Constraints on Lepton Universality Violation from Rare B Decays

The LHCb collaboration has very recently released a new study of B + → K + (cid:96) + (cid:96) − and B → K ∗ 0 (cid:96) + (cid:96) − ( (cid:96) = e, µ ) decays, testing lepton universality with unprecedented accuracy using the whole Run 1 and 2 dataset. In addition, the CMS collaboration has recently reported an improved analysis of the branching ratios B ( d,s ) → µ + µ − . While these measurements oﬀer, per se, a powerful probe of New Physics, global analyses of b → s(cid:96) + (cid:96) − transitions also rely on the assumptions about nonperturbative contributions to the decay matrix elements. In this work, we perform a global Bayesian analysis of New Physics in (semi)leptonic rare B decays, paying attention to the role of charming penguins which are diﬃcult to evaluate from ﬁrst principles. We ﬁnd data to be consistent with the Standard Model once rescattering from intermediate hadronic states is included. Consequently, we derive stringent bounds on lepton universality violation in | ∆ B | = | ∆ S | = 1 (semi)leptonic processes.

Since the first collisions in 2010, the Large Hadron Collider (LHC) allowed for a tremendous step forward in the electroweak (EW) sector of the Standard Model (SM) of Particle Physics -culminated with the discovery of the Higgs boson [1,2] -while it has also excited the community with a few interesting hints of Physics Beyond the SM (BSM).In particular, the LHCb collaboration provided the first statistically relevant hint for Lepton Universality Violation (LUV) in flavor-changing neutral-current (FCNC) processes [3], measuring the ratio R K ≡ Br(B + → K + µ + µ − )/Br(B + → K + e + e − ) in the dilepton invariant-mass range q 2 ∈ [1, 6] GeV 2 .These hints have been confirmed by subsequent measurements, always by the LHCb collaboration, namely R K [4], R K * [5,6], R K S and R K * + [7].
In this Letter we provide a reassessment of NP effects in b → s µ + µ − transitions in view of the experi- FIG. 1. Example of charming-penguin diagrams contributing to the B → K ( * ) + − amplitude.Diagram (a) represents the class of charming-penguin amplitudes related to c − c state that subsequently goes into a virtual photon, see refs.[43,[45][46][47][48].
Diagram (b) and (c) represent the kind of contributions from rescattering of intermediate hadronic states, at the quark and meson level respectively.The phenomenological relevance of rescattering for the SM prediction of the B → K ( * ) + − decays has been recently considered in ref. [38].
mental novelties discussed above.Adopting the modelindependent language of the Standard Model Effective Theory (SMEFT) [82,83], we present an updated analysis of |∆B| = |∆S| = 1 (semi)leptonic processes and show that current data no longer provide strong hints for NP.Indeed, updating the list of observables considered in our previous global analysis [38] with the results in eqs.( 1) and ( 2), the only remaining measurements deviating from SM expectations and not affected by hadronic uncertainties are the LUV ratios R K S and R K * + [7], for which a re-analysis by the LHCb collaboration is mandatory in view of what discussed in [54,55].
The anatomy of the B → K ( * ) + − decay can be characterized in terms of helicity amplitudes [24,84], that in the SM at a scale close to the bottom quark mass m b can be written as: with λ = 0, ± and C SM 7,9,10 the SM Wilson coefficients of the semileptonic operators of the |∆B| = |∆S| = 1 weak effective Hamiltonian [85][86][87], normalized as in ref. [41].The naively factorizable contributions to the above amplitudes can be expressed in terms of seven q 2 -dependent form factors, V 0,± , T 0,± and S [88,89].At the loop level, non-local effects parametrically not suppressed (neither by small Wilson coefficients nor by small CKM factors) arise from the insertion of the following four-quark operator: that yields non-factorizable power corrections in H λ V via the hadronic correlator h λ (q 2 ) [26,30,90], receiving the main contribution from the time-ordered product: with j µ em (x) the electromagnetic (quark) current.This correlator receives two kinds of contributions.The first corresponds to diagrams of the form of diagram (a) in Fig. 1, where the initial B meson decays to the K ( * ) plus a cc state that subsequently goes into a virtual photon.This contribution has been studied in detail in the context of light-cone sum rules in the regime q 2 4m 2 c in [43]; in the same reference, dispersion relations were used to extend the result to larger values of the dilepton invariant mass.While the operator product expansion performed in ref. [43] was criticized in ref. [29], and multiple soft-gluon emission may represent an obstacle for the correct evaluation of this class of hadronic contributions [30,40,91,92], refs.[45,46] have exploited analyticity in a more refined way than [43].In those works the negative q 2 region -where perturbative QCD is supposed to be valid -has been used to further constrain the amplitude.Building on these works, together with unitarity bounds [47], ref. [48] found a very small effect in the large-recoil region.
The second kind of contribution to the correlator in eq. ( 4) originates from the triangle diagrams depicted in Fig. 1 (b), in which the photon can be attached both to the quark and antiquark lines and we have not drawn explicitly the gluons exchanged between quark-antiquark pairs.An example of an explicit hadronic contribution of this kind is depicted in Fig. 1 (c). 1 The D s D * pair is produced by the weak decay of the initial B meson with low momentum, so that no color transparency argument holds and rescattering can easily take place.Furthermore, the recent observation of tetraquark states in e + e − → K(D s D * + D * s D) by the BESIII collaboration [94] confirms the presence of nontrivial nonperturbative dynamics of the intermediate state.
One could think of applying dispersive methods also to this kind of contributions, but the analytic structure of triangle diagrams is quite involved, depending on the values of external momenta and internal masses.A dispersion relation in q2 of the kind used in refs.[43,[45][46][47][48], based on the cut denoted by (1) in Fig. 1 (b), could be written if the B invariant mass were below the threshold for the production of charmed intermediate states.
However, when the B invariant mass raises above the threshold for cut (2), an additional singularity moves into the q 2 integration domain, requiring a nontrivial deformation of the path (see for example the detailed discussion in ref. [95]).Another possibility would be to get an order-of-magnitude estimate of contributions as the one in Fig. 1 (c) using an approach similar to ref. [93].
To be conservative, and in the absence of a firstprinciple calculation of the diagrams in Fig. 1, we adopt a data-driven approach based on the following parameterization of the hadronic contributions, inspired by the expansion of the correlator of eq. ( 4) as originally done in ref. [24], and worked out in detail in ref. [92]: + q 2 + h This parameterization -while merely rooted on a phenomenological basis -has the advantage of making transparent the interplay between hadronic and possible NP contributions.Indeed, the coefficients h − and h − have the same effect of a lepton universal shift due to NP in the real part of the Wilson coefficients C 7 and C 9 , respectively.Consequently, the theoretical assumptions on the size of these hadronic parameters crucially affect the extraction of NP contributions to C 7,9 from global fits.Within the SM, the new measurements in eqs.( 1)-( 2) do not affect the knowledge of the h λ coefficients; the most up-to-date data-driven extraction of the hadronic parameters introduced in eq. ( 5) can be found in Table 1 of ref. [38].See the Appendix for further details regarding the hadronic parameterization employed in the data driven approach.
Moving to the analysis of NP, current constraints from direct searches at the LHC reasonably suggest in this context that BSM physics would arise at energies much larger than the electroweak scale.Then, a suitable framework to describe such contributions is given by the SMEFT, in particular by adding to the SM the following dimension-six operators: where in the above τ A=1,2,3 are Pauli matrices, a sum over A is understood, L i and Q i are SU (2) L doublets, e i and d i singlets, and flavor indices are defined in the basis where the down-quark Yukawa matrix is diagonal.For concreteness, we normalize SMEFT Wilson coefficients to a NP scale Λ NP = 30 TeV and we only consider NP contributions to muons. 3 The matching between the weak effective Hamiltonian and the SMEFT operators implies the following contributions to the SM operators and to the chirality-flipped ones denoted by primes [98]: with ).As evident from the above equation, operators O LQ (1,3)   2223 always enter as a sum.Hence we denote their Wilson coefficient as C LQ 2223 .We perform a Bayesian fit to the data in refs.[13, 17-23, 49, 54, 55, 99-105] employing the HEPfit code [106,107].For the form factors and input parameters, we follow the same approach used in our previous refs.[30, 38, 40-42, 91, 92].In particular, we use the same inputs as in ref. [38], with the only exception of CKM parameters, which have been updated according to the results of ref. [51].We compute B → K ( * ) + − and B s → φ + − decays using QCD factorization [108].
As already mentioned discussing Fig. 1, a global analysis of b → s + − transitions can be sensitive to hadronic contributions that are difficult to compute from first principles and that can yield important phenomenological effects.Therefore, in what we denote below as data driven scenario, we assume a flat prior in a sufficiently large range for the h (0,1,2) ± and h (0,1) 0 parameters, which are then determined from data simultaneously with the NP coefficients. 4To clarify the phenomenological relevance of charming penguins, we compare the results of the data For both panels, we show the p.d.f. in green and orange on the basis of the hadronic approach adopted in the global analysis (see the text for more details).For both panels, we show 68% and 95% probability regions in green and orange on the basis of the hadronic approach adopted in the global analysis (see the text for more details).
driven approach against what we denote instead as model dependent treatment of hadronic uncertainties, in which we assume that the contributions generated by the diagrams in Fig. 1 (b) (or (c)) are negligible and that the correlator in eq. ( 4) is well described by the approach of refs.[43][44][45][46][47][48], yielding a subleading effect to the hadronic effects computable in QCD factorization.See the Appendix for further details regarding the parameterization of hadronic contributions employed in the model dependent approach.
In both approaches to QCD long-distance effects, we obtain a sample of the posterior joint probability density function (p.d.f.) of SM parameters, including form factors, and, in the data driven scenario, h λ parameters, together with NP Wilson coefficients.From each posterior p.d.f.we compute the highest probability density intervals (HPDIs), which represent our best knowledge of the model parameters after the new measurements.We also perform model comparison using the information criterion [109], defined as: where the first and second terms are the mean and variance of the log-likelihood posterior distribution.The first term measures the quality of the fit, while the second one is related to the effective degrees of freedom involved, penalizing more complicated models.Models with smaller IC should then be preferred [110].While the posterior    We show 68% and 95% probability regions in green and orange on the basis of the hadronic approach adopted in the global analysis (see the text for more details).
two NP Wilson coefficients are reported in Fig. 2, while the corresponding numerical results for the 95% HPDIs are reported in the first row of Tables I and II.As anticipated above, no significant preference for NP is seen in the data driven scenario, while NP contributions are definitely needed in the model dependent scenario, with a clear preference for C NP 9,µ = 0. Figure 3  Next, we consider NP models in which right-handed b → s transitions arise.In the weak effective Hamiltonian, we allow for nonvanishing C NP 9,µ and C ,NP 9,µ or C ,NP 10,µ .In particular, in Fig. 4 we present the results of the fit in the C NP 9,µ − C ,NP 10,µ case, which we considered in ref. [41] as the best fit one in view of the deviation from one of the ratio R K /R K * [111].With the current experimental situation, this is not the case anymore, and C ,NP 10,µ is again strongly constrained by BR(B s → µ + µ − ).In the SMEFT, we consider nonvanishing C LQ For both panels, we show the 68% and 95% probability regions in green and orange on the basis of the hadronic approach adopted in the global analysis (see the text for more details).
− ) and Re(h − ) in a SM fit in the "data driven" scenario.Darker (lighter) regions correspond to 68% (95%) probability.Notice that according to our hadronic parameterization given in eq. ( 5), Re(h − ) can be reinterpreted as a lepton universal NP contribution, C NP 9,U .
ing penguin contributions.Eventually, notice that the allowed ranges for NP coefficients are much larger in the data driven scenario since the uncertainties on charming penguins leak into the determination of NP Wilson coefficients.
Before concluding, we comment briefly on the possibility of a lepton universal NP contribution to C 9 , that we denote here C NP 9,U , affecting only absolute BRs and angular distributions of b → s + − decays, but leaving LUV ratios as in the SM.This possibility was already discussed in detail in ref. [38], and the experimental situation has not changed since then.Therefore, we just summarize here the main findings of ref. [38] for the reader's convenience.Performing a fit to experimental data within the SM in the data driven scenario, one finds that several h λ parameters are determined to be different from zero at 95% probability, supporting the picture of sizable rescattering in charming penguin amplitudes (see Table 1 in ref. [38]).In particular, there is an interesting correlation between Re(h − ), as is evident from Fig. 7. 5 Data definitely require a nonvanishing combination of the two parameters; if charming penguins are treated à la [43][44][45][46][47][48], Re(h − ) is identified with a lepton universal contribution C NP 9,U , leading to an evidence of NP inextricably linked to the assumptions on charming-penguin amplitudes. 5To identify Re(h − ) as C NP 9,U , we work in the flavour SU (3) F symmetric limit, in which the same hadronic contribution affects both B → K * and Bs → φ transitions (see the Appendix for further details); moreover, for the sake of simplicity, we focus only on these two channels and do not take in consideration additional correlations with other hadronic parameters that similarly mimic the effect of C NP 9,U in B → K transitions.
Summarizing, we performed a Bayesian analysis of possible LUV NP contributions to b → s + − transitions in view of the very recent updates on BR(B (d,s) → µ + µ − ) by the CMS collaboration [49] and on R K and R K * by the LHCb collaboration [54,55].As pointed out in refs.[24, 26, 30, 38, 40-42, 91, 92], the NP sensitivity of these transitions is spoilt by possible long-distance effects, see Fig. 1.Thus, in the data driven scenario we determined simultaneously hadronic contributions, parameterized according to eq. ( 4), and NP Wilson coefficients, finding no evidence for LUV NP.Conversely, evidence for NP contributions is found if charming penguins are assumed to be well described by the approach of refs.[43][44][45][46][47][48], as reported in Tables I and II.
Finally, we considered the case of a lepton universal NP contribution to C 9 , which is phenomenologically equivalent to the effect of h − in our data driven analysis, confirming our previous findings in ref. [38]: in the context of the data driven approach, we found several hints of nonvanishing h i λ parameters, but no evidence of a nonvanishing Re(h (1) − ) −C NP 9,U ; evidence for C NP 9,U only arises in the model dependent scenario in which all genuine hadronic contributions are phenomenologically negligible.Future improvements in theoretical calculations and in experimental data will hopefully allow clarifying this last point.

APPENDIX
In this Appendix, we give further details regarding the parameterizations employed for the hadronic contributions in the data driven and model dependent approaches in each of the two main decays investigated in this work, namely B → K * and B → K , and how these approaches are related to each other.Concerning the third process discussed in this work, namely B s → φ , we work under the assumption of SU (3) F symmetry, i.e., we consider the same hadronic contributions to B → K * and B s → φ .This choice is justified by the fact that it is not possible with current data to single out any SU (3) F -breaking effect from B s → φ , see our previous work in ref. [38] for a detailed analysis on this matter.Starting from the model dependent approach in the B → K * mode, we follow the definition of ref. [43] and give the hadronic contributions as helicitydependent shifts in C 9,i : In our fits, all the involved parameters are considered real according to the way they have been defined and computed in ref. [43], namely by performing a Wick rotation to the Euclidean space in order to compute the light cone sum rule.In particular, they are considered flatly distributed according to the ranges given in Table 2 of the same reference, for q2 = 1.As discussed in TABLE III.68% HPDI for the hadronic contribution |∆C9,1(q 2 )| entering in B → K * and B → φ transitions at different values of q 2 , both in the data driven and the model dependent approaches.In the last column, we also report the expected size of the contributions coming from QCDF.
ref. [30], the relation between this parameterization and the one employed for the data driven approach is given by: where we have introduced the helicity functions h λ (q 2 ).These functions have been defined in such a way that, in the helicity amplitudes shown in Eq. ( 5), the coefficients h − and h − have the same effect of a NP lepton universal |∆C 9,2 (q 2 )| data driven model dependent QCDF TABLE IV. 68% HPDI for the hadronic contribution |∆C9,2(q 2 )| entering in B → K * and B → φ transitions at different values of q 2 , both in the data driven and the model dependent approaches.In the last column, we also report the expected size of the contributions coming from QCDF.
shift in the real part of C 7 and C 9 , namely − q 2 +h (2) − q 2 +h (0) + q 2 + h + q 4 + O(q 6 ) , − q 2 +h (0) 0 Notice that, compared to h ± , h 0 enters the decay amplitude with an additional factor of q 2 , which is the reason why we keep only two terms in its expansion.In our fits, the parameters h (i) λ are allowed to be complex, and we consider the following prior ranges for both their real and imaginary parts: − ∈ [0, 4] , h − ∈ [0, 10 −4 ] , h + ∈ [0, 0.0005] , h Such ranges have been chosen with the only requirement that increasing them would not alter the results of our fits, and are representative of our current ignorance within the data driven approach, where we refrain , both in the data driven and the model dependent approaches.In the last column, we also report the expected size of the contributions coming from QCDF.
ourselves from introducing any kind of theory bias other than the choice of the parameterization.The direct comparison of the fitted results for the hadronic parameters in the two different scenarios is, for several reasons, a non trivial task.Indeed, as explained above, the hadronic contributions are differently parametrized in the two approaches, with no trivial way to directly relate a set of parameters to the other, due in particular to the presence of form factors in Eqs. ( 11)- (12).Moreover, strong correlations as the one shown in Fig. 7 would have to be taken into account, in order to perform a fair comparison among the two scenarios.Finally, it is also important to remember that while in the model dependent approach the hadronic parameters are taken real, this is not the case for the data driven case where they are allowed to be complex.Nevertheless, it is still possible to circumvent all these issues in order to perform a meaningful comparison among the two approaches, by simply confronting the obtained values for |∆C 9,i |, in a similar fashion to what we did graphically in Figure 3 of ref. [38].To this end, we report here the values obtained from the fitted value of the hadronic parameters for the three |∆C 9,i | in Tables III-V, for values of q 2 ranging from 1 to 8 GeV 2 , both in the data driven approach and in the model dependent one.As a reference, we show also the expected size of the contributions coming from QCDF.
Concerning the B → K mode, for the model dependent approach we include only the non-factorizable effects coming from hard-gluon exchanges, being the soft-|∆C 9 (q 2 )| data driven QCDF TABLE VI. 68% HPDI for the hadronic contribution |∆C9(q 2 )| entering in B → K transitions at different values of q 2 in the data driven approach.In the last column, we also report the expected size of the contributions coming from QCDF.
gluon induced terms subleading as found in ref. [44], and O(10%) of the (already small) ones introduced for the B → K * mode and described by eq. ( 9).On the other hand, in the data driven approach we apply the same rationale used behind eq. ( 11) and define h B→K (q 2 ) = q 2 m 2 B V L (q 2 )h (1) B→K q 4 + O(q 6 ) . ( Once again, the parameters h B→K are allowed to be complex, and in our fits we allow the following prior ranges for both their real and imaginary parts: B→K ∈ [0, 0.0002] . Also in this case the ranges have been chosen only taking care that they are large enough in order not to affect the results of our fits.The particularly large range for h (1) B→K is due to its strong correlation to C NP 9 , see Figure 5 of ref. [38].Similarly to what done for the B → K * transition, we report in Table .VI the fitted values for |∆C 9 (q 2 )|.Since, as we stated above, in the model dependent approach we do not include the soft-gluon effects, negligible in this scenario, we report in the table only the fitted values for this hadronic correction in the data driven approach, together with the expected size of the contributions coming from QCDF.

FIG. 2 .
FIG. 2. Left panel: Posterior p.d.f. for the NP coefficient C NP 9,µ .Right panel: Posterior p.d.f. for the SMEFT Wilson coefficient C LQ 2223 .For both panels, we show the p.d.f. in green and orange on the basis of the hadronic approach adopted in the global analysis (see the text for more details).

ΛFIG. 3 .
FIG. 3. Left panel: Joint posterior p.d.f. for C NP 9,µ and C NP 10,µ .Right panel: Joint posterior p.d.f. for the SMEFT Wilson coefficients C LQ 2223 and C Qe 2322 .For both panels, we show 68% and 95% probability regions in green and orange on the basis of the hadronic approach adopted in the global analysis (see the text for more details).

FIG. 4 .
FIG.4.Joint posterior p.d.f. for C NP 9,µ and C NP 10,µ .We show 68% and 95% probability regions in green and orange on the basis of the hadronic approach adopted in the global analysis (see the text for more details).
displays the allowed regions in the C NP 9,µ − C NP 10,µ and C LQ 2223 − C Qe 2322 planes, while the corresponding HPDIs are reported in the second row of Tables I and II respectively.Again, no evidence for NP is seen in the data driven case, while clear evidence for a nonvanishing C NP 9,µ appears in the model dependent approach.Deviations from zero of C NP 10,µ are strongly constrained by BR(B s → µ + µ − ), corresponding to the strong correlation C LQ 2223 ∼ C Qe 2322 seen in the right panel of Fig. 3.
FIG. 5. Two-and one-dimensional marginalized joint p.d.f. for the set of SMEFT Wilson coefficients C LQ 2223 , C Qe 2322 , C ed 2223 and C Ld 2223 .For both panels, we show the 68% and 95% probability regions in green and orange on the basis of the hadronic approach adopted in the global analysis (see the text for more details).

FIG. 6 .FIG. 7 .
FIG. 6. Correlation matrix of the Wilson coefficients of the SMEFT operators studied in this work under the "data driven" (left panel, orange) and the "model dependent" (right panel, green) approaches to hadronic uncertainties in our global analysis.

TABLE I .
HPDI for the Wilson coefficients of the low-energy weak Hamiltonian in all the considered NP scenarios along with the corresponding ∆IC.White rows correspond to results obtained in the data driven scenario, while model dependent scenario results are shaded in gray.See the text for the definition of the two scenarios.

TABLE II .
Same as Tab.I for SMEFT Wilson coefficients.
2223, belonging to the SMEFT.The p.d.f.s for the