Precise limits on the charge-2/3 U1 vector leptoquark

Arvind Bhaskar,1, ∗ Diganta Das,1, 2, † Tanumoy Mandal,3, ‡ Subhadip Mitra,1, § and Cyrin Neeraj1, ¶

1Center for Computational Natural Sciences and Bioinformatics,International Institute of Information Technology, Hyderabad 500 032, India

2Department of Physics and Astrophysics, University of Delhi, Delhi 110 007, India3Indian Institute of Science Education and Research Thiruvananthapuram, Vithura, Kerala, 695 551, India

(Dated: August 16, 2021)The U1 leptoquark is known to be a suitable candidate for explaining the semileptonic B-

decay anomalies. We derive precise limits on its parameter space relevant for the anomaliesfrom the current LHC high-pT dilepton data. We consider an exhaustive list of possible B-anomalies-motivated simple scenarios with one or two new couplings that can also be usedas templates for obtaining bounds on more complicated scenarios. To obtain precise limits,we systematically consider all possible U1 production processes that can contribute to thedilepton searches, including the resonant pair and single productions, nonresonant t-channelU1 exchange, as well as its large interference with the Standard Model background. Wedemonstrate how the inclusion of resonant production contributions in the dilepton signalcan lead to appreciably improved exclusion limits. We point out new search channels of U1that can act as unique tests of the flavour-motivated models. The template scenarios can alsobe used for future U1 searches at the LHC. We compare the LHC limits with other relevantflavour bounds and find that a TeV-scale U1 can accommodate both RD(∗) and RK(∗) anomalieswhile satisfying all the bounds.

CONTENTS

I. Introduction 2

II. The U1 leptoquark model 4RD(∗) scenarios 5RK(∗) scenarios 10

III. Production modes and decays 13Pair production 14Single production 16Nonresonant production and interference 17

IV. Recast of dilepton data 18ATLAS ττ search 19CMS µµ search 19

V. Exclusion limits 21

VI. Summary and conclusions 28

Acknowledgments 29

A. Cross section parametrization for the `` signal processes 29

B. Limits estimation: χ2 tests 31

References 32

∗ arvind.bhaskar@research.iiit.ac.in† diganta.das@iiit.ac.in‡ tanumoy@iisertvm.ac.in§ subhadip.mitra@iiit.ac.in¶ cyrin.neeraj@research.iiit.ac.in

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I. INTRODUCTION

The concept of lepton flavour universality, a key prediction of the Standard Model (SM), seemsto be in tension with the present experimental measurements of some semileptonic B-meson de-cays [1–11]. Differences between theoretical predictions and experimental measurements, hintingtowards the existence of some physics beyond the SM (BSM), have been observed in the RD(∗) andRK(∗) observables:

RD(∗) =B(B→ D(∗)τν)

B(B→ D(∗) ˆν)and RK(∗) =

B(B→ K(∗)µ+µ−)B(B→ K(∗)e+e−)

. (1)

We use ˆ to denote the light charged leptons, e or µ and B(x→ y) for the x→ y decay branchingratio (BR). The experimental values of RD and RD∗ exceed their SM predictions by 1.4σ and 3.1σ ,respectively [12–15] (combined excess of 3.1σ in RD(∗) , according to the 2019 world averages [16]),whereas, the RK and RK∗ measurements [17, 18] are smaller than the theoretical predictions byabout 3.1σ [19, 20].

A TeV-scale vector leptoquark (vLQ), a color-triplet vector boson with nonzero lepton andbaryon numbers, is considered to be a suitable candidate to address these anomalies in the lit-erature [21–50].1 It is shown in [37] that a charge-2/3 weak-singlet vLQ, U1 ≡ (3,1,2/3), canresolve both RD(∗) and RK(∗) anomalies simultaneously. If the vLQ is really responsible for theseanomalies, it is then essential to scrutinise its parameter space that can address the anomalies si-multaneously while satisfying all relevant experimental bounds. In the literature, various flavourand collider data have already been used in this context. However, we find that even though alot of emphasis has been put on obtaining regions of parameter space that are either ruled out orfavoured by the observed anomalies and other flavour data, relatively less attention has been paidto obtain precise bounds from the Large Hadron Collider (LHC) data.

It is known that the regions of parameter spaces favoured by the flavour anomalies in variousleptoquark (LQ) models are already in tension with the high-pT dilepton data [29, 41, 76–81]. Inthis paper, we specifically investigate the case of U1 and argue that the bounds from the LHC datamight actually be underestimated and, in some regions of the parameter space, the data could bemore constraining than what has been considered so far if one systematically computes all relevantprocesses and considers the latest direct search limits. As we see, different production processes ofU1 contribute to the dilepton or monolepton plus missing energy (MET) signals affecting variouskinematic distributions. When incorporated in the statistical analysis, they can give strong boundson the unknown LQ-q-` (where ` can be any charged lepton) couplings together. However, whilemost of these processes contribute constructively to the signal, a significant contribution (in fact,the most dominant one, in some cases) comes from the nonresonant t-channelU1 exchange processthat interferes destructively with the SM background. Hence, there is a competition among the U1

production processes, which are highly sensitive to the U1 parameters. Usually, the contributionof the resonant production processes (i.e., pair and single productions) to the `` or `+ /ET signalsare ignored assuming that it would give only minor corrections. However, we find, especially inthe lower mass region, that the resonant productions’ effect on the exclusion could be significant.In this paper, we systematically put together all the sources of resonant and nonresonant dileptonevents in our analysis and obtain robust and precise limits on the U1 parameters to date.

To contribute to RD(∗) , a U1 must couple to the third-generation lepton(s) and, second andthird-generation quarks [see Fig. 1(a), assuming that it does not alter the denominators in Eq. (1)]and to contribute to the RK(∗) observables, it should couple to the second-generation leptons [seeFig. 1(b)]. Within the SM, the b→ cτν decay is mediated by a tree-level charge current interaction,and the neutral current b→ sµ+µ− decay occurs through a loop. However, the U1 LQ can mediate

1 See Refs. [51–75] and the references therein for other recent phenomenological studies on LQs.

2

B

D(∗)

b

τ

c

ν

U1

(a)

B

K(∗)

b

µ−

s

µ+

U1

(b)

B

D(∗)

bν

c

τ

W

(c)

B

K(∗)

b

µ−

s

µ+W−

W+t

ν

(d)

FIG. 1. Representative leading order diagrams showing the B→ D(∗)τν and B→ K(∗)µ+µ− decays: thetree level U1 contribution to (a) B→ D(∗)τν and (b) B→ K(∗)µ+µ−, and (c) and (d) the corresponding SMprocesses, respectively.

both the flavour-changing transitions, b → cτν and b → sµ+µ− at the tree level, as shown inFig. 1. Here, we adopt a bottom-up approach and construct all possible minimal or next-to-minimalscenarios within the U1 LQ model with one or two new couplings at a time that can accommodateeither the RD(∗) or the RK(∗) anomalies. These scenarios can be used as templates to obtain boundson more complicated scenarios (as explained in Ref. [81] for the S1 LQ).

There is another motivation for considering various minimal scenarios with different couplingcombinations. An effective field theory suitable for describing the outcomes of low-energy exper-iments is not well suited for high-energy collider experiments where some of the heavy degreesof freedom are directly accessible. The SM-like Wilson operator, OvL = [cγµPLb]

[τγµPLν

] playsthe most important role in the RD(∗) observables. However, by looking only at this operator, itis not obvious that the `` data would lead to strong bounds and the interference between thenew physics and the SM background processes would play the prominent role in determining thebounds. Scenarios with very different LHC signatures can lead to the same effective operator (wediscuss such an example later). Hence, even though these two scenarios would look similar inlow-energy experiments, the limits from LHC would be different.

In the case of the scalar LQ S1, we have seen the dilepton data putting stronger bounds thanthe monolepton plus MET data [81]. Hence, in this paper, we consider only the dilepton (ττ andµµ [82, 83]) data to put bounds on the regions of U1 parameter space relevant for the RD(∗) andRK(∗) observables. Unlike the existing bounds on LQ masses from their pair production searchesat the LHC, the bounds thus obtained are model dependent (i.e., they depend on unknown cou-plings). However, for large new couplings they become more restrictive than the pair productionones. We obtain the LHC bounds for various scenarios with different coupling structures and showthat they are competitive and complimentary to other flavour bounds. Also, these bounds areindependent of other known theoretical constraints on the U1 parameter space. Obtaining themrequires a systematic consideration of different LQ signal processes at the LHC (including their

3

TABLE I. Summary of LQ mass exclusion limits from recent direct searches by the CMS (ATLAS) Collabora-tion. We recast some of the recent scalar searches (marked with “∗”) for better limits on U1 than the onesfor vLQs.

Integrated Scalar LQ Vector LQ, κ = 0 Vector LQ, κ = 1

Luminosity [fb−1] Mass [GeV] Mass [GeV] Mass [GeV]LQ→ tν (B = 1.0) [85, 87] 35.9 (36.1) 1020 (992) 1460 1780

LQ→ qν (B = 1.0) [85] 35.9 980 1410 1790

LQ→ bν (B = 1.0) [85, 87] 35.9 (36.1) 1100 (968) 1475 1810

LQ→ bτ /tν(B = 0.5) [88] 137 950 1290 1650

LQ→ bτ (B = 1.0) [87] ∗ (36.1) (1000) − −LQ→ µ j (B = 1.0) [86] ∗ (139) (1733) − −LQ→ µc (B = 1.0) [86] (139) (1680) − −LQ→ µb (B = 1.0) [86] ∗ (139) (1721) − −

interference with the SM backgrounds which plays the dominant role in determining the bounds).Here, for systematics, we largely follow the analysis of Ref. [81] (where a similar analysis wasdone for a S1-type scalar LQ that can alleviate the RD(∗) anomalies).

Before we proceed further, we review the direct detection bounds on LQs that couple withsecond- and third-generation fermions. Assuming the extra gluon-U1 coupling κ = 0 (we followthe same convention as [84]), a recent LQ pair production search at the CMS detector has excludedvLQs with masses below 1460 GeV for B(LQ→ tν) = 1 [85]. For a vLQ decaying to a light quarkand a neutrino with 100% BR, the mass exclusion limit is at 1410 GeV. In the case where it decaysto a bottom quark and a neutrino, the limit goes to 1475 GeV. If the vLQ decays to a top quark+ a neutrino and a bottom quark + τ with equal BRs, then the mass points below 1115 GeV areexcluded. For κ = 1, the limits go up [85]. Pair produced scalar LQs decaying to a light quarkand a neutrino with branching ratio unity can be excluded up to 980 GeV. A scalar LQ decaying toa b-quark and a neutrino with a 100% branching ratio can be excluded up to 1100 GeV [85]. TheATLAS experiment searched for scalar LQs decaying to the following final states, µc, µ + a lightquark, and µb [86]. The exclusion limits from these channels and the above are summarised inTable I.

The paper is organized as follows. In the next section, we introduce the U1 LQ model and therelevant scenarios. In Section III, we describe its LHC phenomenology. In Section IV, we discussthe dilepton search and their recasts. In Section V, we present the numerical results, and finally,in Section VI, we conclude.

II. THE U1 LEPTOQUARK MODEL

The interaction between U1 and the SM quarks and leptons can be expressed as [89–92],

L ⊃ xLL1 i j Qi

γµU µ

1 PLL j + xRR1 i j di

RγµU µ

1 PR`jR +H.c., (2)

if we ignore the diquark interactions which are severely constrained by the proton decay bounds.Here, Qi and L j denote the SM left-handed quark and lepton doublets, respectively and di

R and ` jR

are the down-type right-handed quarks and leptons, respectively. The indices i, j = {1, 2, 3} standfor quark and lepton generations; i.e., xLL

1 i j and xRR1 i j are 3×3 matrices in flavour space. In general,

these matrices are complex. We, however, simply assume them to be real since the LHC wouldbe mostly insensitive to their complex natures. Global fits to experimental data with complex

4

couplings are similar to the fits obtained with real couplings, albeit with slightly greater signifi-cance [93, 94]. Hence, predictions for flavor observables with complex couplings are expected tobe similar to the ones obtained with purely real couplings. Moreover, since we are interested inonly those U1 scenarios that can accommodate the RD(∗) and RK(∗) anomalies, we further simplifythe xLL

1 i j and xRR1 i j matrices by setting all the components that do not participate directly in these

decays to zero. We refer to any type of neutrinos simply as ν , i.e., without any flavour index as thiswould not affect our LHC analysis. As the b→ cτν and b→ sµ+µ− decays involve independentcouplings, we analyse the RD(∗)- and RK(∗)-anomalies-motivated scenarios separately.2

RD(∗) scenarios

In the SM, the b→ cτν transition is a tree-level charged-current-mediated process and the La-grangian responsible for it can be written as

LSM =−4GF√2

Vcb OVL =−4GF√

2Vcb [cγ

µPLb][τγµPLντ

]. (3)

New physics can generate additional contributions to the b→ cτν transition in the form of four-fermion operators. The most general form of the Lagrangian can be written as [95]

L ⊃−4GF√2

Vcb [(1+CVL)OVL +CVROVR +CSLOSL +CSROSR +CTROTR ] , (4)

where the Wilson coefficient corresponding to an operator Oi is denoted as Ci. The operators havethree different Lorentz structures:

• Vector: OVL = [cγµPLb]

[τγµPLν

]OVR = [cγµPRb]

[τγµPLν

]• Scalar:

OSL = [cPLb] [τPLν ]

OSR = [cPRb] [τPLν ]

• Tensor: OTL = [cσ µνPLb][τσµνPLν

].

From Fig. 1 we see that the cνU1 and bτU1 couplings have to be nonzero for U1 to contributein the b→ cτν process. We make the following flavour Ansatz for simplicity:

xLL1 =

0 0 00 0 λ L

230 0 λ L

33

, xRR1 =

0 0 00 0 00 0 λ R

33

. (5)

Given the Ansätze of the five operators listed above, only OVL and OSL can be generated by U1,i.e., C U1

VR= C U1

SR= C U1

TR= 0. Note that the simplified assumption of several zeros in the coupling

matrices are purely phenomenological. This may not be strictly valid in some specific models,e.g., in the models in Refs. [66, 71] where the LQ induced flavour structures are parametrised byFroggatt-Nielsen charges.

2 From here onwards, we refer to the RD(∗) - and RK(∗) -anomalies-motivated scenarios simply as RD(∗) and RK(∗) scenariosfor brevity.

5

TABLE II. Bounds on the RD(∗) scenarios.

Observable Experimentally Allowed Range SM Expectation Ratio Value

RD 0.340±0.027±0.013 [16] 0.299±0.003 [12] rD 1.137±0.101

RD∗ 0.295±0.011±0.008 [16] 0.258±0.005 [16] rD∗ 1.144±0.057

FL(D∗) 0.60±0.08±0.035 [10, 11] 0.46±0.04 [98] fL(D∗) 1.313±0.198

Pτ (D∗) −0.38±0.51+0.21−0.16 [97] −0.497±0.013 [95] pτ (D∗) 0.766±1.093

B(B→ τν) < (1.09±0.24)×10−4 [99] (0.812±0.054)×10−4 [100]B(Bc→ τν) < 10% [101]

The nonzero coefficients, CVL and CSL can be written in terms of the cνU1 and bτU1 couplings,

C U1VL

=1

2√

2GFVcb

λ Lcν

(λ L

bτ

)∗M2

U1

C U1SL

= − 12√

2GFVcb

2λ Lcν

(λ R

bτ

)∗M2

U1

. (6)

The actual relationship of λ Lcν and λ

L/Rbτ

with λ L23 and λ

L/R33 , defined in Eq. (5), varies from scenario

to scenario. We can express the ratios, rD(∗) =RD(∗)/RSMD(∗) in terms of the nonzero Wilson coefficients

as [96],

rD ≡RD

RSMD≈∣∣∣1+C U1

VL

∣∣∣2 +1.02∣∣∣C U1

SL

∣∣∣2 +1.49 Re[(1+C U1

VL)C U1∗

SL

], (7)

rD∗ ≡RD∗

RSMD∗≈∣∣∣1+C U1

VL

∣∣∣2 +0.04∣∣∣C U1

SL

∣∣∣2−0.11 Re[(1+C U1

VL)C U1∗

SL

]. (8)

There are two other observables where nonzeroC U1VL

andC U1SL

would contribute to – the longitudinalD∗ polarization FL(D∗) and the longitudinal τ polarization asymmetry Pτ(D∗). They have beenmeasured by the Belle Collaboration [10, 11, 97]. For our purpose, we can express FL(D∗) andPτ(D∗) as [96],

fL(D∗)≡FL(D∗)

FSML (D∗)

≈ 1rD∗

{|1+C U1

VL|2 +0.08 |C U1

SL|2−0.24 Re

[(1+C U1

VL)C U1∗

SL

]}, (9)

pτ(D∗)≡Pτ(D∗)

PSMτ (D∗)

≈ 1rD∗

{|1+C U1

VL|2−0.07 |C U1

SL|2 +0.22 Re

[(1+C U1

VL)C U1∗

SL

]}. (10)

A nonzero C U1VL

and C U1SL

would also contribute to leptonic decays Bc→ τν and B→ τν as,

B(Bc→ τν) =τBcmBc f 2

BcG2

F |Vcb|28π

m2τ

(1− m2

τ

m2Bc

)2∣∣∣∣∣1+C U1

VL+

m2Bc

mτ(mb +mc)C U1

SL

∣∣∣∣∣2

, (11)

B(B→ τν) = B(B→ τν)SM

∣∣∣∣1+C U1VL

+m2

B

mτ(mb +mu)C U1

SL

∣∣∣∣2 (12)

where τBc is the lifetime of the Bc meson, fBc is its decay constant, and B(B→ τν)SM is the branch-ing ratio within the SM. The LEP data put a constraint on the Bc → τν branching ratio [101]as, B(Bc → τν) < 10%. The experimental upper bound on the B→ τν decay is given as [99]B(B → τν) < (1.09± 0.24)× 10−4, and the corresponding SM branching ratio is estimated tobe [100] B(B→ τν)SM = (0.812± 0.054)× 10−4. The current bounds on these observables are

6

b

Bs

s

µ µ

s

b

U1

U1

Bs

(a)

B

K(∗)

bℓ−

s

ℓ+

U1

γτ

(b)

FIG. 2. (a) A representative diagram showing the U1 contribution to Bs-Bs mixing and (b) a U1-mediatedphoton penguin diagram contributing to b→ s`+`−.

summarized in Table II. Wherever applicable, we also consider constraints from Bs-Bs mixing [seeFig. 2(a)] through the effective Hamiltonian,

Heff = (C SMbox +C U1

box)(sLγαbL)(sLγαbL) (13)

where the SM contribution, C SMbox , and theU1 contribution, C U1

box, which generically depends on newcoupling(s) as ∼ λ 4, are given as

C SMbox =

G2F

4π2 (VtbVts)2M2

W S0(xt), (14)

C U1box =

λ 4

8π2M2U1

. (15)

In Eq. (15), the generation indices and possible Cabibbo-Kobayashi-Maskawa (CKM) elementshave been omitted as they depend on the scenario that we are interested in. The loop functionis the Inami-Lim function [102], S0(xt ≡ m2

t /m2W )∼ 2.37 [103]. The UT f it Collaboration gives the

following bounds on the ratio C U1box/C

SMbox [100]:

0.94 <

∣∣∣∣∣1+ C U1box

C SMbox

∣∣∣∣∣< 1.29. (16)

Additionally, whenever λ Lbτand λ L

sτ are simultaneously nonzero, they contribute to another lepton-flavour-universal operator in a log-enhanced manner through an off-shell photon penguin diagramas [see Fig. 2(b)]

L ⊃−4GF√2(VtbV ∗ts)C

univ9 Ouniv

9 (17)

where

Ouniv9 =

α

4π(sLγαbL)

( ¯γα`) and C univ

9 =− 1VtbV ∗ts

λ Lsτ

(λ L

bτ

)∗3√

2GFM2U1

log(m2b/M2

U1). (18)

We consider the 2σ limits from the global fits to the b→ sµ+µ− data [104–106] as−1.27≤C univ9 ≤

−0.51.

We now consider different scenarios with different combinations of the three couplings λ L23,

λ L33 and λ R

33. As indicated in the Introduction, these scenarios may not always appear very differ-ent from each other if we look at them only from the perspective of effective operators but their

7

LHC phenomenology are different. As a result, the bounds from the LHC data differ within thescenarios. We elaborate this point further shortly.� Scenario RD1A: In this scenario, only λ L

23 is assumed to be nonzero. This directly generates thefollowing two couplings: cνU1 and sτU1. We assume that the U1 interaction is aligned with thephysical basis of the up-type quarks. The interactions with the physical down-type quarks are thenobtained by rotating them with the CKMmatrix (i.e., by considering mixing among the down-typequarks) [81]. This way, an effective bτU1 coupling of strength V ∗cbλ L

23 is generated. The interactionLagrangian now reads as

L ⊃ λL23[cLγµνL + sLγµτL)]U

µ

1 ,

= λL23[cLγµνL +(V ∗cd dL +V ∗cssL +V ∗cbbL)γµτL)]U

µ

1 (19)giving

C RD1AVL

=1

2√

2GF

(λ L

23)2

M2U1

, C RD1ASL

= 0. (20)

This implies the observables, RD(∗) , FL(D∗), Pτ(D∗), and B(B(c)→ τν) would receive contributionsfromU1. Due to the off-shell photon-penguin diagram shown in Fig. 2, there will be a log-enhancedlepton-universal contribution to the b→ s`+`− transition [36]:

C univ9 =−VcbV ∗cs

VtbV ∗ts

(λ L

23)2

3√

2GFM2U1

log(m2b/M2

U1). (21)

This scenario would lead to a nonzero contribution to the Bs-Bs mixing coefficient as

C U1box =

|Vcb|2|Vcs|2(λ L23)

4

8π2M2U1

. (22)

The dominant decay modes of U1 in this scenario are U1 → cν and U1 → sτ+, and both of themshare almost 50% BR.� Scenario RD1B: In this scenario, only λ L

33 is assumed to be nonzero, thus generating the bτU1

and tνU1 couplings. Assuming the U1 interaction to be aligned with the physical basis of thedown-type quarks, we generate cνU1 coupling Vcbλ L

33 through the mixing in the up-type quarks.The interaction Lagrangian is given by

L ⊃ λL33[tLγµνL + bLγµτL]U

µ

1

= λL33[(VubuL +VcbcL +VtbtL)γµνL)+ bLγµτL]U

µ

1 , (23)and the contributions to the Wilson coefficients are given by

C RD1BVL

=1

2√

2GF

(λ L

33)2

M2U1

, C RD1BSL

= 0. (24)

Like in Scenario RD1A, the observables RD(∗) , FL(D∗), Pτ(D∗), and B(B(c) → τν) would receivecontribution from U1 in this case too. Here, the dominant decay modes of U1 are U1 → tν andU1→ bτ+ with 50% BR each.� Scenario RD2A: In this scenario, we assume λ L

23 and λ L33 to be nonzero, and the interaction of

U1 is aligned with the physical basis of the down-type quarks. The interaction Lagrangian can bewritten as

L ⊃ [λ L23(cLγµνL + sLγµτL)+λ

L33(tLγµνL + bLγµτL)]U

µ

1

= [λ L23(VusuLγµνL +VcscLγµνL +VtstLγµνL + sLγµτL)

+λL33(VubuLγµνL +VcbcLγµνL +VtbtLγµνL + bLγµτL)]U

µ

1 , (25)

8

where, in the second step, we have assumed mixing among the up-type quarks. In the absence ofλ R

23, in this case, C RD2AVL

is the only nonzero Wilson coefficient, i.e.,

C RD2AVL

=1

2√

2GFVcb

(VcsλL23 +Vcbλ L

33)λL33

M2U1

, C RD2ASL

= 0 . (26)

In addition to the contribution to the RD(∗) , FL(D∗), Pτ(D∗), andB(B(c)→ τν) processes, we considerthe lepton flavour-universal contribution

C univ9 =− 1

VtbV ∗ts

λ L23λ L

33

3√

2GFM2U1

log(m2b/M2

U1). (27)

In this scenario, the Bs-Bs mixing coefficient would receive a contribution from U1

C U1box =

(λ L

23)2 (

λ L33)2

8π2M2U1

. (28)

Here, U1 can decay to cν , sτ+, tν and bτ+ final states with comparable BRs.� Scenario RD2B: Here, both λ L

23 and λ R33 are nonzero. Ignoring possible CKM-suppressed cou-

plings, the interaction Lagrangian is given byL ⊃ [λ L

23(cLγµνL + sLγµτL)+λR33bRγµτR]U

µ

1

= [λ L23(VusuLγµνL +VcscLγµνL +VtstLγµνL + sLγµτL)+λ

R33bRγµτR]U

µ

1 (29)where, once again in the second step we have assumed mixing among the up-type quarks. Thisgives the following contribution to CSL:

C RD2BVL

= 0 , C RD2BSL

=− Vcs√2GFVcb

λ L23λ R

33

M2U1

. (30)

Here, RD(∗) , FL(D∗), Pτ(D∗), and B(B(c)→ τν) would receive contributions from U1. The dominantdecay modes of U1 are U1→ cν , U1→ sτ+, and U1→ bτ+. Note that even though λ L

33 = 0 in thisscenario, a small CVL can be generated from effective λ L

33 coupling if, instead of up-type quarkmixing, one assumes mixing in the down sector (like in Scenario RD1A).� Scenario RD3: All the three free couplings λ L

23, λ L33, and λ R

33 are free to vary. Assuming mixingin the up-type quark sector, the interaction Lagrangian is given by

L ⊃ [λ L23(cLγµνL + sLγµτL)+λ

L33(tLγµνL + bLγµτL)+λ

R33bRγµτR]U

µ

1

= [λ L23(VusuLγµνL +VcscLγµνL +VtstLγµνL + sLγµτL)

+λL33(VubuLγµνL +VcbcLγµνL +VtbtLγµνL + bLγµτL)+λ

R33bRγµτR]U

µ

1 . (31)This Lagrangian contributes to CVL and CSL as

C RD3VL

=1

2√

2GFVcb

(Vcbλ L

33 +VcsλL23)

λ L33

M2U1

, C RD3SL

=− 1√2GFVcb

(Vcbλ L

33 +VcsλL23)

λ R33

M2U1

. (32)

The lepton flavour-universal contribution through the off-shell photon penguin diagram is

C univ9 =− 1

VtbV ∗ts

λ L23λ L

33

3√

2GFM2U1

log(m2b/M2

U1). (33)

The contribution of U1 to the Bs-Bs mixing coefficient is given as

C U1box =

(λ L

23)2 (

λ L33)2

8π2M2U1

. (34)

In this scenario, U1 dominantly decays to cν , sτ+, tν , and bτ+ final states.

9

RK(∗) scenarios

A general Lagrangian for b→ sµ+µ− transition can be written as [107, 108]

L ⊃ 4GF√2

VtbV ∗ts ∑i=9,10,S,P

(CiOi +C ′i O

′i) (35)

where the Wilson coefficients are evaluated at µren = mb. The operators are given by

O9 =α

4π(sLγαbL)(µγ

αµ) , O ′9 =

α

4π(sRγαbR)(µγ

αµ) ,

O10 =α

4π(sLγαbL)(µγ

αγ5µ) , O ′10 =

α

4π(sRγαbR)(µγ

αγ5µ) ,

OS =α

4π(sLbR)(µµ) , O ′S =

α

4π(sRbL)(µµ) ,

OP =α

4π(sLbR)(µγ5µ) , O ′P =

α

4π(sRbL)(µγ5µ)

where α is the fine-structure constant. Keeping the RK(∗) observables in mind, we make the fol-lowing simple Ansatz:

xLL1 =

0 0 00 λ L

22 00 λ L

32 0

; xRR1 =

0 0 00 λ R

22 00 λ R

32 0

. (36)

The U1 contribution to the Wilson coefficients can be written in terms of the bµU1 and sµU1 cou-plings in general as

C U19 = −C U1

10 =π√

2GFVtbV ∗tsα

λ Lsµ(λ

Lbµ)∗

M2U1

C U1S = −C U1

P =

√2π

GFVtbV ∗tsα

λ Lsµ(λ

Rbµ)∗

M2U1

C ′ U19 = C ′ U1

10 =π√

2GFVtbV ∗tsα

λ Rsµ(λ

R∗bµ)

M2U1

C ′ U1S = C ′ U1

P =

√2π

GFVtbV ∗tsα

λ Rsµ(λ

L∗bµ)

M2U1

. (37)

Like in the RD(∗) scenarios, the relationship between {λ L/Rsµ ,λ

L/Rbµ} with {λ L/R

22 ,λL/R32 } would depend

on the particulars of the scenario we consider. The relevant global fits of the Wilson coefficientsto the b→ sµ+µ− data are taken from Refs. [105, 106, 109] and are listed in Table III.� Scenario RK1A: In this scenario, only λ L

22 is nonzero. This generates the sµU1 coupling. ThebµU1 coupling is generated via CKM mixing in the down-quark sector (as in Scenario RD1A andScenario RD1B). The interaction Lagrangian can be written as

L ⊃ λL22[cLγµνL +(V ∗cd dL +V ∗cssL +V ∗cbbL)γµ µL)]U

µ

1 . (38)

This Lagrangian contributes to the following coefficients:

C RK1A9 =−C RK1A

10 =πVcbV ∗cs√

2GFVtbV ∗tsα(λ L

22)2

M2U1

. (39)

10

TABLE III. Global fits of relevant combinations ofWilson coefficients in b→ sµµ observables [105, 106, 109].

Combinations Best fit 1σ 2σ Corresponding scenarios

CU19 =−CU1

10 −0.44 [−0.52,−0.37] [−0.60,−0.29] RK1A, RK1B, RK2ACU1

S =−CU1P −0.0252 [−0.0378,−0.126] [−0.0588,−0.0042] RK2B

C ′ U19 = C ′ U1

10 +0.06 [−0.18,+0.30] [−0.42,+0.55] RK1C, RK1D, RK2DC ′ U1

S = C ′ U1P −0.0252 [−0.0378,−0.126] [−0.0588,−0.0042] RK2C

The contribution to the Bs-Bs mixing coefficient is

C U1box =

|Vcb|2|Vcs|2(λ L22)

4

8π2M2U1

. (40)

The dominant decay modes ofU1 in this case areU1→ cν andU1→ sµ+ with almost 50% BR each.� Scenario RK1B: Only λ L

32 is nonzero. The interaction Lagrangian is given by

L ⊃ λL32[tLγµνL +(V ∗td dL +V ∗ts sL +V ∗tbbL)γµ µL)]U

µ

1 . (41)

The relevant Wilson coefficients are given by

C RK1B9 =−C RK1B

10 =π√

2GFα

(λ L32)

2

M2U1

, (42)

and the contribution to the Bs-Bs mixing coefficient is given as

C U1box =

|Vtb|2|Vts|2(λ L32)

4

8π2M2U1

. (43)

Here, the sµU1 coupling is V ∗ts-suppressed. The coupling λ L32 alone, however, cannot explain the

R(∗)K anomalies. From Table III we see that the anomalies need a negative C9, whereas the r.h.s. of

Eq. (42) is always positive (even if we consider a complex λ L32). The dominant decay modes of U1

in this case are U1→ tν and U1→ bµ+, and they share almost 50% BR each.� Scenario RK1C: In this scenario, we assume only λ R

22 to be nonzero. The interaction Lagrangianis given by,

L ⊃ λR22[(Vcd dR +VcssR +VcbbR)γµ µR]U

µ

1 . (44)

The nonzero Wilson coefficients from Eq. (37) are

C ′RK1C9 = C ′RK1C

10 =πV ∗cbVcs√

2GFVtbV ∗tsα(λ R

22)2

M2U1

, (45)

and the contribution to the Bs-Bs mixing coefficient is

C U1box =

|Vcb|2|Vcs|2(λ R22)

4

8π2M2U1

. (46)

Here, the bµU1 coupling is V ∗cb suppressed. In this scenario, the U1→ sµ+ decay mode has almost100% BR.� Scenario RK1D: We assume λ R

32 to be nonzero and the rest of the couplings to be SM-like. Theinteraction Lagrangian is given by

L ⊃ λR32[(Vtd dR +VtssR +VtbbR)γµ µR]U

µ

1 (47)

11

where the sµU1 coupling is Vts suppressed. The nonzero Wilson coefficients are

C ′RK1D9 = C ′RK1D

10 =πV ∗tbVts√

2GFVtbV ∗tsα

(λ R32)

2

M2U1

. (48)

In this scenario, the U1→ bµ+ decay mode is dominant with almost 100% BR. The contribution tothe Bs-Bs mixing coefficient is given as

C U1box =

|Vtb|2|Vts|2(λ R32)

4

8π2M2U1

. (49)

� Scenario RK2A: In this scenario, two couplings, namely, λ L22 and λ L

32 are nonzero. The interac-tion Lagrangian is given by

L ⊃ [λ L22(cLγµνL + sLγµ µL)+λ

L32(tLγµνL + bLγµ µL)]U

µ

1 . (50)

Here, we have not shown the CKM-suppressed couplings. The Wilson coefficients getting thedominant contributions are

C RK2A9 =−C RK2A

10 ≈ π√2GFVtbV ∗tsα

λ L22λ L

32

M2U1

. (51)

The contribution to the Bs-Bs mixing coefficient is

C U1box =

(λ L22)

2(λ L32)

2

8π2M2U1

. (52)

In this scenario, the dominant decay modes for U1 are bµ+, sµ+, cν , and tν .� Scenario RK2B: In this scenario, only λ L

22 and λ R32 are nonzero. The interaction Lagrangian is

given by

L ⊃ [λ L22(cLγµνL + sLγµ µL)+λ

R32bRγµ µR]U

µ

1 (53)

Here, once again, the CKM-suppressed couplings are ignored. The Wilson coefficients getting thedominant contributions are

−C RK2BP = C RK2B

S ≈√

2π

GFVtbV ∗tsαλ L

22λ R32

M2U1

. (54)

For this scenario, Bs-Bs mixing is not relevant. The dominant decay modes of U1 are bµ+, sµ+,and cν .� Scenario RK2C: Only λ R

22 and λ L32 are nonzero. Ignoring the CKM-suppressed couplings, we get

the following interaction Lagrangian:

L ⊃ [λ R22sRγµ µR +λ

L32(tLγµνL + bLγµ µL)]U

µ

1 (55)

The Wilson coefficients getting the dominant contributions are

C ′RK2CP = C ′RK2C

S ≈√

2π

GFVtbV ∗tsαλ R

22λ L32

M2U1

. (56)

In this case also Bs-Bs mixing is not relevant. The dominant decay modes of U1 are bµ+, sµ+, andtν .

12

� Scenario RK2D: Only λ R22 and λ R

32 are nonzero. Ignoring the CKM-suppressed couplings, we get

L ⊃ (λ R22sRγµ µR +λ

R32bRγµ µR)U

µ

1 . (57)

The Wilson coefficients getting the dominant contributions are

C ′RK2D9 = C ′RK2D

10 ≈ π√2GFVtbV ∗tsα

λ R22λ R

32

M2U1

. (58)

The contribution to the Bs-Bs mixing coefficient is given as

C U1box =

(λ R22)

2(λ R32)

2

8π2M2U1

. (59)

The dominant decay modes of U1 are bµ+ and sµ+.� Scenario RK4: All couplings are nonzero. In this scenario, the interaction Lagrangian is givenby,

L ⊃[λ

L22(cLγµνL + sLγµ µL)+λ

L32(tLγµνL + bLγµ µL)+λ

R22sRγµ µR +λ

R32bRγµ µR

]U µ

1 , (60)

and the dominant contributions to the Wilson coefficients can be read from Eq. (37). The maindecay modes of U1 are cν , tν , sµ+, and bµ+.

Our selection of scenarios motivated by the R(∗)K anomalies is not exhaustive. For example, we do

not consider any three-coupling scenarios. (One can define RK3X scenarios by taking combinationsof three couplings at a time for completeness. We, however, skip the three-coupling-RK(∗) scenariossince they would not add anything significant to our study.) The single-coupling scenarios can bethought of as templates that can help us read bounds on scenarios where more than one couplingsare nonzero [63, 81]. In Table III, we show the relevant global fits for the one- and two-couplingscenarios. We have summarised the couplings that contribute to the RD(∗) and RK(∗) observables indifferent scenarios in Table IV.

As mentioned earlier, one of the reasons for considering the RD(∗) and RK(∗) scenarios is that they canhave different signatures at the LHC. We are now in a position to illustrate that point further. Let usconsider the first two RD(∗)-motivated one-coupling scenarios – Scenario RD1A and Scenario RD1B.In both cases, CVL receives a nonzero contribution proportional to the square of an unknown newcoupling (either λ L

23 or λ L33). Hence, from an effective theory perspective, these two look almost the

same. However, the dominant decay modes of U1 in these two scenarios are different – in the firstone, they areU1→ cν andU1→ sτ, whereas in the second one, they areU1→ tν andU1→ bτ.3 As aresult, aU1 can produce t+ /ET or τ+b signatures in the second scenario, as opposed to the jet+ /ET

or τ + jet signatures in the first one. Not only that, in the first scenario, aU1 can be produced via c-or s-quark-initiated processes, as compared to the b-quark-initiated processes in the second one.Hence, in these two scenarios, U1 would have different single production processes. Moreover,since the b-quark parton distribution function (PDF) is smaller than the second-generation ones,U1 production cross sections would be higher in Scenario RD1A than those in Scenario RD1B.Hence, one needs to analyse the LHC bounds for the scenarios differently.

III. PRODUCTION MODES AND DECAYS

We now explore the possible LHC signatures of the minimal scenarios with only one free couplingand the next-to-minimal scenarios with more than one nonzero couplings we constructed in the

3 From here on, unless necessary, we do not distinguish between particles and their antiparticles as it is not importantfor the LHC analysis.

13

TABLE IV. Summary of the coupling combinations that contribute to the RD(∗) and RK(∗) observables indifferent one-, two- and multi-coupling scenarios.

RD(∗) scenarios λ Lcν λ L

bτλ R

bτRK(∗) scenarios λ L

sµ λ Lbµ

λ Rsµ λ R

bµ

RD1A λ L23 V ∗cbλ L

23 − RK1A V ∗csλL22 V ∗cbλ L

22 − −RD1B Vcbλ L

33 λ L33 − RK1B V ∗tsλ L

32 V ∗tbλ L32 − −

RK1C − − VcsλR22 Vcbλ R

22

RK1D − − VtsλR32 Vtbλ R

32

RD2A VcsλL23 +Vcbλ L

33 λ L33 − RK2A λ L

22 λ L32 − −

RD2B VcsλL23 − λ R

33 RK2B λ L22 − − λ R

32

RK2C − λ L32 λ R

22 −RK2D − − λ R

22 λ R32

RD3 Vcbλ L33 +Vcsλ

L23 λ L

33 λ R33 RK4 λ L

22 λ L32 λ R

22 λ R32

g

g

g U1

U1

(a)

q

q

ℓ

U1

U1

(b)

qℓ

U1

g U1

(c)

qℓ

U1

q ℓ

(d)

FIG. 3. Representative Feynman diagrams for various U1 production processes: (a) gluon-initiated pairproduction, (b) quark-initiated pair production, (c) single production, and (d) t-channel (nonresonant)production. The q`U1 vertices (λ) are marked with red colour.

previous section. There are different ways to produce U1 at the LHC (see Fig. 3) – resonantly(through pair and single productions) and nonresonantly (through t-channelU1 exchange). Below,we briefly discuss various production channels and the subsequent decaymodes ofU1 that can arisein the flavour-motivated scenarios. We also discuss how different production modes with similarfinal states can contribute to the exclusion limits.

Pair production

We have classified the RD(∗) scenarios with the three free couplings, λ L23, λ L

33, and λ R33. In Sce-

nario RD1A (where only λ L23 is nonzero), U1→ sτ and U1→ cν are the main decay modes of U1

with roughly equal (about 50%) BRs. In this case the pair production of U1 leads to the follow-ing final states (we ignore the CKM-suppressed effective couplings in the discussions on the LHCphenomenology of U1 as they do not play any important role):

pp→

U1U1 → sτ sτ ≡ ττ +2 jU1U1 → sτ cν ≡ τ + /ET +2 jU1U1 → cν cν ≡ /ET +2 j

(61)

where j denotes a light jet or a b-jet. Among the three channels, the second one (i.e., τ + /ET +2 j)has almost two times the cross section of the first or the third (a factor of 2 comes from combina-torics), but due to the presence of missing energy, it is not fully reconstructable (or, is difficult to

14

reconstruct). As a result, both the first and second channels have comparable sensitivities. How-ever, the sensitivity of the third channel, /ET +2 j, is very poor because of the two neutrinos in thefinal state. So far, these channels with cross-generation couplings have not been used in any LQsearch at the LHC.

In Scenario RD1B (where only λ L33 is nonzero), the pair production of U1 mostly leads to the

following final states:

pp→

U1U1 → bτ bτ ≡ ττ +2 jU1U1 → bτ tν ≡ τ + /ET + jt + jU1U1 → tν tν ≡ /ET +2 jt

. (62)

Here, jt represents a fat-jet originating from a top quark decaying hadronically (one can alsoconsider the top quark’s leptonic decay modes with lower cross section). It is possible to tag the(boosted) top-jets with sophisticated jet-substructure techniques and thus improve the second andthird channels’ prospects. The symmetric /ET +2 jt channel has been considered in Refs. [58, 110].The asymmetric channel, the one with single τ, one top-jet, and missing energy (τ + /ET + jt +b), has started receiving attention only very recently [88]. Due to the factor of 2 coming fromcombinatorics, this channel has a bigger cross section. Hence, its unique final state might act as asmoking-gun signature for this type of scenarios (i.e., ones with non-negligible λ L

33).If only λ R

33 is nonzero, U1 cannot resolve the RD(∗) anomalies anymore as it is not possible togenerate the necessary couplings in that case. Here, U1 entirely decays through the U1→ bτ modeand contributes to the bτ bτ ≡ ττ +2 j final state [85].

When two or more couplings are nonzero simultaneously (Scenario RD2A, Scenario RD2B andScenario RD3) with comparable strengths, numerous possibilities arise (Reference [63] discussesthis in the context of scalar LQ searches). It is then possible to have all the final states shown inEqs. (61) and (62). One can have more asymmetric channels like pp→ U1U1 → sτbτ etc. Thestrength of any particular channel would depend on the couplings involved in production (if wedo not ignore the small t-channel lepton exchange) as well as the BRs involved (the dependenceof the pair production signal on multiple couplings is made explicit in Appendix A).

The RK(∗) scenarios have similar signatures with muons in the final states. When only λ22 isnonzero (Scenario RK1A), we can easily obtain the possible final states by replacing τ → µ inEq. (61). In Scenario RK1B, the possible final states are obtained by replacing τ → µ in Eq. (62).In Scenario RK1C, the BR of the U1→ sµ decay is 100% leading to the process, U1U1→ sµ sµ ≡µµ + 2 j. Similarly, in Scenario RK1D, the BR of the U1→ bµ decay is 100% leading to the sametwo-muon+two-jet final states through the U1U1 → bµ bµ ≡ µµ + 2 j process. Like the RD(∗) sce-narios with more than one nonzero couplings, these scenarios also lead to numerous interestingpossibilities [63]. The LHC is yet to perform searches for LQs in most of the asymmetric channelsand some of the symmetric channels.

In Table V, we have summarized the possible final states from U1 pair production and thefraction ofU1 pairs producing the final states in the one- and two-coupling scenarios. The fractionsdepend on combinatorics and the relevant U1 BRs. (Here, we have ignored the possible minorcorrection due the the mass differences between different final states, i.e., assumed all final stateparticles are much lighter than U1.) For example, in Scenario RD1A, since β (U1→ sτ) ≈ β (U1→cν)≈ 50%, only 25% of the produced U1 pairs would decay to either ττ +2 j or /ET +2 j, whereas,as explained above, 50% of them would decay to the τ + /ET + 2 j final state. Interestingly, wesee that even in some two-coupling scenarios the fractions corresponding to the ττ/µµ +2 j finalstates are constant irrespective of the relative magnitudes of the couplings – for example, it is 25%in Scenario RD2A or 100% in Scenario RK2D. This is interesting, because in the presence of twononzero couplings, one normally expects the fraction corresponding to a particular final state todepend on their relative strengths. This, of course, happens because we sum over the possibleflavours of the jets. Moreover, we show that it is possible to parametrise all final states with just

15

TABLE V. Effect of branching ratios on different final states generated from the pp→U1U1 process in var-ious one and two-coupling scenarios. Here, we show the possible final states and the fraction of U1 pairsproducing them. One multiplies the pair production cross section with the fractions shown in the table toestimate its contribution to various channels in the narrow width approximation. Here, 0≤ ξ ≤ 1

2 is a freeparameter. We have ignored the mass differences among the daughter particles.Nonzero couplings Signatures

ττ +2 j τ + /ET +2 j /ET +2 j τ + /ET + jt + j /ET +2 jt /ET + jt + j

λ L23 (Scenario RD1A) 0.25 0.50 0.25 − − −

λ L33 (Scenario RD1B) 0.25 − − 0.50 0.25 −

λ R33 1.00 − − − − −

λ L23,λ

L33 (Scenario RD2A) 0.25 ξ ξ 2 1

2 −ξ( 1

2 −ξ)2

2ξ( 1

2 −ξ)

λ L23,λ

R33 (Scenario RD2B) ( 1

2 +ξ)2

2( 1

4 −ξ 2) ( 12 −ξ

)2 − − −µµ +2 j µ + /ET +2 j /ET +2 j µ + /ET + jt + j /ET +2 jt /ET + jt + j

λ L22 (Scenario RK1A) 0.25 0.50 0.25 − − −

λ L32 (Scenario RK1B) 0.25 − − 0.50 0.25 −

λ R22 (Scenario RK1C) 1.00 − − − − −

λ R32 (Scenario RK1D) 1.00 − − − − −

λ L22,λ

L32 (Scenario RK2A) 0.25 ξ ξ 2 1

2 −ξ( 1

2 −ξ)2

2ξ( 1

2 −ξ)

λ L22,λ

R32 (Scenario RK2B) ( 1

2 +ξ)2

2( 1

4 −ξ 2) ( 12 −ξ

)2 − − −λ R

22,λL32 (Scenario RK2C) ( 1

2 +ξ)2 − − 2

( 14 −ξ 2) ( 1

2 −ξ)2 −

λ R22,λ

R32 (Scenario RK2D) 1.00 − − − − −

one free parameter (ξ ). Such simple parametrisations could guide us in future U1 searches at theLHC.

Note that the model dependence of the pair production ofU1 appears in two places. One occursthrough the free parameter κ present in the kinetic terms (igsκU†

1µT aU1νGa µν) [84, 92]. The pair

production cross section depends on κ. The other occurs in the contribution of the t-channellepton/neutrino exchange. The amplitudes of these diagrams grow as λ 2, and the cross sectiongrows as λ 4. Although the λ dependence of the pair production is negligible for small λ values, itcan become significant for larger couplings. As we see later, the pair production channels produce arelativelyminor contribution to the final exclusion limits. Therefore, we take a benchmark value forκ by setting κ = 0 in our analysis. However, we keep the λ -dependent terms in the pair productioncontributions (see Appendix A).

Single production

In the single-production channels, a U1 is produced in association with other SM particles. Thereare two types of single productions of our interest: (a) where a U1 is produced in association witha lepton, i.e., U1µ, U1τ or U1ν and (b) where a U1 is produced with a lepton and a jet, i.e., U1µ j,U1τ j or U1ν j. One has to be careful while computing the second type of process as the set ofFeynman diagrams for them might overlap with the pair production ones when the lepton-jet pairoriginates in a LQ decay. We keep the two types of single production contributions in our anal-ysis by carefully avoiding any double-counting with the pair production contribution [111–113].Single productions of U1 are fully model-dependent processes; they depend on the coupling λ aswell as κ [84]. Like the pair production, the single production processes can also be categorisedinto symmetric and asymmetric channels [63]. In Scenario RD1A, we have the following single

16

production channels:

pp→

U1τ +U1τ j → (sτ)τ +(sτ)τ j ≡ ττ +n jU1ν +U1ν j → (cν)ν +(cν)ν j ≡ /ET +n jU1τ +U1τ j → (cν)τ +(cν)τ j ≡ τ + /ET +n jU1ν +U1ν j → (sτ)ν +(sτ)ν j ≡ τ + /ET +n j

. (63)

Notice that the single production processes produce similar final states as the pair production. Inthe above equation, the first and the second channels are symmetric, whereas the third and theforth are asymmetric. In the τ + /ET +n j final state, both pp→U1τ +U1τ j and pp→U1ν +U1ν jcontribute. This channel also has not been considered for LQ searches so far. Scenario RD1B is verysimilar to Scenario RD1A and gives the same final states if we treat the b-jet as a light jet. If onlyλ R

33 is nonzero, U1 decays only to bτ. Thus this scenario only leads to the (bτ)τ +(bτ)τ j ≡ ττ +n jfinal state.

The possible final states in case of Scenario RK1A can be obtained by replacing τ → µ inEq. (63). In Scenario RK1B, we have some interesting signatures from boosted top quarks inthe final states,

pp→

U1µ +U1µ j → (bµ)µ +(bµ)µ j ≡ µµ +n jU1µ +U1µ j → (tν)µ +(tν)µ j ≡ µ + /ET + jt +n jU1ν +U1ν j → (bµ)ν +(bµ)ν j ≡ µ + /ET +n jU1ν +U1ν j → (tν)ν +(tν)ν j ≡ /ET + jt +n j

. (64)

These final states can also come from pair production in Scenario RK1B. In Scenario RK1C, theU1→ sµ decay has 100% BR, and it leads to the process U1µ +U1µ j→ (sµ)µ +(sµ)µ j ≡ µµ +n j.In Scenario RK1D, the U1→ bµ decay mode has 100% BR. It leads to the process U1µ +U1µ j→(bµ)µ +(bµ)µ j ≡ µµ +n j.

Nonresonant production and interference

A U1 can be exchanged in the t-channel and give rise to both dilepton and lepton+missing-energyfinal states [see e.g., Fig. 3(d)]. As the cross sections of the nonresonant production grows asλ 4, this channel becomes important for large values of the new couplings. Especially when themass of the U1 is large, the nonresonant production contributes more than the resonant pair andsingle productions. There is a possibility of large interference of the nonresonant processes withthe SM backgrounds like pp→ γ/Z(W )→ `` (`+ /ET ). The interference contribution grows as λ 2

but the contribution can be significant due to the large SM background. For U1, the interferenceis destructive in nature. However, depending on the parameter/kinematic region we consider, thecross section of the exclusive pp→ `` (`+ /ET ) process can be bigger or smaller than the SM-onlycontribution [57] because the total nonresonant contribution, including the term proportional toλ 4 and the destructive λ 2 term, can be both positive or negative.

In Fig. 4, we show the parton-level cross sections of various production modes of U1 as a functionof MU1 . In Figs. 4(a) and 4(b) the cross sections have been obtained by setting κ=0 and the newcouplings, λ L

23 = 1 and λ L22 = 1 respectively. The pair production cross section is the same in both

figures as it is insensitive to the λ couplings. As expected, the single production cross sections aremore significant at higher mass values. Processes likeU1τ j,U1µ j,U1ν j are generated after ensuringthat no more than one onshell LQ contributes to the cross section to avoid contamination fromthe pair production process. The nonresonant LQ production cross section does not depend verystrongly on the LQ mass. With nonzero λ L

23 and λ L22, we now have the possibility of producing U1

17

10−6

10−5

10−4

10−3

10−2

10−1

100

1 1.5 2 2.5 3

σ(p

b)

MU1 (TeV)

U1U1U1τU1τjU1νU1νj

τττν

(a)

10−6

10−5

10−4

10−3

10−2

10−1

100

1 1.5 2 2.5 3

σ(p

b)

MU1 (TeV)

U1U1U1µU1µjU1νU1νjµµµν

(b)

FIG. 4. Parton-level cross sections of various production modes of U1 LQ as functions of MU1 . These crosssections are computed at the 13 TeV LHC for benchmark couplings, λ L

23 = 1 (left) and λ L22 = 1 (right) with

κ = 0. Here, j stands for all light jets including the b-jet. A basic generation-level cut, pT > 20 GeV is appliedon the jets and leptons.

(that couples to the third-generation fermions) through charm- and/or strange-initiated processesat the LHC.

There are some phenomenological consequences of having more than one coupling. The pres-ence of multiple couplings affects the BRs. For example, we see from Table V that BRs for one-coupling scenarios are different from those in two coupling ones. Then, different single and non-resonant production (including its interference with the SM background) processes may or maynot become significant depending on the strength of various couplings. All these can significantlyaffect the exclusion limits.

IV. RECAST OF DILEPTON DATA

From the different production mechanisms of U1 discussed in the previous section, it is evidentthat pair, single and nonresonant productions can give rise to dilepton (``+ jets) and/or monolep-ton plus missing-energy, `+ /ET + jets signatures. However, as pointed out in Ref. [81] for S1 LQ,the bounds on the LQ model parameter space from the dilepton resonance search data are morestringent. Therefore, apart from the direct search bounds, we rely only on the resonant dileptonsearches (pp→ Z′→ ``) [82, 83] and recast the bounds in terms of U1 parameters for various sce-narios. Note that the number of jets are not restricted in those searches, and hence all productionmodes of U1 with ``+ jets final states would contribute in the exclusion limits. As shown in [81],the interference of the t-channel U1 exchange process with the SM background play the leadingrole in determining the exclusion limits. However, pair, single, and nonresonant productions alsocontribute non-negligibly, especially in the lower mass region. Since, the kinematics of differentU1 contributions to the ``+ jets channel are different from those of the resonant dilepton produc-tion (pp→ Z′→ ``), recasting is nontrivial, especially when multiple new couplings are present.Possible interference among different signal processes complicate the recasting further. We sys-tematically take care of all these factors in our analysis. We explain our method in Appendix A.

18

ATLAS ττ search

The ATLAS Collaboration searched for a heavy particle decaying to two taus at the 13 TeV LHCwith 139 fb−1 integrated luminosity [82]. The analysis comprised events categorised on the basisof two modes of τ decays. In the first, one has both τs decaying hadronically (τhadτhad). In thesecond, one tau decays hadronically and the other leptonically (τhadτlep). We provide an outlineof the basic event selection criteria for the ττ channel.

• The τhadτhad channel has

– at least two hadronically decaying τ ’s with no additional electrons or muons,– two τhad ’s with pT > 65 GeV. They should be oppositely charged and separated in the

azimuthal plane by |∆φ(pτ1T , pτ2

T )| > 2.7 rad.– pT of leading τ must be > 85 GeV.

• The τlepτhad channel has one τhad and only one `= e or µ such that

– the hadronic τ has pT > 25 GeV and |η(τhad)| < 2.5(excluding 1.37 < |η| < 1.52),– if `= e, then |η| < 2.47 (excluding 1.37 < |η| < 1.52) and if `= µ then |η| < 2.5,– the lepton has pT(`)> 30 GeV with azimuthal separation from the τhad , |∆φ(p`T, pτhad

T )|> 2.4.

– the transversemass on the selected lepton andmissing transversemomentum, mT(p`T, /ET)

> 40 GeV.– If `= e, to reduce the background from Z→ ee events with an invariant mass for τ− `

pair between 80 GeV and 110 GeV are rejected.

The transverse mass is defined as

mT(pAT, pB

T) =

√2pA

T pBT

{1− cos∆φ(pA

T, pBT)}. (65)

The analysis also make use of the total transverse mass defined as

mtotT (τ1,τ2, /ET) =

√m2

T (pτ1T , pτ2

T )+m2T (pτ1

T , /ET)+m2T (pτ2

T , /ET), (66)

Here, τ2 in the τlepτhad channel represents the lepton. We use the distribution of the observed andthe SM events with respect to mtot

T presented in the analysis.

CMS µµ search

A search for nonresonant excesses in the dilepton channel was performed by the CMS experimentat a centre-of-mass energy of 13 TeV corresponding to a integrated luminosity of 140 fb−1 [83].The event selection criteria that we use in our analysis can be summarised as

• In the dimuon channel, the requirement is that both of the muons must have |η |< 2.4 andpT > 53 GeV. The invariant mass of the muon pair is mµµ > 150 GeV.

We use the distribution of the observed and the SM events with respect to the invariant mass ofthe muon pair, mµµ to extract bounds.

19

TABLE VI. The table displays the cross section (σ) in fb, efficiency (ε) in % and number of events (N )

surviving the cuts applied in the dilepton searches from various production processes. The superscripts areexplained in Appendix A. The negative signs in the interference contributions signify destructive interfer-ence.

Pair production Single production t-channel LQ InterferenceMass(Tev) σ p ε p N p σ s εs N s σnr4 εnr4 N nr4 σnr2 εnr2 N nr2

Contribution to ττ signal [82]λ

L23 = 1 (Scenario RD1A)

1.0 40.87 2.33 8.59 58.80 3.30 35.07 70.57 7.22 183.33 -232.63 3.17 -266.211.5 1.39 1.50 0.19 3.91 2.74 1.93 14.94 7.00 37.77 -104.31 3.34 -125.622.0 0.08 1.01 0.01 0.44 2.50 0.20 5.04 7.25 13.19 -58.79 3.28 -69.57λ

L33 = 1 (Scenario RD1B)

1.0 35.67 1.69 5.43 29.00 2.57 13.46 20.20 6.21 45.26 -75.02 3.08 -83.411.5 1.17 1.09 0.11 1.72 2.16 0.67 4.31 6.22 9.68 -33.62 2.88 -33.012.0 0.06 0.81 0.00 0.17 1.98 0.06 1.39 6.27 3.15 -18.97 2.88 -19.71λ

R33 = 1

1.0 35.67 1.74 22.45 29.18 2.43 25.62 20.17 6.45 46.97 -27.4 3.32 -32.831.5 1.17 1.10 0.46 1.69 1.88 1.15 4.31 6.47 10.06 -12.31 3.27 -14.542.0 0.06 0.84 0.02 0.17 1.57 0.10 1.39 6.33 3.18 -6.94 3.26 -8.17

Contribution to µµ signal [83]λ

L22 = 1 (Scenario RK1A)

1.0 40.89 71.88 265.27 58.68 72.66 769.52 70.40 62.77 1595.21 -233.00 42.73 -3594.151.5 1.39 64.44 8.10 3.91 71.35 50.30 15.20 64.33 352.97 -105.00 42.59 -1614.372.0 0.08 52.62 0.36 0.44 70.15 5.60 5.00 64.22 115.92 -58.80 43.08 -914.54λ

R22 = 1 (Scenario RK1B)

1.0 38.91 71.74 1007.69 58.29 72.36 1522.36 70.43 62.69 1593.99 -82.52 49.17 -1464.791.5 1.32 64.18 30.64 3.81 68.62 94.40 15.21 64.20 352.57 -37.33 49.09 -661.522.0 0.07 52.50 1.36 0.42 63.79 9.78 5.00 64.53 116.48 -21.0 48.62 -368.53λ

L32 = 1 (Scenario RK1C)

1.0 35.67 71.59 230.45 28.93 72.74 379.76 20.00 63.49 458.17 -75.30 39.10 -1062.871.5 1.17 64.46 6.78 1.72 72.33 22.44 4.29 64.58 100.49 -33.70 39.82 -484.392.0 0.06 52.47 0.29 0.17 71.77 2.22 1.41 64.90 33.04 -19.00 40.12 -275.17λ

R32 = 1 (Scenario RK1D)

1.0 35.67 71.75 923.90 29.04 72.37 758.73 20.05 63.73 461.36 -26.29 45.77 -434.431.5 1.17 64.60 27.19 1.69 69.28 42.27 4.29 64.43 99.74 -11.84 46.32 -197.942.0 0.06 52.00 1.14 0.17 65.35 3.95 1.41 65.37 33.25 -6.69 46.64 -112.60

We implement the above cuts in our analysis codes after validating them with the efficiencies givenin the experimental papers. As explained in Ref. [81], we generated pp→ Z′→ `` events for valida-tion and compared our cut efficiencies (ε) with the experimental efficiencies×detector acceptanceto ensure they agree with each other. In Table VI, we show the production cross sections, cut ef-ficiencies, and number of events surviving the cuts for different signal contributions for the RD(∗)-motivated and RK(∗)-motivated one-coupling scenarios, respectively. We obtain these numbers bysetting the concerned coupling to unity. There are a few points to note here. Pair productionis, in general, insensitive to new physics couplings. However, a mild sensitivity arises due to themodel-dependent t-channel lepton exchange diagram that contributes to the pair production [seeFig. 3(b)]. In Scenario RD1A where only λ L

23 is nonzero, the pair production cross section is 40.87

20

fb for MU1 = 1 TeV, whereas in Scenario RD1B, it is 35.67 fb. This is because the t-channel leptonexchange contribution is larger in Scenario RD1A. In this scenario, the second-generation quarkscontribute in the initial states with PDFs bigger than the b-PDF contributing in Scenario RD1B.A similar minor difference can be seen between Scenario RK1A and Scenario RK1B. In ScenarioRK1A, the process cc→U1U1 through a neutrino exchange is present, but it is absent in ScenarioRK1B causing the minor difference. The single production cross sections are relatively larger inscenarios where the second-generation quarks appear in the initial states than those where onlyb-quarks can appear.

The cut efficiencies for different production modes for RK(∗) scenarios are generally much highercompared to RD(∗) scenarios. This is mainly because the selection efficiency of the τ in the finalstate is much lower compared to the muons. For instance, in the RK(∗) scenarios, the efficiency forpair production processes ε p can be as high as 71% for MU1 = 1 TeV, whereas for RD(∗) scenarios, itis only∼ 2%. The hadronic BR of τ is∼ 64%, and the τ-tagging efficiency is about 60%. Combiningjust these two factors we get a factor of 0.642×0.62 ∼ 1/7 reduction in the efficiency for the twoτhad ’s in the pair production final state. Note that all of pair and single productions, and t-channelU1 exchange contributes positively towards the dilepton signal, whereas the signal-backgroundinterference contribute negatively as it is destructive in nature. The minus signs in σnr2 and N nr2

indicate the destructive nature of the interference.

Before presenting our results, we list the publicly available HEP packages used at various stagesof our analysis.• Lagrangian and model files: The Lagrangian terms defined in the previous section are im-

plemented in the FeynRules package [114] to obtain the UFO model files [115].

• Event generation: Using the UFO model files, we generate signal events using the MadGraph5Monte-Carlo event generator [116] at the leading order (LO). The NNPDF2.3LO PDFs [117]are used with default dynamical scales.4 Higher-order QCD corrections for the vLQ are notconsidered in this analysis as they are not available in the literature.

• Showering and hadronization: This is performed by passing the parton-level events toPythia6 [118]. We use the MLMmatching scheme [119, 120] (up to two additional jets) withvirtuality-ordered Pythia showers to avoid double counting of the matrix-element partonswith parton showers.

• Detector simulation: We use Delphes 3.4.2 [121] (with ATLAS and CMS cards) to performthe detector simulations. The jet-clusterings are done using the FastJet package [122]. Weuse the anti-kT algorithm [123] with the radius parameter R = 0.4.

V. EXCLUSION LIMITS

There are three free couplings λ L23,λ

L33, and λ R

33 that take part in the RD(∗) scenarios. We show thecurrent exclusion limits on these couplings taken one at a time in Fig. 5(a) from the latest LHCττ resonance search data [82]. Similarly, the RK(∗) scenarios have four free couplings in total:λ L

22,λL32, λ R

22, and λ R32. We use the latest CMS µµ resonance search data [83] to obtain exclusion

limits on these couplings by considering one of them at a time as shown in Fig. 5(b). These are the95% (2σ) confidence level (CL) exclusion limits. To obtain the limits, we set all other couplingsexcept the one under consideration to zero. The method we follow is the same as the one used inRef. [81] and is elaborated on in Appendix B. From the left plot, we see that the limit on λ L

23 is

4 The NNPDF2.3LO PDF for the heavy quarks might have considerable uncertainties. However, our results, i.e., thelimits on the U1 parameters, are largely insensitive to these.

21

(a) (b)

FIG. 5. The 2σ LHC exclusion limits on the couplings participating in the (a) RD(∗) and (b) RK(∗) scenarios.The regions above these lines are excluded. These exclusion limits are obtained by recasting the dileptonsearch data [82, 83] with a combination of all possible U1 production processes that can contribute to thedilepton final states.

(a) (b)

FIG. 6. These exclusion plots are similar to the ones in Fig. 5 but they are obtained only for the nonresonantproduction and its interference with the SM background. Comparing with Fig. 5, they show that in thelower mass region, the contribution of the resonant U1 productions is significant. However, in the highermass region, the nonresonant production and its interference with the SM background determine the limits.

more severe than on λL/R33 . This is because, for nonzero λ L

23, there is a s-quark-initiated contributionto the t-channel U1 exchange that interferes with the SM ss→ γ∗/Z∗→ ττ process. In the case ofnonzero λ

L/R33 , the process is b-quark-initiated and, therefore, is suppressed by the small b-PDF.

Among the offshell photon and Z-boson contributions to the signal-background (SB) interference,the second one dominates. Similarly, one can also understand the relatively weaker limits on λ

L/R32

in Fig. 5(b).Among the four sources of dilepton events [the pair and single production, t-channel U1 ex-

change, and the SB interference], different processes play the dominating roles in deciding thelimits in different mass ranges. We observe an interesting role switch in these plots. In the highmass region where the limits on λ reach high values (& 1), the resonant productions are relativelyless important (also see Fig. 4)], and the nonresonant productions play the determining roles.However,for MU1 . 1.5 TeV, mainly the resonant productions determine the limits, and all the λ -

22

(a) (b)

(c) (d)

FIG. 7. The 2σ exclusion limits from the LHC and the preferred regions by the flavour anomalies. The purpleregions are excluded at 2σ by the ATLAS ττ data [82]. The magenta and blue regions depict the excludedmass ranges from direct searches. (a) Scenario RD1A: Only λ L

23 is nonzero. The FlavD region (yellow),defined in Eq. (67), is favoured by the RD(∗) anomalies. The light orange region is favoured by Bs-Bs mixing.The direct detection mass limit (magenta) is from the CMS search in the ccνν channel [85]. (b) ScenarioRD1B: Only λ L

33 is nonzero. The FlavD9 region (green) agrees with all the constraints in FlavD except C univ9

[Eq. (18)] since the corresponding operator, Ouniv9 , cannot be generated in this scenario. The magenta

region here depicts the limits from the very recent CMS combined search in the tbτν + tτν channel [88].We have recast the observed events data to extrapolate the limit for λ L

33 > 2.5. There is no Bs-Bs mixingin this scenario. (c) Scenario RD2A: λ L

33 = 0.5 (benchmark choice), and λ L23 is free. The magenta and blue

regions are the limits from the direct searches at CMS [85, 88]. (d) Scenario RD2B: λ R33 = 0.5 (benchmark

choice) and λ L23 is free. The blue dashed line shows the limits from the ATLAS bbττ direct search data [87].

The dashed line implies that the recast has been done from a scalar LQ direct search. The FlavD9B region(olive green), defined in Eq. (69), agrees with all the constraints in the FlavD9 region without the constraintsfrom the B(c)→ τν decays. The light orange colour shows the region preferred by B(c)→ τν .

dependent contributions are small. This switch of roles can also be inferred from Fig. 6 wherewe show the same limits as in Fig. 5 but ignore the resonant contributions in the dilepton signal.Comparing Figs. 5 and 6 we see that for MU1 . 2.5 TeV, the limits can vary significantly dependingon whether one considers the resonant productions or not. Since the nonresonant processes donot depend on the U1 branching ratios, for a low mass U1 the limits thus obtained are strictly validwhen the branching ratio of the decay U1→ τ j/τb mediated by the coupling λ is small. However,the limits in Fig. 5 are obtained assuming only one coupling is nonzero, i.e., maximum branchingratio for the U1 → τ j/τb decay mediated by the coupling λ . For a very heavy U1 (& 3 TeV) theresonant productions become negligible, and hence, the limits of Figs. 5 and 6 converge. We also

23

(a) (b)

(c) (d)

FIG. 8. The regions favoured by the flavour observables (yellow, green, and orange) and allowed by the LHCdata (blue). We consider two benchmark mass values, MU1 = 1.5 TeV and 2 TeV. (a) and (c) Scenario RD2A(λ R

33 = 0): FlavD (yellow) and FlavD9 (green) are the regions preferred by the flavour anomalies with andwithout the Ouniv

9 operator, respectively. The light orange region is favoured by the Bs-Bs mixing. (b) and(d) Scenario RD2B (λ L

33 = 0): The light orange color marks the regions preferred by the B(c)→ τν decay.FlavD9B (olive green) shows the region preferred by the flavour anomalies except Ouniv

9 and the B(c)→ τν

decay.

observe that in the low mass region (MU1 ∼ 1 TeV) where the limit is determined almost solely bythe pair production, the data are badly fitted by theU1 events (i.e., χ2

min/d.o. f .� 1). This suggeststhat the dimuon data disfavour pair production of a ∼ 1 TeVU1 – this is similar to but independentof the direct LQ search limits.

We plot the relevant direct search limits from the j j+ /ET and tt + /ET channels [85] togetherwith the limits on the RD(∗) scenarios with one coupling in Figs. 7(a) and 7(b). We also showthe parameter regions that are favoured by the RD(∗) observables and consistent with the relevantflavour observables (as discussed in Section II) in the same plots. We see that in Scenario RD1A,the region marked as FlavD, which is defined as

FlavD≡ the region allowed by {RD(∗) +FL(D∗)+Pτ(D∗)+B(B(c)→ τν)+C univ9}, (67)

is in tension with the Bs-Bs mixing data and is independently and entirely excluded by the ττ data.The tension between FlavD and the Bs-Bs mixing data arises since the Bs-Bs mixing data favours asmaller C U1

VL(via C U1

box which roughly goes as the square of C U1VL

) than the RD(∗) observables. ScenarioRD1B does not contribute to Eq. (17) and hence, cannot accommodate a nonzero C univ

9 . We markthe parameter region favoured by the RD(∗) observables in this scenario as FlavD9 which stands for

24

(a) (b)

(c) (d)

FIG. 9. The regions excluded at 2σ by the CMS µµ search data [83] in the minimal RK(∗) scenarios (violet).FlavK (yellow), defined in Eq. (70), is the region favoured by the global fits to the b→ sµµ and Bs-Bsmixing.(a) Scenario RK1A: Only λ L

22 is nonzero. (b) Scenario RK1B: Only λ L32 is nonzero. Since, λ L

32 alonecannot explain the RK(∗) anomalies, we only show the region allowed by the Bs-Bs mixing data. (c) ScenarioRK1C: Only λ R

22 is nonzero. (d) Scenario RK1D: Only λ R32 is nonzero. The magenta dashed lines denote

the recast exclusion limits from the ATLAS direct search for pair production of scalar LQs in the µµ + j j/bbchannels [86].

the region allowed by all the constraints included in FlavD except C univ9 , i.e.,

FlavD9 ≡ the region allowed by {RD(∗) +FL(D∗)+Pτ(D∗)+B(B(c)→ τν)}. (68)

From Fig. 7(b), we see that for this strictly one-coupling scenario, the entire FlavD9 is excludedby the latest ττ data.

If we look at the two-coupling scenarios, the situation somewhat improves. For example, inFigs. 7(c) and 7(d), we show a projection of the three-dimensional parameter space of ScenarioRD2A and Scenario RD2B, respectively. In Fig. 7(c), we keep λ L

33 = 0.5 and let λ L23 and MU1 vary

(Scenario RD2A). We see that a good part of the FlavD9 region (note that the FlavD is a subregionwithin FlavD9) survives the LHC bounds but a large part of it remains in conflict with the Bs-Bs

mixing data. However, a small part of FlavD9 does agree with Bs-Bs mixing and survives the LHCbounds. We show two more projections of the parameter space of Scenario RD2A in Figs. 8(a) and8(c) – we let λ L

23 and λ L33 vary and keep the mass of U1 fixed. There we show the region allowed by

the LHC data and the relevant flavour regions for MU1 = 1.5 (2.0) TeV, respectively. In absence ofλ L

33, Scenario RD2B cannot accommodate the allowed C univ9 , and, in this case, the region favoured

by the RD(∗) observable stands in conflict with the B(B(c)→ τν) constraint – see the region marked

25

(a) (b)

(c) (d)

FIG. 10. The regions favoured by the RK(∗) observables (yellow and green) and allowed by the LHC data(blue). The blue regions are obtained by recasting the CMS µµ search data [83]. (a) and (d) In ScenarioRK2A and Scenario RK2D, the FlavK region (yellow) depicts the allowed regions by the b→ sµµ globalfits and Bs-Bs mixing. (b) and (c) In Scenario RK2B and Scenario RK2D, there is no restriction from theBs-Bs mixing. FlavKB (green), defined in Eq. (71), stands for the region preferred by the b→ sµµ globalfits alone.

as FlavD9B for the region allowed by all the constraints included in FlavD9 except B(B(c)→ τν),i.e.,

FlavD9B ≡ the region allowed by {RD(∗) +FL(D∗)+Pτ(D∗)} . (69)From Fig. 7(d) [where we keep λ R

33 = 0.5 and let λ L23 and MU1 vary (Scenario RD2B)], we see that

the entire FlavD9B region is ruled out by the LHC data. This can also be seen from the two couplingplots in Figs. 8(b) and 8(d).

In Fig. 9, we compare the bounds on λ L22,λ

L32, λ R

22 and λ R32 from the CMS µµ data [83] with the

regions favoured by the RK(∗) anomalies and allowed by the Bs-Bs mixing data marked asFlavK≡ the region favoured by

{the global fits to b→ sµµ data+Bs-Bs mixing} (70)

in the one-coupling scenarios [except for Scenario RD2B where, as already pointed out, λ L32 alone

cannot explain the RK(∗) anomalies. Hence, in Fig. 9(b) we only show the region allowed by theBs-Bs mixing data]. In these plots, we also show the recast limits from the recent pair productionsearch by the ATLAS Collaboration that effectively rule out U1 masses almost up to 2 TeV. Toobtain these limits we have recast the recent ATLAS search for scalar LQ in the µµ + j j/bb channelobtained with 139 fb−1 of integrated luminosity [86].5 We see the LHC µµ data are much less5 We show the recast ATLAS limits [86] only with dashed lines because, strictly speaking, the search was optimisedfor a scalar LQ. In our recast for U1, we have assumed that the selection efficiencies remain unchanged when oneswitches from the pair production of scalar LQs to that of the vector ones.

26

(a) (b)

(c) (d)

FIG. 11. Examples of regions in the parameter space of a 1.5 TeV U1 surviving the LHC limits and simul-taneously accommodating the RD(∗) and RK(∗) anomalies. The FlavK (yellow), FlavKB (green) and the blueregions are identical to the ones in Fig. 10. When we recast the ATLAS search in the µµ + bb/ j j chan-nels [86], the presence of the additional couplings, λ L

23 = 0.006 and λ L33 = 0.93 [allowed by the LHC and

flavour data, see Fig. 8(a)], relaxes the exclusion limits, thus allowing the otherwise excluded mass valuein the RK(∗) two-coupling scenarios.

restrictive on the FlavK regions than the ττ data on the FlavD regions. This is mainly becausethe magnitudes of these couplings required to explain the RK(∗) anomalies are much smaller thanthose in the RD(∗) scenarios. We also note that the direct search mass exclusion limits are weakerin the scenarios with left-type couplings (i.e., Scenario RK1A and Scenario RK1B) than those withright-type couplings (Scenario RK1C and Scenario RK1D). This is because the decay U1→ µb/µ jhas 100% BR in the right-type coupling scenarios instead of the 50% in the left-type ones. Therecast limits imply that a 1.5 TeV U1 is ruled out in all the two-coupling scenarios. Hence weconsider a 2 TeV U1 in the two-coupling scenarios in Fig. 10. There we show the regions allowedby the LHC data along with the FlavK regions in Scenario RK2A and Scenario RK2D and FlavKB

regions in Scenario RK2B and Scenario RK2C. In the last two scenarios, the constraints from Bs-Bs

mixing data are not applicable, and the FlavKB regions are just the ones favoured by the global fitof b→ sµµ data

FlavKB ≡ FlavK+ the region exclusively disfavoured by Bs-Bs mixing≡ the region favoured by the global fits to b→ sµµ data. (71)

The recast ATLAS scalar search limits however does not entirely rule out a 1.5 TeV U1 solutionfor the RK(∗) anomalies. To see this, one needs to make β (U1 → µb/µ j) . 0.25 by introducingadditional nonzero coupling(s). For example, in Fig. 11 we show that for λ L

23 = 0.006 and λ L33 = 0.93

27

[a point we chose randomly from the FlavD9 region in Fig. 8(a) that agrees with Bs-Bs mixing andis allowed by the LHC data] , all the two-couplings RK(∗) scenarios survive the recast bounds forMU1 = 1.5 TeV. This is interesting, as it explicitly shows four possible parameter choices for whicha 1.5 TeV U1 can account for both the RD(∗) and RK(∗) anomalies. These choices are, of course, onlyillustrative, not unique or special.

VI. SUMMARY AND CONCLUSIONS

In this paper, we have derived precise limits on a flavour-anomalies-motivated U1 LQ model usingthe latest LHC and flavour data. We started with a generic coupling texture forU1 (with seven freecouplings) that can contribute to the RD(∗) and RK(∗) observables. Taking a bottom-up approach,suitable for obtaining bounds from the existing LHC searches, we constructed all possible one- andtwo-coupling scenarios that can accommodate either the RD(∗) or RK(∗) anomalies. In particular, weconsidered two one-coupling and two two-coupling scenarios that can give rise to the bτU1 andcνU1 couplings required by the RD(∗) observables. Similarly, we considered four one-coupling andfour two-coupling scenarios contributing to the b→ sµ+µ− transition. We recast the current LHCdilepton searches (ττ and µµ) [82, 83] to obtain limits on the U1 couplings for a range of MU1

in these scenarios. We also looked at the bounds from the latest direct LQ searches from ATLASand CMS. Whenever needed, we recast the latest scalar LQ searches in terms of U1 parameters asthese were found to give better limits than the existing ones. Put together, our results give the bestlimits on the U1 parameters currently available from the LHC. These bounds are independent ofand complementary to other flavour bounds.

Previously, the high-pT dilepton data were used to put limits on theU1 couplings. Most of theseanalyses, however, focused only on the nonresonant t-channel U1 exchange process. However, wefound that this process interferes destructively with the SM background, and, in most cases, it isthis interference that plays the prominent role in setting the exclusion limits, especially for a heavyU1. Also, other resonant production processes, namely the pair and the single productions of U1,can also contribute significantly to the high-pT dilepton tails. We have shown the differences thatthe inclusion of resonant production processes can make on the exclusion limits in Figs. 5 and 6.The limits we obtained are robust as they depend only on a few assumptions about the under-lying model. They are also precise as all the resonant and nonresonant contributions includingthe signal-background interference are systematically incorporated in our statistical recast of thedilepton data. The low mass regions are bounded by the direct pair production search limits thatdepend only on the BRs. When U1 is heavy, the limits mostly come from the nonresonant processand its interference with the SM background that depend only on the value of the coupling(s)involved, not on the BRs.

We found that in the minimal (with one free coupling) or some of the next-to-minimal (withtwo free couplings) scenarios, the parameter spaces required to accommodate the RD(∗) anomaliesare already ruled out or in tension with the latest LHC data (see e.g. Figs. 7 and 8). In somescenarios, the regions favoured by the anomalies are in conflict with other flavour bounds but inthe RD1B minimal scenario, a part of the parameter space survives the LHC bounds [see Fig. 7(b)]that can explain the RD(∗) anomalies. We found that a good part of the parameter space requiredto explain the RK(∗) anomalies survives the dilepton bounds, except the recent ATLAS search forscalar LQ in the µµ + j j/bb channel [86] put some pressure for MU1 . 2 TeV (see Figs. 9 and 10).

Our method for obtaining bounds is generic. It is possible to extend our analysis to scenarioswith more nonzero couplings and/or additional degrees of freedom by considering our scenarios astemplates. As an example, we showed the bounds on a combined scenario with three free couplings(λ L

33, λL/R32 , λ

L/R22 ) that can accommodate both the RD(∗) and RK(∗) anomalies simultaneously with a

1.5 TeVU1 in Fig. 11 (even though a simple recast of the recent ATLAS LQ search in the µµ + j j/bbchannel rules out a U1 of mass 1.5 TeV that decays to these final states exclusively [86]. One

28

should therefore keep in mind that a U1 LQ with mass . 2 TeV is still allowed and can resolve theB-anomalies). To obtain limits on other general scenarios, one can use the parametrization andmethod elaborated in Appendices A and B.

We also identified some possible new search channels of U1 that have not been considered sofar. Our simple parametrization of various possible scenarios in terms of a few parameters canserve as a guide for the future U1 searches at the LHC. It can also be used for interpreting theresults of future bottom-up experimental searches of vLQs.

ACKNOWLEDGMENTS

A.B. and S.M. acknowledge support from the Science and Engineering Research Board (SERB),DST, India, under Grant No. ECR/2017/000517. D.D. acknowledges the DST, Government ofIndia for the INSPIRE Faculty Fellowship (Grant No. IFA16-PH170). T.M. is supported by theintramural grant from IISER-TVM. C.N. is supported by the DST-Inspire Fellowship.

Appendix A: Cross section parametrization for the `` signal processes

It is not straightforward to obtain precise LHC exclusion limits from the dilepton data when mul-tiple couplings are nonzero simultaneously. This is mainly because different couplings contributeto different topologies with the same final states. In multi-coupling scenarios, the presence of sub-stantial signal-background interference and/or signal-signal interference complicates the picturefurther. All these possibilities are present in the U1 scenarios considered here. Therefore, a discus-sion on a systematic approach to properly take care of these complications might be useful for thereader. Below, we discuss the method we have used for multi-coupling scenarios in the context ofU1. However, this method is not limited to U1 and ττ or µµ final states but can be adapted easilyfor any BSM scenarios wherever needed.� Pair production: As mentioned in Section III, the pair production of U1 is not fully model-independent. It depends on two parameters - κ, parametrizing the new kinetic terms, and λ , thegeneric coupling for `qU1 interactions. In our analysis, we have set κ = 0. The dependence onλ enters in the pair production through the t-channel lepton exchange diagrams. If n differentnew couplings (λi with i = {1,2,3, . . .n}) are contributing, the total cross section for the processpp→U1U1 can be expressed as

σp (MU1 ,λ ) = σ

p0 (MU1)+n

∑i

λ2i σ

p2i (MU1)+

n

∑i≥ j

λ2i λ

2j σ

p4i j (MU1) (A1)

where the sums go up to n. The σ px functions on the r.h.s. depend only on the mass of U1. Here,σ p0 (MU1) is the λ -independent part determined by the strong coupling constant. This part canbe computed taking λi → 0 for all the new couplings. The σ

p2i (MU1) terms originate from the

interference between the QCD-mediated model-independent diagrams and the t-channel leptonexchange diagrams. The σ

p4i j (MU1) terms are from the pure t-channel lepton exchange diagrams.

For a particular MU1 , there are n unknown σp2i and n(n+ 1)/2 unknown σ

p4i j functions that

we need to find out. For that, we compute σ p for n(n+ 3)/2 different values of λi and solve theresulting linear equations. We repeat the same procedure for different mass points. We can nowget σ p(MU1 ,λ ) for any intermediate value of MU1 either from numerical fits to direct evaluation.

In the presence of kinematic selection cuts, different σ px(MU1) parts contribute differently tothe surviving events. Hence, the overall cut efficiency for the pair production process ε p dependson both MU1 and λ . The total number of surviving events from the pair production process passing

29

through some selection cuts can, therefore, be expressed as

N p = σp ◦ ε

p (MU1 ,λ )×B2(MU1 ,λ )×L

=

{σ

p0× εp0 +

n

∑i

λ2i σ

p2i × ε

p2i +

n

∑i≥ j

λ2i λ

2j σ

p4i j × ε

p4i j

}×B2(MU1 ,λ )×L (A2)

where all ε px depend only on MU1 . Here L is the integrated luminosity, and B(MU1 ,λ ) is therelevant branching ratio (of the decay mode of U1 that contributes to the signal) which can beobtained analytically. The ε px(M) functions can be obtained by computing the fraction of eventssurviving the selection cuts while computing the σ px(MU1) functions.� Single production: As discussed earlier, single production of U1 usually contains two types ofcontributionsU1x andU1yz (where x,y,z are SM particles). The amplitudes ofU1x type of processesare always proportional to λ . But U1yz amplitudes can have both linear and cubic terms in λ .Therefore, themost generic form of the single production process pp→U1x+U1yz can be expressedas

σs(M,λi) =

n

∑i

λ2i σ

s2i (MU1)+

n

∑i≥ j≥k

λ2i λ

2j λ

2k σ

s6i jk(MU1). (A3)

The σ sx(M) functions can be obtained following the same method used for pair production. Wecan express the total number of single production events as

N s = σs ◦ ε

s (MU1 ,λ )×B(MU1 ,λ )×L

=

{∑

iλ

2i σ

s2i (MU1)× ε

s2i (MU1)+ ∑

i≥ j≥kλ

2i λ

2j λ

2k σ

s6i jk(MU1)× ε

s6i jk(MU1)

}×B(MU1 ,λi)×L. (A4)

� Nonresonant production: The t-channel U1 exchange processes fall in this category. There aretwo different sources of nonresonant contributions one has to consider. One is from the signal-SMbackground interference and is proportional to λ 2

i . The other is from the signal-signal interferenceand hence is quartic in λ . We can express the total nonresonant pp→ xy cross section as

σnr(MU1 ,λ ) =

n

∑i

λ2i σ

nr2i (MU1)+

n

∑i≥ j

λ2i λ

2j σ

nr4i j (MU1). (A5)

Note that σnr(MU1 ,λ ) can be negative when the signal-background interference is destructive.Indeed, this is the case we observe for U1. By introducing the ε(MU1) functions, the total numberof surviving events can be written as

N nr = σnr ◦ ε

nr (MU1 ,λ )×L

=

{n

∑i

λ2i σ

nr2i (MU1)× ε

nr2i (MU1)+

n

∑i≥ j

λ2i λ

2j σ

nr4i j (MU1)× ε

nr4i j (MU1)

}×L. (A6)

Notice, no BR appears in the above equation. A negative σnr(MU1 ,λ )makesN nr a negative numberas presented in Table VI.

For an illustration, we show the observed µµ data [83] and the corresponding SM contributions inFig. 12. In the lower panel we show the different signal components for MU1 = 1.5 TeV and λ L

22 = 1(Scenario RK1A).

30

FIG. 12. The observed Mµµ distribution and the corresponding SM contributions from Ref. [83]. Theerrors are obtained using Eq. (B3). (Lower panel) The different signal components for a typical choiceof parameters, MU1 = 1.5 TeV and λ L

22 = 1 (Scenario RK1A). The number of U1 signal events are denotedby NU1 [Eq. (B2)] and N p+s = N p +N s denotes the events from pair and single production processes.Among the nonresonant contributions, N nr2 denotes the SM-BSM interference and N nr4 is the pure BSMnonresonant part.

Appendix B: Limits estimation: χ2 tests

To obtain the limits on the parameter space in various U1 scenarios, we recast the LHC ττ andµµ search data [82, 83]. In particular, we perform χ2 tests to estimate the limits on parametersfrom the transverse (invariant) mass distribution of the ττ (µµ) data. The method is essentiallythe same as the one used in Ref. [81] for S1 LQ. Here we briefly outline the steps.

1. For each distribution, the test statistic can be defined as

χ2 = ∑

i

(N i

T (MU1 ,λ )−N iD

∆N i

)2

(B1)

where the sum runs over the corresponding bins. Here, N iT (MU1 ,λ ) stands for expected

(theory) events, and N iD is the number of observed events in the ith bin. The number of

theory events in the ith bin can be expressed

N iT (MU1 ,λ ) = N i

U1(MU1 ,λ )+N i

SM

=[N p(MU1 ,λ )+N s(MU1 ,λ )+N nr(MU1 ,λ )

]+N i

SM. (B2)

Here, N iU1

and N iSM are the total signal events fromU1 and the SM background in the ith bin,

respectively. The total signal events are composed of N p, N s, and N nr from Eqs. (A2),(A4), and (A6), respectively. The details on how to calculate N i

U1for different scenarios is

sketched in Appendix A. For the error ∆N i, we use

∆N i =

√(∆N i

stat)2

+(∆N i

syst)2 (B3)

31

where ∆N istat =

√N i

D and we assume a uniform 10% systematic error, i.e., ∆N isyst = δ i×N i

D

with δ i = 0.1.

2. In every scenario, for some discrete benchmark values of MU1 = MbU1

we compute the mini-mum of χ2 as χ2

min(MbU1) by varying the couplings λ .

3. In one-coupling scenarios (like Scenario RD1A, Scenario RK1A, etc.), we obtain the 1σ and2σ confidence level upper limits on the coupling at MU1 =Mb

U1from the values of λ for which

∆χ2(MbU1,λ ) = χ2(Mb

U1,λ )−χ2

min(MbU1) = 1 and 4, respectively.

In two-coupling scenarios (like Scenario RD2A, Scenario RK2A, etc.), we do the same, ex-cept we obtain the 1σ and 2σ limits (contours) from the 2-variable limits on ∆χ2; i.e., wesolve ∆χ2(Mb

U1,λ1,λ2) = χ2(Mb

U1,λ1,λ2)−χ2

min(MbU1) = 2.30 and 6.17, respectively.

Similarly, we can obtain the limits for the scenarios with n(≥ 2) free couplings by using then-variable ranges for ∆χ2.

4. We obtain the limits for arbitrary values of MU1 by interpolating the limits for the benchmarkmasses.

[1] BaBar collaboration, J. P. Lees et al., Evidence for an excess of B→ D(∗)τ−ντ decays, Phys. Rev. Lett.109 (2012) 101802, [1205.5442].

[2] BaBar collaboration, J. P. Lees et al., Measurement of an Excess of B→ D(∗)τ−ντ Decays andImplications for Charged Higgs Bosons, Phys. Rev. D88 (2013) 072012, [1303.0571].

[3] LHCb collaboration, R. Aaij et al., Test of lepton universality using B+→ K+`+`− decays, Phys. Rev.Lett. 113 (2014) 151601, [1406.6482].

[4] LHCb collaboration, R. Aaij et al., Test of lepton universality with B0→ K∗0`+`− decays, JHEP 08(2017) 055, [1705.05802].

[5] LHCb collaboration, R. Aaij et al., Measurement of the ratio of branching fractionsB(B0→ D∗+τ−ντ)/B(B0→ D∗+µ−νµ), Phys. Rev. Lett. 115 (2015) 111803, [1506.08614].[Erratum: Phys. Rev. Lett.115,no.15,159901(2015)].

[6] LHCb collaboration, R. Aaij et al., Measurement of the ratio of the B0→ D∗−τ+ντ andB0→ D∗−µ+νµ branching fractions using three-prong τ-lepton decays, Phys. Rev. Lett. 120 (2018)171802, [1708.08856].

[7] LHCb collaboration, R. Aaij et al., Test of Lepton Flavor Universality by the measurement of theB0→ D∗−τ+ντ branching fraction using three-prong τ decays, Phys. Rev. D97 (2018) 072013,[1711.02505].

[8] Belle collaboration, M. Huschle et al., Measurement of the branching ratio of B→ D(∗)τ−ντ relativeto B→ D(∗)`−ν` decays with hadronic tagging at Belle, Phys. Rev. D92 (2015) 072014,[1507.03233].

[9] Belle collaboration, Y. Sato et al., Measurement of the branching ratio of B0→ D∗+τ−ντ relative toB0→ D∗+`−ν` decays with a semileptonic tagging method, Phys. Rev. D 94 (2016) 072007,[1607.07923].

[10] Belle collaboration, S. Hirose et al., Measurement of the τ lepton polarization and R(D∗) in the decayB→ D∗τ−ντ , Phys. Rev. Lett. 118 (2017) 211801, [1612.00529].

[11] Belle collaboration, S. Hirose et al., Measurement of the τ lepton polarization and R(D∗) in the decayB→ D∗τ−ντ with one-prong hadronic τ decays at Belle, Phys. Rev. D97 (2018) 012004,[1709.00129].

[12] D. Bigi and P. Gambino, Revisiting B→ D`ν , Phys. Rev. D94 (2016) 094008, [1606.08030].[13] F. U. Bernlochner, Z. Ligeti, M. Papucci and D. J. Robinson, Combined analysis of semileptonic B

decays to D and D∗: R(D(∗)), |Vcb|, and new physics, Phys. Rev. D95 (2017) 115008, [1703.05330].[erratum: Phys. Rev.D97,no.5,059902(2018)].

[14] D. Bigi, P. Gambino and S. Schacht, R(D∗), |Vcb|, and the Heavy Quark Symmetry relations betweenform factors, JHEP 11 (2017) 061, [1707.09509].

32

[15] S. Jaiswal, S. Nandi and S. K. Patra, Extraction of |Vcb| from B→ D(∗)`ν` and the Standard Modelpredictions of R(D(∗)), JHEP 12 (2017) 060, [1707.09977].

[16] HFLAV collaboration, Y. Amhis et al., Averages of b-hadron, c-hadron, and τ-lepton properties as ofsummer 2016, Eur. Phys. J. C77 (2017) 895, [1612.07233]. We have used the Spring 2019averages fromhttps://hflav-eos.web.cern.ch/hflav-eos/semi/spring19/html/RDsDsstar/RDRDs.html.For regular updates see https://hflav.web.cern.ch/content/semileptonic-b-decays.

[17] LHCb collaboration, R. Aaij et al., Search for lepton-universality violation in B+→ K+`+`− decays,Phys. Rev. Lett. 122 (2019) 191801, [1903.09252].

[18] LHCb collaboration, R. Aaij et al., Test of lepton universality in beauty-quark decays, 2103.11769.[19] G. Hiller and F. Kruger, More model-independent analysis of b→ s processes, Phys. Rev. D69 (2004)

074020, [hep-ph/0310219].[20] M. Bordone, G. Isidori and A. Pattori, On the Standard Model predictions for RK and RK∗ , Eur. Phys.

J. C76 (2016) 440, [1605.07633].[21] R. Alonso, B. Grinstein and J. Martin Camalich, Lepton universality violation and lepton flavor

conservation in B-meson decays, JHEP 10 (2015) 184, [1505.05164].[22] L. Calibbi, A. Crivellin and T. Ota, Effective Field Theory Approach to b→ s``(

′), B→ K(∗)νν andB→ D(∗)τν with Third Generation Couplings, Phys. Rev. Lett. 115 (2015) 181801, [1506.02661].

[23] S. Fajfer and N. Košnik, Vector leptoquark resolution of RK and RD(∗) puzzles, Phys. Lett. B 755(2016) 270–274, [1511.06024].

[24] R. Barbieri, G. Isidori, A. Pattori and F. Senia, Anomalies in B-decays and U(2) flavour symmetry,Eur. Phys. J. C 76 (2016) 67, [1512.01560].

[25] D. Bečirević, N. Kosnik, O. Sumensari and R. Zukanovich Funchal, Palatable Leptoquark Scenariosfor Lepton Flavor Violation in Exclusive b→ s`1`2 modes, JHEP 11 (2016) 035, [1608.07583].

[26] S. Sahoo, R. Mohanta and A. K. Giri, Explaining the RK and RD(∗) anomalies with vector leptoquarks,Phys. Rev. D95 (2017) 035027, [1609.04367].

[27] B. Bhattacharya, A. Datta, J.-P. Guévin, D. London and R. Watanabe, Simultaneous Explanation ofthe RK and RD(∗) Puzzles: a Model Analysis, JHEP 01 (2017) 015, [1609.09078].

[28] M. Duraisamy, S. Sahoo and R. Mohanta, Rare semileptonic B→ K(π)l−i l+j decay in a vectorleptoquark model, Phys. Rev. D95 (2017) 035022, [1610.00902].

[29] D. Buttazzo, A. Greljo, G. Isidori and D. Marzocca, B-physics anomalies: a guide to combinedexplanations, JHEP 11 (2017) 044, [1706.07808].

[30] N. Assad, B. Fornal and B. Grinstein, Baryon Number and Lepton Universality Violation inLeptoquark and Diquark Models, Phys. Lett. B 777 (2018) 324–331, [1708.06350].

[31] L. Calibbi, A. Crivellin and T. Li, Model of vector leptoquarks in view of the B-physics anomalies, Phys.Rev. D 98 (2018) 115002, [1709.00692].

[32] M. Blanke and A. Crivellin, B Meson Anomalies in a Pati-Salam Model within the Randall-SundrumBackground, Phys. Rev. Lett. 121 (2018) 011801, [1801.07256].

[33] A. Greljo and B. A. Stefanek, Third family quark–lepton unification at the TeV scale, Phys. Lett. B782 (2018) 131–138, [1802.04274].

[34] S. Sahoo and R. Mohanta, Impact of vector leptoquark on B→ K∗l+l− anomalies, J. Phys. G45(2018) 085003, [1806.01048].

[35] J. Kumar, D. London and R. Watanabe, Combined Explanations of the b→ sµ+µ− and b→ cτ−ν

Anomalies: a General Model Analysis, Phys. Rev. D 99 (2019) 015007, [1806.07403].[36] A. Crivellin, C. Greub, D. Müller and F. Saturnino, Importance of Loop Effects in Explaining the

Accumulated Evidence for New Physics in B Decays with a Vector Leptoquark, Phys. Rev. Lett. 122(2019) 011805, [1807.02068].

[37] A. Angelescu, D. Bečirević, D. Faroughy and O. Sumensari, Closing the window on single leptoquarksolutions to the B-physics anomalies, JHEP 10 (2018) 183, [1808.08179].

[38] J. Aebischer, A. Crivellin and C. Greub, QCD improved matching for semileptonic B decays withleptoquarks, Phys. Rev. D 99 (2019) 055002, [1811.08907].

[39] B. Chauhan and S. Mohanty, Leptoquark solution for both the flavor and ANITA anomalies, Phys. Rev.D 99 (2019) 095018, [1812.00919].

[40] B. Fornal, S. A. Gadam and B. Grinstein, Left-Right SU(4) Vector Leptoquark Model for FlavorAnomalies, Phys. Rev. D 99 (2019) 055025, [1812.01603].

[41] M. J. Baker, J. Fuentes-Martín, G. Isidori and M. König, High- pT signatures in vector–leptoquarkmodels, Eur. Phys. J. C 79 (2019) 334, [1901.10480].

33

[42] C. Hati, J. Kriewald, J. Orloff and A. Teixeira, A nonunitary interpretation for a single vectorleptoquark combined explanation to the B-decay anomalies, JHEP 12 (2019) 006, [1907.05511].

[43] C. Cornella, J. Fuentes-Martin and G. Isidori, Revisiting the vector leptoquark explanation of theB-physics anomalies, JHEP 07 (2019) 168, [1903.11517].

[44] L. Da Rold and F. Lamagna, A vector leptoquark for the B-physics anomalies from a composite GUT,JHEP 12 (2019) 112, [1906.11666].

[45] K. Cheung, Z.-R. Huang, H.-D. Li, C.-D. Lü, Y.-N. Mao and R.-Y. Tang, Revisit to the b→ cτν

transition: in and beyond the SM, 2002.07272.[46] P. Bhupal Dev, R. Mohanta, S. Patra and S. Sahoo, Unified explanation of flavor anomalies, radiative

neutrino masses, and ANITA anomalous events in a vector leptoquark model, Phys. Rev. D 102 (2020)095012, [2004.09464].

[47] S. Kumbhakar and R. Mohanta, Investigating the effect of U1 vector leptoquark on b→ uτν mediatedB decays, 2008.04016.

[48] S. Iguro, M. Takeuchi and R. Watanabe, Testing Leptoquark/EFT in B→ D(∗)lν at the LHC,2011.02486.

[49] C. Hati, J. Kriewald, J. Orloff and A. Teixeira, The fate of vector leptoquarks: the impact of futureflavour data, 2012.05883.

[50] J. Alda, J. Guasch and S. Penaranda, Anomalies in B mesons decays: A phenomenological approach,2012.14799.

[51] T. Mandal, S. Mitra and S. Seth, Pair Production of Scalar Leptoquarks at the LHC to NLO PartonShower Accuracy, Phys. Rev. D93 (2016) 035018, [1506.07369].

[52] D. Das, C. Hati, G. Kumar and N. Mahajan, Towards a unified explanation of RD(∗) , RK and (g−2)µ

anomalies in a left-right model with leptoquarks, Phys. Rev. D 94 (2016) 055034, [1605.06313].[53] P. Bandyopadhyay and R. Mandal, Vacuum stability in an extended standard model with a

leptoquark, Phys. Rev. D95 (2017) 035007, [1609.03561].[54] U. K. Dey, D. Kar, M. Mitra, M. Spannowsky and A. C. Vincent, Searching for Leptoquarks at IceCube

and the LHC, Phys. Rev. D98 (2018) 035014, [1709.02009].[55] P. Bandyopadhyay and R. Mandal, Revisiting scalar leptoquark at the LHC, Eur. Phys. J. C78 (2018)

491, [1801.04253].[56] U. Aydemir, D. Minic, C. Sun and T. Takeuchi, B-decay anomalies and scalar leptoquarks in unified

Pati-Salam models from noncommutative geometry, JHEP 09 (2018) 117, [1804.05844].[57] S. Bansal, R. M. Capdevilla, A. Delgado, C. Kolda, A. Martin and N. Raj, Hunting leptoquarks in

monolepton searches, Phys. Rev. D 98 (2018) 015037, [1806.02370].[58] A. Biswas, D. Kumar Ghosh, N. Ghosh, A. Shaw and A. K. Swain, Collider signature of U1 Leptoquark

and constraints from b→ c observables, J. Phys. G47 (2020) 045005, [1808.04169].[59] R. Mandal, Fermionic dark matter in leptoquark portal, Eur. Phys. J. C78 (2018) 726, [1808.07844].[60] A. Biswas, A. Shaw and A. K. Swain, Collider signature of V2 Leptoquark with b→ s flavour

observables, LHEP 2 (2019) 126, [1811.08887].[61] J. Roy, Probing leptoquark chirality via top polarization at the Colliders, 1811.12058.[62] A. Alves, O. J. P. Éboli, G. Grilli Di Cortona and R. R. Moreira, Indirect and monojet constraints on

scalar leptoquarks, Phys. Rev. D99 (2019) 095005, [1812.08632].[63] U. Aydemir, T. Mandal and S. Mitra, Addressing the RD(∗) anomalies with an S1 leptoquark from

SO(10) grand unification, Phys. Rev. D101 (2020) 015011, [1902.08108].[64] K. Chandak, T. Mandal and S. Mitra, Hunting for scalar leptoquarks with boosted tops and light

leptons, Phys. Rev. D100 (2019) 075019, [1907.11194].[65] W.-S. Hou, T. Modak and G.-G. Wong, Scalar leptoquark effects on B→ µν decay, Eur. Phys. J. C79

(2019) 964, [1909.00403].[66] M. Bordone, O. Catà and T. Feldmann, Effective Theory Approach to New Physics with Flavour:

General Framework and a Leptoquark Example, JHEP 01 (2020) 067, [1910.02641].[67] R. Padhan, S. Mandal, M. Mitra and N. Sinha, Signatures of R2 class of Leptoquarks at the upcoming

ep colliders, Phys. Rev. D 101 (2020) 075037, [1912.07236].[68] A. Bhaskar, D. Das, B. De and S. Mitra, Enhancing scalar productions with leptoquarks at the LHC,

Phys. Rev. D 102 (2020) 035002, [2002.12571].[69] P. Bandyopadhyay, S. Dutta and A. Karan, Investigating the Production of Leptoquarks by Means of

Zeros of Amplitude at Photon Electron Collider, 2003.11751.[70] L. Buonocore, U. Haisch, P. Nason, F. Tramontano and G. Zanderighi, Lepton-Quark Collisions at

the Large Hadron Collider, Phys. Rev. Lett. 125 (2020) 231804, [2005.06475].

34

[71] M. Bordone, O. Cata, T. Feldmann and R. Mandal, Constraining flavour patterns of scalarleptoquarks in the effective field theory, 2010.03297.

[72] A. Greljo and N. Selimovic, Lepton-Quark Fusion at Hadron Colliders, precisely, 2012.02092.[73] U. Haisch and G. Polesello, Resonant third-generation leptoquark signatures at the Large Hadron

Collider, 2012.11474.[74] P. Bandyopadhyay, S. Dutta and A. Karan, Zeros of Amplitude in the Associated Production of Photon

and Leptoquark at e-p Collider, 2012.13644.[75] A. Crivellin, D. Müller and L. Schnell, Combined Constraints on First Generation Leptoquarks,

2101.07811.[76] A. Greljo, G. Isidori and D. Marzocca, On the breaking of Lepton Flavor Universality in B decays,

JHEP 07 (2015) 142, [1506.01705].[77] D. A. Faroughy, A. Greljo and J. F. Kamenik, Confronting lepton flavor universality violation in B

decays with high-pT tau lepton searches at LHC, Phys. Lett. B 764 (2017) 126–134, [1609.07138].[78] N. Raj, Anticipating nonresonant new physics in dilepton angular spectra at the LHC, Phys. Rev. D 95

(2017) 015011, [1610.03795].[79] I. Doršner, S. Fajfer, D. A. Faroughy and N. Košnik, The role of the S3 GUT leptoquark in flavor

universality and collider searches, JHEP 10 (2017) 188, [1706.07779].[80] D. Bečirević, I. Doršner, S. Fajfer, N. Košnik, D. A. Faroughy and O. Sumensari, Scalar leptoquarks

from grand unified theories to accommodate the B-physics anomalies, Phys. Rev. D 98 (2018) 055003,[1806.05689].

[81] T. Mandal, S. Mitra and S. Raz, RD(∗) motivated S1 leptoquark scenarios: Impact of interference onthe exclusion limits from LHC data, Phys. Rev. D99 (2019) 055028, [1811.03561].

[82] ATLAS collaboration, G. Aad et al., Search for heavy Higgs bosons decaying into two tau leptons withthe ATLAS detector using pp collisions at

√s = 13 TeV, Phys. Rev. Lett. 125 (2020) 051801,

[2002.12223]. HEPData link: https://www.hepdata.net/record/ins1782650.[83] CMS collaboration, A. M. Sirunyan et al., Search for resonant and nonresonant new phenomena in

high-mass dilepton final states at√

s = 13 TeV, 2103.02708.HEPData link: https://www.hepdata.net/record/ins1849964.

[84] A. Bhaskar, T. Mandal and S. Mitra, Boosting vector leptoquark searches with boosted tops, Phys. Rev.D 101 (2020) 115015, [2004.01096].

[85] CMS collaboration, A. M. Sirunyan et al., Constraints on models of scalar and vector leptoquarksdecaying to a quark and a neutrino at

√s = 13 TeV, Phys. Rev. D98 (2018) 032005, [1805.10228].

[86] ATLAS collaboration, G. Aad et al., Search for pairs of scalar leptoquarks decaying into quarks andelectrons or muons in

√s = 13 TeV pp collisions with the ATLAS detector, JHEP 10 (2020) 112,

[2006.05872].[87] ATLAS collaboration, M. Aaboud et al., Searches for third-generation scalar leptoquarks in

√s = 13

TeV pp collisions with the ATLAS detector, JHEP 06 (2019) 144, [1902.08103].[88] CMS collaboration, A. M. Sirunyan et al., Search for singly and pair-produced leptoquarks coupling

to third-generation fermions in proton-proton collisions at√

s = 13 TeV, 2012.04178.[89] W. Buchmuller, R. Ruckl and D. Wyler, Leptoquarks in Lepton - Quark Collisions, Phys. Lett. B191

(1987) 442–448. [Erratum: Phys. Lett.B448,320(1999)].[90] J. Blumlein, E. Boos and A. Pukhov, Leptoquark pair production at ep colliders, Mod. Phys. Lett. A9

(1994) 3007–3022, [hep-ph/9404321].[91] J. Blumlein, E. Boos and A. Kryukov, Leptoquark pair production in hadronic interactions, Z. Phys.

C76 (1997) 137–153, [hep-ph/9610408].[92] I. Dorsner, S. Fajfer, A. Greljo, J. F. Kamenik and N. Kosnik, Physics of leptoquarks in precision

experiments and at particle colliders, Phys. Rept. 641 (2016) 1–68, [1603.04993].[93] J. Alda, J. Guasch and S. Penaranda, Some results on Lepton Flavour Universality Violation, Eur.

Phys. J. C 79 (2019) 588, [1805.03636].[94] A. K. Alok, B. Bhattacharya, D. Kumar, J. Kumar, D. London and S. U. Sankar, New physics in

b→ sµ+µ−: Distinguishing models through CP-violating effects, Phys. Rev. D 96 (2017) 015034,[1703.09247].

[95] M. Tanaka and R. Watanabe, New physics in the weak interaction of B→ D(∗)τν , Phys. Rev. D 87(2013) 034028, [1212.1878].

[96] S. Iguro, T. Kitahara, Y. Omura, R. Watanabe and K. Yamamoto, D∗ polarization vs. RD(∗) anomaliesin the leptoquark models, JHEP 02 (2019) 194, [1811.08899].

[97] Belle, Belle-II collaboration, K. Adamczyk, Semitauonic B decays at Belle/Belle II, in 10th

35

International Workshop on the CKM Unitarity Triangle, 1, 2019. 1901.06380.[98] S. Bhattacharya, S. Nandi and S. Kumar Patra, b→ cτντ Decays: a catalogue to compare, constrain,

and correlate new physics effects, Eur. Phys. J. C 79 (2019) 268, [1805.08222].[99] Particle Data Group collaboration, M. Tanabashi et al., Review of Particle Physics, Phys. Rev. D 98

(2018) 030001.[100] UTfit collaboration, M. Bona, Latest results for the Unitary Triangle fit from the UTfit Collaboration,

PoS CKM2016 (2017) 096.[101] A. Akeroyd and C.-H. Chen, Constraint on the branching ratio of Bc→ τν from LEP1 and

consequences for R(D(∗)) anomaly, Phys. Rev. D 96 (2017) 075011, [1708.04072].[102] T. Inami and C. S. Lim, Effects of Superheavy Quarks and Leptons in Low-Energy Weak Processes

KL→ µµ, K+→ π+νν and K0↔ K0, Prog. Theor. Phys. 65 (1981) 297. [Erratum: Prog.Theor.Phys.65, 1772 (1981)].

[103] L. Di Luzio, M. Kirk and A. Lenz, Updated Bs-mixing constraints on new physics models for b→ s`+`−

anomalies, Phys. Rev. D 97 (2018) 095035, [1712.06572].[104] M. Algueró, B. Capdevila, A. Crivellin, S. Descotes-Genon, P. Masjuan, J. Matias et al., Emerging

patterns of New Physics with and without Lepton Flavour Universal contributions, Eur. Phys. J. C 79(2019) 714, [1903.09578]. [Addendum: Eur.Phys.J.C 80, 511 (2020)].

[105] J. Aebischer, W. Altmannshofer, D. Guadagnoli, M. Reboud, P. Stangl and D. M. Straub, B-decaydiscrepancies after Moriond 2019, Eur. Phys. J. C 80 (2020) 252, [1903.10434].

[106] M. Algueró, B. Capdevila, S. Descotes-Genon, J. Matias and M. Novoa-Brunet, b→ s`` global fitsafter Moriond 2021 results, in 55th Rencontres de Moriond on Electroweak Interactions and UnifiedTheories, 4, 2021. 2104.08921.

[107] C. Bobeth, M. Misiak and J. Urban, Photonic penguins at two loops and mt dependence ofBR[B→ Xsl+l−], Nucl. Phys. B 574 (2000) 291–330, [hep-ph/9910220].

[108] C. Bobeth, A. J. Buras, F. Kruger and J. Urban, QCD corrections to B→ Xd,sνν , Bd,s→ `+`−,K→ πνν and KL→ µ+µ− in the MSSM, Nucl. Phys. B 630 (2002) 87–131, [hep-ph/0112305].

[109] W. Altmannshofer, C. Niehoff, P. Stangl and D. M. Straub, Status of the B→ K∗µ+µ− anomaly afterMoriond 2017, Eur. Phys. J. C 77 (2017) 377, [1703.09189].

[110] N. Vignaroli, Seeking leptoquarks in the tt plus missing energy channel at the high-luminosity LHC,Phys. Rev. D99 (2019) 035021, [1808.10309].

[111] T. Mandal, S. Mitra and S. Seth, Single Productions of Colored Particles at the LHC: An Example withScalar Leptoquarks, JHEP 07 (2015) 028, [1503.04689].

[112] T. Mandal and S. Mitra, Probing Color Octet Electrons at the LHC, Phys. Rev. D87 (2013) 095008,[1211.6394].

[113] T. Mandal, S. Mitra and S. Seth, Probing Compositeness with the CMS ee j j & ee j Data, Phys. Lett.B758 (2016) 219–225, [1602.01273].

[114] A. Alloul, N. D. Christensen, C. Degrande, C. Duhr and B. Fuks, FeynRules 2.0 - A complete toolboxfor tree-level phenomenology, Comput. Phys. Commun. 185 (2014) 2250–2300, [1310.1921].

[115] C. Degrande, C. Duhr, B. Fuks, D. Grellscheid, O. Mattelaer and T. Reiter, UFO - The UniversalFeynRules Output, Comput. Phys. Commun. 183 (2012) 1201–1214, [1108.2040].

[116] J. Alwall, R. Frederix, S. Frixione, V. Hirschi, F. Maltoni, O. Mattelaer et al., The automatedcomputation of tree-level and next-to-leading order differential cross sections, and their matching toparton shower simulations, JHEP 07 (2014) 079, [1405.0301].

[117] R. D. Ball et al., Parton distributions with LHC data, Nucl. Phys. B867 (2013) 244–289,[1207.1303].

[118] T. Sjostrand, S. Mrenna and P. Z. Skands, PYTHIA 6.4 Physics and Manual, JHEP 05 (2006) 026,[hep-ph/0603175].

[119] M. L. Mangano, M. Moretti, F. Piccinini and M. Treccani, Matching matrix elements and showerevolution for top-quark production in hadronic collisions, JHEP 01 (2007) 013, [hep-ph/0611129].

[120] S. Hoeche, F. Krauss, N. Lavesson, L. Lonnblad, M. Mangano, A. Schalicke et al., Matching partonshowers and matrix elements, in HERA and the LHC: A Workshop on the implications of HERA for LHCphysics: Proceedings Part A, 2006. hep-ph/0602031.

[121] DELPHES 3 collaboration, J. de Favereau, C. Delaere, P. Demin, A. Giammanco, V. Lemaître,A. Mertens et al., DELPHES 3, A modular framework for fast simulation of a generic colliderexperiment, JHEP 02 (2014) 057, [1307.6346].

[122] M. Cacciari, G. P. Salam and G. Soyez, FastJet User Manual, Eur. Phys. J. C72 (2012) 1896,[1111.6097].

36

[123] M. Cacciari, G. P. Salam and G. Soyez, The anti-kt jet clustering algorithm, JHEP 04 (2008) 063,[0802.1189].

37