Low-energy Effects of Lepton Flavour Universality Violation

The persisting anomalous data in semileptonic B-decays point towards New Physics models exhibiting large sources of Lepton Flavour Universality Violation. In this work we generalise previous studies by considering frameworks which include an enlarged set of semileptonic four-fermion operators invariant under the SM gauge group, with New Physics affecting mainly the third generation. We derive the low-energy effective Lagrangian including the leading electroweak corrections, mandatory to obtain reliable predictions. As a particularly interesting case, we analyse the scenario where the dominant New Physics effects are encoded in the Wilson coefficient C_9, as favoured by global fit analyses of b ->s data. We find that also in this case the stringent experimental bounds on Z-pole observables and tau decays challenge a simultaneous explanation of charged and neutral-current non-standard data.


Introduction
In the last few years, various experimental collaborations observed indications of Lepton Flavour Universality Violation (LFUV) in semileptonic B decays. Although such indications are not yet conclusive, the overall pattern of deviations from the Standard Model (SM) predictions is very coherent. The anomalous data refer to i) charged-current transitions b → c ν with τ /e and τ /µ LFUV [1][2][3][4] and ii) neutral-current transitions b → s ¯ with µ/e LFUV [5,6]. Interestingly enough, global fit analyses for the angular distributions of the B 0 → K * 0 µ + µ − decay reported anomalies which are consistent with LFUV data [7][8][9]. From a theoretical point of view, it would be desirable to explain both the charged-and neutral-current anomalies within a coherent extension of the SM [10][11][12][13][14][15][16][17][18]. A first step towards this goal is represented by an effective theory where the effects of New Physics (NP) are described by four-fermion operators involving left-handed currents, (s L γ µ b L )(μ L γ µ µ L ) and (c L γ µ b L )(τ L γ µ ν L ), which are related by the SU (2) L gauge symmetry [19,20]. A crucial ingredient of such a theory requires that NP couples much more strongly to the third generation than to the first two, since (c L γ µ b L )(τ L γ µ ν L ) is induced already at the tree level in the SM while (s L γ µ b L )(μ L γ µ µ L ) arises only at loop-level. The latter requirement is realized, for instance, if NP is coupled only to the third fermion generation in the interaction basis. Couplings to lighter generations are generated after electroweak symmetry breaking by the misalignment between the mass and the interaction bases through small flavour mixing angles [21].
Hence, a minimal framework addressing the B-anomalies consists of an effective Lagrangian defined above the electroweak scale and containing gauge-invariant semileptonic operators involving purely left-handed fermions of the third generation. Assuming such starting point, in [22,23] the low-energy effective Lagrangian including leading electroweak corrections was derived. The most striking effects found were large corrections to the leptonic couplings of the W and Z vector bosons and the generation of a purely leptonic effective Lagrangian. The resulting LFUV in Z and τ decays and τ Lepton Flavour Violating (LFV) contributions turned out to challenge a simultaneous explanation of charged-and neutralcurrent anomalies. Although this conclusion applies under certain assumptions, our main message was that including electroweak corrections is mandatory when addressing the Banomalies with NP at the TeV scale. Another important challenge that one has to face is the lack of signals in direct production at LHC of any mediators responsible of the four-fermion interactions invoked to explain the B-anomalies [24,25].
In this paper we make a step forward compared to [22,23]. In particular, we consider both purely left-handed operators (V − A) × (V − A) as well as operators with right-handed currents of the form (V + A) × (V + A) and (V ± A) × (V ∓ A). This effort is justified by the fact that many NP models, proposed to accommodate B-anomalies, exhibit the operators considered here 1 . Moreover, as we will discuss in the following, such enlarged operator basis will allow us to consider one of the most favoured solutions to the neutral-current anomalies, with dominant NP effects encoded in the low-energy Wilson coefficient C 9 [7][8][9].
The paper is organised as follows. In section 2, we present the theoretical framework and construct the low-energy effective Lagrangian including electroweak corrections in the leading logarithm approximation. In section 3, we examine the phenomenological implications of our setup, discussing both tree-level and loop-induced low-energy observables. In section 4, we focus on the scenario where the dominant NP effects are encoded in the Wilson coefficient C 9 , providing a numerical analysis. Our conclusions are presented in section 5.

Theoretical framework
We assume that strong and electroweak interactions at the scale Λ m W are described by the effective Lagrangian where the NP contribution is given by and the semileptonic operators Q i are defined in table 1, where primed fields indicate fields in the interaction basis. We denote the Wilson coefficients at the scale Λ by (3) q (Λ)] 3333 and so on. Notice that (2) assumes that NP couples only to third generation fermions. Couplings to light generations will arise when switching from the interaction to the mass basis after electroweak symmetry breaking, as we will describe shortly. Such an assumption is motivated by the need of generating a hierarchy between NP effects in chargedand neutral-current semileptonic B-decays, as suggested by experimental data. We move to the mass basis, denoted by unprimed fields, by means of the unitary transformations where we work in the approximation of massless neutrinos. To keep track of the flavour structure of the Lagrangian, we define the following matrices in flavour space where λ and Γ are both projectors with trace equal to one, and the λ matrices are related by being the quark mixing matrix. Hereafter we will omit the subscript CKM for simplicity. In the mass basis the Lagrangian L 0 NP reads: From this expression we can read the independent parameters of our setup, namely the five Wilson coefficients C i and the matrices λ e , λ d , Γ e and Γ d .

Leptonic operators
Semileptonic operators Vector operators Hadronic operators Table 1: SU (2) L × U (1) Y invariant operators involved in the renormalization group evolution of L 0 NP from Λ to the EW scale. We adopt the same notation as in [28].
Following the same steps of ref. [22,23], we include RGE electroweak effects in leading logarithmic approximation. The operators involved in the running from Λ to the EW scale are displayed in table 1. We find that the effective Lagrangian at the scale m EW < µ < Λ is given by L = L SM + L 0 NP + L eff , where L eff describes the contribution induced by RGE and can be written as Explicitly we have where L = log Λ µ , the sum over repeated flavour indices is understood and the results are expressed in the interaction basis. In the above expressions, we have sistematically included both gauge and top yukawa interactions, exploiting the results of [29,30] 2 . Instead, we have neglected down-quark and leptons yukawas since their effects are very small.
After the breaking of the electroweak symmetry, δL V induces modifications of the W and Z couplings to fermions. The full Z and W Lagrangian reads: where and c W = cos θ W . These expressions include the SM contribution and the NP contribution, encoded in the deviations ∆g f L,R and ∆g q/ . For the Z couplings we have while for W couplings we find 2 Notice that QCD interactions do not renormalise the quark currents V ± A analysed here. We see that RGE effects induce flavour and flavour universality violating interactions, which are absent in the SM. We have explicitly checked that the dependence on the unphysical scale µ cancels when physical quantities are computed. For W and Z decays, this approximately amounts to make use of L Z,W in eq. (11) in the tree-level approximation by replacing µ with the electroweak scale. At the scale µ = m EW we match the effective Lagrangian L eff with a new Lagrangian L EW eff obtained by integrating out the W , Z bosons and the top quark. For the vector bosons W and Z we work at the tree-level. Disregarding the purely hadronic contribution, we get: The operators Q i and their coefficients ξ i are listed in the tables 2,3,4 and 5.
Below the electroweak scale only the residual electromagnetic gauge symmetry is relevant to our discussion, and the effective theory consists of a combination of U (1) em -invariant operators whose Wilson coefficients run under the effect of QED interactions only. By lowering the scale µ we first cross the bottom quark mass threshold, then the charm one. When crossing a threshold we integrate out the corresponding quark and match the theory to a new one. At the scale µ ≈ 1 GeV we get the following result for the effective Lagrangian L QED eff : where the U (1) em -invariant operators Q em i and their coefficients δξ i are collected in tables 6, 7, 8 and 9.

Observables
This section addresses the phenomenological consequences of Lagrangian (2), making use of the RGE-improved low-energy effective field theory (EFT) derived in the previous section. The NP contribution to the observables is parametrised in terms of the free parameters of L 0 NP , namely the five C i and the matrices λ e , λ d , Γ e and Γ d . In order to simplify our phenomenological analysis, we assume real entries in λ e/d and Γ e/d , negligible mixing with the first generation in the matrices λ e/d and Γ e/d , λ     Beyond semileptonic B-decays, we focus on fully leptonic processes and leptonic decays of the Z vector boson as they are the only processes that compete with semileptonic B-decays in constraining our NP parameter space. The structure of this section is as follows. In section 3.1, we discuss how to address both charged-and neutralcurrent B anomalies within our framework. In section 3.2, we discuss the most relevant tree-level phenomenology connected with the B anomalies. In section 3.3, we proceed to study observables in the leptonic sector receiving large contributions at loop-level. In section 3.4, a global numerical analysis is performed in a phenomenologically relevant scenario, where NP affects dominantly the Wilson coefficient C 9 .

The B anomalies
The most significant measurements related to charged-and neutral-current B-anomalies are: where = e, µ, which follow from the HFAG averages [46] of Babar [1], Belle [3], and LHCb data [2], combined with the SM predictions [32,33], and based on combination of LHCb data [6] with the SM expectation R µ/e K ( * ) = 1.00 ± 0.01 [34]. We recall that b → s semileptonic transitions are conventionally described by means of the effective Lagrangian L NC eff 4 : 3 The largest mixing arises from λ e 23 ≈ 0.3. In our numerical analysis we will let |λ e 23 | and |Γ e 23 | vary up to 0.5 by using complete formulae. 4 In our analysis, the inclusion of dipole operators is not necessary as they provide negligible effects.
where λ t bs = V tb V * ts and the operators O i are given by As to the charged-current transition b → c ν, we address it using the effective Lagrangian L CC eff , defined as In our framework B anomalies receive NP contributions at tree level. These contributions can be computed explicitly by matching the low-energy Lagrangians in eqs. 23, 25 with the NP Lagrangian L 0 NP 5 . As a result, we find where subleading RGE terms have been neglected. We remind that NP should contribute dominantly to charged-current transitions compared to the neutral-current ones, since in the SM the former arise at the tree-level while the latter at one-loop. This can be achieved in our framework by assuming a hierarchy between λ d 33 λ e 33 and λ d 23 λ e 22 , which control B → D ( * ) τ ν and B → Kµ + µ − , respectively.

B → K ¯
Keeping only linear terms in NP contributions, R µ/e K ( * ) can be written in our framework as [35] R µ/e K 5 Strictly speaking L NC eff and L CC eff should be matched to the Lagrangian obtained by running the Wilson coefficients down to µ = m B , but RGE induced terms are generally negligible with respect to tree-level ones. This is true unless accidental cancellations among parameters take place, which we exclude.
13 where (C 9 NP ) ee , (C 9 NP ) ee , (C 10 NP ) ee and (C 10 NP ) ee can be neglected because λ e 11 = 0. Remembering that C 9 SM ≈ −C 10 SM ≈ 4.2 [36], we find the numerical expressions and dots stand for smaller contributions. From (28) and the current experimental results (21) and (22), we argue that a simultaneous explanation of R µ/e K and R µ/e K * requires the condition Keeping only linear NP contributions and neglecting λ e 11 and λ e 22 with respect to λ e 33 , we find Then, using the relation λ ud = V CKM λ d , we end up with the following expression As a result, in order to accommodate the R τ / D ( * ) anomaly, we need C 3 < 0 and C 3 ∼ O(1), for Λ = 1 TeV.

Tree-level semileptonic phenomenology
Our framework predicts a set of deviations in leptonic and semileptonic B-decays which are strictly related to the anomalies discussed so far. Since dominant effects occur at tree level, the inclusion of quantum effects is not relevant here.

B → ν
A charged-current process closely related to B → D ( * ) ν is the decay B → ν. We define the related LFUV observable, R τ / Bτ ν , as where γ ≈ 70 • . Since Belle II aims to measure R τ /µ Bτ ν with a 5% accuracy, it is likely that R τ /µ Bτ ν will provide a strong constraint to the present framework.

B → K ( * ) νν
Another important process is B → Kνν, which is strictly related to the neutral-current anomaly. We consider the observable R νν K , defined as which is subject to the experimental constraint R νν K < 4.3 [37,38]. In our framework R νν K can be expressed as By expanding the numerator and using the property ij |λ e ij | 2 = 1 and i λ e ii = 1, we find Since C ν SM ≈ −6.4 [38,39], we get the numerical result

B s → µμ
NP contributions for the observable R µ/e K can also enter the process B s → µμ. In particular, NP effects for B s → µμ are encoded by the following expression to be compared with the current experimental measurement and SM prediction for the branching ratio of this process [40,41]:

Lepton-flavour violating B decays
In our model, LFV decays like B s → τ ± µ ∓ and B → Kτ ± µ ∓ are generated at the tree level. Their branching ratios are given by [42] where the factor of two in the above expressions accounts for the final state τ ± µ ∓ = τ + µ − + τ − µ + . As we will see shortly, loop-induced τ LFV decays are typically better probes of our scenario than LFV B-decays.

One-loop phenomenology
Electroweak corrections induce two main effects. First, Z and W couplings to fermions are modified with respect to the SM. Second, as we can see from eq. (18) and related tables, a purely leptonic Lagrangian is also generated at low energies. As a consequence, we expect LFV and LFUV effects in Z, W and τ observables.

Z-pole observables
The NP modifications to Z couplings arising in our setup, see eq. (15), explicitly break both LFV and LFUV. The consequent deviations of Z-pole observables from SM expectations are tightly constrained by LEP measurements of the Z decay widths, left-right and forwardbackward asymmetries. We recall the definition of the axial and vector couplings and we consider the observables v τ /v e and a τ /a e , which quantify the universality of Z couplings to charged leptons. In our framework they read leading to the following estimates to be compared with the LEP bounds [43] v τ v e = 0.959 (29) a τ a e = 1.0019 (15) .
Another important observable is the number of neutrinos N ν , which is extracted from the invisible Z width. Taking the NP modification of Z couplings to neutrinos into account, N ν can be approximated by while the experimental bound reads N ν = 2.9840 ± 0.0082 [43] . Electroweak quantum corrections give rise also to the LFV decay mode Z → µ ± τ ∓ . However, we have explicitly checked that in our model its branching ratio, typically of order 10 −7 , is always well below the current experimental bound B(Z → µ ± τ ∓ ) exp ≤ 1.2 × 10 −5 . At the loop-level also the W ± couplings to leptons are modified with respect to their SM expectations. However, the constraints on our model parameters arising from Z-pole observables are much stronger and therefore, hereafter, we neglect W ± decays.

Purely leptonic effective Lagrangian
The effective low-energy Lagrangian (18) contains a purely leptonic Lagrangian L eff . Taking into account the explicit values of the ξ i and δξ i for leptonic operators, and omitting terms manifestly respecting LFU in charged leptons, we can write it as where f = {ν L , e L , e R } and g f SM is the Z coupling to the f field in the SM. The coefficients c e t , c e γ , c e t , c e γ are given by Notice that, in all observables analysed in this work but R τ / 1,2 τ (see eq. 47), we systematically neglected corrections to the Fermi constant. Their inclusion would amount to replace G 0 where G F is the value extracted from the muon decay rate measurement. Numerically, such correction is below the 0.1% level and therefore safely negligible since G 0 F ≈ G F (1 − 0.004 λ e 22 C 3 /Λ 2 (TeV)) with λ e 22 1. Lagrangian (45) manifestly generates both LFV and LFUV processes. Given the hierarchy in λ e ij and Γ e ij , NP effects are maximized in transitions involving the third generation. As a consequence, we focus on τ decays such as τ → νν and τ → 3µ.

τ → νν
LFU breaking effects in τ → νν (with 1,2 = e, µ) are described by the observables which are subject to the strong experimental constraints R τ /µ τ = 1.0022 ± 0.0030 and R τ /e τ = 1.0060±0.0030 [44]. Taking into account the correlation of these measurements, the combined constraint reads In our setup the effective Lagrangian describing e α → e βνj ν i is given by where Notice that the SM contribution to e α → e βνj ν i is accounted for by the first term of (50). The ratio R τ / τ can be expressed in terms of these coefficients as follows Working linearly in the NP contribution, we find that

τ → 3µ
One of the most studied LFV processes generated by L eff is the decay τ → 3µ, which is forbidden in the SM. The only contribution is given by L eff where c ( ) Adapting the formula given in ref. [45] we find Keeping only the Yukawa contribution, which is typically the dominant one, we end up with the following numerical estimate to be compared with the current experimental bound B(τ → 3µ) 1.2 · 10 −8 [46].

Numerical analysis
In this section, we focus on a phenomenologically relevant scenario where only (C 9 NP ) µµ is non-vanishing. This can be achieved by imposing the following conditions Taking the NP scale to be Λ ≈ 1 TeV, the free parameters in this setup are C 1 , C 3 , λ d

23
and λ e 23 where |λ e,d 23 | ≤ 0.5 [22,23]. We can further restrict the bounds on λ e 23 because the non-observation of LFUV in R µ/e D implies that |λ e 22 | ≈ |λ e 23 | 2 ≤ 0.1 [10]. As to C 1,3 , we assume |C 1,3 | ≤ 3. Given (57), we obtain the following expressions for B-physics observables On the other hand, Z-pole observables simplify to a τ a e = 1 + 0.007 λ e 33 C 3 Λ 2 (TeV 2 ) Finally, for τ decays, we obtain the following estimates It is interesting to observe that the ratio a τ /a e depends exclusively on the Wilson coefficient C 3 of the charged-current operator. Choosing |λ d 23 | V cb in order to avoid too much fine tuning when reproducing the CKM matrix, there is a strong correlation among R τ / D ( * ) , a τ /a e and R τ / 1,2 τ . In particular, it turns out that the NP room left to R τ / D ( * ) is significantly reduced after taking into account all existing bounds. This can be clearly seen in the graph displayed in fig. 1, which shows the allowed regions for R µ/e K and R τ / D ( * ) after imposing the experimental bounds on Z-pole and τ observables at 2σ level 6 . Altough all observables receiving NP contribution at one loop impose strong bounds on B anomalies, Z-pole observables set the stringest limits, forcing δR τ / D ( * ) to be 0.05. Like in [22,23], we conclude that current data on τ and Z-pole observables challenge a simultaneous explanation of the present values of R µ/e K ( * ) and R τ / D ( * ) , when NP above the electroweak scale mainly affects the operator O 9 and the third generation.
In the plot of fig. 2 we analyse the correlation between the branching ratios of LFV decays, B → Kτ µ and τ → 3µ. The graph shows that the loop-induced process τ → 3µ is a much more sensitive probe of the considered scenario than the tree level observable B → Kτ µ, due to the current and expected future experimental resolution. Exp. bounds

Conclusions
The persisting and coherent anomalous data in semileptonic B-decays point towards New Physics scenarios with large sources of Lepton Flavour Universality Violation. If this is the case, one would expect other non-standard effects to show up in low-and/or high-energy observables. The experimental signatures of specific scenarios able to accommodate these anomalies have been discussed extensively in the recent literature. On the other hand, the importance of including electroweak corrections in scenarios with left-handed semileptonic operators defined at the scale Λ v was stressed in Ref. [22,23]. In this work, by assuming that New Physics mainly affects the third generation, we have generalised the analysis of [22,23] by considering an effective theory involving both purely left-handed operators (V −A)×(V −A) and operators with right-handed currents of the form (V + A) × (V + A) and (V ± A) × (V ∓ A). In this framework, we have derived the low-energy effective Lagrangian by means of the running and matching procedure outlined in [22,23]. As in the previous analysis, we find that the dominant effects concern the corrections to the leptonic couplings of the W and Z vector bosons as well as the generation of a purely leptonic effective Lagrangian. Then we focused on a phenomenologically favoured setup where the dominant New Physics effects are encoded in the low-energy Wilson coefficient C 9 [7][8][9]. As our numerical analysis shows, also in this case the inclusion of electroweak corrections are mandatory to obtain reliable predictions. In particular, we confirm and reinforce the conclusion that the stringent experimental bounds on Z-pole observables and τ decays severely reduce the New Physics room for a simultaneous explanation of charged and neutral-current non-standard data.