Assessing lepton-flavour non-universality from $B\to K^*\ell\ell$ angular analyses

The $B\to K^*\mu\mu$ decay exhibits deviations with respect to Standard Model expectations and the measurement of the ratio $R_K$ hints at a violation of lepton-flavour universality in $B\to K\ell\ell$ transitions. Both effects can be understood in model-independent fits as a short-distance contribution to the Wilson coefficient $C_{9\mu}$, with some room for similar contributions in other Wilson coefficients for $b\to s\mu\mu$ transitions. We discuss how a full angular analysis of $B\to K^*ee$ and its comparison with $B\to K^*\mu\mu$ could improve our understanding of these anomalies and help confirming their interpretation in terms of short-distance New Physics. We discuss several observables of interest in this context and provide predictions for them within the Standard Model as well as within several New Physics benchmark scenarios. We pay special attention to the sensitivity of these observables to hadronic uncertainties from SM contributions with charm loops.


Introduction
In recent years, several deviations from the Standard Model (SM) have arisen in B-physics observables, with the experimental confirmation of the anomaly [1] in the B → K * µµ observable P 5 [2][3][4][5], several tensions in branching ratios for b → sµµ transitions [6][7][8] and evidence for the violation of lepton flavour universality (LFU) in different observables (R K , R(D), R(D * )) [9][10][11][12][13] 1 . Global analyses of the deviations in b → s transitions point towards a large additional contribution to the Wilson coefficient C 9µ of the semileptonic operator in the effective Hamiltonian [17] for b → sµµ, as initially discussed in Ref. [1] and later confirmed by several works [18][19][20][21][22][23]. Even though such a contribution to C 9µ in b → sµµ appears as a rather economical way of explaining a large set of deviations with respect to SM expectations, theory predictions for some b → sµµ observables may also get a better agreement with data once additional contributions are allowed in other Wilson coefficients (such as C 9 µ or C 10µ ) [21]. On the other hand, B → K * ee observables and the R K ratio suggest that b → see transitions agree well with the SM [24], pointing to explanations with New Physics (NP) models with a maximal violation of LFU, affecting only muon and not electron modes. These hints of lepton flavour non-universality (LFNU) have triggered a lot of theoretical activity [25][26][27][28][29][30][31][32][33][34][35][36][37][38][39].
As discussed in several works [40][41][42][43][44][45], long-distance SM contributions from diagrams involving charm loops enter the computation of b → s processes, acting as additional contributions to the Wilson coefficient C 9 . These contributions are process-dependent and they must be estimated through different theoretical methods according to the dilepton invariant mass q 2 . The latest estimates of these contributions [40,41] have been included in the global fits for B → K * µµ [21][22][23], providing the consistent picture described above. In particular, bin-by-bin fits indicate that the data agrees well with a single, processindependent contribution to C 9µ , independent of the dimuon invariant mass, and present only in muon modes, as expected from a short-distance (NP) flavour-non-universal contribution. In order to confirm this pattern, it would be very desirable to design observables probing: a) only the short-distance part of C 9 , b) other Wilson coefficients, such as C 10 , which do not receive long-distance contributions from the SM, c) the amount of lepton-flavour non-universality between electron and muon modes. 1 The observable P 2 [14,15] exhibited also a coherent deviation in the bin [2,4.3] with the 1 fb −1 dataset [3]. Given the large experimental error in the 3 fb −1 dataset in the bin [2.5,4] due to F exp L 1 in that bin [4], it was not possible to confirm nor disprove this deviation. It would be very desirable to collect more data in that bin and in particular to measure F L with a higher precision. In fact, a recent analysis by the Babar collaboration [16] hints at a deviation in the same region.
In all cases, hadronic uncertainties should remain controlled: while non-universality is a smoking-gun-signal of NP (the SM predictions being very precise), the measurement of the effect is affected by the same hadronic uncertainties as the individual b → s modes.
The purpose of this article is to investigate which observables can be built that match these criteria, once a full angular analysis of B → K * ee, with an accuracy comparable to that of B → K * µµ, is available. If the most obvious quantity consists in comparing branching ratios though the ratio R K * (similar to R K ) (see Ref. [21] for predictions for these ratios for different NP scenarios), it is also interesting to consider other ratios probing the violation of LFU using the angular coefficients J i describing the whole angular kinematics of these decays. In this note, we will discuss observables that can measure LFNU in B → K * . Some of them are variations around the basis of optimised observables introduced in Refs. [2,15] and others can be built directly by combining angular coefficients from muon and electron modes. We will discuss the advantages of these observables in the context of hadronic uncertainties, and provide predictions in the SM and in several benchmark scenarios corresponding to the best-fit points obtained in our recent global analysis of b → s modes [21].
We begin with a presentation of the observables of interest in Section 2. In addition to observables naturally derived from the angular coefficients J i and the optimised observables P ( ) i , we consider other observables, namely B i and M (and B i , M ) which have a reduced sensitivity to charm contributions in some NP scenarios. In Section 3 we present our predictions in the SM and in several NP benchmark points, illustrating how these observables can help in discerning among NP scenarios and how (in)sensitive they are with respect to hadronic uncertainties. We present our conclusions in Section 4. In the appendices we discuss the dependence of M and M observables on charm contributions, we recall the definition of binned observables, and we provide further predictions for the various observables within the different benchmark scenarios.
2 B → K * observables assessing lepton flavour universality 2.1 Observables derived from J i , P i and S i We want to exploit the angular analyses of both B → K * µµ and B → K * ee decays in order to build observables that will probe the violation of LFU, the short-distance part of C 9µ and/or the other Wilson coefficients, with limited hadronic uncertainties. Natural combinations are 2 where P i should be replaced by P i for Q i=4,5,6,8 . B i and B i differ mostly at very low q 2 and become almost identical for large q 2 , where β = 1 − 4m 2 /q 2 1 for both electrons and muons. The optimised observables P ( ) i have already a limited sensitivity to hadronic uncertainties [2,15,21,46,47], contrary to the angular averages S i [2,15,[47][48][49][50]. We thus expect the Q i observables to exhibit a correspondingly low sensitivity to hadronic uncertainties. 3 Moreover, these observables are protected from long-distance charm-loop contributions in the SM.
A measurement of Q i different from zero would point to NP in an unambiguous way, confirming the violation of LFU observed in R K . A second step would then consist in identifying the pattern of NP, which requires to separate the residual hadronic uncertainties (in particular, charm-loop contributions) from the NP contributions. The set of observables Q i , T i and B k ( B k ) can be particularly instrumental at this second stage, with a sensitivity to the various Wilson coefficients depending on the particular angular coefficients considered.
We have already investigated this sensitivity [21,46,47], but we would like to highlight the difference of behaviour in the case of two of the relevant observables P 4 and P 5 , directly related to Q 4 and Q 5 respectively. Both LHCb and Belle collaborations [3,5,8] observed the same pattern, i.e., a significant deviation from the SM for P 5 for q 2 between 4 and 8 GeV 2 and a result consistent with the SM within errors for P 4 . This behaviour is expected in the presence of NP in the Wilson coefficient C 9 . From the large-recoil expressions of A L,R ⊥, ,0 (see Eqs. (3.8)-(3.10) of Ref. [54]) one finds that the right-handed amplitudes |A R 0,⊥,|| | ∝ (C eff 9 + C 10 ) + ... are suppressed compared to the left-handed ones in the SM, due to the approximated cancellation C eff 9 + C 10 0. This cancellation is not so effective in the presence of a negative NP contribution to C 9 , and A R 0, , |A R ⊥ | increase while |A L 0, |, A L ⊥ decrease. Both effects add up coherently in the numerator of due to the relative minus sign, and the effect is to reduce the value of |P 5 | in the region far up from the photon pole, in agreement with the experimental observation. In P 4 ∝ Re(A L 0 A L * + A R 0 A R * ), however, an increase in the right-handed amplitudes will compensate a decrease in the left-handed ones, due to the relative positive sign. For this reason, no deviation is expected in P 4 in the presence of NP in C 9 (but in the absence of right-handed currents). The same mechanism is at work for Q 4 and Q 5 .
As discussed in Sec 2.3.1 of Ref. [21], LHCb currently determines the polarisation fraction F T and F L using a simplified description of the angular kinematics. This means that these two quantities are actually measured from J 1c rather than J 2s and J 2c respectively. Both determinations are equivalent in the massless limit, and therefore this only has a limited impact, apart from the first bin [0.1,0.98]. In order to interpret the actual measurements more precisely, we define theP i observables involvingF T andF L , as measured currently by LHC: and we will provide predictions for both Q i andQ i observables, in order to illustrate the differences in the first bin, as well as the insensitivity of the effect in higher bins.
In the case of the S i , the consideration of the T i ratio is also natural, but unfortunately these quantities are quite sensitive to hadronic uncertainties. They depend on soft form factors even in the large recoil limit due to lepton mass effects at very low q 2 , related to differences between muon and electron contributions in the normalization. Finally, the ratios B i that are soft-form-factor independent at leading order in the large-recoil limit will be shown to complement the observables Q i in an interesting way.

Observables with reduced sensitivity to charm effects
In the presence of NP, all observables Q i , T i and B i are in principle affected by longdistance charm loop contributions in C 9 , both transversity-independent and transversitydependent. We define these two terms in the following way: transversity-independent long-distance charm corresponds to an identical contribution to all B → K * transversity amplitudes, whereas transversity-dependent contributions differ for each amplitude. Both of them are expected to exhibit a q 2 -dependence in general. The explicit computation of charm-loop contributions performed in Ref. [40] using light-cone sum rules indicates that they are transversity-dependent, in agreement with general expectations that such hadronic contributions are different for different external hadronic states (including different K * helicities). It is interesting to investigate these issues by considering specific observables with different sensitivity to transversity-dependent and independent long-distance charm contributions, as well as to LFNU New Physics.
One can think of exploiting the angular coefficients in electron and muon modes in order to build observables only sensitive to some of the Wilson coefficients, and in some cases, insensitive to transversity-independent long-distance charm contributions. It is easy to check that in the large-recoil limit and in the absence of right-handed or scalar operators, four angular coefficients exhibit a linear sensitivity to C 9 . Taking the results from Refs. [15,54] we have: whereŝ = q 2 /m 2 B andm b = m b /m B , ξ ⊥ and ξ || correspond to the soft form factors [17], and the ellipses indicate terms suppressed in the large-recoil limit (including terms of order m 2 /q 2 ). If we limit ourselves to real NP contributions, it is interesting to consider B 5 and B 6s (and B 5 and B 6s ) in Eq. (1), as well as a combination of them in the form 4 .
By construction, B 5 and B 5 have a pole at the position of the zero of J e 5 in the SM (around q 2 = 2 GeV 2 ) and B 6s , B 6s have a pole at the position of the zero of A FB in the SM (around q 2 = 4 GeV 2 ). We expect large uncertainties for these observables in the corresponding bins. On the contrary, M is well behaved in the same bins, but it will have large uncertainties when B 5 B 6s . In this sense, the observable M is well suited for NP scenarios and energy regions that yield very different contributions to B 5 and B 6s . While the B i have a value in the SM slightly different from zero (specially the first bin) due to β µ /β e kinematic effects, the B i observables vanish by construction in the SM. 5 Even more interesting is the case of M , constructed in the same spirit as B i , i.e. to cancel the dependence of the angular coefficients on β . Its first bin can be accurately predicted even in the presence of NP, while its M counterpart suffers from large uncertainties in that bin. In the next section we will discuss some NP scenarios and show how these set of observables can become instrumental to disentangle them.
Let us write C ie = C i and C iµ = C i + δC i for i = 9, so that δC i measure the LFU violation, whereas C ie can include LFU NP effects. Furthermore, for i = 9 we take C 9e = C 9 + ∆C 9 and C 9µ = C 9 + δC 9 + ∆C 9 where ∆C 9 is a long-distance charm contribution. In order to illustrate the relevant aspects of the various observables, within this Section we will give analytic formulas assuming the contribution ∆C 9 is transversity independent and neglecting imaginary parts. But all our numerical evaluations will be based on complete expressions, as computed in Ref. [21] where transversity-dependent charm contributions are included following Ref. [40], and imaginary parts are properly accounted for. We see that δC 7,7 = 0 6 and δC 9 are directly related to short-distance physics, while ∆C 9 comes from long-distance contributions from cc loops where the lepton pair is created by an electromagnetic current, and thus identical for C 9e and C 9µ . Any δC i = 0 indicates the presence of LFNU New Physics.
In the large-recoil limit and in the absence of right-handed or scalar operators, we have: where M 0 , ∆M , A ( ) and B ( ) are defined in App. A, and the ellipsis denote again terms neglected in Eqs. (7) and (8) and suppressed in the large-recoil limit. The difference between the muon and electron masses relative to q 2 , induces a non-vanishing SM value for the B i observables at low q 2 . B i are exactly zero in the SM, and can be obtained from Eqs. (10), (11) in the limit β → 1. Note that the B i observables always have a residual charm dependence ∆C 9 in the denominator in the presence of NP. From Eq. (12), M appears sensitive to the muon-electron mass difference via ∆M , A and B, and the last two terms introduce a sensitivity to charm effects through ∆C 9 . Moreover, the first bin of M is very sensitive to this mass difference and will be affected by very large uncertainties in some NP scenarios. On the contrary, M is blind to such mass effects. In addition, if there is no NP in δC 10 then M becomes also insensitive to transversity-independent charm effects at leading order and at large recoil. This means that M is particularly clean at low q 2 (where large-recoil expressions are relevant), especially in the presence of NP in δC 9 . For larger values of q 2 and/or in the presence of NP in C 10 , subleading charm effects are present and will enlarge the uncertainties, even though the impact of NP on this observable remains very large. M at low q 2 will turn out to be very efficient to disentangle NP scenarios.
We have the following behaviour for δC 9 = 0: For B 5 and B 6s , the limit of very small q 2 is equivalent to δC 9 = 0, and M is not well predicted in this limit (subleading effects dominate the computation). This is however not a problem in the current context where global analyses point towards a large NP 6 C 7 includes both the SM C eff 7 plus possible LFU NP (the same applies to C 9 ). contribution to C 9 . On the other hand, if δC 10 = 0, we have 7 B 5 and B 6s contain then a residual charm sensitivity through ∆C 9 , while M is totally free from this transversity-independent long-distance charm at leading order. This is a very specific property of M which is independent of transversity-independent charm contributions in the presence of New Physics in C 9 only. Transversity-dependent charm effects are kinematically suppressed at very low q 2 in these observables as it will be shown later on.
In the case where both δC 9 and δC 10 are non-zero, a precise interpretation of these observables requires a more detailed study (including an assessment of all cc contributions to C 9 ). We see therefore that some of these observables will have a limited sensitivity to charm-loop contributions in some cases (SM, NP only in C 9µ ), but not in other cases (NP also in C 10,µ for instance).
As a conclusion, the behaviour of B 5 ( B 5 ), B 6s ( B 6s ) and M ( M ) in specific q 2 -regions should provide powerful tests of physics beyond the SM, with a limited sensitivity to hadronic uncertainties.
3 Predictions in the SM and in typical NP benchmark scenarios

Observables and scenarios
The above discussion assumed that one can determine exactly the value of the angular coefficients J i differentially in q 2 . This is in principle possible using the method of amplitudes in Ref. [56] even if for electrons it could be particularly difficult. The other methods (likelihood fit and method of moments) lead to binned observables, where the cancellations advocated above hold only in an approximate way, for bins small enough so that the angular coefficients do not exhibit steep variations. The modifications due to binning for the predictions of observables were described in detail in Ref. [46], and are also recalled in App. B for the observables described above. They will obviously have an impact on the previous discussion concerning the cancellation of hadronic uncertainties, which will then be only approximate.
In order to illustrate the interest of the various observables, in addition to the SM, we consider several NP benchmark scenarios corresponding to the best-fit points for hypotheses with a large pull in the global analysis of Ref. [21] (with NP contributions in b → sµµ but not in b → see). We follow the same approach as in Ref. [21] and compute the various observables following the definition of binned observables in App. B. The results are shown in App. C and in Figs. 1-8. In the SM, Q i , T i and B i are expected to be close to zero, as shown in App. C. The binned observables B 5 and B 6s are actually different from zero due to the kinematic factors β 2 µ and β 2 e in the transversity amplitudes -one could imagine measuring the binned values of J 5,6s /β 2 and checking that the values for both lepton flavours are indeed identical. The difference between β µ and β e becomes less relevant for large q 2 (above 2.5 GeV 2 ), leading to B 5 and B 6s decreasing in magnitude and getting closer to each other. In the same region, M becomes larger as it involves the difference B 5 − B 6s in the denominator. In the presence of NP affecting differently C 9µ and C 9e , B 5 and B 6s are different over the whole kinematic range. In the SM, the binned version of M is charm dependent due to β µ /β e terms. In the presence of LFNU in C 9 , it is interesting to focus instead on the observable M , which is not affected by lepton-mass effects and is essentially charm independent at very low-q 2 . If there are NP contributions in other Wilson coefficients, the situation becomes more complicated concerning the charm dependence of the observables. In the remainder of this Section we will identify patterns based on the set of Q i andQ i , and we will describe a very promising test based on B 5 , B 6s and M .
The observablesQ i (see Figs. 1-8) show specific patterns for the different scenarios considered here: • Scenario 1: C NP 9µ = −1.1. BothQ 2 andQ 5 are affected significantly, especially the latter. The most interesting region is q 2 6GeV 2 , taking into account that these observables receive essentially no charm contributions in the SM. No deviation should be observed inQ 1 orQ 4 in the same region within this scenario (see the discussion in Section 2 concerning the sensitivity of P 4 to C 9 ).
Within this scenarioQ 2 andQ 5 show milder deviations, especially in the bin 6-8 GeV 2 where they are expected to be SM-like (contrary to Scenario 1). Indeed, the constraint from B s → µµ on C 10µ reduces the allowed size of the deviation in C 9µ in this particular scenario. On the contrary,Q 4 could be particularly interesting in the region below 6 GeV 2 with a q 2 -dependence rather different from Scenario 1. No deviation is expected inQ 1 .
• Scenarios 3 and 4: C NP 9µ = −C 9µ = −1.07 and C NP 9µ = −C 9µ = −1.18, C NP 10µ = C 10µ = 0.38 respectively. Both scenarios are quite difficult to distinguish using these observables. They have implications in all four relevant observablesQ 1,2,4,5 .   The behaviour ofQ 2 andQ 5 is similar to Scenario 1, making the three scenarios difficult to disentangle when looking only to these observables.Q 1 , which is designed to test the presence of right-handed currents, is affected significantly. Finally,Q 4 both at very low-and large-q 2 (but within the large recoil region) could be useful if accurate measurements are obtained. In particular, above 6 GeV 2 this observable is only sensitive to right-handed currents [57].
The same discussion applies to the observables Q i . We note thatQ i (Q i ) in the bin [6][7][8], which have no charm uncertainties in the SM, may play a central role in disentangling the first two scenarios.
These observables are quite complementary to R K * , for which we provide predictions in App. C. Indeed, the value of R K * is very similar (within uncertainties) in the first two scenarios, whereas a larger suppression is expected for the other scenarios at moderately large q 2 , illustrating the complementarity with theQ i (Q i ) observables. For completeness we also present predictions for the observables T i in the same appendix.

B and B observables
We also give predictions for the B i observables in App. C and in Figs. 1-8 [4,6] for B 6s ) and cannot be predicted accurately. All scenarios give very similar predictions, apart from the first bin of B 5 and the two first bins of B 6s .
The first bin of these observables is predicted accurately both in the SM and in the presence of NP. Not only it is insensitive to form factors in the large-recoil limit at leading order, but it is also protected from long-distance charm contributions due to a kinematical suppression of the charm-dependent contribution at low q 2 (see also Ref. [57]). The analysis of this bin in the SM and in the scenarios presented above is particularly interesting. As explained in the previous section, the SM predictions B SM 5 = −0.155 ± 0.003 and B SM 6s = −0.121 ± 0.001 are only different from zero due to β µ /β e effects integrated over the bin. This can be checked through the corresponding prediction for the B i observables, which are free from these effects and equal to zero in the SM. In the case of a negative NP contribution to C 9µ , both B 5 and B 6s receive a positive contribution that pushes them towards zero in the first bin. If there is a positive NP contribution in C 10µ , the contribution to both observables is negative and large (of size C NP 10µ /C 10µ ). In summary, a contribution close to zero will favour a scenario with NP only in C 9µ < 0, whereas values of B 5 and B 6s lower than the SM will signal NP in C 10µ (NP in C 9µ is better discriminated by other observables). In both cases B 5 and B 6s are almost equal, while a contribution to C 10µ would break this degeneracy. The second bin of B 6s exhibits a similar pattern (above the SM in Scenario 1, below in Scenario 2).
The same discussion applies to B i , which have a similar behaviour in those bins, the only difference being that they are centered around zero (SM prediction). For instance, the first bin of B 5 and B 6s in the Scenario 1 (Scenario 2) receives a positive (negative) contribution. The second bin of B 6s follows the same rules as B 6s .
The low-recoil behaviour of the B i and B i observables is particularly interesting because it points to large deviations that cannot be seen easily in the Q i observables. Unfortunately, they are not useful in distinguishing Scenarios 1 and 2, except if compared together with the corresponding Q i at low recoil, which show a slightly different behaviour in that region.

M and M observables
M is also an interesting observable to get information on the existence of NP contributions and identifying their nature. This can be seen from the results in App. C and Figs. 1-8 by looking at the third bin, where it can be noted that this observable can help to disentangle Scenario 2 from Scenarios 1 and 3, thus testing for the presence of NP in C 10µ .
However in the first bin, where B 5 B 6s , M is poorly predicted. In these region it proves instead very useful to exploit the alternative observable M , where effects related to β are removed. This observable then gives additional information in discerning between Scenario 2 and Scenarios 1 and 3. The effects in this first bin can also be confirmed by looking at the second bin (notice that M is well defined in its second bin even if B 5 has a pole in its second bin).

Hadronic uncertainties
The observables presented here, specially Q i , B i and B i , are built to be very accurate in the SM, and almost insensitive to long-distance charm contributions. Moreover, whether NP is present or not, these observables are built to have no dependence on soft form factors at leading order in the large-recoil limit. In the presence of NP, these observables become again sensitive to charm-loop contributions, but in a very specific way that we discuss now.
Let us first recall that we introduced the observablesQ i in order to provide predictions taking into account how LHC measures F L currently. Here the cancellation of soft form factors between numerator and denominator is not fully operative and these observables are thus sensitive to soft form factors arising in J 1c but suppressed by powers of m 2 /q 2 . This explains why the errors ofQ i are larger (but still small in most of the bins) than for Q i . The observables T i exhibit a residual sensitivity to soft form factors in most of the bins. Finally, the observable M suffers from large uncertainties when B 5 B 6s , even though it is designed to have no dependence on soft form factors at leading order in the large-recoil limit.
Concerning long-distance charm-loop contributions, the most interesting observables are B i ( B i ) and M ( M ). In the analytic expressions provided in Section 2.2, we have assumed that the charm contribution ∆C 9 entered all transversity amplitudes in the same way. One can generalize the expressions for B 5,6s and M in Eqs. (10,11) and allow for transversity-dependent charm contributions ∆C ⊥, ,0 9 (q 2 ) associated to each amplitude: e 2(C 10 + δC 10 )δC 9ŝ C 10 (2C 7mb (1 +ŝ) + (2C 9 + ∆C 9,0 + ∆C 9,⊥ )ŝ) (18) e 2(C 10 + δC 10 )δC 9ŝ C 10 4C 7mb + (2C 9 + ∆C 9,⊥ + ∆C 9, )ŝ (19) M = (2C 10 δC 9ŝ + δC 10 (2C 7mb (1 +ŝ) + (2C 9 + 2δC 9 + ∆C 9,⊥ + ∆C 9,0 )ŝ)) 2C 10 (C 10 + δC 10 )δC 9 2C 7mb (ŝ − 1) + (∆C 9,0 − ∆C 9, )ŝ ŝ × 2C 10 δC 9ŝ + δC 10 4C 7mb + (2C 9 + 2δC 9 + ∆C 9,⊥ + ∆C 9, )ŝ (20) The corresponding expressions for the B 5,6s are obtained in the limit β → 1. In the case of NP only in δC 9 they simplify to (2C 7mb + (C 9 + (∆C 9 + ∆C ⊥ 9 )/2)ŝ) The observable M was designed to cancel exactly a transversity-independent charm contribution ∆C 9 at leading order in the large recoil limit, which occurs in the denominator of the B i observables. The above expressions indicate that for B i , all the long-distance charm dependence is contained in the denominator, and its numerical impact is somehow reduced by a large C 9 , which explains their reduced sensitivity to ∆C 9 (this is even more efficient at very low q 2 due to the photon pole). In the case of M , C 9 cancels, leaving only the photon pole to tame the sensitivity to transversity-dependent charm-loop contributions. For this reason at higher q 2 values, where the photon pole contribution is smaller, the sensitivity to this transversity-dependent charm contribution is maximal in M as can be seen in App. C and in Figs. 1-8. In addition, looking at Eq. (20), it is interesting to note that M is sensitive to charm contributions only if a) there is LFNU New Physics in C 10 or right-handed operators, or b) there are transversity-dependent charm-loop contributions (such that ∆C 0 9 = ∆C 9 ). We should finally comment on the fact that our predictions do not include any evaluation of Bremsstrahlung effects. Naively one expects these effects to be of order α log(m 2 e /m 2 µ ) ∼ 8% [38]. Part of these effects are taken into account at the level of the experimental analysis by means of a Montecarlo simulation with PHOTOS [58], which accounts for soft-photon emission from the leptons. Other contributions (e.g., real emission from the mesons, virtual photons) should still be estimated by separating in the theoretical computations the radiative corrections already implemented experimentally and those to be estimated theoretically (see Refs. [59,60] for a discussion of this issue in the context of K 4 decays). Such a work goes far beyond the present note, but the impact of such effects should be expected to be of a few percent.

Discussion and conclusion
The recent LHCb and Belle results on b → s transitions, with the anomalies observed in some angular observables such as P 5 (B → K * µµ), and the hints of LFNU in B → K have raised a considerable interest for these processes. In the present article we have discussed how angular analyses of B → K * ee and B → K * µµ decay modes can be combined to understand better the pattern of anomalies observed and to get a solid handle on the size of some SM long-distance contributions.
We have proposed different sets of observables comparing B → K * ee and B → K * µµ, discussing their respective merits. A first set of observables is obtained directly from the observables that have been introduced for B → K * µµ, namely Q i (related to the optimised observables P i ), T i (related to the angular averages S i ) and B i (related to the angular coefficients J i ), measuring in each case the differences between muon and electron modes.
We have discussed further the merits of the observables B 5 and B 6s which are built from angular coefficients exhibiting only a linear dependence on C 9 at large recoil. In principle, this allows us to disentangle the contributions coming from NP in C 9 and C 10 , with a clean separation between lepton-flavour dependent (NP) and lepton-flavour universal (NP or SM long-distance) contributions to C 9 . We have also built an observable M which exhibits very interesting features: in the presence of LFNU NP in C 9 or C 10 only, the large-recoil expression for M is independent of long-distance LFU contributions (in particular transversity-independent charm contributions) and provides clean signals of NP. It proves also interesting to consider B 5 and B 6s , built from angular coefficients divided by appropriate powers of β , thus removing some kinematic effects affecting B 5 and B 6s at very low q 2 .
We have then considered the situation for binned observables, and we have provided predictions for the SM and for several benchmark points inspired by our recent global analysis of b → s transitions. We can summarise our findings as follows. First, the Q i observables are efficient to separate several NP scenarios where NP enter only b → sµµ transitions due to very different q 2 dependences in the large-recoil region. Second, the observables B 5,6s and B 5,6s at very large and low recoils provide further information, as NP in different muon Wilson coefficients will affect these observables significantly.
Finally, the M observable at low q 2 proves particularly clean and efficient in identifying and interpreting NP in muon modes, with a limited sensitivity to charm contributions. These observables provide complementary information compared to the measurement of the ratio R K * that is expected very soon from the LHCb collaboration.
In view of these results, we are looking forward to the next measurements to be performed at LHCb and Belle-II. We expect their analysis to be decisive in determining the exact origin of the anomalies currently observed in b → s modes.

B Definition of binned observables
The binned observables are defined following the same rules as in Ref. [46]: where P i and S i correspond to the observables defined in Ref. [46] with = e or µ.

C Predictions for the observables in various benchmark scenarios
Our predictions are obtained following Ref. [21]. We quote two uncertainties, the second corresponding to the charm contributions, the first to all other sources of uncertainties. Bars denote predictions affected by a very large uncertainty (presence of a pole).