Disentangling Lepton Flavour Universal and Lepton Flavour Universality Violating Effects in $b\to s\ell^+\ell^-$ Transitions

In this letter we propose a strategy for discerning if new physics in the Wilson coefficient ${\cal C}_{9\mu}$ is dominantly lepton flavour universality violating or if it contains a sizable lepton flavour universal component (${\cal C}_9^{\rm U}$). Distinguishing among these two cases, for which the model independent fit of the related scenarios exhibits similar pulls w.r.t. the Standard Model, is crucial to advance our understanding of the $B$ anomalies. We first identify the origin of the degeneracy of these two cases and point out the key observables that can break it. In particular, while the observables measured so far that test lepton flavour universality exhibit similar dependencies to all the relevant Wilson coefficients, the forthcoming measurement of $Q_5=P_{5}^{\prime\mu}-P_{5}^{\prime e}$ is particularly sensitive to ${\cal C}_{9\mu}-{\cal C}_{9e}$. In fact, if $Q_5$ were found to be small (i.e. close to its Standard Model value), this would imply a small ${\cal C}_{9\mu}-{\cal C}_{9e}$ but a sizable ${\cal C}_{9}^{\rm U}$, given the preference of global fits for a large negative new physics contribution in ${\cal C}_{9\mu}$. We discuss the possible origins of ${\cal C}_9^{\rm U}$, in particular how it could originate from new physics. Here, a promising scenario, that could even link $b\to s \ell^+\ell^-$ to $R_{D^{(*)}}$, is the one in which ${\cal C}_9^{\rm U}$ is generated from a tau loop via an off-shell photon penguin diagram. This setup predicts the branching ratios of $B_s\to\tau^+\tau^-$ and $B\to K^{(*)}\tau^+\tau^-$ to lie within the reach of LHCb, CMS and Belle II. Alternatively, in case of a non-observation of these tauonic processes, we show that the most natural possibility to generate ${\cal C}_9^{\rm U}$ is a $Z^\prime$ with partially lepton flavour universal couplings.


I. INTRODUCTION AND MOTIVATION
In b → s + − transitions, intriguing hints for physics beyond the Standard Model (SM) have been uncovered. In particular, the accumulated evidence for new physics (NP) obtained by combining many different channels points convincingly towards a breakdown of the SM at the (multi) TeV scale. These global fits are, in principle, capable of identifying which specific NP patterns describe data best. However, it was found in Refs. [1,2] that several different NP patterns (or scenarios) are preferred over the SM hypothesis with a similar significance of more than 7σ (see also Refs. [3][4][5][6]). In fact, since the appearance of the first anomalies in the angular observables, most prominently P µ 5 in B → K * µ + µ − [7,8], and later on in the ratios R K ( * ) [9][10][11][12], a main goal of the model-independent global fits was to identify which pattern of NP can account best for data. However, with each new update (see Ref. [13] for details and Ref. [14] for future progress) the pull with respect to the SM for the already preferred scenarios increased while no clear discrimination among them arose. In this context, the Wilson coefficient C 9µ of the operator (sγ µ P L b) (μγ µ µ) plays the prominent role because its NP contribution is sizable in all scenarios that describe data well. Nonetheless, it is not yet clear if the NP contribution entering C 9µ is dominantly related to muons, i.e. if it has a large lepton flavour universality violating (LFUV) component, or if it contains a sizable lepton flavour universal (LFU) term.
In fact, all patterns that describe data well can be basically classified in two groups depending on the size of the LFUV component of C 9µ (or inversely of the LFU one). Therefore, distinguishing among these two possibilities (small or sizable LFUV component) is crucial to advance our understanding of which model could lie behind the B anomalies. However, while refining the measurements of the ratios R X (with X = K, K * , K S , K * + , φ) is of utmost importance to establish if LFU is violated in Nature, it will not help in the near future in making progress towards disentangling between these two cases.
In this paper we will show that the inclusion of a different kind of LFUV observable (other than the R X ones) is decisive to break the degeneracy among the scenarios with large and small LFUV component in C 9µ to finally make progress in the identification of the pattern of NP arXiv:2205.15212v2 [hep-ph] 14 Jul 2022 realized in Nature. Indeed, the use of a C 9 -dominated observable is a more promising strategy in the short term to answer this question, rather than collecting more and more statistics for R X -type observables 1 .
Currently, the best candidate for a C 9 -dominated observable testing LFU that can discriminate among the leading scenarios is a measurement of Q 5 = P µ 5 −P e 5 [15]. We will lay out a path based on this observable and b → sτ + τ − processes that can be implemented in the near future to clarify the composition of C 9µ . In particular, the proposed strategy will help identifying the nature of the NP contributions entering C 9µ , i.e. how big the LFU component is, using the plausible direct link with b → sτ + τ − processes such as B B→K ( * ) τ + τ − and B Bs→τ + τ − [16,17], motivated by the R D ( * ) anomalies. As a byproduct we will also be able to quantify if some marginal, so far unknown, extra hadronic contribution affects C 9µ .
The structure of this paper is as follows. In Sec. II we discuss the anatomy of the Wilson coefficient C 9µ . Sec. III describes how to determine the LFUV and LFU piece of this coefficient, while Sec. IV discusses the possible origins of the LFU component. Finally, Sec. V shows that Q 5 classifies scenarios in two groups to conclude in Sec. VI.

II. SETTING THE STAGE
The weak effective Hamiltonian relevant to the processes considered here is [18] where the semileptonic operators we will focus on are (see Ref. [19][20][21] for details and the definition of the rest of operators) The Wilson coefficient C 9 , that plays a prominent role in this discussion, has a rather complicated structure compared to the other Wilson coefficients. On the one hand, it always appears in observables in combination with the Wilson coefficients of the four-quark operators O 1,..,6 (with O i = 4P i and P i defined in Ref. [19]) and hence an "effective" superscript is included to denote this combination (see Ref. [22]). On the other hand, the nonlocal contributions from cc loops enter the same amplitude structures as O 9 (and O 7 ). Therefore, they can be naturally recast as a q 2 (dilepton invariant mass), helicity and process dependent contribution accompanying the perturbative SM term C SM 9µ pert . We thus write (for the B → K * + − case) where C cc B→K * 9µ j = s j C cc B→K * 9µ j KMPW stands for the one softgluon non-factorizable leading long-distance cc contribution. This contribution is transversity amplitude dependent, therefore we introduce the subscript j for each amplitude j =⊥, ||, 0 [23] of the B → K * + − decay 2 . Finally, a s j nuisance parameter is included and allowed to vary from -1 to 1 for each amplitude independently due to the theoretical difficulty to ascertain the relative phase between the long and short distance contributions and in order to be very conservative. For a discussion at length on our treatment of the charm-loop contributions we refer the reader to Refs. [25,26].
A first step in the analysis of C NP 9µ , in order to fix our notation for the next sections, is to split the NP piece of C NP 9µ into two: where V (U) stands for the LFUV (LFU) contribution, respectively. Finally, if some hypothetical additional hadronic effect beyond the long-distance charm one already included 3 exists, it would in general be process and q 2 dependent, like the long-distance charm contribution, and manifest itself in inconsistent values for C NP 9µ when comparing its determination from different channels 4 . However, if it were, for unknown reasons, q 2 , process and helicity independent, such a hypothetical contribution could hide inside C U 9 mimicking a universal NP contribution.
In the following sections we will propose a strategy based on identifying specific observables (i.e. Q 5 and b → sτ + τ − processes) able to disentangle the LFUV from the LFU piece in Eq. (7) and bound the possible size of extra "universal" hadronic contributions. In turn, the addition of those observables in the global fits will break present degeneracies among scenarios with large and small C V 9µ , providing a guideline to identify the NP pattern realized in Nature.
First of all, note that a non-zero C V 9µ must undoubtedly be of NP origin given that the SM gauge interactions do not discriminate between the lepton families. Since due to the measurements of R K and R K * all scenarios that allow for a very good fit to data require of C V 9µ to some degree, it is clear that also P µ 5 (or other b → sµ + µ − observables) must necessarily contain a NP contribution. But how can one determine the size of C V 9µ ? For this purpose one should focus on observables that I) can disentangle the LFUV piece from the LFU one and II) are C 9 -dominated. Condition I is fulfilled by all observables measuring LFUV, however, not all LFUV observables are equally instrumental (or useful) in separating the two pieces (i.e. do not fulfill condition II) because of the different relative weights of the Wilson coefficients entering the LFUV observables.
In order to illustrate this, we have computed the expressions for R K , R K * as semi-analytic functions of the Wilson coefficients in the [1.1,6] GeV 2 bin where S K = 0.03 [(C NP 9 + C 9 ) 2 + (C NP 10 + C 10 ) 2 ] and S K * = 0.03 [(C NP 9 − C 9 ) 2 + (C NP 10 − C 10 ) 2 ] + 0.02 [C NP 9 C 9 +C NP 10 C 10 ] are subleading quadratic terms 5 . Expanding the observables in Eq. (8) in the limit of small values for the NP Wilson coefficients naturally give rise to an expression in terms of C V iµ = C iµ − C ie (analogous to Eq.(7)) and subleading crossed terms C V iµ C U j . These expressions show that while these observables are fundamental to establish the presence of LFUV NP, they are not particularly useful to pin down one among the different currently preferred scenarios. This is clear from the expression of R K , where C NP 9µ and C NP 10µ have approximately the same weight, such that e.g. C NP 9µ = −1 results in nearly the same prediction as C NP 9µ = −C NP 10µ = −0.5. Furthermore, even though in the case of R K * the sensitivity to C NP 9µ and C NP 10µ differs by near a factor of two, this is not enough to determine C V 9µ precisely due to the hadronic uncertainties affecting R K ( * ) in presence of NP.
These considerations clearly imply that we cannot discern from the R K ( * ) observables C V 9µ due to the similar dependence on C V 10µ (and also on primed Wilson coefficients) that they exhibit; i.e. the R K ( * ) ratios do not fulfill condition II. Instead, a LFUV observable that fulfills condition II is Q 5 (evaluated on the same bin [1.1,6] GeV 2 ): with Q SM 5 = −0.01 and where, contrary to R X observables, the subleading contribution includes also linear terms where the last three terms correspond to small cross products of LFUV and LFU contributions. Eq. (9) shows that Q 5 , contrary to R K ( * ) , breaks the approximate degeneracy of the two scenarios (C NP 9µ versus C NP 9µ = −C NP 10µ ) as the dominant contributions in Q 5 originate from C V 9µ and C V 10 µ (or combined asC 10µ playing a negligible role as can be seen from S Q5 . Moreover, the structure of Q 5 and the preference of the global fits [1] for a large negative C V 9µ with a small (also) negative C V 10 µ , implies that a measurement of Q 5 , even in presence of right-handed currents (RHCs), provides a lower bound on the absolute value of |C V 9µ |. For instance, the best fit point of the 1D scenario with only C NP 9µ is C NP 9µ = −1.01, while in the 2D scenario with C NP 9µ and C 10 µ it was found [1] whereC V 9µ is obtained using The left plot in Fig. 1 shows Q 5 versus C V 9µ , i.e. the value of C V 9µ that can be inferred from a measurement of Q 5 in absence of any other source of NP.
In summary, even in presence of RHCs contributing to C V 10 µ , Q 5 provides a lower bound (or a measurement in as a function of C NP 9µ (blue band). The current experimental value from the LHCb collaboration [8] is indicated as an orange horizontal band. absence of them) on C V 9µ . Note that in the future a precise measurement of the observables P 1 (with = e, µ) [29][30][31] can help to identify the presence of RHCs. This observable is approximately zero in the large-recoil region in absence of RHCs. Therefore a deviation observed in P 1 will indicate non-zero primed Wilson coefficients.
The next step is to identify the LFU piece C U 9 using P µ 5 and in the future also P e 5 . For this we calculate P µ 5 in the [1.1,6] GeV 2 bin which follows from Q 5 with obvious substitutions: where P µ SM 5 = −0.44 and Our central point here is to show that one can determine a lower bound on C U 9 as well using a similar strategy as for Q 5 . Eq. (13) can be approximated to whereC 9µ = C NP 9µ − 0.8C 10 µ . Then if we have data available only from Q 5 and P µ 5 we obtain immediately an upper bound on the universal piece from 6 : 6 In the same way, we can obtain, using the definitions ofC 9µ andC V 9µ , not just a bound, but the total amount of LFU from where the terms on the r.h.s. correspond to the measurement of the NP piece entering P µ 5 and Q 5 using Eq. (12) and Eq. (15), respectively. Notice that in the absence of universal RHC contributions the bound turns into a measurement. Instead, if experiments also provided data on the electronic angular observable P e 5 we could directly get where the Wilson coefficients for electrons are obtained from Eq. (15) by replacing µ with e, where a preference from global fits for a negative sign in both coefficients (C U 9 and C U 10 ) is also required (see scenario 7 and 11 in Ref. [1]).
Note that in contrast to C V 9µ , which can only originate from NP, the LFU piece, as pointed out earlier could in principle contain, together with the NP contribution, some residual higher-order hadronic effect. For this reason we have to proceed differently for the LFU piece, identifying and measuring separately all possible sources of LFU NP. The difference between C U 9 obtained following Eq. (16) and Eq. (17), and the one obtained from the different NP sources (called from now on C NP U 9 ), provides the best strategy to identify the actual amount (if any) of an unknown hadronic contribution.

IV. POSSIBLE ORIGINS OF C U 9
Assuming Q 5 is found to be small, this would imply also a small C V 9µ but a sizeable C U 9 , given the present preference of global fits for a large negative C NP 9µ . Then the question of a possible origin of the LFU contribution arises since most models considered in the literature generate purely LFUV effects (see Refs. [32][33][34][35] for some exceptions). There are three (non-exclusive) possibilities how one could obtain an effect in C U 9 . First of all, it could originate, as discussed above, from some unknown and intricate hadronic effect. Second, it could be generated via a direct LFU NP contribution, and finally it could be induced indirectly from a NP effect via renormalization group effects originating from loops involving light SM particles.
A. Direct NP contribution NP above the EW scale is in general chiral (i.e. it couples to left-handed or right-handed SM fermions). Therefore, a single new particle (with the exception of a Z ) does not generate solely a contribution to C 9 . Furthermore, in models containing leptoquarks (LQs), or new scalars and fermions contributing at the loop level, in order to get a LFU effect, several generations of such new particles are necessary to avoid the stringent bounds from charged lepton flavour violating processes generated otherwise such as µ → eγ [36][37][38]. Therefore, a Z boson with vectorial leptons couplings seems to be the most natural possibility to generate C NP U 9 , in particular since such couplings allow for simple gauge anomaly free charge assignments. For example, 3L µ + L e − 4L τ gives a (partially) LFU effect, it can describe data very well and the necessary couplings to b − s can be induced by vector-like quarks [39,40].

B. Tau Loops
Four-fermion operators of the typesγ µ P L bf γ µ f , with f being any light SM fermion, can naturally generate C NP U 9 via an off-shell photon penguin leading to mixing into O U 9 [41]. However, if f = u, d, LHC bounds on di-jet searches [42,43] rule out such a possibility. Furthermore, for f = s, c a sizable effect is disfavoured by these constraints, while for f = e, µ one generates direct LFUV effects dominant over the mixing induced one. Therefore, we are left with f = b, τ . Given that any treelevel NP model generatingsγ µ P L bbγ µ b also gives rise to B s −B s mixing, the possible size of this contribution is significantly limited. Therefore, the most unconstrained scenario capable of providing the largest effect in C NP U 9 are tau leptons. In fact, one can find the following modelindependent relation originating from the mixing of O τ τ 9 into O U 9 with Λ being the NP scale. Again, since beyond the SM physics is chiral, at the scale Λ also C τ τ 10 is in general present. In fact, because leptoquarks are the only particles that can generate a large C τ τ 9 without being in conflict with other processes (like B s −B s mixing and/or LHC searches), we have either C τ τ 9 = C τ τ 10 in case of the S 2 LQ [38] or C τ τ 9 = −C τ τ 10 for U 1 [17] of S 1 + S 3 [44] unless several representations are combined. 7 Therefore, Eq. (18) allows for a direct correlation between b → sτ + τ − processes and C NP U 9 if we assume the dominance of C τ τ 9 = ±C τ τ 10 at the matching scale and neglect the SM contribution to b → sτ + τ − processes. Using we show the relation between C NP U 9 and these decays in Fig. 2. One can see that in order for C NP U 9 to be sizable, these processes are significantly enhanced (by orders of magnitude compared to the SM) such that they are within the reach of LHCb and Belle II. Therefore, a measurement of B Bs→τ + τ − and/or B B→K ( * ) τ + τ − will give us information on the size of C NP U 9 (assuming the absence of scalar currents and RHCs).

C. Connection to R D ( * )
Furthermore, in case of C τ τ 9 = −C τ τ 10 , i.e. if NP is left-handed, a model independent connection within the SMEFT formalism, based on the leptoquark analysis of Ref. [17], between the anomalies in R D ( * ) and a tauloop contribution to b → s + − can be established [16]: Requiring the absence of dangerously large effects in b → sνν, and given a generic flavour structure, we have 8 where X D ( * ) = R D ( * ) /R SM D ( * ) . We assumed a real NP effect in R D ( * ) , i.e. with the same phase as the SM contribution. Using Eq. (18), this translates into which implies a link between C NP U 9 and X D ( * ) , under the given assumptions, as shown in Fig. 3 9 . 7 Note that U 3 alone does not work as it induces dangerously large effects in B → K * νν. 8 In case of partial alignment, i.e. a non-generic flavour structure [46], the right-handed side of Eq. (20) should be multiplied by a factor 1/(1 + V cb C also has a small q 2 dependent contribution to B → K ( * ) + − matrix elements [47]. for different values of the NP scale Λ (blue Λ = 1 TeV, orange Λ = 2 TeV and green Λ = 10 TeV). Right: Branching ratio of B → K * τ + τ − as a function of C NP U 9 . A similar plot can be made for B → Kτ + τ − using Eq. (19).

Name of Hypothesis
Definition Best fit   [21,45] and updated in Ref. [1], as well as their best fit points.

D. Unknown hadronic contribution
The resulting amount of C NP U 9 generated via a direct NP contribution, tau-loops or all other possible NP sources informs us also of the maximal size of any hypothetical extra hadronic contribution that could have been hidden inside C U 9 . This contribution is given by the difference between C U 9 and C NP U 9 .
In order to get now a rough idea of the size of such hypothetical hadronic quantity that we will refer as ∆ and given the current absence of a precise measurement of Q 5 from LHCb, we can proceed in the following way. Let's assume that Scenario 8 or Scenario-U (see Table I for definition), which predicts a sizable C U 9 and currently gives the best fit to data of all scenarios is the right scenario implemented in Nature. Assuming in addition the link with R D ( * ) , the difference between C U 9 obtained from the global fit (see Table I), without using the connection with R D ( * ) , and C NP U 9 obtained using Eq. (21) for Λ = 2 TeV gives us an idea of an upper bound on an unknown hadronic contribution. With present data, in Scenario-U this amounts roughly to a value for ∆ ∼ −0.28, which is a rather small effect (≈ 6.8%) compared to the SM value of C 9µ , especially considering that the NP contribution to this coefficient is of order 20% of its SM value. But more interesting is the fact that this quantity depends on the NP scale (Λ) 10 . This implies that a larger NP scale (Λ) leads also to a larger C NP U 9 (given a fixed R D ( * ) ) and in turn ∆ becomes even smaller (in absolute value) up to ∆ ∼ −0.13 for Λ = 10 TeV.  Table I for explicit definitions of hypotheses and scenarios in terms of Wilson coefficients. The scenario C NP 10µ is included as an example of a non-preferred NP scenario (smaller PullSM) that can be easily disfavoured with a precise measurement of Q As we have discussed in the previous section, Q 5 is a unique discriminator of the presence of a LFUV piece in C NP 9µ . So far, only Belle has provided a measurement of this observable [48] (using an average of neutral and charged decays to increase the statistics), albeit with very large error bars, The ten most strongly preferred NP scenarios, i.e. with Pull SM above 6.5σ, (see Fig. 4) can be classified in two groups: • Scenarios with C V 9µ −1: C NP 9µ , {C NP 9µ , C 9 µ }, {C NP 9µ , C 10 µ }, Hypotheses I and V (or Scenario-R), Scenarios 10, 11, 13 (see Table I for definitions). They all predict Q 5 ≈ +0.3. The largest Pull SM was found in Ref. [1] for Scenario-R (see Table I). In this scenario the contribution of C NP 9µ is counterbalanced by the contribution of the primed coefficients (RHCs) in observables like R K .
• Scenarios where the contribution to C NP 9µ contains both a LFUV and a LFU piece with the absolute value of the former being smaller compared to the previous case: Scenario-U and Scenario 6. The reason why the absolute value of the LFUV piece is smaller than in the previous case is the link to C NP 10µ , resulting in a bound from B Bs→µ + µ − . The largest Pull SM is found for Scenario-U (also called Scenario 8) predicting Q 5 ≈ +0.1.
This clearly indicates that while R K is fundamental to establish the violation of LFU, finding Q 5 close to its SM value would by no means be contradictory to the violation of LFU in R K . In other words, it is essential to consider the pair R K and Q 5 (but also other R X observables). Therefore, a measurement of Q 5 close to +0.3 improves by a factor of 2 (see Table III), some difference of around 0.7σ appears between the two options. The corresponding results using only LFUV observables are also displayed in Table II and Table III.

VI. CONCLUSIONS AND OUTLOOK
In summary, a combined measurement of P µ 5 , Q 5 , R X (with X = K, K * , K S , K * + , φ) and b → sτ + τ − processes (once all data is available), together with R D ( * ) /R SM D ( * ) , can provide us not only with a clear signal of NP, but also will inform us about its structure. In particular, this will tell us the size of C U 9 and can possibly even uncover the origin of this Wilson coefficient. In this context, Q 5 is of utmost importance to make progress by breaking the present degeneracy among the scenarios that currently explain data best.  Each of those observables plays a specific but complementary role: R X observables determine the breaking of LFU, P µ 5 gives the size of NP (and possible unknown hadronic contributions) within C 9µ , Q 5 determines the amount of LFUV in C NP 9µ . B Bs→τ + τ − , if found to be very large, is a clear NP signal, and if linked to R D ( * ) /R SM D ( * ) , determines the amount of LFU NP in C NP 9µ . In turn a large LFU NP implies the smallness of any unknown hadronic contribution (beyond those already included in our conservative setup outlined in Sec. II). The resulting proposed procedure for determining the origin of NP is shown graphically in Fig. 5.