [50],inwhichwe of the scale and PDF uncertainties. Comparing the four input virtual corrections calculated with the FeynArts, subfigures, we observe that depending on the process, the – FormCalc [51] and COLLIER [52 55] packages. The NNLL PDFs, the magnitude of the Yukawa couplings, and mLQ, corrections are evaluated with two independent in-house the considered corrections can influence the predictions in Monte Carlo codes. different, often contrasting, ways.

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ð−1=3Þ ð1=3Þ ð5=3Þ ð−5=3Þ FIG. 1. S1 S1 (left) and R2 R2 (right) production at the 13 TeV LHC, using the NNPDF3.1 (upper row) and CT18 (lower row) PDF sets. In the top panels of the subfigures, we present cross section predictions at the NLO-QCD (magenta dotted), NLO w=t-channel (blue dashed) and NLO w=t-channelþNNLL (red solid) accuracy. The associated scale and PDF uncertainties are also displayed (middle panels). In the lower panels, we show ratios of the NLO w=t-channel, NLO w=t-channelþNNLL, NLO w=t-channel calculated using NNLO PDFs (olive dash-dotted) and NLO-QCD þ NNLL (turquoise dotted) results to the NLO-QCD cross section.

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ð−4=3Þ ð4=3Þ FIG. 2. NLO w=t-channelþNNLL total cross section for S3 S3 production at the 13 TeV LHC as a function of the S3 mass ¼ ðyLLÞ ¼ −ðyLLÞ mLQ mS3 and the Yukawa couplings 3 22 3 32 (all other Yukawa couplings being set to 0). We present predictions obtained with the NNPDF3.1 (left) and CT18 (right) PDF set.

Although providing a positive correction, the size of the 40% of the NLO-QCD result for the pair production of t-channel contributions depends very differently on mLQ 2TeVR2 leptoquarks, whereas the complete NLO for the two processes (blue dashed curves). On the contrary, w=t-channelþNNLL one is only of about 20%. In contrast, NNLL effects, which we estimate through the ratio of the results obtained with CT18 densities exhibit the opposite NLO-QCD þ NNLL to the NLO-QCD cross sections both behavior, the cross sections being typically enhanced calculated with the same NLO PDF set (turquoise dotted when switching from NLO to NNLO PDFs. The t-channel curves), are independent of the process and PDF choice. and soft-gluon resummation pieces are of comparable size and thus equally contribute to the combined correction. As expected, this ratio is bigger than 1 for all mLQ Therefore a precise knowledge of the cross section requires considered, and grows with increasing mLQ, i.e., approach- ing the production threshold. calculating all classes of corrections. This behavior is vastly modified by the interplay of In the middle panels of the four subfigures of Fig. 1,we t-channel contributions, PDF effects and soft-gluon cor- focus on scale and PDF uncertainties. We distinguish the rections, all entering the NNLL results matched with NLO impact of scale variations (second panel) from the one w=t-channel (red solid curves). The effect of evaluating the originating from the PDF determination (third panel). NLO w=t-channel cross sections with NNLO PDF sets Our results show that soft-gluon resummation leads to a instead of NLO sets is illustrated by the difference between significant reduction of the scale uncertainties from around – the corresponding ratios to NLO QCD predictions (blue 10% (for the NLO predictions) to about 1% 2% for mLQ dashed vs olive dash-dotted curves). For the NNPDF3.1 values ranging up to slightly above the current exclusion PDF set (upper row), this effect diminishes the cross limits. The reduction might however be underestimated due sections, offsetting the increase stemming from the to the chosen method for scale uncertainty evaluation. This NNLL contributions. As a consequence, the NLO w=t- calls for a more comprehensive study. Correspondingly, the þ channelþNNLL results deliver a positive correction of total theoretical error for our final NLO w=t-channel NNLL about 10%–20% with respect to the NLO-QCD predictions predictions is dominated by its PDF component. The size of ∈ ½1 2 the PDF error is however strongly dependent on the PDF for mLQ ; TeV. Similarly for the NLO w=t-channel result, the full NLO w=t-channelþNNLL correction exhib- choice. For instance, results derived with NNPDF3.1 exhibit, ∼ 1 for mLQ TeV, PDF errors smaller or comparable in its the opposite behavior with increasing mLQ in the S1 magnitude to the size of the perturbative corrections, whilst (upper left) and R2 (upper right) cases. When CT18 PDFs are used instead (lower row), the at higher mLQ values, the PDF error becomes significantly corrections are larger and reach a magnitude of about bigger. In comparison, PDF errors obtained with the CT18 – set are larger for small m values, but do not grow as quickly 20% 50%, the impact this time increasing with mLQ for LQ both processes. Furthermore, in the NNPDF3.1 case, for higher masses. Still, the PDF errors turn out to be of the various contributions to the total correction are often much same order as the full perturbative corrections for large mLQ bigger than the correction itself. For example, the correc- values. Those large PDF errors at high masses hence obscure tion due to including t-channel diagrams reaches up to the accuracy of the predictions. However, as more LHC data

115017-4 SCALAR LEPTOQUARK PAIR PRODUCTION AT HADRON … PHYS. REV. D 101, 115017 (2020) will be analyzed, one can expect a substantial improvement as large as suggested by the recent B-anomalies) impor- of the PDF knowledge, in particular in the large Bjorken-x tantly affect the total rates, in potentially contrasting and regime, so that the PDF errors associated with predictions sizable ways. This emphasizes the necessity of including all relevant for high-mass system production will be signifi- contributions whose calculation has been pioneered in this cantly reduced. work. While the perturbative series exhibits smaller scale In Fig. 2, we calculate the NLO w=t-channelþNNLL uncertainties, the precision of the predictions is limited by ð−4=3Þ ð4=3Þ the poor PDF knowledge in the large Bjorken-x regime total cross section for S3 S3 production, and study its dependence on the leptoquark mass and Yukawa coupling relevant for the production of high-mass systems. In light strength. We consider both the NNPDF3.1 (left) and CT18 of our findings, we recommend the usage of NLO w=t- (right) PDF sets. At small values of the Yukawa coupling, channel+NNLL cross sections, to be taken together with the dominant production mechanism is QCD driven so that the correspondingly reduced scale uncertainties and PDF errors extracted from the envelope spanned by computa- the cross section solely depends on mLQ. On the contrary, as the coupling approaches 1, the t-channel contributions tions, left for future work, performed with different PDF sets. This follows the strategy outlined in various recom- become more relevant and the total rate significantly – increases. This behavior is mostly independent of the mendations for LHC cross section calculations [59 63]. chosen PDF set, CT18 predictions being slightly less The computer codes used in this work are available upon sensitive to the t-channel diagrams. request.

IV. SUMMARY ACKNOWLEDGMENTS We have significantly advanced the precision of scalar We are grateful to V. Hirschi, O. Mattelear and H. S. leptoquark pair production cross section computations. Shao for their help with MG5_aMC technical issues related First, we have included all contributions to the process, to mixed-order computations, as well as to M. Krämer for both the QCD ones and those involving the t-channel insightful discussions, and D. Bečirević, D. Guadagnoli exchange of a lepton, at NLO QCD. Second, we have and R. Ruiz for useful comments on the manuscript. resummed soft-gluon radiation in the threshold regime to This work has been supported in part by the DFG NNLL accuracy. Grant No. KU3103/2. We furthermore acknowledge The t-channel contributions, threshold resummation, the support by the state of Baden-Württemberg through adopted parton densities and benchmark scenario (in bwHPC and the DFG Grant No. INST 39/963-1 FUGG particular when the leptoquark Yukawa couplings are taken (bwForCluster NEMO).

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