Collider signature of Leptoquark with flavour observables
Abstract
The Leptoquark model has been instrumental in explaining the observed lepton flavour universality violating charged () and neutral current () anomalies that have been the cause for substantial excitement in particle physics recently. In this article we have studied the role of the vector leptoquark in explaining the neutral current () anomalies and . Moreover, we have performed a thorough collider search for this Leptoquark using () final state at the Large Hadron Collider. From our collider analysis we maximally exclude the mass of the Leptoquark up to 2340 GeV at 95% confidence level for the 13 TeV Large Hadron Collider for an integrated luminosity of 3000 . Furthermore, a significant portion of the allowed parameter space that is consistent with the neutral current () observables is excluded by collider analysis.
I Introduction
The discovery of the Higgs boson in 2012 by the CMS Chatrchyan:2012xdj () and ATLAS Aad:2012tfa () collaborations is definitely one of the greatest achievements of Large Hadron Collider (LHC). Unfortunately, it has not been able to detect signatures corresponding to any new physics (NP) particles till date. Experimental measurements of observables related to physics have, however, exhibited deviations of a few from their Standard Model (SM) expectations hinting towards the existence^{1}^{1}1Apart from such deviations, nonzero neutrino mass, signatures for the existence of dark matter, observed baryon asymmetry etc. also concur to the fact that BSM physics is indeed a reality of nature. of beyond SM (BSM) physics. physics experiments at LHCb, Belle and Babar have pointed at intriguing lepton flavour universality violating (LFUV) effects. To that end, flavour changing neutral current^{2}^{2}2Experimental signatures are also present for LFUV via charge current semileptonic transition processes. For example the ratios average () and Aaij:2017tyk () show significant deviations from their corresponding SM predictions. (FCNC) processes such as have drawn much attention due to anomalies that have been observed recently at the LHCb and Belle experiments. A deviation of 2.6 has been observed in with a value of Aaij:2014ora () from the corresponding SM prediction ( DescotesGenon:2015uva (); Bordone:2016gaq ()) for the integrated dilepton invariant mass squared range . LHCb has reported a deviation in at the level of and for the two ranges [0.0451.1] (called lowbin) and [1.16.0] (called centralbin) with values Aaij:2017vbb () and Aaij:2017vbb () respectively. The corresponding SM predictions are Capdevila:2017bsm () and DescotesGenon:2015uva (); Bordone:2016gaq () respectively.
In order to explain the above mentioned anomalies we have selected a particular extension of the SM consisting of several hypothetical particles that mediate interactions between quarks and leptons at treelevel. Hence, these particles are known as Leptoquarks (LQs). Such particles can appear naturally in several extensions of the SM (e.g., composite models Schrempp:1984nj (), Grand Unified Theories Georgi:1974sy (); Pati:1973rp (); Dimopoulos:1979es (); Dimopoulos:1979sp (); Langacker:1980js (); Senjanovic:1982ex (); Cashmore:1985xn (); Pati:1974yy (), superstringinspired models Green:1984sg (); Witten:1985xc (); Gross:1984dd (); Hewett:1988xc () etc). Considerable amount of work regarding LQs have been done both from the point of view of their diverse phenomenological aspects Davidson:1993qk (); Hewett:1997ce (); Nath:2006ut (), and specific properties Shanker:1981mj (); Shanker:1982nd (); Buchmuller:1986iq (); Buchmuller:1986zs (); Hewett:1987yg (); Leurer:1993em (); Leurer:1993qx (); Aaltonen:2007rb (); Dorsner:2014axa (); Allanach:2015ria (); Evans:2015ita (); Li:2016vvp (); Diaz:2017lit (); Dumont:2016xpj (); Faroughy:2016osc (); Greljo:2017vvb (); Baumgartel:2014dqa (); Aad:2015caa (); Aaboud:2016qeg (); Sirunyan:2017yrk (); Sirunyan:2018nkj (); Dorsner:2017ufx (); Allanach:2017bta (); Crivellin:2017zlb (); Hiller:2017bzc (); Buttazzo:2017ixm (); Calibbi:2017qbu (); Sahoo:2016pet (); Altmannshofer:2017poe (); Biswas:2018snp (). Furthermore, several articles Sakaki:2013bfa (); Popov:2016fzr (); Chen:2017hir (); Alok:2017sui (); Crivellin:2017zlb (); Assad:2017iib (); Aloni:2017ixa (); Wold:2017wdj (); Muller:2018nwq (); Hiller:2018wbv (); Biswas:2018jun (); Fajfer:2018bfj (); Monteux:2018ufc (); Kumar:2018kmr (); Crivellin:2018yvo () that explain the different flavour anomalies with different versions of LQ models exist in the literature.
In connection to the above, we consider the vector leptoquark (VLQ) that is of interest in our current article. It is capable of mediating the observables at tree level, due to its electromagnetic charge^{3}^{3}3A LQ cannot mediate the charged current observables, as opposed to a LQ which can mediate both the charged and neutral current processes. . We provide bounds on the parameter space for the VLQ subject to constraints due to the observables . Furthermore, we have used the latest experimental value CMS:2014xfa () of the branching fraction for the decay as another constraint in our analysis while the SM prediction for the same decay is Bobeth:2013uxa (). We provide constraints on the real and imaginary parts for the allowed values of the coupling products with respect to the mass of the VLQ up to (corresponding to the experimental errors for these observables).
The LQs being potential candidates in explaining the flavour anomalies, it is only relevant that one investigates the production and decay signatures of these LQs at colliders. There exist several articles Dorsner:2014axa (); Allanach:2015ria (); Greljo:2017vvb (); Evans:2015ita (); Diaz:2017lit (); Dorsner:2017ufx () in the literature that have been dedicated to collider studies of LQs, but in most of the cases these studies have been performed on scalar LQs. The collider studies for vector LQs are limited in number Aaltonen:2007rb (); Diaz:2017lit (); Dorsner:2018ynv (); Sirunyan:2018ruf (); Biswas:2018snp (). Our current interest for this article being the VLQ, it is imperative that one probes this LQ at the current or future collider experiments. To the best of our knowledge, the present article is the first which deals with the collider prospects of the VLQ^{4}^{4}4The VLQ belongs to the antifundamental representation of the part of the SM gauge group Dorsner:2018ynv (). Hence, there is no available model file for this VLQ. Therefore, we believe this to be the first article which deals with collider prospects of VLQ after proper implementation of the model in FeynRules Alloul:2013bka (). at the LHC. We study signatures corresponding to this VLQ for final states at the LHC with the centre of momentum energy TeV. Although the ATLAS collaboration has also looked at the same final state ATLAS:2017hbw () but they have searched for the Rparity violating scalar top partners at the 13 TeV LHC. Their exclusion limit, depending on the branching fractions of the scalar top to bottom and electron/muon, is set from 600 GeV to 1500 GeV. Using several interesting kinematic variables we maximize the signal event with respect to relevant SM backgrounds. From our collider analysis and depending on the SM bilinear couplings with VLQ we exclude the mass of this VLQ up to 2140 GeV and 2340 GeV for the two bench mark values of integrated luminosities 300 and 3000 respectively at the 95% confidence level (C.L.).
The paper is organised as follows. We briefly discuss the Lagrangian for the VLQ and set the notations in section II. In section III we show the flavour analysis of transition observables mediated by the VLQ. Section IV is dedicated to the collider analysis for with final states. Finally, we discuss our results and conclude in section V.
Ii Effective Lagrangian of vector Leptoquark
LQs are special kinds of hypothetical particles that carry both lepton (L) and baryon (B) number. Consequently they couple to both leptons and quarks simultaneously. Furthermore, they possess colour charge and fractional electromagnetic charges. However, unlike the quarks they are either scalars or vectors. For further discussions regarding all LQ scenarios, one can look into the review Dorsner:2016wpm (). Due to the above distinguishable properties, these LQs have several phenomenological implications with respect to the other BSM particles. In general there are twelve LQs, among them six are scalars () and the rest () transform vectorially under Lorentz transformations. In the current article we will focus only on VLQ in order to explain the anomalies. The Lagrangian which describes the interaction for the VLQ with the SM fermion bilinear is given as Dorsner:2016wpm ():
(1) 
To avoid the constraint due to the proton decay from VLQ, we set the corresponding couplings for diquark interactions to zero. Under the SM gauge group the VLQ transforms as . represents the left handed quark doublet, denotes for the left handed lepton doublet, stands for the right handed down type quark singlet and is the right handed charged lepton. Left (right) handed gauge coupling constants are represented by with the fermion generation indices .
Iii flavour Signatures
We closely follow ref. Kosnik:2012dj () in the following discussion about the operator basis relevant to decays and the expressions for the observables. The effective dimension six Hamiltonian at the mass scale of the quark is written as Kosnik:2012dj (); Grinstein:1988me ()
(2)  
where . The VLQ contributes to the following twoquark, twolepton operators:
(3) 
and their corresponding “primed” counterparts. The chiraly flipped “primed” operators are obtained by an exchange in the above operators. Where represents the unit for electromagnetic charge, is the strong coupling constant and . The fourquark operators and the radiative penguin operators are provided in ref. Bobeth:1999mk (). The decay amplitudes for the transition in terms of the effective Wilson coefficients (WCs) evaluated at the scale are provided in Buras:1993xp ().
The theoretical expression for the branching fraction corresponding to the decay reads Kosnik:2012dj ()
(4) 
In the above , , and are denoted as the masses of meson, bottom quark and charged lepton respectively. is the Fermi constant, represents the life time while stands for the decay constant of meson. It is evident from eq. 4, that the is only sensitive to the contributions due to the differences between operators with left and righthanded quark currents, , and .
In contrast to the case for , the decay width for receives contributions from , , , and . The tensor operators have small contributions in LQ models Kosnik:2012dj (). The corresponding decay width reads Bobeth:2007dw ()
(5) 
where
(6) 
and are defined as:
where
Here
(7)  
The functions , for are defined as:
(8)  
(9) 
The form factors , and have been obtained from ref. Altmannshofer:2014rta () where the authors perform a combined fit to the lattice computation Bouchard:2013pna () and light cone sum rules (LCSR) predictions at Ball:2004ye (); Bartsch:2009qp (), using the parametrization and conventions of Bouchard:2013pna ().
WCs corresponding to the operators related to the VLQ (eq. 3) that contribute to a transition are Kosnik:2012dj ():
(10) 
It is evident that of the eight relevant WCs, only four are independent, which we take to be , , and . We numerically solve for these four WCs subject to the experimental values of observables (, , and ) provided in introduction. These solutions translate to constraints on the model parameters for the VLQ model. We find that these observables can impose constraints on the mass versus coupling product plane for the VLQ. These constraints are displayed in fig. 1 for the real and imaginary parts of the coupling product. In general, however, constraints on individual couplings cannot be derived from flavour physics alone since it is the product of the couplings that enter the individual WCs (viz. eq. 10). The bands correspond to the experimental errors for the measured observables.
Fig. 1(a) displays the variation of the real and imaginary parts of the coupling product with respect to the mass of the LQ . The variation for the real (imaginary) part is due to the real (imaginary) part of the solution for the WC with respect to the experimental observables given in introduction. The real part of has a unique solution, resulting in the single brown band close to the horizontal axis in fig. 1(a). However, the imaginary part of has two sets of solutions which are symmetric with respect to 0, and hence translate into the blue bands symmetric with respect to to the horizontal axis. Similarly, the real and imaginary parts for the WC translate into fig. 1(b). The unique negative solution for the real part translates into the wide brown band and the solutions for the imaginary part give rise to the blue bands symmetric to the horizontal axis for the coupling product . For a benchmark value GeV, the ranges for the real and imaginary parts of these coupling products are:
The cases 1(c) and 1(d) are a little different from the cases discussed above. 1(c) arises due to , both of whose real and imaginary part have two solutions, one positive and one negative, at both the higher and lower limits considering experimental errors. However, the regions for these solutions overlap, and hence get broad brown and blue bands both above and below the horizontal axis for each of the real and imaginary parts of the coupling product . Similarly, the different sets of solutions for the WC translate into fig. 1(d) for the coupling product . These solutions do not overlap as in the case of 1(c), and hence we get distinct bands corresponding to the real and imaginary parts of the corresponding coupling product. As in the former cases, we provide values for these coupling products for the benchmark value GeV:
Iv Collider Analysis
In this section we study the collider prospects of VLQ at the LHC. We look for signals where the VLQ decays into a bottom quark () and a lepton () with a branching ratio that depends on the corresponding coupling. We vary the coupling of to quark and from 0.1 to 0.9. As a result, the branching ratio varies from to for individual light leptonic channels. For further simplicity, we assume the coupling of to both lepton and bottom quark to be equal while that to the rest of the quarks and leptons is fixed at 0.1. Hence, the signal we consider from VLQ pair production is two jets with GeV and and two light leptons with GeV and . The dominant backgrounds from the SM processes are + jets, + jets and + jets. Furthermore, the SM process which contribute subdominantly are + jets and + jets. The SM processes like + jets, + jets, + jets and + jets contribute mildly to this analysis because we tag two jets in the final states. We therefore do not consider these backgrounds in our present analysis.
Both the signal and background processes in this analysis have been generated using Madgraph5 Alwall:2014hca () with the default parton distribution functions NNPDF3.0 Ball:2014uwa (). The VLQ model file used in this analysis is obtained from FeynRules Alloul:2013bka (). The parton level events generated from Madgraph5 are then passed through Pythia8 Sjostrand:2014zea () for showering and hadronization. The backgrounds and signal events are matched properly using the MLM matching scheme Hoche:2006ph (). The detector level simulation is done using Delphes(v3) deFavereau:2013fsa () and the jets are constructed using fastjet Cacciari:2011ma () with anti jet algorithm with radius and GeV. The crosssection corresponding to the background processes that have been used in this analysis are provided in table 1. The signal crosssection is calculated from Madgraph at LO (leadingorder).
Background process  crosssection (pb) 

(NNLO + NNLL)  815.96 Czakon:2011xx () 
(NLO + NNLL)  71.7 Kidonakis:2015nna () 
(NLO)  0.6448 (0.8736) Maltoni:2015ena () 
(NLO)  16.91 Cascioli:2014yka () 
We have utilized some interesting kinematic variables which efficiently discriminate the signal and background events and maximizes the signal reach at the LHC. These variables are Konar:2008ei (); Konar:2010ma (); Swain:2014dha (), transverse momentum of the lepton, invariant mass of the jet and lepton, and the invariant mass of two jets and the dilepton. In addition, we also make use of the dilepton invariant mass to handle the backgrounds involving the boson. The kinematic variable was originally proposed in order to measure the mass scale of new physics produced at the LHC. It is defined as the minimum partonic CM energy that is consistent with the final state measured momenta and the missing transverse energy of the event. Mathematically, this variable is defined as,
(11) 
where is the sum of the masses for the “invisible” particles. is the total energy and the total longitudinal momentum of the “visible” particles. In this analysis we take two jets and two leptons as our “visible” particles and use their momenta for calculating . Since the signal we consider here does not involve any invisible particle, the missing energy in each event is very small and can solely be attributed to mismeasurement. is also taken to be zero due to the same reason. As per our expectations, peaks at twice the mass of the leptoquark as shown by the red dashed distribution in fig. 2 (top panel right plot). The VLQ mass, for this representative plot, is taken to be 1 TeV.
Similarly, the other variables like the invariant mass of the two jets and the two leptons (), and of one jet and corresponding lepton () are also very efficient in separating the signal from the backgrounds. While the invariant mass peaks at the at twice the mass of the VLQ, the variable peaks at mass of the VLQ (1 TeV) as expected. Since the lepton from the VLQ is highly boosted, we also have utilized the lepton transverse momenta, , as a discriminating variable.
With the above variables we have done a cut based analysis where the following cuts are employed to maximize the signal significance,

GeV,

GeV,

GeV,

,

GeV.
After implementing the above cuts, we have calculated the signal significance using the following formula,
(12) 
Here represent the number of signal (background) events for a given luminosity after implementing the cuts mentioned above. Eq. 12 allows us to exclude the mass of the VLQ up to 2140 GeV for the coupling at C.L. for 13 TeV LHC with 300 of integrated luminosity. This limit is reduced to a value as low as 1.6 TeV for (displayed in fig. 3 with red band) at C.L. for 300 of integrated luminosity. As is evident from the figure, the exclusion limit can go up to 2340 GeV for 3000 at C.L. for and for the limit is 1.8 TeV which is represented by the blue band. Note that one might expect a better limit by limiting the other coupling(s) to a very small value (which, as mentioned earlier we have taken to be 0.1) so that the considered channel will get branching ratio. However, that limit as we have checked, is marginally better than for because even in this case also the branching ratio approaches . Hence, in this analysis, the limits that we have obtained for VLQ in mass and coupling plane from the collider study in conjunction with the flavour physics constraints are more or less optimal.
As discussed earlier in sec. III using the WCs of the flavour physics observables like , , and etc. one can get constraint in the VLQ mass and coupling product plane which is demonstrated in fig. 3 by the gray region. The coupling in the vertical axis is obtained by choosing one value of the coupling as . It is however not possible for us to put constraints on the imaginary part of individual couplings from a combined collider and flavour point of view, since the collider analysis inherently assumes the couplings to be real. We find that part of the allowed parameter space for the real part of the coupling (corresponding to a value of ) is disallowed by the collider constraints. However, the parameter space due to flavour contraints from is retained completely. We hence conclude that the values for that fall within the yellow band in fig. 3 represent the allowed parameter space upto w.r.t the mass of the VLQ for all collider and flavour constraints taken together^{5}^{5}5A similar analysis can also be done for . However, from fig. 1 it is clear that one will not obtain common points for the real part of such a coupling after requiring from the flavour analysis alone (see figs. (b)b and (d)d). Moreover, most of the allowed parameter space for such a scenario will correspond to negative values of and hence will have no intersection with the contraints due to the collider analysis. This will hence provide no further insight as to the allowed parameter space for such a coupling and hence we refrain from showing the corresponding plot..
V Conclusion
We consider a component of the VLQ of electromagnetic charge which mediates neutral current processes at tree level. We use the , , and data along with their errors in order to solve for the involved Wilson coefficients, and, in turn, provide constraints on the product of coupling with respect to the mass of the VLQ. Simultaneously, we probe this VLQ at 13 TeV LHC via final state. For a reliable collider analysis we have accounted for several relevant SM background processes. Using different interesting kinematic variables and with judicious cut selections we maximise the signal significance with respect to the SM backgrounds. Our collider study reveals that it is possible to maximally exclude the mass of the VLQ up to 2340 GeV at 95% C.L. at the 13 TeV LHC for an integrated luminosity 3000 . In addition, our collier study reduces chunk of parameter space that is consistent with the neutral current observables in the coupling and VLQ mass plane for a fixed value of .
Acknowledgements.
AS would like to thank Ilja Dorsner for discussions regarding the implementation of the model file for leptoquark in Feynrules. AKS acknowledges the support received from Department of Science and Technology, Government of India under the fellowship reference number PDF/2017/002935 (SERB NPDF). We also would like to thank Prof. Dillip Ghosh and Nivedita Ghosh for our initial collaboration and useful discussions regarding collider analysis.References
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