$R_{K^{(*)}}$ and $R(D^{(*)})$ anomalies resolved with lepton mixing

In a recent paper arXiv:1706.08437, we have advanced a minimal resolution of some of the persistent anomalies in semileptonic $B$-decays. These include the neutral-current observables $R_K$ and $R_{K^*}$, as well as the charged-current observables $R(D)$ and $R(D^*)$. Recently, it has been observed that the semileptonic decays of the $B_c$ meson also hint at a similar type of anomaly. In this longer version, we discuss in detail why, if the anomalies are indeed there, it is a challenging task to explain the data consistently in terms of a simple and compelling new physics scenario. We find that the minimal scheme to achieve a reasonable fit involves the inclusion of just two (or, at worst, three with a possible symmetry relationship between their Wilson coefficients) new current-current operators, constructed in terms of the flavour eigenstates, augmented by a change of basis for the charged lepton fields. With only three unknown parameters, this class of models not only explain all the anomalies (including that in $B_c \to J/\psi \, \ell \nu$) to a satisfactory level but also predict some interesting signatures, like $B\to K\mu\tau$, $B_s\to \tau\tau$, or $B\to K$ plus missing energy, that can be observed at LHCb or Belle-II.


Introduction
The last few years have seen some intriguing hints of discrepancies in a few charged-as well as neutralcurrent decays of B-mesons, when compared to the expectations within the Standard Model (SM). While the fully hadronic decay modes are subject to large (and, in cases, not-so-well understood) strong interaction corrections, the situation is much more under control for semileptonic decays, where the dominant uncertainties come from the form factors and quark mass values. Even these uncertainties are removed to a large extent if one considers ratios of similar observables. Thus, the modes b → c ν and b → s + − are of great interest. While, individually, none of these immediately calls for the inclusion of New Physics (NP) effects (given the current significance levels of the discrepancies), viewed together, they strongly suggest that some NP is lurking around the corner (see, e.g., Ref. [2]). Most interestingly, the pattern also argues convincingly for some NP that violates lepton-flavour universality (LFU), a cornerstone of the SM. The violation is to a level that cannot be simply explained by the inclusion of right-handed neutrino fields and the consequent neutrino masses; indeed, the strength is only a little over an order of magnitude below that generated from the weak scale dynamics.
As we have just mentioned, relative partial widths (or, equivalently, the ratios of branching ratios (BR)) are particularly clean probes of physics beyond the SM, on account of the cancellation of the leading uncertainties inherent in individual BR predictions. Of particular interest to us are the ratios R(D) and R(D * ) pertaining to charged-current decays, defined as with = e or µ, the ratio R J/ψ defined as and analogous ratios for the neutral-current sector For the K * mode, the discrepancy is visible not only in the ratios of binned differential distribution for muon and electron mode, but also in some angular asymmetries in B → K * µµ which we discuss later.
The SM estimates for these decays are already quite robust. With the major source of uncertainty being the form factors, they cancel out in ratios like R(D ( * ) ), R J/ψ , or R K ( * ) . The values of R(D) and R(D * ) as measured by BABAR [3], when taken together, exceed SM expectations by more than 3σ. While the Belle measurements lie in between the SM expectations and the BABAR measurements and are consistent with both [4], their result on R(D * ) [5], with the τ decaying semileptonically, agrees with the SM expectations only at the 1.6σ level 1 . Similarly, the first measurement by LHCb [7] is also 2.1σ above the SM prediction. Considering the myriad results together, and including the correlations, the tension between data and SM is at the level of 4.1σ [9].
While the data on R(D) and R(D * ) lie above the SM predictions, those on R K and R K * are systematically below the expectations. A similar shortfall has been observed in the q 2 ∈ [1 : 6] GeV 2 bin for the decay B s → φµµ, which is again mediated by the process b → sµµ. However, no appreciable discrepancy is found between the data on the purely leptonic decay B s → µ + µ − and the radiative decay B → X s γ, and the corresponding SM expectations. The same is true for the mass difference ∆M s and mixing phase φ s measurements for the B s system. The pattern of deviations is thus a complicated one and, naively at least, seemingly contradictory. These, therefore, pose an interesting challenge to the model builders: how to incorporate all the anomalies in a model with the least number of free parameters?
While both model-dependent and model-independent search strategies for NP based on R K and/or R(D ( * ) ) data are being discussed in the literature for a while now [10,11], the subject has recently received a further impetus from the announcement of the apparent deficit in R K * [12]. There has been a flurry of activity trying to explain the R K and R K * anomalies within the context of such models, or in a model-independent framework [13]. However, only a handful of them try to explain both R K ( * ) and R(D ( * ) ) anomalies together [14]. Unfortunately, either the models are not minimal in nature, or the fits are not very good. Following our earlier rather concise work [1] of a bottom-up model-independent explanation of both the anomalies, we will explain, in this paper, the framework in detail, and also expand the earlier study significantly.
Rather than advocating a particular model, we shall assume a very phenomenological approach. Virtually all the data can be explained in terms of an effective Lagrangian, operative at low-energies, and different from that obtained within the SM. However, with 'minimality' being a criterion, we would like to follow the principle of Occam's razor and not introduce an arbitrarily large number of new parameters, as would be the case with a truly anarchic effective theory. The issue of operator mixing and possible generation of new operators at the scale m b is also nontrivial, as shown in Ref. [15]. Rather than considering the complete set of dimension-6 current-current operators and obtaining the best fit to all data, our approach could be deemed as an attempt for an inspired guess for the minimal set of operators that still produces a more than satisfactory fit, yet with small values of the Wilson coefficients (WC). If the NP scale is not beyond a few TeVs, the aforementioned operator mixing would be small and the WCs of any new operator generated as a quantum corrections would be tiny. The best fit for the WCs, thus obtained, would presumably pave the way for inspired model building. We will show how the relationship between the NP WCs may hold the key for a yet unknown flavour dynamics.
As the reader would have noticed, we have considered only the operators built out of vector and axial-vector currents alone. This is solely because our aim has been to correlate the charged-current and the neutral-current anomalies and demonstrate that they may have originated from the same source. While the charged-current anomalies in R(D) and R(D * ) can possibly be addressed with scalar and/or tensor current operators while evading the strong constraints arising from the lifetime of the B c meson [16], such operators are not of much help for R K ( * ) . For example, it has widely been discussed in the literature that if the only new operators are those constructed with (pseudo)scalar currents, the explanation for R K would be incompatible with the data on B s → + − . Similarly, the viability of tensor operators in the case of the neutral current anomalies is discussed in Ref. [11]. Once one deviates from the assumption of minimality, it is indeed possible to have different Lorentz structures of the NP operators. While this may actually occur in some specific NP models, in the absence of a well-motivated ultraviolet-complete theory, such an Ansatz would militate against the spirit of Occam's razor.
The rest of the paper is arranged as follows. In Sec. 2, we recount and discuss the experimental situation. This is followed, in Sec. 3, by a discussion of the microscopic dynamics that lead to such processes. Sec. 4 discusses a model that could have explained the data but falls short narrowly. This is followed, in Sec. 5, by a discussion of more realistic scenarios. Sec. 6 contains our results, and finally we conclude in Sec. 7.

The data : a brief recounting
We begin by briefly reviewing the experimental measurements and theoretical predictions for the observables of interest. We also take this opportunity to review some further processes that would turn out to have important consequences in our attempt to explain the anomalies.
• As already mentioned, discrepancies in the measurements of the observables R(D) and R(D * ) have been seen, over the last several years, in multiple experiments such as LHCb, BABAR and Belle. In Table. 1, taken from Ref. [17], we summarize the measurements along with the corresponding SM predictions.
R(D) and R(D * ) exceed their respective SM predictions by 2.3σ and 3.4σ. Taking the correlation into effect, the discrepancy is at the level of 4.1σ [9]. Thus, SM prediction 0.300 ± 0.008 [18] 0.252 ± 0.003 [19] BABAR (Isospin constrained) 0.440 ± 0.058 ± 0.042 0.332 ± 0.024 ± 0.018 [3] Belle ( [9]. At this point, let us mention that recently two groups have published their own calculations of R(D * ) SM , namely, 0.260 ± 0.008 [20] and 0.257 ± 0.005 [21]. While these estimates reduce the tension slightly, to 2.3σ [20] and 2.9σ [21] respectively, note that the error bars are somewhat bigger. Given that the consequent changes in numerical estimates are minimal, we will continue to use the results of Ref. [19] in the main, but would also indicate the change the results that would be wrought by the use of the new calculations 2 .
• Recently, the LHCb collaboration has observed the hint of another discrepancy [23] for the same quark-level transition b → cτ ν but in the B c meson system. With only the spectator quark changing, the SM analysis would be very similar to that for R(D * ), with the main modification accruing from the change in the phase-space factors. Much the same would be true for a large class of new physics scenarios, wherein the tensorial structure of the effective four-fermion operators remain essentially unchanged 3 . Analyzing 3 fb −1 data, The LHCb Collaboration found The SM prediction [24][25][26] includes the uncertainties coming from the B c → J/ψ form factors and, thus, is quite robust. Given the relatively low production cross section for the B c meson, the uncertainty in the measurement is still large and the discrepancy is just below the 2σ level.
With the accumulation of more data, not only would the statistical uncertainties reduce, even the systematics are expected to come down on account of a better understanding of the same. While the present level of the discrepancy is not a very significant one, it is interesting to note that it points in the same direction as the other charged-current decays.
2 It might be tempting to consider an average of the three calculations of R(D * )SM. However, this cannot be effected in a straightforward manner as, on the one side, some of the theoretical errors are correlated, while, on the other some of the theoretical inputs are at slight variance. On the other hand, Ref. [22] shows that the soft electromagnetic corrections enhance the SM prediction for R(D) by 3-5%. This reduces the tension with the SM slightly, but we have not taken this into account for our analysis, as a corresponding analysis for R(D * ) is not yet available. 3 Clearly, if the dominant new physics effect emanates from an operator different from that within the SM (as, for example, may happen in a theory with a light charged scalar), this would no longer be true.
• For the neutral current transitions (pertaining to b → s + − ), we have [12,27]: The SM predictions for both R K and R central K * are virtually indistinguishable from unity [28], whereas for R low K * it is ∼ 0.9 (due to the finite lepton mass effect). These calculations are very precise with negligible uncertainties associated with them. Thus the measurements of R K , R low K * and R central K * , respectively, correspond to 2.6σ, 2.1σ and 2.4σ deviations from the SM predictions.
• Another hint of deviation (at a level of more than 3σ), for a particular neutral-current decay mode is evinced by B s → φµµ [29][30][31].
where q 2 = m 2 µµ . Intriguingly, the q 2 region where this measurement has relatively low error (and data is quoted) is virtually the same as that for R K and R central K * . This measurement, thus, suggests strongly that the discrepancies in R K and R K * have been caused by a depletion of the b → sµ + µ − channel, rather than an enhancement in b → se + e − . This is further vindicated by the long-standing P 5 anomaly [32] in the angular distribution of B → K * µµ, which again occurs in the central q 2 region, with the mismatch between data and SM prediction being more than 3σ.
• Such a conclusion, though, has to be tempered with the data for the corresponding two-body decay, viz. B s → µ + µ − . The theory predictions are quite robust now, taking into account possible corrections from large ∆Γ s , as well as next-to-leading order (NLO) electroweak and nextto-next-to-leading order (NNLO) QCD corrections. The small uncertainty essentially arises from the corresponding Cabibbo-Kobayashi-Maskawa (CKM) matrix elements and the decay constant of B s . The recent measurement of LHCb at a significance of 7.8σ [33,34] shows an excellent agreement between the data and the SM prediction: and hence puts very strong constraints on NP models, in particular on those incorporating (pseudo-)scalar currents (this has also been noted in Ref. [35]). Of course, as the data shows, a small depletion ∼ 20% in the rates is still allowed, and, indeed, to be welcomed. NP operators involving only vector currents do not affect this channel, but scalar, pseudoscalar, and axial vector operators do.
For future reference, we list here three further rare decay modes.
• Within the SM, the decay modes b → sν i ν i are naturally suppressed, owing to the fact that these are generated by either an off-shell Z-mediated diagram 4 coupling to a flavour-changing quark-vertex, or through box-diagrams. Interestingly, the current upper bounds (summed over all three neutrinos) as obtained by the Belle collaboration [36], viz.
(both at 90% C.L) are not much weaker than what is expected within the SM, namely [37] BR( In other words, we have BR exp • The purely leptonic mode B s → τ τ is yet to be observed but LHCb put a 95% confidence level bound on the BR [38]: The SM prediction, of course, is way too small, (7.73 ± 0.49) × 10 −7 [34].
• The limits on the rare lepton flavour violating modes B → Kµτ [39] are Finally, the direct production of τ + τ − pairs at the LHC was well-measured for both the 8 TeV and the 13 TeV runs [40]. This serves to constrain the WCs of any effective theory giving rise to a τ + τ − final state Refs. [41,42].

Operators relevant to the observables
Having delineated, in the preceding section, the observables of interest, we now proceed to the identification of the operators, within the SM and beyond, responsible for effecting the transitions. As the scale of NP surely is above the electroweak scale, we will talk in terms of effective current-current operators that, presumably, are obtained by integrating out the heavy degrees of freedom, and not confine ourselves within any particular model. To be precise, the genesis of these operators will be left to the model builders.
Given the fact that, even within the SM, the neutral-current decays under consideration occur only as loop effects, several current-current operators would, in general, be expected to contribute to a given four-fermion amplitude. However, as we shall soon see, certain structures have a special role. To this end, we introduce a shorthand notation:

The b → cτ ν τ transition
This proceeds through a tree-level W -exchange in the SM. If the NP adds coherently to the SM, one can write the effective Hamiltonian as where the NP contribution, parametrized by C NP , vanishes in the SM limit. Using Eq. (1), we can write Thus, to explain the data, one needs either small positive values, or large negative values, of C NP .
On the other hand, if the NP involves an operator with a different Lorentz structure, or with different field content (like b → cτ ν µ ), the addition would be incoherent in nature, and the relative phase of C NP is immaterial. More crucially, though, this would alter the functional form of the differential width. An identical situation arises for R J/ψ .

The
Responsible for the FCNC decays B → K ( * ) µ + µ − and B s → φµ + µ − , within the SM, this transition proceeds, primarily, through two sets of diagrams, viz. the "penguin-like" one (driven essentially by the top quark) and the "box" (once again, dominated by the top). It is, thus, convenient to parametrize the ensuing effective Hamiltonian as where the relevant operators are The WCs, matched with the full theory at m W and then run down to m b with the renormalisation group equations at the next-to-next-to-leading logarithmic (NNLL) accuracy [43], are given in the SM as C 7 = −0.304 , C 9 = 4.211 , C 10 = −4.103 . If the NP operators are made only with (axial)vector currents, one can denote the modified WCs as The consequent normalized differential branching fraction for the B → Kµ + µ − decay in terms of the dimuon invariant mass squared, q 2 , is given by where is the famed phase-space factor. The functions F i depend on the WCs and q 2 dependent form factors f 0,+,T of the B → K transition [44], namely The differential distribution for the B → V mode, where V stands for a K * or φ meson, can be expressed in terms of certain angular coefficients I i [45] as The coefficients I i are functions of the transversity amplitudes A L,R λ,t where λ denotes the three states of polarisations of the meson V , and L and R denote the left and right chirality of the lepton current, respectively. We refer to Appendix A for the detailed expressions of I i and A L,R λ,t .

The b → sνν transition
Quite akin to the preceding case, this transition (which governs the B → K ( * ) νν decay) proceeds through both Z-penguin and box diagrams. Unless NP introduces right-handed neutrino fields, the low energy effective Hamiltonian may be parametrised by [37] where α em is the fine structure constant and C NP ν denotes the NP contribution. Including NLO QCD corrections and the two loop electroweak contribution, the SM WC is given by C SM L = −X t /s 2 w where X t = 1.469 ± 0.017 [37,46].

The two-body decay rates
The branching fraction for B 0 s → + − , where is any charged lepton, can be written, at the leading order, as while that for B − c → τ − ν τ is given by where m B , τ B and f B are the mass, lifetime and decay constant of the relevant B meson respectively. We assume an identical operator structure leading to coherent addition of the amplitudes.

Semi-realistic scenarios
In the spirit of effective theories, a "model" in our discussions would correspond to only a combination of (at most) two four-fermi operators at the scale m b . We, expressly, do not venture to obtain the ultraviolet (UV) completion thereof, leaving this for future studies. We also assume that the NP scale is low enough, maybe at a few TeV, so that higher-loop corrections do not generate any new operator of significant strength, leading to, e.g., the purely leptonic decay τ → 3µ. Our aim, thus, is to identify the smallest set of couplings that are phenomenologically viable. To begin with, we present a model that we call semi-realistic, because while it can explain all other anomalies, it is inconsistent with the constraint coming from the decay B s → τ + τ − .
This instructive exercise would allow us to pinpoint the structure that NP must lead to so as both explain the anomalies as well as satisfy all other constraints. We will follow this, in the next section, by constructing appropriate realistic scenarios.
Clearly, any such operator should, at the very least, respect the (gauged) symmetries of the SM. On the other hand, since the data explicitly calls for lepton-flavour violation, the latter cannot be a symmetry of the NP operator(s). Such a violation can accrue from a plethora of NP scenarios, such as models of (gauged) flavour, leptoquarks (or, within the supersymmetric paradigm, a breaking of R-parity) etc. Let us here investigate a structure that could, in principle, be motivated from such theories.
The data in question calls for the effect of NP to be concentrated in effective operators straddling the second and third generations. On the other hand, any such effective Hamiltonian accruing from a NP scale higher than the electroweak scale (as it must be, on account of the colliders, past and present, not having seen such resonances), can only be written in terms of the weak-interaction eigenstates [47]. The breaking of the electroweak symmetry, aided by non-diagonal Yukawa couplings, would, in general, induce extra operators even if we started with a single one. While some of these effects would be trivial (and aligned with the usual CKM rotations), this is not necessarily true for all. We will exploit this in striving to explain all the anomalies starting from a minimal set. In particular, our operators will involve second and third generation quark fields, to account for the b → c and b → s transitions, but only the third generation lepton fields, with the charged lepton state appropriately rotated to give rise to the mass eigenstates of µ and τ leptons.

The could-have-been model
Let us consider the following set of operators: where A 1,2 are the 5 WCs, and Q i and L i denote the usual i-th generation (and weak-eigenstate) SU (2) L quark and lepton doublet fields respectively. The subscript '3' and '1' denote that the currents are SU (2) L triplets and singlets, respectively, whereas the factor of √ 3 has been introduced explicitly to account for the Clebsch-Gordan coefficients. It should be noted that only the second and third generation quark doublets and the third generation lepton doublet alone are involved in Eq. (25), as mentioned earlier. Considering the simplest of field rotations for the left-handed leptons from the unprimed (flavour) to the primed (mass) basis, namely τ = cos θ τ + sin θ µ , ν τ = cos θ ν τ + sin θ ν µ .
terms with the potential to explain the b → sµµ anomalies are generated.
Note that the small value of A 1 means that at the matching scale of the effective theory, A 1 could very well be zero, making this a two-parameter model (A 2 and sin θ). The fit would have improved significantly if we exclude R low K * from our analysis. The BRs for B → Kµτ and B → K * µτ [48] increase to 2.1 × 10 −5 and 3.6 × 10 −5 respectively, just below the current observed limit (see Eq. (12)).
The NP contribution to the WCs C NP 9 , C NP 10 and C NP come out to be Not only is this completely consistent with the global fit results for b → s transitions, but also provides an explanation for the well known P 5 [32] anomaly as it calls for an (axial)vector contribution to the muon mode with similar coupling strength [30].
We come, finally, to the decay mode that rules out this model, namely B s → τ + τ − . The theoretical expression for this mode is given in Eq. (23). It can be ascertained that the typical value of the BR as predicted within Model I is an order of magnitude higher than the LHCb limit quoted before [38]. Indeed, the only way the B s → τ + τ − constraint can be satisfied is to tweak the values of the WCs to an extent that the best fit value for R K * is nearly 2σ away from the global average, thereby negating the very essence of the effort. Depending on the structure of the UV-complete theory, one may even have a similar large contribution, in stark contrast to the data, to the mass difference ∆M s for the B s system. This issue is discussed at a later stage. At the same time, let us note that the LHCb collaboration has, very recently, announced measurement of R(D * ) through the 3-prong decay of the τ [49], with this particular data being consistent with the SM expectations at about 1σ. More importantly, while the global average of R(D * ) reduces, its deviation from the SM value actually increases marginally to ∼ 3.4σ [49] on account of the improved precision 6 . This movement of the global average allows for the adoption of smaller values for A i , thereby reducing the tension with B s → τ + τ − . Indeed, if R(D * ) were to be entirely dominated by this new measurement, or the BR for B s → τ + τ − were found to be larger than the LHCb limit, the tension would be reduced to acceptable levels and this model would have worked fine.
Before we proceed further, let us mention that one can construct similar models, like Model II : Model III : while all these models satisfy every constraint and yield similar χ 2 min , they all fail to keep BR(B s → τ + τ − ) within the allowed limit.
We, nonetheless, endeavour below to identify a scenario that is not dependent on wishful thinking, as aforementioned.

Realistic models
The strong constraints from ∆M s and B s → τ + τ − , as seen in the preceding section, emanated from the fact that, in each of the cases, we generated a substantial C NP 10 , namely a NP contribution to the operator O 10 for the tauonic mode (in Eq. (17)) e 2 16π 2 (sγ µ P L b) (τ γ µ γ 5 τ ) . Indeed, we consistently had C NP 10 = −C NP 9 . While the operator O 9 has only a vanishing contribution to the decay, the situation is very different for O 10 . Although the latter contribution suffers a chirality suppression, it is still substantial for B s → τ + τ − . Thus, consistency with this mode would require C NP 10 to be small.
On the other hand, a substantial change in R K ( * ) would require at least one of C NP 10 and C NP 9 to be substantial. In other words, we must break the relation C NP 10 = −C NP 9 , which was a consequence of working, thus far, with left-handed currents alone. Learning from the lessons of the preceding section, we now consider simple variants of the models already introduced.

Consider the following set of operators
where the new WC A 5 (note that the factor of √ 2 is a Clebsch-Gordan factor) parametrizes the strength of the right-handed tauonic current. In terms of the component field, this reduces to where we have introduced the shorthand notation and the terms in the second line of eqn. (34) are irrelevant for the processes in hand. Clearly, A 5 3A 1 /4 would suppress any NP contribution to B s → τ + τ − . Simultaneously, this will automatically generate a tiny contribution to B → K ( * ) µ + µ − , comparable with the SM contribution, without needing to tune the leptonic mixing angle to an unnaturally small value. Note that we have already imposed the symmetry that had led to the suppression of the B → K ( * ) νν modes.
While the introdution of the right-handed current opens the possibility of introducing an independent mixing matrix for the right-handed leptons, we eschew this in the interest of having the fewest parameters in the NP sector. This would also be the case for the other model discussed below.

Model V
The analogous change in Model II and in Model III would, interestingly, result in the same set of operators as leading to with the first line containing all terms of relevance. Once again, the (symmetry) relation between the coefficients of the first two operators helps evade constraints from B → K ( * ) νν. Moreover, the very structure of Eq. (37) suggests that A 5 A 1 would be preferred by B s → τ + τ − .

Results
The fitting of NP operators progress in exact analogy with that in Sec. 4.1, with the relaxation of the condition C NP 10 = −C NP 9 . Whereas the χ 2 (defined in Eq. (27)) is still minimized for | sin θ| 0.018, such a solution would not simultaneously satisfy both of BR(B + → K + µ − τ + ) as well as BR(B s → τ + τ − ) and yet lead to a satisfactory solution for the other variables. Consequently, the best fit is obtained for a slightly different value, namely 7 | sin θ| 0.016 (38) applicable to each case. We now have χ This shows a marked improvement 8 from the SM value of χ 2 /d. o. f. = 6.1. It should be noted that even this low value of χ 2 min /d. o. f. = 1.7 is dominated by a single measurement, namely, R low K * . Indeed, for the best fit points of the NP parameter space, we have R low K * = 0.83 which is a little more than 1σ way from the LHCb measurement. This is quite akin to the various other studies in the literature [13] who too have concluded that an agreement to this experimental value to better than 1σ is not possible if the NP contribution can be expressed just as a modification of the SM Wilson coefficients. Rather, it necessitates the introduction of a new dynamical scale altogether. Having determined sin θ as above, we choose, for illustrative purposes, to relax the conditions imposed by BR(B + → K + µ − τ + ) as well as BR(B s → τ + τ − ). This would allow us to choose the best fit values for the parameters in each of the two cases. These are displayed in Table 2.
Best fit points Model IV Model V |sinθ| 0.016 0.016 Table 2: Central values of the fitted parameters for the two models.
In Fig. 1, we depict the 95% and 99% C.L. bands, in the plane of the WCs, around the new best fit point. Also shown are the regions in the parameter space allowed by the upper limits on BR(B s → τ + τ − ) (the orange shaded region) and BR(B + → K + µ − τ + ) (the yellow shaded region). The former, as expected, is truly restrictive. The overlap region, thus, denotes the viable parameter space at a given confidence level.
The observables at the overlap region, corresponding to a least value of χ 2 /d. o. f. 2.6, are given by R K = 0.80 , From Fig. 1, one might be led to think that the models IV and V are consistent only at 95% C.L. or worse, but this is deceptive. It should be noted that the contours shown in Fig. 1 are not drawn around the absolute minimum of χ 2 , which, in any case, is incompatible with other data, namely, B + → K + µ − τ + and B s → τ + τ − . Yet, the χ 2 values corresponding to the overlap region, namely ∼ < 15 (18) for the 95% (99%) C.L. bands, are much better than that obtained within the SM, which is ∼ 49. In other words, the improvement is remarkable.
It has been shown in Ref. [50] that the inclusion of left-handed NP current in the b → cτ ν τ transition to explain the R(D ( * ) ) anomalies, does not jeopardize the lifetime of the B c meson significantly, although it opens up an annihilation mode B c → τ ν for its decay. Our results for NP coefficients correspond to a modification of ∼ 3% of the lifetime and is well within the allowed limit.
We also make a couple of strong predictions which should be tested in LHCb in near future. First, for both the models under consideration, the similar ratio in B s → φ + − mode, namely, R φ ≡ BR(B s → φµ + µ − )/BR(B s → φe + e − ) should be less than unity and is predicted to be 0.83 for the range q 2 ∈ [1 : 6] GeV 2 . Second, as the allowed region almost saturates the bounds arising from the modes BR(B s → τ + τ − ) and BR(B + → K + µ − τ + ), they should also be observable in near future, and so should be B → K * τ µ. Apart from this, several anomalous top decay channels may be probed at the LHC or the next generation e + e − collider. Each of these predictions provides independent modes to both test and falsify the scenarios proposed.

Caveat: Quantum corrections and new operators
While the issue of too large a BR(B s → τ + τ − ) faced by the semi-realistic scenarios discussed in Sec. 4 was taken care of 9 in the models proposed in Sec. 5, a further issue remains.
The operators that we discussed can also, in principle, generate new contributions to B s -B s mixing. Consider, e.g. the τ -loop diagram contributing to the effective ∆B = ∆S = 2 operator. The amplitude is formally a quadratically divergent one. Thus, to calculate it one needs to introduce a cutoff Λ which could be estimated by parametrizing A i = a i /Λ 2 with a i ∼ O (1). In other words, the fit dictates Λ to be a few TeV at most. Using Λ as a cutoff would, naively, generate a WC that only scales with Λ −2 . In other words, we seemingly have [51] ζ Accepting this, the experimental measurement of the mass difference ∆M s imposes a rather strong constraint, namely, A 1 ∼ < 0.1 TeV −2 [51], and this limit falls linearly with increasing Λ. This apparent conflict with our fit results can be evaded if there are more contributions to the B s -B s mixing. A trivial example is provided by postulating a "tree-level" operator with a form identical to that above and a coefficient with a sign opposite to ζ above. A more interesting alternative could be to appeal to some yet-to-be-discovered symmetry of the full theory that cancels out, at least approximately, all the quantum corrections to the existing set of dimension-6 operators that may lead to a sizable ∆M s . Either solution could, of course, be termed a slightly fine-tuned one.
Before we delve deeper into this problem, it behoves us to consider the very calculation of ζ indicated above. One criticism is that the result is dependent on the regularisation prescription and a different one could have resulted in a markedly different ζ. A more subtle issue pertains to the very nature of such calculations in an effective theory. Indeed, an effective Lagrangian is presumably the result of either having integrated out the heavy fields in a more fundamental theory or having incorporated (some of) the quantum corrections to yield an effective action. If the local Lagrangian under consideration is to be thought of as the lowest order approximation in an expansion, further quantum corrections due to the UV physics alone can only result in corrections to the WCs and not generate any new terms in the Lagrangian 10 . Such new terms should arise only when quantum corrections to low-energy physics are taken into account.
In the present context, if say the operator O V were the result of a Z exchange, then the operator of Eq. (40) should have been generated at the same level as O V . Suppressing B s -B s mixing would, then, require that the bsZ coupling be far smaller than the τ τ Z one. Calculating the τ -loop would, then, be unnecessary and largely meaningless. On the other hand, imagine that the operator O V was generated as a combined effect of a slew of coloured fields (with the displayed form being the result of a final Fierz rearrangement). In such a case, the operator of Eq. (40) would be generated only when mixed loops involving both these coloured (and heavy) fields as well as the SM bosons. Intricately woven with this are dependence on the light masses and the analogue of the GIM cancellations 11 . With the attendant additional suppression, by a factor of α wk m 2 light /Λ 2 where m light is the typical mass of the SM fields, the consequent value of ζ is small enough for our effective Lagrangian to be in consonance with B s -B s mixing. In other words, this constraint should be considered as only an indicative one, perhaps pointing to the nature of the UV completion.
Thus, we are brought back to the assertion implicit in the entire discussion of this paper, namely that there has to exist some symmetry in the UV-complete theory that ensures that the discussed operators (in a given scenario) constitute the entire set appearing at the lowest order in the said EFT. The Wilson coefficients of any other four-fermion operator generated as a result of quantum corrections must, necessarily, be suppressed by at least α wk m 2 light /Λ 2 . Such a suppression, of course, would render the scenario safe from the perspective of ∆M s . Very similar to the discussion above is the case for τ → 3µ, putatively generated by a quark loop.

Conclusion
In this paper, we identify the minimal extension of the SM in terms of effective four-fermion operators that can explain two sets of anomalies: R K and R K * on the one hand, as well as R(D), R(D * ) and R J/ψ on the other. Explaining both sets at a single stroke has been challenging for two reasons: (i) there is a deficiency in the former case but an excess in the latter, and (ii) because they involve different leptons, viz. muons for the first pair and τ s for the second.
The final state leptons, though, can be related by postulating a small rotation of the original charged lepton field involved in the dimension-6 operator(s). With the very inclusion of a flavour-nonuniversal operator, such a rotation is no longer a trivial one (as is the case with the SM). With the neutrino flavour in such decays not being observed, only the incoherent sum over states is a measurable quantity. As for excesses in both the neutral-and charged-current processes, these are related by the usual SU (2) L symmetry.
Based on these principles, we have formulated several scenarios with a minimal set of dimension-6 gauge and Lorentz invariant operator. The "models" have at most three parameters, namely the WCs corresponding to the effective operators (two or less) and the lepton mixing angle.
Taking all the data into account, we find that two such operators are enough to get an acceptable fit. For the best fit points, all the observables, barring R low K * and BR(B s → φµµ) in the low-q 2 bin, are consistent within 1σ. (For the latter, the agreement is better than 2σ.) Even for the standout observable (for which the data is still not of great quality), the disagreement is only slightly worse than 1σ. In addition, we can also explain the observed suppression in the low-q 2 bins for the decay B s → φµµ.
A strong prediction of our analysis is that either B → Kµτ and/or B s → τ + τ − will be close to discovery, and one should look for such channels in LHCb as well as Belle-II. At the same time, we do theory as the EFT. Had the charm-quark been absent in the UV-theory but Cabibbo mixing present, the integrating out of the W -and the Z-fields would not only have generated the usual CC and NC interactions, but also large FCNC terms in the EFT. The reintroduction of the c, before the integration, removes the FCNC to the lowest order, but retains it at a higher order, and renders the FCNC proportional to m 2 c . In other words, since a symmetry in the UV-theory had forbidden the generation of a particular term in the EFT, its subsequent generation, which could occur only through the participation of the light fields, bore an imprint of the masses of the light fields (mc in this context). not attempt to probe the origin of these new operators; while a Z or a vector leptoquark may do the job, this is left for the model builders.
Although we start with a simplistic scenario with two operators in each case, it is conceivable that a hitherto unknown symmetry relates the two unknown WCs. Indeed the choices A 2 = A 1 /2, applicable for Models I and II, are strikingly simple, and conceivably, may arise from some unidentified flavour dynamics. Such a symmetry, if exact, would lead to a vanishing NP contribution (at the tree level in the effective theory) to the b → sνν amplitude. However, quantum corrections would be expected to break this symmetry. Although such rates would be small, they should still be visible at Belle-II.
Model I is interesting from a different perspective. The best-fit value of the WC A 1 is two-orders of magnitude below the other, namely A 2 . Indeed, this is one case where a single operator does almost as well as two together, or in other words, only two parameters for the new physics are required here. This is a remarkably simple solution to all of the disparate set of anomalies that confront us. A slight modified version of this case has been discussed in Ref. [1], where the bounds from B s → τ τ are well under control. Most interestingly, the scale of the new (flavour) physics is suggested to be a few TeVs at best, rendering the situation extremely attractive for the current run of the LHC.
It has to be noted, though, that the recent LHCb bound on B s → τ + τ − rules out the simplest of the scenarios. While this measurement is crucially dependent on the use of neural networks etc. (as the τ s are yet to be fully reconstructed) and the consequent uncertainties, it is worthwhile to investigate if the suppression of the C NP 10 contribution that it calls for, can be accommodated in such scenarios. We find that this can indeed be done without the introduction of additional parameters, but at the cost of introducing an additional gauge invariant operator, whose WC is envisaged to be related with the other WCs by some yet-to-be-discovered symmetry. While the χ 2 worsens marginally, it is still miles better than that in the SM. The models have a few generic predictions, like the possibility of observing B → K ( * ) µτ or B s → τ + τ − in near future, and possibly B → K ( * ) plus missing energy. They will definitely be checked within the next couple of years at LHCb, and Belle-II will be able to make precision studies on these observables. The other features of the scenarios, including the possibility of direct observation of TeV-scale resonances at the LHC, remain unaltered.