Precision predictions for scalar leptoquark pair-production at hadron colliders
Christoph Borschensky, Benjamin Fuks, Anna Kulesza, Daniel Schwartländer
SScalar leptoquark pair production at hadron colliders
Christoph Borschensky, ∗ Benjamin Fuks,
2, 3, † Anna Kulesza, ‡ and Daniel Schwartl¨ander § Institute for Theoretical Physics, University of T¨ubingen,Auf der Morgenstelle 14, 72076 T¨ubingen, Germany Sorbonne Universit´e, CNRS, Laboratoire de Physique Th´eorique et Hautes ´Energies, LPTHE, F-75005 Paris, France Institut Universitaire de France, 103 boulevard Saint-Michel, 75005 Paris, France Institute for Theoretical Physics, WWU M¨unster, D-48149 M¨unster, Germany
We revisit scalar leptoquark pair production at hadron colliders. Apart from QCD contribu-tions, we include the lepton t -channel exchange diagrams relevant in the light of the recent B -flavoranomalies. We evaluate all contributions at next-to-leading order in QCD and resum, in the thresh-old regime, soft-gluon radiation at next-to-next-to-leading-logarithmic accuracy. All corrections arefound equally relevant. Our predictions consist of the most precise leptoquark cross section calcu-lations available to date and are necessary for the best exploitation of leptoquark LHC searches. Introduction – Many extensions of the Standard Model(SM) predict the existence of scalar leptoquarks [1–8], i.e. scalar bosons coupling to a quark and a lepton simultane-ously. Evidence for their existence is consequently vastlysearched for at the LHC. However, none of the recent AT-LAS [9, 10] and CMS [11–15] analyses find any hint forthese leptoquarks, so that their mass is now constrainedto be larger than 1–1.5 TeV. Recently, scalar leptoquarkshave gained a significant interest as they may providean explanation [16–22] for the B -meson anomalies [23–30] and address [31] the discrepancy between theoreticalpredictions [32] and experimental measurements [33] ofthe anomalous magnetic moment of the muon ( g − µ .In this context, favored scenarios generally feature largelepton-quark-leptoquark Yukawa couplings y .The most stringent bounds originating from LHC di-rect searches for leptoquark pair production and decayare extracted by assuming that leptoquarks are solelyproduced via strong interactions. In other words, non-QCD diagrams involving lepton t -channel exchanges of O ( y ) are neglected. In the associated limit setting pro-cedure, signal cross sections evaluated at next-to-leading-order (NLO) accuracy in the strong coupling α s [11–15],sometimes also supplemented by logarithmic thresholdcorrections [9, 10], are used. Thus the predictions includecontributions at O ( α s ) and O ( α s ), or possibly of higherorder in α s , but are independent of y [34, 35]. Bearing inmind the B -anomalies and ( g − µ motivation, the limitsmay thus be incorrectly estimated.In this paper, we perform for the first time a full NLO-QCD cross section calculation for scalar leptoquark pairproduction at hadron colliders, in which we include boththe QCD and t -channel contributions. Hadronic produc-tion of heavy systems, which is the case considered here,inevitably probes partonic center-of-mass energies closeto the production threshold given by twice the leptoquark ∗ [email protected] † [email protected] ‡ [email protected] § d [email protected] mass m LQ . In this limit, radiative corrections are dom-inated by soft-gluon emissions, manifesting themselvesas large logarithmic terms that must be consistently re-summed to all orders [36–39]. We report here threshold-resummed results at next-to-next-to-leading-logarithmic(NNLL) accuracy and showcase predictions obtained bymatching them to our new NLO results. In the follow-ing, we first present the considered theoretical frameworkand provide brief technical computational details. Wethen show an illustrative selection of results that under-lines how all considered corrections affect the results incomparable and significant ways. Our predictions, whichare the most precise to date, are hence required to de-rive limits consistently, in particular when assessing theinfluence of the leptoquark Yukawa couplings. Theoretical framework – We focus on a simplifiedmodel in which the SM is supplemented by several speciesof scalar leptoquarks S , ˜ S , R , ˜ R and S . Inspiredby standard naming conventions [40, 41], these lepto-quarks lie in the ( , ) − / , ( , ) − / , ( , ) / , ( , ) / and ( , ) − / representations of the SM gauge group re-spectively, and we target their Yukawa interactions in-volving exactly one lepton and quark. The latter arecollected in the Lagrangian: L int . = y RR1 ¯ u c R (cid:96) R S † + y LL1 ( ¯ Q c L · L L ) S † + ˜y RR1 ¯ d c R (cid:96) R ˜ S † + y LR2 ¯ e R Q L R † + y RL2 ¯ u R ( L L · R ) + ˜y RL2 ¯ d R ( L L · ˜ R )+ y LL3 (cid:0) ¯ Q c L · σ k L L (cid:1) ( S k ) † + H . c . . (1)In this expression, all flavor indices are suppressed forclarity, σ k stands for the Pauli matrices and the dotfor the invariant product of two fields lying in the(anti)fundamental representation of SU (2). The Q L and L L spinors denote the SM weak doublets of left-handedquarks and leptons, and u R , d R and (cid:96) R are the corre-sponding weak singlets. Moreover, the y / ˜y couplings are3 × y ij / ˜ y ij referring to the quark generation and thesecond one to the lepton generation in the gauge basis.The calculations reported in this work concern scalarleptoquark pair production and include fixed order con-tributions at leading order (LO) and NLO in QCD. In a r X i v : . [ h e p - ph ] J un contrast with previous work [34, 35, 42, 43], we notonly consider the QCD components at O ( α s ) and O ( α s ),but also include the t -channel lepton exchange contribu-tions at O ( y ) and O ( y α s ) as well as the O ( y α s ) and O ( y α s ) interference of the t -channel diagrams with theQCD ones. The full NLO-accurate predictions are col-lectively coined “NLO w/ t -channel” in the following, incontrast to the pure QCD ones that we refer to as the“NLO-QCD” predictions. The NLO w/ t -channel crosssections are then additively matched with the resummedNNLL soft-gluon contributions, resulting in cross sec-tion predictions at NLO w/ t -channel+NNLL accuracy.Threshold resummation is performed in Mellin space (see e.g. [39]) and involves one-loop matching coefficients [44].To ensure the correctness of the results, we performthe calculations in two independent ways: We first im-plement the above model into FeynRules [45], whichwe jointly use with
NLOCT [46] and
FeynArts [47] torenormalize the bare Lagrangian of eq. (1) at O ( α s ).We then generate a UFO model file [48] that we useto evaluate fixed-order LO and NLO predictions withinthe MG5 aMC framework [49]. The latter are cross-checked with results obtained within the
Powheg-Box framework [50], in which we input virtual correctionscalculated with the
FeynArts , FormCalc [51] and
Col-lier [52–55] packages. The NNLL corrections are evalu-ated with two independent in-house Monte Carlo codes.
Scalar leptoquark pair production at the LHC – Wepresent selected predictions for scalar leptoquark pairproduction at the 13 TeV LHC for the three most com-monly discussed types of scalar leptoquarks in the con-text of the flavor anomalies: the SU (2) L singlet state S (denoted by S ( − / due to its electric charge of − / R and the triplet state S . More specifi-cally, in the last two cases, we consider the pair produc-tion of the R mass eigenstate of electric charge of 5 / R (5 / ) and the one of the S mass eigenstateof electric charge of − / S ( − / ). In all ourcalculations, we treat the leptoquark mass m LQ as a freeparameter and assume the CKM matrix to be diagonal.While the determination of a scenario compatible withflavor constraints and Z -pole observables is desirable [56],this goes beyond the scope of this study. We consider in-stead benchmarks motivated by ref. [21]. The values ofthe Yukawa couplings found in this study were obtainedin a fit to low-energy observables and did not involve con-straints from direct searches for leptoquarks at the LHC.Given that the description of lepton flavor university-violating observables involves both leptoquark couplingsand masses, optimally one should aim at a global fit basedon direct and indirect constraints, in which case the cal-culations presented in this work will play a crucial role.For S S ∗ production, we adopt a minimal flavor ansatzfor the leptoquark Yukawa couplings, ( y LL1 ) = − . y LL1 ) = 3 with all other y LL1 elements set to 0. For R R ∗ production, we similarly consider as the only non-vanishing coupling ( y RL2 ) = 1 .
5, a value still allowedby direct exclusion bounds [21], while for S S ∗ produc- tion, we adopt ( y LL3 ) = − ( y LL3 ) , keeping the actualcoupling value free and setting all other couplings to 0.Our results are obtained by convoluting the partonicresults with two different sets of parton distribution func-tions (PDFs), NNPDF3.1 [57] and CT18 [58]. Unlessstated otherwise, NLO sets are employed for NLO-QCDand NLO w/ t -channel predictions, while NNLO sets areused for NLO+NNLL calculations. We set the renor-malization ( µ R ) and factorization ( µ F ) scales equal to acommon value µ = µ R = µ F . The central scale choice µ = µ is fixed to µ = m LQ , and scale uncertainties areestimated by varying µ by a factor of 2 up and down.In fig. 1, we present cross section predictions for S ( − / S (1 / (left column) and R (5 / R ( − / (right col-umn) production, both for the NNPDF3.1 (upper row)and CT18 (lower row) parton densities. We estimate therelative importance of the various corrections studied inthis work with respect to NLO-QCD predictions, andassess the size of the scale and PDF uncertainties. Com-paring the four subfigures, we observe that depending onthe process, the PDFs, the magnitude of the Yukawa cou-plings, and m LQ , the considered corrections can influencethe predictions in different, often contrasting, ways.Although providing a positive correction, the size ofthe t -channel contributions depends very differently on m LQ for the two processes (blue dashed curves). On thecontrary, NNLL effects, which we estimate through theratio of the NLO-QCD+NNLL to the NLO-QCD crosssections both calculated with the same NLO PDF set(turquoise dotted curves), are independent of the processand PDF choice. As expected, this ratio is bigger than 1for all m LQ considered, and grows with increasing m LQ , i.e. , approaching the production threshold.This behavior is vastly modified by the interplay of t -channel contributions, PDF effects and soft-gluon correc-tions, all entering the NNLL results matched with NLOw/ t -channel (red solid curves). The effect of evaluatingthe NLO w/ t -channel cross sections with NNLO PDFsets instead of NLO sets is illustrated by the differencebetween the corresponding ratios to NLO QCD predic-tions (blue dashed vs. olive dash-dotted curves). For theNNPDF3.1 PDF set (upper row), this effect diminishesthe cross sections, offsetting the increase stemming fromthe NNLL contributions. As a consequence, the NLO w/ t -channel+NNLL results deliver a positive correction ofabout 10–20% with respect to the NLO-QCD predictionsfor m LQ ∈ [1 ,
2] TeV. Similarly for the NLO w/ t -channelresult, the full NLO w/ t -channel+NNLL correction ex-hibits the opposite behavior with increasing m LQ in the S (upper left) and R (upper right) cases.When CT18 PDFs are used instead (lower row), thecorrections are larger and reach a magnitude of about 20–50%, the impact this time increasing with m LQ for bothprocesses. Furthermore, in the NNPDF3.1 case, vari-ous contributions to the total correction are often muchbigger than the correction itself. For example, the cor-rection due to including t -channel diagrams reaches upto 40% of the NLO-QCD result for the pair production − − σ (cid:18) pp → S ( − / S (1 / (cid:19) (fb) √ S = 13 TeV, NNPDF 3.1 (cid:16) y LL (cid:17) = − . (cid:16) y LL (cid:17) = 3NLO w/ t -channel + NNLLNLO w/ t -channelNLO-QCD . . . ( σ X ± ∆ µ σ X ) /σ X . m S < µ R , µ F < m S . . . ( σ X ± ∆ PDF σ X ) /σ X (symmetric uncertainty) m S (GeV)0 . . . . . . σ X /σ NLO-QCD
NLO w/ t -channel (NNLO PDFs)NLO-QCD + NNLL (NLO PDFs) − − σ (cid:18) pp → R (5 / R ( − / (cid:19) (fb) √ S = 13 TeV, NNPDF 3.1 (cid:16) y RL (cid:17) = 1 . t -channel + NNLLNLO w/ t -channelNLO-QCD . . . ( σ X ± ∆ µ σ X ) /σ X . m R < µ R , µ F < m R . . . ( σ X ± ∆ PDF σ X ) /σ X (symmetric uncertainty) m R (GeV)0 . . . . σ X /σ NLO-QCD
NLO w/ t -channel (NNLO PDFs)NLO-QCD + NNLL (NLO PDFs) − − σ (cid:18) pp → S ( − / S (1 / (cid:19) (fb) √ S = 13 TeV, CT18 (cid:16) y LL (cid:17) = − . (cid:16) y LL (cid:17) = 3NLO w/ t -channel + NNLLNLO w/ t -channelNLO-QCD . . . ( σ X ± ∆ µ σ X ) /σ X . m S < µ R , µ F < m S . . . ( σ X ± ∆ PDF σ X ) /σ X (symmetric uncertainty) m S (GeV)0 . . . . σ X /σ NLO-QCD
NLO w/ t -channel (NNLO PDFs)NLO-QCD + NNLL (NLO PDFs) − − σ (cid:18) pp → R (5 / R ( − / (cid:19) (fb) √ S = 13 TeV, CT18 (cid:16) y RL (cid:17) = 1 . t -channel + NNLLNLO w/ t -channelNLO-QCD . . . ( σ X ± ∆ µ σ X ) /σ X . m R < µ R , µ F < m R . . . ( σ X ± ∆ PDF σ X ) /σ X (symmetric uncertainty) m R (GeV)0 . . . . σ X /σ NLO-QCD
NLO w/ t -channel (NNLO PDFs)NLO-QCD + NNLL (NLO PDFs) FIG. 1. S ( − / S (1 / (left) and R (5 / R ( − / (right) production at the 13 TeV LHC, using the NNPDF3.1 (upper row) andCT18 (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 -channel+NNLL (red solid) accuracy. The associated scaleand PDF uncertainties are also displayed (middle panels). In the lower panels, we show ratios of the NLO w/ t -channel, NLOw/ t -channel+NNLL, NLO w/ t -channel calculated using NNLO PDFs (olive dash-dotted) and NLO-QCD+NNLL (turquoisedotted) results to the NLO-QCD cross section. m S (GeV)0 . . . . . . . (cid:16) y LL (cid:17) = − (cid:16) y LL (cid:17) σ (cid:18) pp → S ( − / S (4 / (cid:19) (fb) √ S = 13 TeV, NNPDF 3.1 . . − − m S (GeV)0 . . . . . . . (cid:16) y LL (cid:17) = − (cid:16) y LL (cid:17) σ (cid:18) pp → S ( − / S (4 / (cid:19) (fb) √ S = 13 TeV, CT18 . − FIG. 2. NLO w/ t -channel+NNLL total cross section for S ( − / S (4 / production at the 13 TeV LHC as a function of the S mass m LQ = m S and the Yukawa couplings ( y LL3 ) = − ( y LL3 ) (all other Yukawa couplings being set to 0). We presentpredictions obtained with the NNPDF3.1 (left) and CT18 (right) PDF set. of 2 TeV R leptoquarks, whereas the complete NLOw/ t -channel+NNLL one is only of about 20%. In con-trast, results obtained with CT18 densities exhibit theopposite behavior, the cross sections being typically en-hanced when switching from NLO to NNLO PDFs. The t -channel and soft-gluon resummation pieces are of com-parable size and thus equally contribute to the combinedcorrection. Therefore a precise knowledge of the crosssection requires calculating all classes of corrections.In the middle panels of the four subfigures of fig. 1,we focus on scale and PDF uncertainties. We distin-guish the impact of scale variations (second panel) fromthe one originating from the PDF determination (thirdpanel). Our results show that soft-gluon resummationleads to a significant reduction of the scale uncertaintiesfrom around 10% (for the NLO predictions) to about 1–2% for m LQ values ranging up to slightly above the cur-rent exclusion limits. The reduction might however beunderestimated due to the chosen method for scale un-certainty evaluation. This calls for a more comprehensivestudy. Correspondingly, the total theoretical error for ourfinal NLO w/ t -channel+NNLL predictions is dominatedby its PDF component. The size of the PDF error is how-ever strongly dependent on the PDF choice. For instance,results derived with NNPDF3.1 exhibit, for m LQ ∼ m LQ values, the PDF error becomes significantly bigger. Incomparison, PDF errors obtained with the CT18 set arelarger for small m LQ values, but do not grow as quicklyfor higher masses. Still, the PDF errors turn out to beof the same order as the full perturbative corrections forlarge m LQ values. Those large PDF errors at high masseshence obscure the accuracy of the predictions. However,as more LHC data will be analyzed, one can expect asubstantial improvement of the PDF knowledge, in par- ticular in the large Bjorken- x regime, so that the PDFerrors associated with predictions relevant for high-masssystem production will be significantly reduced.In fig. 2, we calculate the NLO w/ t -channel+NNLL to-tal cross section for S ( − / S (4 / production, and study itsdependence on the leptoquark mass and Yukawa couplingstrength. We consider both the NNPDF3.1 (left) andCT18 (right) PDF sets. At small values of the Yukawacoupling, the dominant production mechanism is QCDdriven so that the cross section solely depends on m LQ .On the contrary, as the coupling approaches 1, the t -channel contributions become more relevant and the to-tal rate significantly increases. This behavior is mostlyindependent of the chosen PDF set, CT18 predictionsbeing slightly less sensitive to the t -channel diagrams. Summary – We have significantly advanced the pre-cision of scalar leptoquark pair production cross sectioncomputations. First, we have included all contributionsto the process, both the QCD ones and those involvingthe t -channel exchange of a lepton, at NLO QCD. Second,we have resummed soft-gluon radiation in the thresholdregime to NNLL accuracy.The t -channel contributions, threshold resummation,the adopted parton densities and benchmark scenario(in particular when the leptoquark Yukawa couplings aretaken as large as suggested by the recent B -anomalies)importantly affect the total rates, in potentially contrast-ing and sizable ways. This emphasizes the necessity ofincluding all contributions whose calculation has beenpioneered in this work. While the perturbative seriesexhibits smaller scale uncertainties, the precision of thepredictions is limited by the poor PDF knowledge inthe large Bjorken- x regime relevant for the productionof high-mass systems. In light of our findings, we rec-ommend the usage of NLO w/ t -channel+NNLL crosssections, to be taken together with the correspondinglyreduced scale uncertainties and PDF errors extractedfrom the envelope spanned by computations, left for fu-ture work, performed with different PDF sets. This fol-lows the strategy outlined in various recommendationsfor LHC cross section calculations [59–63]. The computercodes used in this work are available upon request. Acknowledgments – We are grateful to V. Hirschi,O. Mattelear and H.S. Shao for their help with
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