Constraints on leptoquarks from lepton-flavour-violating tau-lepton processes

JHEP04(2022)165 Published for SISSA by Springer Received:November 19, 2021 Revised:March 2, 2022 Accepted:April 4, 2022 Published:April 27, 2022 Constraints on leptoquarks from lepton-flavour-violating tau-lepton processes Tomáš Husek,a,b Kevin Monsálvez-Pozoaand Jorge Portolésa aInstituto de Física Corpuscular, CSIC — Universitat de València, Apt. Correus 22085, E-46071 València, Spain bDepartment of Astronomy and Theoretical Physics, Lund University, Box 43, SE-22100 Lund, Sweden E-mail: [email protected],[email protected], [email protected] Abstract: Leptoquarks are ubiquitous in several extensions of the Standard Model and seem to be able to accommodate the universality-violation-driven B-meson-decay anomalies and the (g−2)µdiscrepancy interpreted as deviations from the Standard Model predictions. In addition, the search for lepton-flavour violation in the charged sector is, at present, a major research program that could also be facilitated by the dynamics generated by leptoquarks. In this article, we consider a rather wide framework of both scalar and vector leptoquarks as the generators of lepton-flavour violation in processes involving the tau lepton. We single out its couplings to leptoquarks, thus breaking universality in the lepton sector, and we integrate out leptoquarks at tree level, generating the corresponding dimension-6 operators of the Standard Model Effective Field Theory. In ref. [1] we obtained model-independent bounds on the Wilson coefficients of those operators contributing to lepton-flavour-violating hadron tau decays and `–τconversion in nuclei, with `=e, µ. Hence, we use those results to translate the bounds into the couplings of leptoquarks to the Standard Model fermions. Keywords: Beyond Standard Model, Effective Field Theories ArXiv ePrint: 2111.06872 Open Access,c The Authors. Article funded by SCOAP3.https://doi.org/10.1007/JHEP04(2022)165 JHEP04(2022)165 Contents 1 Introduction 1 2 Leptoquark Lagrangian 4 3 The integration of leptoquarks 6 3.1 Four-fermion operators 7 3.2 Dipole operator and Cγ9 4 Results 10 4.1 Dictionary for the CLFV effective basis 11 4.2 Scalar leptoquarks 12 4.2.1 Four-fermion constraints 12 4.2.2 Dipole operator and Cγconstraints 13 4.3 Vector leptoquarks 15 5 Conclusions 17 A Identification of the SMEFT operator basis 18 B Leading contribution to Cγfrom leptoquarks 19 C Single Yukawas of vector leptoquarks 22 D Translating the bounds 22 1 Introduction The Standard Model (SM) of particle physics is a very successful quantum field theory, which describes the dynamics of the strong interaction as well as the unified electromagnetic and weak interactions — the electroweak (EW) theory. While the SM has passed a number of elaborate experimental tests over a broad range of energies, it is believed already for decades that it does not provide us with the final and complete picture of reality as expected from a fundamental theory. There are purely theoretical reasons to think like that: besides the fact that the SM contains many a priori unknown parameters, there are indications for the unification of the strong and EW forces and the common underlying structure of all fermions, which form the so-called families or generations. From the phenomenological point of view, there are several phenomena which cannot be explained within its framework. For instance, the SM does not provide a viable dark-matter candidate, fails to predict the observed matter-antimatter asymmetry in the Universe, and, as is, it does not strive to – 1 – JHEP04(2022)165 unambiguously incorporate the tiny (though nonzero) masses of neutrinos. Hence, one of the major goals of contemporary particle physics is to look beyond the SM (BSM) for possible explanations of these and other shortcomings. The effects of BSM phenomena on the dynamics of SM particles, arising at energy scales higher than the EW scale (ΛEW), can be encoded in terms of the Standard Model Effective Field Theory (SMEFT) [2,3], in particular in the Wilson coefficients (WCs) — low-energy constants standing in front of the monomials in such an effective Lagrangian. These coefficients can be related to parameters of particular BSM models and could be also determined from experimental results. Within the rich palette of BSM scenarios, a very well motivated class of theories predicts the existence of leptoquarks (LQs) — electrically charged bosons (with spin S= 0,1) which transform as triplets under SU(3)Cand can turn quarks into leptons and vice versa. They naturally emerge in Grand Unified Theories, where strongly non-interacting leptons are accommodated into the same multiplets as quarks: they first appeared in the Pati-Salam model [4,5], and right after in theories based on other symmetry groups, such as the simplest SU(5) in the case of the Georgi-Glashow model [6], SO(10) [7,8], or further on in superstring-inspired E6models [9,10]. They were as well predicted in technicolor and other related models based on the dynamically generated symmetry breaking [11–13], or in models with composite fermions [14–16] or extended scalar sectors [17,18]. At the same time, the persisting existence of several anomalies — discrepancies between the SM theoretical predictions of observables and their experimental values — signals possible effects of new physics (NP), and leptoquarks present themselves as relevant NP candidates, being able to address one or more of these deviations, depending on the chosen model. The discrepancies that have drawn more attention in the recent literature are RD(∗)[19–26], RK[27–34] together with very recent RK0 Sand RK∗+[35], the so-called Banomalies [36], and the anomalous magnetic moment (g−2) of the muon [37,38]. Effects of LQs in these and other processes are extensively studied and parameterised via effective field theory (EFT) frameworks; see e.g. ref. [39] and references therein. More recent updates on the role of leptoquarks in Banomalies and constraints on their couplings to ordinary matter can be found in refs. [40–62], while the (g−2)µdiscrepancy is addressed in refs. [40–44,57–59,61–73]. A detailed analysis of low-energy signals of scalar leptoquarks is presented in ref. [67]. Finally, for current and expected limits from collider searches see refs. [46,48,49,53,62,74–87]. Besides the notorious anomaly-related issues, leptoquarks have also been considered to address other BSM problems like the generation of neutrino masses through one [40, 55,73,88–90], two [40,43,58,64,91] and three loops [41]. Furthermore, their role as mediators between the SM sector and dark matter candidates is studied in refs. [92,93], their implications for baryogenesis are considered in ref. [94], and the ANITA anomalous events are explained using particular leptoquark models in refs. [91,95]. Their existence might also offer a hint on why there are exactly three generations of matter or why there is the same number of quark and lepton species, the consequence of which is the fact that the currents associated with the SM gauge symmetries are non-anomalous — free of Adler-Bell-Jackiw axial anomalies [96–98]. – 2 – JHEP04(2022)165 Following the above reasoning, leptoquarks belong, at present, among the most promising NP contributions. However, despite the immense experimental effort, they have not been directly observed yet. In this work, we address another interesting BSM phenomenon, namely processes exhibiting charged-lepton-flavour violation (CLFV). This effect, while absent in the SM, is expected to happen in presence of massive neutrinos. However, minimal extensions of the SM with light right-handed neutrinos predict tiny CLFV rates, inaccessible to current and mid-term foreseen experiments [99–104]: e.g. Γ(µ→eγ)/Γ(µ→eν¯ν)<10−40. Within the leptoquark framework, CLFV processes can occur at tree level via the exchange of a LQ coupled to (¯ `Γq)and (¯qΓ`0)currents (here, qis short for a quark field, `for a lepton field, and Γthe relevant Dirac tensors), providing enhanced rates for these processes that could be measured at present or future experiments. We will focus on CLFV τ-involved processes since most of the work done in this area has been mainly related to the first and second families (see, for instance, the reviews [105–107]), and the persistence of some charged-currentdriven Banomalies suggests an apparent violation of universality around the third family. Hence, we take the most general couplings of the 5 different types of both scalar and vector leptoquarks to the SM fermions (see, for instance, ref. [39]); in the presence of a righthanded neutrino, that we do not consider, there is an additional type of scalar and vector LQs. Upon integration of those leptoquarks — assuming MLQ ΛEW — the four-fermion (mass-)dimension-6 (D= 6) operators of the SMEFT [2,3] are generated. As commented above, we break universality in the lepton sector by attaching different couplings to the tau lepton and those of the first two families. This is because some works [108,109] point to the aforementioned particularity of the third family and, moreover, the implications related to the first two families in this regard are far less compelling [110]. Moreover, we have also taken into account the τ→`γ decays (with `=e, µ), although their leading contribution arises at one loop. Throughout this procedure, we get the WCs of the D= 6 SMEFT operators expressed in terms of products of a pair of unknown couplings of LQs to SM fermions. In addition, we identify the energy scale of the corresponding SMEFT with the masses of leptoquarks. In ref. [1], we performed a model-independent analysis taking into account current and foreseen experimental data for lepton-favour-violating hadronic τdecays (from Belle [111] and Belle II [112]) and `–τconversion in nuclei, `=e, µ (from NA64 expected sensitivity [113]). As a consequence, we obtained tight bounds on the participating WCs that we can now translate to the products of LQs couplings. Upon general assumptions on those couplings we can also arrive at estimates for the lower bounds on LQ masses. Based on our results, we would like to highlight the strong bounds on LQ masses and couplings that Belle II future results on the hadronic τdecays will be able to establish. The paper is organised in the following way. In section 2, we present the most general CLFV leptoquark Lagrangian based on the SM symmetries accommodating scalars and vectors, and describe the important features of this framework. In section 3, we recover the four-fermion D= 6 SMEFT operators that result from integrating out the LQ fields at tree level. Hence, we give the relations between the WCs and the leptoquark couplings, which we use to constrain the latter in section 4by using our results from ref. [1]. We – 3 – JHEP04(2022)165 point out our main conclusions in section 5. Several technical appendices make easier the understanding of the present work. 2 Leptoquark Lagrangian To systematically explore all possible options, leptoquarks are classified with respect to their spin (scalar or vector) and the way they couple to quarks and leptons based on their transformation properties under the SM gauge group SU(3)C×SU(2)L×U(1)Y: they always transform as colour triplets and range from SU(2) singlets to triplets; leptoquarks with the same SU(2) gauge dimensionality then differ by hypercharge. The electric charge of the leptoquarks is then given, as usual, by Q=I3+Y, where I3stands for the SU(2) generator and Yfor the U(1) hypercharge operator. Leptoquarks have a well-defined fermion number F= 3 B+L, with Band Lbeing the baryon and lepton numbers, respectively. All leptoquark fields are categorised into two sets: |F|= 0,2. In what follows, we use the same notation and conventions as in ref. [39]. Generically, the kinetic and gauge couplings of leptoquarks are described by the Lagrangians LS=X scalars h(DµS)†DµS−M2 SS†Si, LV=X vectors −1 2V† µν Vµν +M2 VV† µVµ+. . . , (2.1) where the field-strength tensor for the vector leptoquarks is Vµν =DµVν−DνVµ. The SM covariant derivative is given by Dµ=∂µ+ig1Y Bµ+ig2IkWk µ+ig3 λA 2GA µ,(2.2) where the λAand Ikare the generators of the SU(3) and SU(2) symmetry groups, respectively, while Yis the LQ hypercharge operator. Note that the Ikdepend on the leptoquark SU(2) representation, e.g. for a leptoquark doublet, Ik=τk/2, with τkbeing the Pauli matrices, while for a triplet we have Ik= (Ik)lm =−iεklm, with εabc being the three-dimensional Levi-Civita pseudotensor (ε123 = 1). In eq. (2.1), the dots in the vector leptoquark Lagrangian correspond to other D= 4 terms that involve additional interactions of the SM gauge fields with the leptoquarks, and self-interactions between the latter. These are allowed by the gauge symmetry (although their couplings are not determined by it) and facilitate the renormalizability of the vector Lagrangian (see ref. [114]), but their explicit form is not essential for further discussion so we do not show them here. Note that we do not consider these vector leptoquarks being gauge bosons of an extended gauge symmetry: the gauge interactions of vector leptoquarks cannot be unambiguously defined due to their uncertain gauge nature, and an ultraviolet completion might be needed [115]. Finally, we do not consider the coupling of leptoquarks to the SM Higgs doublet [62, 116]. The diagonal couplings contribute, together with the mass parameters in the Lagrangian, to the total LQ mass after the spontaneous symmetry breaking has taken place – 4 – JHEP04(2022)165 LQ type SM symmetries Lagrangian S3(¯ 3,3,1/3) YLL 3,ij ¯ QCi,a Lεab (τkSk 3)bc Lj,c L+h.c. R2(3,2,7/6) −YRL 2,ij ¯ui RRa 2εab Lj,b L+YLR 2,ij ¯ei RRa† 2Qj,a L+h.c. ˜ R2(3,2,1/6) −˜ YRL 2,ij ¯ di R˜ Ra 2εab Lj,b L+h.c. ˜ S1(¯ 3,1,4/3) ˜ YRR 1,ij ¯ dC,i R˜ S1ej R+h.c. S1(¯ 3,1,1/3) YLL 1,ij ¯ QCi,a LS1εab Lj,b L+YRR 1,ij ¯uCi RS1ej R+h.c. U3(3,3,2/3) XLL 3,ij ¯ Qi,a Lγµ(τkUk 3,µ)ab Lj,b L+h.c. V2(¯ 3,2,5/6) XRL 2,ij ¯ dCi RγµVa 2,µ εab Lj,b L+XLR 2,ij ¯ QCi,a Lγµεab Vb 2,µ ej R+h.c. ˜ V2(¯ 3,2,−1/6) −˜ XRL 2,ij ¯uC,i Rγµ˜ Va 2,µ εab Lj,b L+h.c. ˜ U1(3,1,5/3) ˜ XRR 1,ij ¯ui Rγµ˜ U1,µ ej R+h.c. U1(3,1,2/3) XLL 1,ij ¯ Qi,a LγµU1,µ Lj,a L+XRR 1,ij ¯ di RγµU1,µ ej R+h.c. Table 1. Classification of leptoquarks based on the representations of matter fields under the SM gauge group SU(3)C×SU(2)L×U(1)Y, and related Lagrangians representing the interactions of leptoquarks with SM quarks and leptons. Only terms potentially responsible for CLFV are shown, and all these terms then constitute the LLQ–SM Lagrangian. The Yukawa-like couplings Yχ1χ2 d,ij and Xχ1χ2 d,ij ,d= 1,2,3, are dimensionless. The hypercharge Yis related to the electric charge Qvia Q= I3+Y, with I3staying for the third SU(2)Lgenerator, the specific realisation of which, as mentioned in the main text, depends on the corresponding leptoquark representation. As is customary, QLand LLstand for the left-handed quark and lepton SU(2) doublets, uR(dR)and eRare the up(down)- type quark and lepton right-handed SU(2) singlets, respectively. Finally, τkare the Pauli matrices ({τk, τl}= 2δkl12) and εstands for the Levi-Civita symbol in two dimensions (ε=iτ2). The letters i, j = 1,2,3denote flavour indices, while a, b, c = 1,2are SU(2)-gauge-group-related indices. and the Higgs acquired a vacuum expectation value. On the other hand, the off-diagonal couplings translate into a mixing of leptoquarks in table 1when diagonalising the total LQ mass matrix and working in the mass basis. However, and as a consequence of the fact that the LQ-Higgs couplings involve two leptoquark fields, the corresponding mixing components behave as O(1/M2 LQ)[62]. Hence, upon integration, the effects of the induced mixing to the four-fermion operators present in our analysis are O(1/M4 LQ), and are thus of higher order than the ones we consider in our work. The most general renormalizable Lagrangian based on the SM symmetries that realises interactions of leptoquarks with fermion pairs contains in total 10 types of leptoquark fields (which extends to 12 if right-handed neutrinos are brought into the picture): 5 scalar and 5 vector ones. For each leptoquark type, the terms potentially responsible for the `–τ conversion and τ→(`+hadrons)decays (with `=e, µ) are shown in table 1, where all possible flavour structures for the Yukawa-like couplings should be taken into account. However, as it was motivated in our previous work [1], we consider an egalitarian flavour – 5 – JHEP04(2022)165 structure in the quark sector: equal entries for all quark flavours in the Wilson coefficient matrices Cij of the SMEFT (see below), where i, j run over all quark flavours. Hence, our LQ Yukawas are quark-flavour-blind, whereas in the lepton sector we allow for flavour violation — although only in the third family, while keeping flavour universality for the first and second families. The LQ Yukawa couplings in table 1, namely Yand X, will be assumed to be real. Our ultra-violet (UV) Lagrangian, at the leptoquark mass scale, is then given by: LUV =LSM +LS+LV+LLQ–SM ,(2.3) with LLQ–SM consisting of the operators from table 1. Among various possible additional interactions, the so-called diquark couplings (i.e. when the LQ is coupled to a quarkantiquark pair) may appear at tree level in the Lagrangian (2.3) (although there are no analogous dilepton couplings to leptoquarks). This entails a possible danger to matter stability, or, in turn, strong bounds on LQ masses or couplings. Therefore, to avoid dealing with the proton-decay issue and since, in any case, diquark couplings do not play any role at tree level in CLFV processes, without any loss of generality we do not consider these couplings in this work and we have not included them in table 1. Although this precludes proton decay in the minimal scenario when assuming (besides the SM content) only one leptoquark species at a time, note that regardless of the presence or absence of diquark couplings, a richer scenario as the one treated here with all leptoquarks considered simultaneously is much more involved [39,117]. Hence, since these kinds of interactions do not affect our CLFV analysis and in order not to dive into this issue any further here, we simply assume, in what follows, that the proton is stable. 3 The integration of leptoquarks The aim of this work is to translate the bounds on the ratio C/Λ2 CLFV (containing the Wilson coefficients (Cs) of the D= 6 operators in the SMEFT and the high-energy scale ΛCLFV) obtained by analysing charged-lepton-flavour-violating τprocesses in ref. [1], into constraints on the couplings and mass scales of the leptoquark Lagrangian described in section 2. Direct searches of leptoquarks have been extensively carried out at the LHC [118]. Lower bounds on their masses depend crucially on their spin (scalar or vector), their weak charges and generations involved, and the supposedly dominant decay products. Generically, we can say that the present status requires MLQ >1–2 TeV, with a slight preference for the higher value [119,120]. Meanwhile, indirect determinations from the B anomalies or lepton-number-violating processes [118] require heavier LQs with masses of at least few TeVs (for O(1) LQ couplings). Hence, it is rather fair to say that MLQ ΛEW and that their contribution to D= 6 SMEFT monomials is hidden into the Wilson coefficients we denote C(or, more generally, into C/Λ2). Assuming, in consequence, that there is such a mass gap between the SM particles and leptoquarks, we can integrate out the (heavy) leptoquark fields to recover the associated D= 6 operators that contribute to the CLFV processes we study. – 6 – JHEP04(2022)165 WC Operator WC Operator C(1) LQ ¯ LpγµLr¯ QsγµQtC(3) LQ ¯ LpγµτILr¯ QsγµτIQt Ceu (¯epγµer) (¯usγµut)Ced (¯epγµer)¯ dsγµdt CLu ¯ LpγµLr(¯usγµut)CLd ¯ LpγµLr¯ dsγµdt CQe ¯ QpγµQr(¯esγµet)CLedQ ¯ Lj per¯ dsQj t C(1) LeQu ¯ Lj perεjk ¯ Qk sutC(3) LeQu ¯ Lj pσµνerεjk ¯ Qk sσµνut CeW ¯ LpσµνerτIϕ WI µν CeB ¯ Lpσµνerϕ Bµν Table 2. Dominant D= 6 SMEFT operators contributing to the CLFV processes generated by leptoquarks. The notation (up to small apparent changes) is the one from ref. [3]. For the family indices, we use p,r,sand t, while jand kare weak isospin indices. For I= 1,2,3,τIare the Pauli matrices, with ε=iτ2, and σµν ≡i 2[γµ, γν]. In the first row, Λdenotes the scale where the new dynamics arises. The operators share the same notation with the associated couplings, substituting simply C→ O, i.e. O(1) LQ and so on. Four-fermion operators are obtained by integrating out, at the leading tree-level contribution, the leptoquark Lagrangian that we described in section 2. The last two operators in the table generate Oγ=cWOeB −sWOeW , which is obtained by integrating out the leptoquarks at the leading one-loop contribution. In our previous work [1], we concluded that the strongest bounds from CLFV processes involving the tau lepton, namely hadronic tau decays, were imposed on the four-fermion operators and the operator responsible for the radiative decay τ→`γ; see table 2for a detailed list of these operators. Indeed, it was the Oγ=cWOeB −sWOeW operator (here and below, cW= cos θWand sW= sin θW, with θWbeing the weak mixing (Weinberg) angle; see also section 4.1) which was getting the strongest bound and has two relevant features. Firstly, it incorporates a Higgs field (as can be seen in table 2). Secondly, it is generated at the one-loop level upon leptoquark integration once the spontaneous symmetry breaking of the electroweak symmetry has taken place. In this article, we will consider the leptoquark contributions to both the four-fermion and Oγoperators at leading order. We thus perform the low-energy matching (MLQ  ΛEW) of the leptoquark Lagrangian LUV from eq. (2.3) with the SM extended by the D= 6 operators (i.e. with the SMEFT Lagrangian) and obtain the relation among the Wilson coefficients and the couplings of the leptoquark framework — after their running with the scale is taken into account. 3.1 Four-fermion operators We start first with the matching giving the four-fermion CLFV operators. In the classical limit (tree level) this can be achieved by using the equations of motion of the integrated fields, as follows from the application of the steepest descent method to determine the path integral of the effective action [121,122]. This procedure gives a non-local Lagrangian. As- – 7 – JHEP04(2022)165 suming that these scalars (mass MS) and vectors (mass MV) are very heavy, in comparison with the energy scale of the effective action, we can make an expansion in momenta in the corresponding solutions for scalars Sand vectors V, producing a local action. Sticking to the first order, we have, in a generic notation, Sd≃Yχ1χ2 d,rs M2 S ¯ ψ0s χ1ψr χ2, S† d≃Yχ1χ2 d,rs M2 S ¯ ψr χ1ψ0s χ2,(3.1) Vµ d≃ −Xχ1χ2 d,rs M2 V ¯ ψ0s χ1γµψr χ2, V µ† d≃ −Xχ1χ2 d,rs M2 V ¯ ψr χ1γµψ0s χ2,(3.2) where drefers to the SU(2) representation (dimensionality) of the LQ field, and repeated indices (chiralities χkand flavours r, s) are summed over. We will assume, in the following, that all the scalar leptoquarks, independently of their SU(2) quantum numbers, have the same mass MS, and analogously for vector leptoquarks (having mass MV). Inserting these relations back into eq. (2.3) and introducing the notation of table 1used in the rest of the paper, we obtain contributions to the effective Lagrangians accounting for effects stemming from the interactions of the scalar and vector LQ fields: Leff S⊃Yχ1χ2 d,ij Yχ3χ4 d,mn M2 S (¯ ψi χ1ψ0j χ2)( ¯ ψ0n χ4ψm χ3), Leff V⊃Xχ1χ2 d,ij Xχ3χ4 d,mn M2 V (¯ ψi χ1γµψ0j χ2)( ¯ ψ0n χ4γµψm χ3). (3.3) Using the above prescription and restoring the SU(2) gauge structures, we end up with a list of effective D= 6 four-fermion operators. In order to recognise the couplings of the SMEFT Lagrangian in table 2(or the modified basis suitable for the numerical analysis from ref. [1]), we have employed Fierz reordering and several relations and identities, as detailed in the appendix A. Hence, considering generically Λ = MLQ for every leptoquark type,1we can identify products of two LQ couplings with the Wilson coefficients. The results are collected in table 3. Leptoquarks can mediate lepton-flavour violation even when all Yukawa couplings are considered equal. However, since we have motivated and assumed an enhancement of this phenomenon for the third lepton family (see section 1), we consider the τ-related Yukawas different (potentially larger) than those for the other two charged leptons, so that the (potentially stronger) limits imposed (from other works) on the firstor secondfamily-related Yukawas do not apply to our case. This fact, together with the flavour considerations (quark-flavour-blind Yukawas) discussed in the previous section, entail a rather simple flavour structure for the Yukawa matrices. In general, Yχ1χ2 d,rs =       yχ1χ2 dyχ1χ2 dyχ1χ2 d τ yχ1χ2 dyχ1χ2 dyχ1χ2 d τ yχ1χ2 dyχ1χ2 dyχ1χ2 d τ       rs , yχ1χ2 d6=yχ1χ2 d τ .(3.4) 1As it will be clear further below, we consider two separate cases taking Λ = MS(Λ = MV) for scalar (vector) leptoquarks. – 8 – JHEP04(2022)165 Above, mtis the mass of the top quark, v=h0|ϕ|0i= (√2GF)−1/2, and Vtb is the corresponding Cabibbo-Kobayashi-Maskawa matrix element. The main bounds coming from the four-fermion operators on the above-appearing Yukawa pairs are yRL 2τyLR 2= 8 C(3) `-τ LeQu .1.1ΛCLFV TeV 2and yLR 2τyRL 2= 2 C(1) τh LeQu .5.8× 10−3ΛCLFV TeV 2(Belle). Since the bound from τdecays constrains the value of yLR 2τyRL 2 by about 3 orders of magnitude stronger than the bound from `–τconversion does for yRL 2τyLR 2, we can, in eq. (4.4), neglect the contribution from the former and use the limits from Belle and Belle II for Cγ/Λ2 CLFV to constrain yRL 2τyLR 2/M2 S. The results are presented in table 5. As we can see, the Yukawa pair yRL 2τyLR 2as well as the corresponding probed scale MSboth receive a stronger constraint than in the previous case (cf. table 4) where this pair was only sensitive to the limits from `–τconversion in nuclei, and was, accordingly, bounded rather weakly. Leptoquark S1/3 1.In this case, we end up with the following matching (see appendix B): Cγ Λ2 CLFV 2 =e2N2 Cm2 tV2 tb 211π4v2M4 SQS1/3 1−3Q¯ t2h(yLL 1τyRR 1)2+ (yRR 1τyLL 1)2i.(4.5) The main bounds coming from the four-fermion operators on the Yukawa pairs involved are yLL 1τyRR 1= 8 C(3) `-τ LeQu .1.1ΛCLFV TeV 2and yRR 1τyLL 1= 2 C(1) τh LeQu .5.8×10−3ΛCLFV TeV 2(Belle). As before, in eq. (4.5), we can thus neglect the contribution of yRR 1τyLL 1(which receives stronger bounds), and use the limits from Belle and Belle II on Cγ/Λ2 CLFV to constrain yLL 1τyRR 1/M2 S. The results are presented in table 5. Again the bound on the dipole operator helps to constrain the otherwise weakly-bounded (cf. table 4) Yukawa pair yLL 1τyRR 1. Finally, note that in both single-leptoquark cases — even though the Yukawa pairs are more constrained from the bound on Cγstemming from τdecays than from `–τconversion limits — the probed energy scale (when assuming |yy0| ≈ 1) is still smaller than the value obtained from the four-fermion bound of 13 TeV and 36 TeV from Belle and Belle II limits, respectively, on yLR 2τyRL 2/M2 S(for R5/3 2) and yRR 1τyLL 1/M2 S(for S1/3 1). This is due to the fact that the same mass enters for all WCs (operators). Leptoquarks R5/3 2+S1/3 1.When both contributing leptoquarks R5/3 2and S1/3 1are considered at the same time, the corresponding matching given in eq. (B.6) — for natural values of the Yukawa pairs |yy0| ≈ 1— is probing MS&4.7TeV stemming from the Belle τdecay limits (using Cγ/Λ2 CLFV .7×10−5TeV−2) or MS&13 TeV stemming from the Belle II limits (using Cγ/Λ2 CLFV .9×10−6TeV−2). However, it does not provide relevant bounds on the Yukawa pairs. 4.3 Vector leptoquarks We will consider the contribution of vector leptoquarks to the four-fermion operators only. Upon their integration, with ΛCLFV =MV, we end up with 11 distinct Yukawa pairs contributing to 8 Wilson coefficients (see table 3). Due to the flavour structure of the Yukawas and the integration of the LQs themselves, most of the Yukawa pairs contribute equally to both the `–τconversion and τdecays. However, there are two pairs (stemming – 15 – JHEP04(2022)165 from the last column in table 3) which are not symmetric (built from two same matrices) and produce couplings in combinations relevant only to one type of process at a time, thus adding an extra effective bound. These relate only to CLedQ through the following relations: 1 2C`-τ LedQ =xLL 1τxRR 1−xRL 2τxLR 2,1 2Cτh LedQ =xLL 1xRR 1τ−xRL 2xLR 2τ.(4.6) Hence, we can take the two linear combinations of Yukawa pairs present in eq. (4.6) and define new independent pairs of couplings, x`-τ 1,2≡xLL 1τxRR 1−xRL 2τxLR 2and xτh 1,2≡ xLL 1xRR 1τ−xRL 2xLR 2τ, and set bounds on them instead. The relations among the Yukawa pairs and the WCs are then straightforward: xLL 3xLL 3τ=1 2(C(3) LQ −C(1) LQ), xRL 2xRL 2τ=CLd , xLR 2xLR 2τ=CQe , ˜xRL 2˜xRL 2τ=CLu , xLL 1xLL 1τ=−1 2(C(1) LQ + 3C(3) LQ), xRR 1xRR 1τ=−Ced ,(4.7) ˜xRR 1˜xRR 1τ=−Ceu , xτh 1,2=1 2Cτh LedQ , x`-τ 1,2=1 2C`-τ LedQ . Therefore, even though we have 11 Yukawa pairs restricted by 9 effective bounds — which implies that not all Yukawa pairs can be constrained independently from bounds on the WCs — by introducing xprocess 1,2, this budget is effectively changed to 9 Yukawa pairs only. Moreover, the constraints coming from `–τconversion are not competitive compared to those stemming from the τdecays, which implies a softer bound on x`-τ 1,2, exceeding (as in the scalar LQ scenario) for e–τconversion the limits suggested by perturbativity considerations; the bounds for the probed mass (with |xx0| ≈ 1) indicate again that the experimental constraints expected at present from the `–τconversion in nuclei are rather weak. The numerical results are given in table 6and follow the same pattern as in the scalar scenario. Note that the main differences arise for the xLL 3xLL 3τ,xRL 2xRL 2τand xRR 1xRR 1τpairs, which are constrained stronger than their scalar analogues. The combination of Yukawa pairs xτh 1,2is also strongly constrained. Alternatively, we can try to get information on single Yukawa couplings. Under the flavour structure established in eq. (3.4), we have 14 couplings in total. The 4 wearing a tilde contribute in pairs in a unique way (i.e. not to other WCs or in combination with other Yukawas) to CLu and Ceu (see eq. (4.7)), and thus the resulting relations among the single Yukawa couplings and Wilson coefficients necessarily depend on other Yukawas. However, the 10 remaining Yukawas contribute in different ways to the rest of Wilson coefficients: there are in total 7 effective WCs once the bounds from `–τconversion and τdecays are distinguished. Thus, one can express 7 out of these 10 single Yukawas in terms of WCs and 3 remaining unconstrained (free) Yukawas. The corresponding relations are given in appendix C, with ˜xRL 2and ˜xRR 1in the former case and xLL 3,xLL 1and xLR 2in the latter chosen to be free. In light of possible non-zero values of these WCs, the (usually stronger) bounds on the eand µ-involved Yukawas (chosen to be free in this setting) can be used to constrain the remaining 7. Finally, limits on single Yukawas can also be obtained with the help of the perturbativity bounds. Taking the upper bounds on |yy0|obtained above, it can be assumed that the bound on the absolute value of the target Yukawa (say y) can be obtained by assuming that |y0|equals the value given by the perturbative limit. – 16 – JHEP04(2022)165 τdecays Upper bounds on |xx0| M2 V [10−3TeV−2]Lower bounds on MV[TeV] Yukawa pair Belle Belle II Belle Belle II |xLL 3xLL 3τ|15 1.7 8.2 25 |xRL 2xRL 2τ|,|xRR 1xRR 1τ|10 1.5 10 26 |xLR 2xLR 2τ|8.3 1.3 11 28 |˜xRL 2˜xRL 2τ|24 2.5 6.5 20 |xLL 1xLL 1τ|22 3.1 6.7 18 |˜xRR 1˜xRR 1τ|17 2.1 7.7 22 |xτh 1,2|3.1 0.42 18 49 `–τconversion Upper bounds on |xx0| M2 V [100TeV−2]Lower bounds on MV[TeV] Yukawa pair e–τ µ–τ e–τ µ–τ |x`-τ 1,2|330 1.5 0.055 0.83 Table 6. Obtained bounds for the vector leptoquark case from the bounds determined in ref. [1]. In the left-hand part of the table, we present upper bounds on the ratio |xx0|/M2 V; these numbers also correspond to upper bounds on the Yukawa pairs |xx0|, assuming MV= 1 TeV. On the right, there are lower bounds on the probed energy scale of the scalar leptoquarks mediating CLFV phenomena (ΛCLFV =MV), considering |xx0| ≈ 1. Again, the strongest bounds found are shown, stemming mostly from the τdecay constraints, except for the last row related solely to `–τconversion. The values are given at the 99.8 % confidence level. 5 Conclusions Leptoquarks are omnipresent in the recent literature that brings up extensions of the Standard Model of particle physics. Although their concept comes from the period where the construction of Grand Unified Theories was at its high spot [4,5], they have been reborn in the last ten years as a possible explanation to the LHCb [36] and muon (g−2) anomalies [37,38,126]. To play this role they should have a mass of few to tens TeVs. At present, there is no experimental evidence of their existence and present bounds by LHC indicate MLQ &1TeV. Learning that nature allows for neutrino mixing, the immediate rationale suggests that lepton flavour violation should be also present in processes with charged leptons. In ref. [1], we carried out a phenomenological model-independent analysis of charged-lepton-flavourviolating processes involving the tau lepton, namely its hadronic decays and `–τconversion in the presence of nuclei (`=e, µ), and established bounds on the Wilson coefficients of the corresponding D= 6 SMEFT operators. In this article, we have merged both lines from above. We have considered that leptoquarks (both scalar and vectors) should be driving the dynamics of charged-lepton-flavour- – 17 – JHEP04(2022)165 violating processes and, in addition, we consider an extra input: there is a breaking of universality related to the third lepton family with respect to the two lighter ones. Upon the integration of the heavy LQs, we get most of CLFV D= 6 SMEFT operators and, accordingly, we can relate the couplings of LQs to fermions with the Wilson coefficients of the SMEFT. We can then combine these relations with our results from ref. [1], providing relevant information (in terms of bounds) on those couplings and LQs masses. Our main results are collected in tables 4,5and 6. They all show bounds on the LQ masses and the Yukawa-like couplings of LQs to SM fermions. The first two tables correspond to scalar leptoquarks and the last one to vector leptoquarks. In addition, table 4 shows the couplings of four-fermion operators, while table 5shows the results related to Cγ. In our analyses on the latter-mentioned WC, we have included the bounds on the processes τ→`γ for `=e, µ (not considered in ref. [1]) and its leptoquark generated leading contribution that appears at the one-loop level in the perturbative expansion. As already commented in ref. [1], we notice the significant improvement arriving with the foreseen bounds from the expected Belle II results in the hadronic decays of the tau lepton. Considering the Belle II prospects, our results show that the bounds on scalar leptoquarks are weaker (MS&10 TeV) than those on vector leptoquarks (MV&20 TeV). The numerical values of these bounds are in the expected region where LQs could help to explain the aforementioned phenomenological anomalies. Acknowledgments This work has been supported in part by Grant No. MCIN/AEI/FPA2017-84445-P and by MCIN/AEI/10.13039/501100011033 Grant No. PID2020-114473GB-I00, by PROMETEO/2017/053 and PROMETEO/2021/071 (GV), and by the Swedish Research Council grants contract numbers 2016-05996 and 2019-03779. A Identification of the SMEFT operator basis In this appendix, for completeness, we list the identities and relations used to identify the D= 6 four-fermion operators of the basis in ref. [3] from those resulting from the integration of the leptoquark fields. Regarding the scalar Fierz identities, we have in our case for anticommuting fields (¯aRbL)(¯cLdR) = (¯aPLb)(¯cPRd) = −1 2(¯aRγµdR)(¯cLγµbL),(A.1) (¯aRbL)(¯cRdL) = (¯aPLb)(¯cPLd) = −1 2(¯aRdL)(¯cRbL) + 1 4(¯aRσµνdL)(¯cRσµνbL),(A.2) with σµν ≡i 2[γµ, γν]and projectors PR,L =1 2(1±γ5). The vector Fierz identities then read (¯aLγµbL)(¯cLγµdL) = (¯aγµPLb)(¯cγµPLd) = (¯aLγµdL)(¯cLγµbL),(A.3) (¯aLγµbL)(¯cRγµdR) = (¯aγµPLb)(¯cγµPRd) = −2(¯aLdR)(¯cRbL).(A.4) – 18 – JHEP04(2022)165 The fields in the charge-conjugation basis are defined as ψC≡CψT=CγT 0ψ∗(and consequently ψC=−ψTC−1), with Cbeing the charge-conjugation operator (in standard representation, C=iγ2γ0). For the left-handed (ψL=PLψ) and right-handed (ψR=PRψ) components of Dirac fields, using C−1γ5C= +γT 5, we then have3 ψL=PLψ , ψR=PRψ , ψL=ψPR, ψR=ψPL,(A.5) ψC L=PRψC, ψC R=PLψC, ψC L=ψCPL, ψC R=ψCPR.(A.6) Using C−1γµC=−γT µ, for anticommutating ψ1and ψ2(and thus including an additional minus sign) we arrive at ψC 1ψC 2= +ψ2ψ1,(A.7) ψC 1γµψC 2=−ψ2γµψ1,(A.8) ψC 1σµνψC 2=−ψ2σµνψ1.(A.9) With any representation in which C†=C−1, we have employed ψC 1Γψ2†=ψ2ΓψC 1, with Γ∈ {1, γµ, σµν}. For the SU(2) indices we make use of the following manipulations: 3 X k=1 (ε·τk)ab(ε·τk)† cd =X k=0,1,3 τk,abτk,cd = 2δadδbc −τ2,abτ2,cd = 2δadδbc +εabεcd ,(A.10) where ε=iτ2, and we have used the completeness relation P3 k=0 τk,abτk,cd = 2δadδbc. Combining εabεcd =δacδbd −δadδbc and P3 k=1 τk,cbτk,da = 2δcaδbd −δcbδda, it is apparent that εabεcd =1 2" 3 X k=1 τk,cbτk,da!−δadδbc#,(A.11) so we find 3 X k=1 (ε·τk)ab(ε·τk)† cd =δacδbd +δadδbc =3 2δadδbc +1 2 3 X k=1 τk,cbτk,da .(A.12) Eqs. (A.11) and (A.12) get handy when rewriting the expressions in terms of the operators from the SMEFT basis [3] we work with. B Leading contribution to Cγfrom leptoquarks We will study the process `1→`2γat the leading one-loop order and driven by scalar leptoquarks (see table 1). Then, by integrating out the LQs we will match our UV theory with the corresponding operators in the SMEFT [3]. This procedure will allow us to identify the leptoquark Lagrangian parameters with the CγWilson coefficient (defined by eq. (4.2)). 3We write ψC L,R ≡(ψL,R)C. – 19 – JHEP04(2022)165 Once the matching is performed, we will be able to translate the bounds on the latter WC obtained in ref. [1] over the relevant leptoquark parameters. The SMEFT operator, after spontaneous symmetry breaking, reads L ⊃ Cγv √2Λ2 CLFV h¯ `2σµνPR`1+¯ `2σµνPL`1iFµν ,(B.1) where ΛCLFV stands for the new-physics energy scale where the CLFV phenomena would take place and Fµν is the photon field-strength tensor. We write the effective Lagrangian generated by scalar leptoquarks giving the `1→`2γ process as L`→`0γ eff =e 2¯ `2iσµνσ`1`2 RPR+σ`1`2 LPL`1Fµν ,(B.2) where σ`1`2 Rand σ`1`2 Lare different loop functions given, in the most general case, by ref. [123] and recast here below in terms of the parameters of our LQ framework. Note that in our previous work [1] no distinction between left and right polarisations was considered, which in turn meant working with symmetric WC flavour matrices, e.g. Cµτ γ=Cτµ γ(omitting quark-flavour indices). However, within the leptoquark framework this is not always the case as it can be seen in the distinction between `–τconversion and hadronic τdecay processes for some of the Yukawa pairs (see sections 4.2 and 4.3). This also happens in this loop computation entailing a distinction between Cτh γand C`-τ γ. Therefore, we made use of the amplitude squared of the `1→`2γprocess to relate both frameworks since no direct matching at the Lagrangian level is possible. We find the relation C`1`2 γ Λ2 CLFV 2 =e2 √2v2X i σ`1`2 L,i 2 +X i σ`1`2 R,i 2,(B.3) with irunning over all contributing leptoquarks, i.e. S1/3 3,R5/3 2and S1/3 1. The superscripts on the WC Cγlabel the specific process for which the matching is computed — either `1=`,`2=τfor `–τconversion in nuclei, or `1=τ,`2=`for the hadronic τdecays, with `=e, µ. As explained in the main text, the S1/3 3leptoquark does not provide a chirality enhancement effect and so its σloop functions depend only on the lepton masses and not on the top-quark mass. From now on, we will consider the limit of massless leptons and neglect thus the S1/3 3contribution. Upon the integration of the leptoquarks present in the diagrams in figure 1— following the so-called “integration by regions” method [127] — we find for the σloop functions4 •R5/3 2 στh L, R5/3 2 =σ`-τ R, R5/3 2 =iNC 16π2 mt M2 S yRL 2τyLR 2Vtb3 2Qt−1 2QR5/3 2, στh R, R5/3 2 =σ`-τ L, R5/3 2 =iNC 16π2 mt M2 S yLR 2τyRL 2Vtb3 2Qt−1 2QR5/3 2, (B.4) 4Note that only diagrams in figures 1a and 1c contribute to the matching, while those in figures 1b and 1d are responsible for the cancellation of the divergences. – 20 – JHEP04(2022)165 12 q(qC) S(S∗) γ (a) 12 q(qC) S(S∗)γ (b) 12 q(qC) S(S∗) γ (c) 12 q(qC) S(S∗) γ (d) Figure 1. Feynman diagrams of the leading-order contribution to the `1→`2γprocess from scalar leptoquarks. •S1/3 1 στh L, S1/3 1 =σ`-τ R, S1/3 1 =−iNC 16π2 mt M2 S yLL 1τyRR 1Vtb3 2Q¯ t−QS1/3 1, στh R, S1/3 1 =σ`-τ L, S1/3 1 =−iNC 16π2 mt M2 S yRR 1τyLL 1Vtb3 2Q¯ t−QS1/3 1. (B.5) This coincides — in the limit of massless leptons — with the results from ref. [123], once the logarithms (stemming from the low-energy behaviour and, hence, not contributing to the matching) are removed. Above, mtis the mass of the top quark, Vtb is the top-bottom entry of the CKM matrix, Qt= 2/3(Q¯ t=−2/3) is the charge of the top (anti-top) quark and QR5/3 2 = 5/3and QS1/3 1 = 1/3are the charges of the R5/3 2and S1/3 1leptoquarks, respectively. Note that στh L,i =σ`-τ R,i holds true only in the case of massless leptons. The total result for the matching, including the contribution from both leptoquarks, is thus (see eq. (B.3)) Cγ Λ2 CLFV 2 =e2N2 Cm2 tV2 tb 211π4v2M4 S ×"(yRL 2τyLR 2)2+(yLR 2τyRL 2)2QR5/3 2−3Qt2+(yLL 1τyRR 1)2+(yRR 1τyLL 1)2QS1/3 1−3Q¯ t2 −2(yRL 2τyLR 2)(yLL 1τyRR 1)+(yLR 2τyRL 2)(yRR 1τyLL 1)QR5/3 2−3QtQS1/3 1−3Q¯ t#.(B.6) Above, we have omitted the superscripts `1`2of Cγsince in the limit of massless leptons we obtain the same matching for hadronic τdecays and `–τconversion in nuclei. Consequently, the bounds from both processes can be applied. – 21 – JHEP04(2022)165 C Single Yukawas of vector leptoquarks The integration of leptoquarks and the following matching to the SMEFT lead to relations between the Wilson coefficients of this EFT and products of Yukawa leptoquark couplings. However, due to the rich variety of contributions of the Yukawas to the WCs, and the current and expected bounds on the latter from CLFV-τprocesses, one can do better than just constraining pairs of Yukawas. Under a choice of a few free Yukawas, the rest can be related to these and the WCs. The most general leptoquark-matter interacting model provides, for vector leptoquarks, fourteen Yukawa couplings. These are reduced, upon LQ integration, to eight different WCs which receive a total of nine bounds from the charged-lepton-flavourviolating τprocesses considered in this work. Therefore, by choosing ˜xRL 2and ˜xRR 1on one hand and xLL 3,xLL 1and xLR 2on the other to be free, we find the trivial relations ˜xRL 2τ=CLu ˜xRL 2 ,˜xRR 1τ=−Ceu ˜xRR 1 ,(C.1) xLL 3τ=C(3) LQ −C(1) LQ 2xLL 3 , xLL 1τ=−C(1) LQ + 3C(3) LQ 2xLL 1 , xLR 2τ=CQe xLR 2 ,(C.2) and two different solutions for xRL 2±=2CedC(1) LQ −C`-τ LedQCτh LedQ + 6CedC(3) LQ −4CLdCQe ∓√A 4C`-τ LedQCQe xLR 2, xRL 2τ±=2CedC(1) LQ −C`-τ LedQCτh LedQ + 6CedC(3) LQ −4CLdCQe ±√A 4Cτh LedQ 1 xLR 2 , xRR 1±=−2CedC(1) LQ +C`-τ LedQCτh LedQ + 6CedC(3) LQ −4CLdCQe ±√A 2Cτh LedQ(C(1) LQ + 3C(3) LQ)xLL 1, xRR 1τ±=2CedC(1) LQ +C`-τ LedQCτh LedQ + 6CedC(3) LQ −4CLdCQe ∓√A 4C`-τ LedQ 1 xLL 1 , (C.3) where either all the positive or negative solutions are to be chosen, with A=C`-τ LedQCτh LedQ −2Ced(C(1) LQ + 3C(3) LQ)+4CLdCQe2−16CLdC`-τ LedQCτh LedQCQe .(C.4) Note that (usually stronger) limits on the five free variables ˜xRL 2,˜xRR 1,xLL 3,xLL 1and xLR 2 can be found elsewhere, since they involve just the eand µleptons. These can then be used, under the assumptions taken in this work, to constrain the rest of the Yukawas. D Translating the bounds In this appendix, we show straightforwardly how, numerically, the WC bounds are translated into limits on Yukawa pairs according to eqs. (4.3) and (4.7). The Wilson coefficients were given in the previous work as normal (Gaussian) probability distributions with mean µand variance σ2, i.e. C=N(µ, σ2). The non-vanishing correlations among these were collected in the covariance matrix Wij. – 22 – JHEP04(2022)165 In general, given a set of functions of the WCs for which we do not know their probability distribution functions (p.d.fs.), namely (zz0)k=Fk(~ C), with ~ C= (CQ1, CQ2, . . . ) containing all WCs considered in this work and where the symbolic notation zz0can stand both for scalar (y)and vector (x)Yukawa pairs, one can approximate the expectation value of Fkand its covariance matrix through EFk(~ C)≃Fk(~µ),(D.1) and Ukm ≡covFkFm≃ n X i,j=1 "∂Fk ∂CQi ∂Fm ∂CQj#~ C=~µ Wij .(D.2) Above, the derivatives should be evaluated at the mean values of the WCs collected within the vector ~µ, which is, in our case, just a zero-valued vector ~µ =~ 0. For the simple case of eqs. (4.3) and (4.7), the resulting combinations of the WCs contributing to the Yukawa pairs zz0=PQaQCQlead again to a Gaussian p.d.f. for the latter, and eqs. (D.1) and eqs. 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