New Physics in \({\varvec{b \rightarrow s \ell ^+ \ell ^}}\) confronts new data on lepton universality
Abstract
In light of the very recent updates on the \(R_K\) and \(R_{K^*}\) measurements from the LHCb and Belle collaborations, we systematically explore here imprints of New Physics in \(b \rightarrow s \ell ^+ \ell ^ \) transitions using the language of effective field theories. We focus on effects that violate Lepton Flavour Universality both in the Weak Effective Theory and in the Standard Model Effective Field Theory. In the Weak Effective Theory we find a preference for scenarios with the simultaneous presence of two operators, a lefthanded quark current with vector muon coupling and a righthanded quark current with axial muon coupling, irrespective of the treatment of hadronic uncertainties. In the Standard Model Effective Field Theory we select different scenarios according to the treatment of hadronic effects: while an aggressive estimate of hadronic uncertainties points to the simultaneous presence of two operators, one with lefthanded quark and muon couplings and one with lefthanded quark and righthanded muon couplings, a more conservative treatment of hadronic matrix elements leaves room for a broader set of scenarios, including the one involving only the purely lefthanded operator with muon coupling.
1 Introduction
The past few years have brought us a thriving debate on the possible hints of New Physics (NP) from measurements of semileptonic B decays. In particular, Flavour Changing Neutral Current (FCNC) decay modes into multibody final states, e.g. \(B\rightarrow K^{(*)}\ell ^+\ell ^\) and \(B_s \rightarrow \phi \, \ell ^+ \ell ^\), bring forth a large number of experimental handles, see e.g. [1], that are extremely useful for NP investigations while also allowing to probe the Standard Model (SM) itself in detail [2, 3, 4, 5, 6]. The inference of what pattern is being revealed by the experimental observations is the crux of the debate.
Two distinct classes of observables characterize these semileptonic decays. The first is the class of angular observables arising from the kinematic distribution of the differential decay widths that have been measured at LHCb [7, 8, 9, 10, 11, 12, 13], Belle [14], ATLAS [15] and CMS [16, 17, 18]. These observables, mostly related to the muonic decay channel, while being sensitive to NP [6, 19, 20, 21, 22] are besieged by hadronic uncertainties [23, 24, 25, 26, 27, 28]. The latter, associated with QCD longdistance effects – hard to estimate from first principles [29, 30] – can saturate the measurements so as to be interpreted as possibly arising from the SM or can obfuscate the gleaning of NP from SM contributions [31, 32, 33]. Therefore, in the absence of a complete and reliable calculation of the hadronic longdistance contributions, a clear resolution of this debate based solely on the present set of angular measurements is hard to achieve. Improved experimental information in the near future [34] concerning, in particular, the electron modes is a subject of current crosstalk between the theoretical and experimental communities [35, 36], and may shed new light on this matter [37, 38, 39]. The second class of observables then becomes crucial to this debate. These are the Lepton Flavour Universality Violating (LFUV) ratios that hold the potential to conclusively disentangle NP contributions from SM hadronic effects. The latter are indeed lepton flavour universal [2, 40]. Several hints in favour of LFUV have surfaced in the past few years in experimental searches at LHCb [41, 42] and Belle [14]. These have led to a plethora of theoretical investigations [43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161], all oriented towards physics Beyond the Standard Model (BSM) able to accommodate such LFUV signals, mainly involving \(Z'\) or leptoquark mediators at scales typically larger than a few TeV and with some peculiar flavour structure needed to avoid clashing with the stringent bounds from meson–antimeson mixing and from other observables. Despite possible modelbuilding challenges, the primary message here is clear: a statistically significant measurement of LFUV effects in FCNCs such as \(b \rightarrow s \ell ^+ \ell ^\) decays would herald the discovery of NP unambiguously [162, 163, 164, 165, 166, 167].

We revisit our approach to QCD power corrections streamlined for efficiently capturing longdistance effects, which are of utmost relevance in the interpretation of the current experimental information on the \(B \rightarrow K^{*} \mu ^+ \mu ^\) channel. We discuss several novelties about our new parameterization of hadronic contributions, recently introduced in [39];

We make use of two distinct Effective Field Theory (EFT) frameworks, namely the \(\varDelta B = 1\) Weak Effective Hamiltonian and the Standard Model Effective Field Theory (SMEFT). The former EFT allows us to obtain a better insight on the dynamics at the decay scale, while the latter can offer a deeper link with BSM interpretations.
2 Theoretical framework
As an introduction to the basic ingredients of our analysis we start by reviewing the standard EFT for \(\varDelta B = 1\) transitions, highlighting the distinction between shortdistance and hadronic contributions. We then move on to LFUV effects in terms of SM gaugeinvariant dimensionsix operators, completing the EFT dictionary useful for understanding the results we present in Sect. 4.
2.1 Short distance vs long distance
The shortdistance physics in Eqs. (4)–(8) is, in general, captured by the Wilson coefficients (WCs), denoted as effective couplings \(C^{(\prime )}\). Within the SM, at the dimensionsix level, semileptonic chiralityflipped and (pseudo)scalar operators can be neglected, however they are potentially relevant for the study of NP effects. In our analysis, we evaluate SM WCs at the scale \(\mu _b = 4.8 \,\mathrm{GeV}\) using stateoftheart QCD and QED perturbative corrections, both in the matching [172, 173, 174] and in the anomalous dimension of the operators involved [174, 175, 176, 177].^{1} We note that the remaining theoretical uncertainty on the SM WCs, at the level of few percent, can be neglected in this work.
In the absence of a unique UV complete model that can potentially be responsible for the measured hints of anomalies in the \(b \rightarrow s\) transitions, the formalism of the effective Hamiltonian is extremely powerful. It allows to study the effects of BSM physics in a modelindependent manner, where the presence of NP effects manifests itself as (leptonflavour dependent) shifts of the WCs with respect to the SM values.^{2} On the basis of previous global analyses which allow for LFUV effects [162, 163, 164, 165, 166, 167, 179, 180, 181, 182, 183], in this paper we allow for NP effects in the WCs of the operators \(Q_{9,10,S,P}^{(\prime ) \, \ell =e,\mu }\). We do not consider the case of NP effects in dipole operator coefficients \(C_7^{(\prime )}\), since such a possibility is severely constrained by the inclusive radiative branching fraction of \(B \rightarrow X_s \gamma \) among other measurements [184] and it is anyway irrelevant for LFUV. Moreover, in the following we also set aside the possibility of NP effects entering in Eq. (5), a case considered in the study by the authors of [73]. Our choice is, once more, primarily driven by our focus on LFUV effects. On more general grounds, one should stress that decoding LFUconserving NP effects in current \(b \rightarrow s \) data – a possibility recently considered in [185] – may be a challenging task [186], especially in light of unknown hadronic contributions.
On the contrary, the theoretical estimate of nonfactorizable terms – denoted here generically by \(h_{\lambda }\) – is not so well under control. For the processes of interest, the largest contribution arises from current–current operators involving charm quarks, specifically \(Q_{2}^{c}\) [23, 24], not parametrically suppressed by CKM factors or small WCs. This charmloop effect stemming out of Eq. (10) is therefore a genuine longdistance contribution: it implies potentially sizable nonperturbative effects involving the charm quark pair with strong phases that are very difficult to estimate. While at low \(q^2\) hardgluon exchanges in the charmloop amplitude can be addressed in the framework of QCD factorization (QCDF) [190], the evaluation of softgluon exchange effects remains, in this context, the toughest theoretical task [191]. A detailed analysis of softgluon exchanges in \(B \rightarrow K\) transitions has been performed in Ref. [24]. There these contributions turned out to be subdominant in comparison with the QCDF estimate of the hardgluon contributions, supporting previous results presented in [23].
For \(B \rightarrow K^*\), the only estimate of \(h_{\lambda }\) currently available is the one carried out in Ref. [23] using LCSR techniques in the single softgluon approximation, valid for \(q^2 \ll 4m_c^2\). The regime of validity of the result is then extended to the whole largerecoil region by means of a phenomenological model based on dispersion relations. While an estimate of the error budget is attempted in Ref. [23], there are potentially large systematic effects, related for instance to the lack of control over strong phases, that are difficult to quantify reliably, in particular when approaching the \(c\bar{c}\) threshold at \(q^2 \sim 4m_c^2\) [31], close to the \(J/\psi \) resonance where quark–hadron duality is questionable even in the heavy quark limit [192]. Note that the same considerations also apply to the case of \(B_{s} \rightarrow \phi \ell ^+ \ell ^\), for which a similar LCSR evaluation of the charmloop effect is still pending, leaving room also for appreciable \(SU(3)_{F}\) breaking effects [26].
Recently, renewed attempts to obtain a better grasp of the nonfactorizable terms have appeared in the literature [27, 28]. Both works turn out to be in agreement with the results from Ref. [23]. However, in Ref. [27] – where \(h_\lambda \) is assumed to be welldescribed as a sum of relativistic BreitWigner functions – the authors found a very similar result to the one in [23] only in the case of vanishing strong phases, while quite different outcomes may be obtained for different assumptions on the same phases. In turn, the authors of Ref. [28] exploited the analytic properties of the amplitudes in order to perform a conformal expansion of \(h_\lambda \), isolating physical poles and ensuring unitarity. They use resonant data and additional theoretical information at negative \(q^2\) to fix the coefficients of the expansion, including estimates of strong phases due to the presence of a second branch cut in the amplitude (generated for instance by intermediate states with two charmed mesons), which represents a challenge for the formalism as well as for the numerical estimate. Despite the quite good agreement with the numerical result presented in [23], the coefficients obtained in [28] for the zexpansion of \(h_\lambda \) point to a poor convergence of the series, casting doubts on the actual \(q^2\) shape of the \(h_\lambda \) functions if more terms were to be included in the expansion.

A phenomenological model driven (PMD) approach, employing LCSR results extrapolated by means of dispersion relations in the whole low\(q^2\) region for the decay;

A phenomenological data driven (PDD) approach, taking into account LCSR results only far from the c\(\bar{c}\) threshold and exploiting the results in Ref. [23] for \(q^2=0,1\) GeV\(^2\), with their phases and \(q^2\) dependence inferred from experimental data.
2.2 The SMEFT perspective
Previous modelindependent analyses of \(b \rightarrow s \ell ^+ \ell ^ \) anomalies have essentially pointed to \(\mathcal {O}(10~\mathrm{TeV})\) NP for \(\mathcal {O}(1)\) effective couplings in order to produce a \(\sim \, 25\)% shift of the SM WC values of the semileptonic operators \(Q_{9V,10A}\). The UV dynamics underlying these NP effects is then expected to exhibit a reasonable mass gap with the SM theory. Hence, a quite natural choice for deeper BSM insights is the gaugeinvariant framework of the SMEFT [194, 195].
Note that even under this assumption, the operators in Eq. (13) may be, in principle, testable in other interesting processes other than \(b \rightarrow s \ell ^+ \ell ^\) transitions. The most notable opportunity may be offered by the channel \(B \rightarrow K^{(*)} \nu \bar{\nu }\) [201], sensitive to the operators composed of weak doublets in both lepton and quark currents. At the present experimental sensitivity [202], this channel turns out to have a relatively mild interplay with \(b \rightarrow s \ell ^+ \ell ^\) measurements [86, 113]. Interestingly, with the advent of more data [34] one may hope to distinguish NP effects of \(O^{LQ^{(3)}}\) from the ones of \(O^{LQ^{(1)}}\) due to an accurately measured lightlepton LFUV ratio in semileptonic \(b \rightarrow c\) transitions [54, 86]. Still, for the purposes of our modelindependent study, \(O^{LQ^{(1,3)}}_{ii23}\), \(i =\{1,2\}\), are indistinguishable, as in Ref. [76]. Without loss of generality, the set of operators in Eq. (13) remains indeed the one primarily sensitive to the measurements considered in this work.
2.3 New Physics effects in \(R_{K}\)
We wish to end this section reviewing the relevance of \(R_{K}\) for NP and its complementarity with other present and possibly forthcoming LFUV measurements, as \(R_{K^{*}}\) and \(R_{\phi }\). This completes the stage setup for the presentation and discussion of the results collected in Sect. 4.
In Fig. 1 we show the impact on \(R_{K}\) in the bin discussed so far of each of the operators considered here in the WET (left panel), see Eq. (18), and in the SMEFT (right panel), see Eq. (19). The range on the xaxis in Fig. 1 covers \(\mathcal {O}(1)\) effects relative to the shortdistance SM contributions. The SM limit is emphasized by the silver dotdashed lines, and the new \(R_{K}\) measurement is represented by the horizontal orange band, drawn according to experimental central value and standard deviation, see Eq. (1).
It is clear from what is depicted for the WET that operators featuring both lefthanded and righthanded \(b \rightarrow s\) currents are eligible for a satisfactory explanation of the measured value of \(R_{K}\). In particular, one cannot distinguish effective couplings related to lefthanded or righthanded \(b \rightarrow s\) currents within operators that have the same leptonic structure, since they constructively interfere in Eq. (18). Moreover, as highlighted in the plot, the NP contribution required to explain the present \(R_{K}\) measurement is now about one fifth of the SM one. Therefore, the linearized limit of Eq. (18) may be a good approximation in order to appreciate how LFUV effects actually probe the \(\mu  e\) combination of the leptonic current. This fact is captured in the plot by the mirrorlike behaviour of redblue and magentacyan line pairs with respect to the SM limit. In the same panel, axial and vectorial leptonic effective couplings turn out also to be mirrorlike as a reflection of the SM result: \(C_{9}^{\text {SM}} \sim C_{10}^{\text {SM}}\).
Similar considerations apply to the case of SMEFT operators with leptonic weak doublets, requiring only about \(15 \%\) of the SM WC value for \(Q_{9V,10A}\) to accommodate \(R_{K}\) within a NP scale of \(\varLambda = 30\) TeV, yielding \(C^{LQ,Ld}_{\ell \ell 23} \sim 0.8\). However, the correlations induced by the \(SU(2)_{L} \times U(1)_{Y}\) gauge symmetry no longer allow a full set of 8 different viable solutions for the \(R_{K}\) anomaly. From the right panel of Fig. 1, NP effects from SMEFT operators featuring exclusively righthanded muonic currents are ruled out, while the electronic counterparts are still available at the expense of larger NP effects, \(\gtrsim 35 \%\) of the SM shortdistance physics for the same \(\varLambda = 30\) TeV. Interestingly, among the RGEinduced set of operators reported in Eq. (17) we can then exclude \(O^{eu}_{2233}\).
Unfortunately, the above qualitative considerations remain subject to several uncertainties. First of all, the longitudinal polarization fraction of \(B \rightarrow K^* \ell ^+ \ell ^\) in the bin of interest is not equal to unity [210]: this fact already makes the \(R_{K^*}\) observable less orthogonal to \(R_{K}\) in the study of NP [163]. Moreover, longitudinal and transverse polarization fractions are sensitive to \(\varLambda _{\text {QCD}}/m_{b}\) power corrections not fully under control [47, 163]. This also suggests an experimental information that would be important to handle in the future: the measurement of \(R_{K^*}^{\text {T,L}}[1.1,6]\), i.e. the ratio of longitudinal and transverse parts of the \(B \rightarrow K^* \ell ^+ \ell ^\) amplitude in the \(q^2\)bin [1.1,6] GeV\(^2\). These quantities would be less sensitive to unknown hadronic effects, and distinctively sensitive to NP effects in \(C^{}_{9,10} \pm C_{9,10}^{\prime }\) combinations [163, 166]. Similar information could be extracted from \(B_{s} \rightarrow \phi \ell ^+ \ell ^\) as well.
Secondly, as already noted at the beginning of Sect. 2, the same angular observables measured in \(B \rightarrow K^{*} \mu ^+ \mu ^\) are also affected by nonfactorizable QCD effects. Only corresponding LFUV combinations as the one proposed in Ref. [35, 36] and recently reanalyzed in [186] may help to disentangle genuine NP effects from hadronic contributions theoretically not wellunderstood. At present, the only available measurement of this sort is given by Belle [14] and it is (unfortunately) of limited statistical significance, but more will certainly come in the next years [34].
In the end, a careful study of \(b \rightarrow s \ell ^+ \ell ^\) anomalies calls for a global analysis that can go well beyond the qualitative picture highlighted above, taking care of all the aforementioned subtleties in a framework where a nontrivial interplay between genuine NP effects and hadronic contributions is allowed. The analysis performed in this study, presented in Sect. 4, is precisely dedicated to make interpretations of the underlying NP scenarios behind current \(b \rightarrow s \ell ^+ \ell ^ \) anomalies as robust as possible.
3 Experimental and theoretical input
In this section we plan to review the baseline of our analysis, the experimental dataset included, and the assumptions made throughout this work. In the present study we perform a global analysis on a comprehensive set of \(b \rightarrow s \ell ^+ \ell ^\) data with stateoftheart theoretical computations, within a Bayesian framework.
We adopt for this matter the public HEPfit package [211], whose Markov Chain Monte Carlo (MCMC) analysis framework employs the Bayesian Analysis Toolkit (BAT) [212]. In our MCMC analysis we vary from a minimum of 60 to a maximum of 80 parameters on a case by case basis. Within the MetropolisHastings algorithm implemented in BAT, we set up, for the scenarios presented in Sect. 4, MCMC runs involving 240 chains with a total of 2.4 million events per run, collected after an equivalent number of prerun iterations.
Values of the WET WCs fit from data in all the considered scenarios along with relative \(\varDelta IC\). The italic values highlight the PMD results when this approach can be used to address the experimental data in a particular scenario. The PDD results are presented for all cases. For the definition of the two approaches, see Sect. 2.1
Mean (rms)  \(\varDelta IC\)  

\( C_{9,\mu }^\mathrm{NP} \)  \(\) 1.20(27)  14 
\(\) 1.21(16)  50  
\( C_{10,e}^\mathrm{NP} \)  \(\) 0.87(24)  15 
\( (C_{9,\mu }^\mathrm{NP},C_{9,e}^\mathrm{NP}) \)  (\(\) 1.61(48), \(\) 0.56(53))  13 
(\(\) 1.28(18), \(\) 0.27(34))  48  
\( (C_{9,\mu }^\mathrm{NP},C_{9,\mu }^{\prime , \mathrm{NP}}) \)  (\(\) 1.61(33), 0.72(34))  17 
(\(\) 1.30(15), 0.53(24))  54  
\( (C_{9,\mu }^\mathrm{NP},C_{10,\mu }^{\prime , \mathrm{NP}}) \)  (\(\) 1.55(32), \(\) 0.44(14))  24 
(\(\) 1.38(16), \(\) 0.37(12))  61  
\( (C_{10,\mu }^\mathrm{NP},C_{9,\mu }^{\prime , \mathrm{NP}}) \)  (0.73(17), \(\) 0.04(24))  17 
\( (C_{10,\mu }^\mathrm{NP},C_{10,\mu }^{\prime , \mathrm{NP}}) \)  (0.75(16), 0.04(17))  16 
\( (C_{9,e}^\mathrm{NP},C_{9,e}^{\prime , \mathrm{NP}}) \)  (1.51(38), \(\) 0.81(37))  10 
\( (C_{9,e}^\mathrm{NP},C_{10,e}^{\prime , \mathrm{NP}}) \)  (1.36(32), 0.87(40))  11 
\( (C_{10,e}^\mathrm{NP},C_{9,e}^{\prime , \mathrm{NP}}) \)  (\(\) 1.06(54), \(\) 0.46(46))  12 
\( (C_{10,e}^\mathrm{NP}, C_{10,e}^{\prime , \mathrm{NP}}) \)  (\(\) 1.01(28), 0.29(29))  12 
Values of the SMEFT WCs fit from data in all the considered scenarios along with relative \(\varDelta IC\). The italic values highlight the PMD results when it can address the experimental data in a particular scenario. The PDD results are presented for all cases. For the definition of the two approaches, see Sect. 2.1
Mean (rms)  \(\varDelta IC\)  

\( C^{LQ}_{2223} \)  0.75(14)  23 
0.79(12)  37  
\( (C^{LQ}_{2223},C^{Qe}_{2322}) \)  (0.78(18), 0.06(32))  21 
(0.94(12), 0.67(17))  50  
\( (C^{LQ}_{1123},C^{Qe}_{2311}) \)  (\(\) 0.51(29), 0.96(70))  12 
\( (C^{LQ}_{2223},C^{ed}_{2223}) \)  (0.74(15), 0.16(33))  21 
(0.81(12), \(\) 0.19(29))  35  
\( (C^{LQ}_{2223},C^{Ld}_{2223}) \)  (0.81(15), \(\) 0.20(15))  22 
(0.80(12), \(\) 0.11(12))  36  
\( (C^{LQ}_{1123},C^{ed}_{1123}) \)  (\(\) 0.08(73), \(\) 2.1(14))  12 
\( (C^{LQ}_{1123},C^{Ld}_{1123}) \)  (\(\) 0.93(27), 0.39(27))  12 
\( (C^{Qe}_{2311},C^{ed}_{1123}) \)  (\(\) 0.2(18), \(\) 1.3(18))  8 
\( (C^{Qe}_{2311},C^{Ld}_{1123}) \)  (1.77(47), \(\) 0.18(24))  9 

All the angular observables and branching ratio information on \(B \rightarrow K^* \mu ^+\mu ^\) from the experimental results obtained by LHCb [10, 13], Belle [14], ATLAS [15] and CMS [16, 17] collaborations. When available, we always take into account experimental correlations between the measurements performed in the same bin. Note that we restrict here only to the largerecoil region, i.e. \(q^{2}\) values below the \(J/\psi \) resonance, excluding measurements in the (theoretically challenging) broadcharmonium region.

\(B \rightarrow K^* e^+e^\) angular observables from LHCb in the available \(q^2\) bin, [0.002, 1.12] GeV\(^2\) [11].

Angular observables and branching ratio of \(B_{s} \rightarrow \phi \mu ^+\mu ^\) provided by LHCb [12].

Branching ratio of \(B_s\rightarrow \mu ^+\mu ^\) measured by LHCb [216], CMS [217], and most recently by ATLAS [218]. Note that we also employ the upper limit on \(B_s\rightarrow e^+e^\) decay reported by HFLAV [219], useful for the study of NP coupled to electrons [83].

Branching ratios for \(B^{(+)} \rightarrow K^{(+)} \mu ^+\mu ^\) decays in the largerecoil region by LHCb [8].

Branching ratios for the radiative decay \(B \rightarrow K^* \gamma \), from HFLAV [219], and for \(B_{s} \rightarrow \phi \gamma \) as measured by LHCb [220]. While we are not going to consider NP effects in dipole operators, these measurements are relevant in our PDD approach.

LFUV ratios including the very recent updates: \(R_{K^*}\) in both \(q^2\) bins, [0.045, 1.1] GeV\(^{2}\) and [1.1, 6] GeV\(^{2}\) [42, 169], and the \(R_K\) measurement [168].
Regarding the nonfactorizable part of the amplitudes, we include hardgluon contributions following what already outlined in detail in our previous work [166], while we proceed here differently for what regards our treatment of softgluon exchanges.
In the PMD approach, we do not expand Eq. (10) in powers of \(q^{2}\), but we directly express it in terms of the phenomenological expression given by Eq. (7.14) of Ref. [23],^{10} and we flatly distribute all the involved parameters according to the ranges reported in table 2 of the same reference. In order to allow for imaginary parts as well, each of the three charmloop amplitudes in Ref. [23] is multiplied by a complex phase, flatly varying each angle within [0, 2\(\pi \)), yielding a total of 12 parameters to describe the nonperturbative hadronic contributions within this approach.
We conclude this section mentioning that the rest of the SM parameters varied in our analysis can be found in table 1 of Ref. [166], while for NP WCs, we adopt in general flat priors in the range [− 10, 10], assuming they are real. Note that some of the NP scenarios here considered showed multimodal p.d.f.s. In such cases we focused on the NP solution closer to the SM limit, identified by \(C_i^\mathrm{NP} = 0\). Finally, we point out that all our findings for the study of the SMEFT in Sect. 4 assume a NP scale set to 30 TeV. In order to read out SMEFT WCs at a different NP scale \(\varLambda \), one needs to rescale the results given in Sect. 4 appropriately.
4 EFT results from the new \(R_{K}\) measurement
In this section we present our results. We perform several fits to the experimental measurements listed in Sect. 3, differentiated by the set of NP WC(s) considered. We employ the PDD approach in all the scenarios examined, while exploring the PMD approach only when it can provide a satisfactory fit to current data, i.e. when NP effects built up from lefthanded \(b \rightarrow s\) currents coupled to vectorlike (purely lefthanded) muonic currents are involved in the WET (SMEFT) formalism. The goodness of the fit is here evaluated by means of the IC, defined in Eq. (21), while the details of the PMD and PDD approaches have been presented in Sect. 2.1.
The primary goal of this analysis consists in the study of the interplay between the new \(R_K\) measurement and NP. In particular, we investigate whether the update of \(R_K\) combined with the current \(R_{K^*}\) measurement can actually have an impact on the viable solutions to the anomalies in \(b \rightarrow s\) transitions allowed by the previous \(R_{K}\) from Run I of LHC. To this end, we report in Tables 1 and 2 the results for the fitted values of the WCs in each of the models scrutinized here, employing the WET and the SMEFT formalism respectively. \(\varDelta IC\) values are also reported in the same table, marking the improvement with respect to the SM, see the discussion following Eq. (21) in Sect. 3. Finally, results for what we retain as key observables for our study are also reported in Tables 3 and 4, differentiating once again scenarios in the WET or in the SMEFT, respectively.
Our main results are illustrated here as follows. The posterior p.d.f.s obtained for NP coefficients are shown in Figs. 2, 3, 4, 5, 6, 7, 8 and 9. Figure 2 refers to scenarios where a single WC is taken into account. Figures 3 and 4 involve fits with two operators at the same time and correspond to two popular benchmarks previously studied in literature. Figures 5, 6, 7 and 8 correspond to 2D scenarios where NP effects in the form of \(b \rightarrow s\) righthanded currents are present. Finally, in Fig. 9 the result for the largest set of SM gaugeinvariant operators probed by current experimental data is presented.
4.1 New Physics in \(b \rightarrow s\) lefthanded currents
 (i)

a vectorial muonic current, i.e. \(C_{9,\mu }^\mathrm{NP}\,\);
 (ii)

a purely lefthanded muonic current, i.e. \(C_{2223}^{LQ}\,\);
 (iii)

an axial electronic current, i.e. \(C_{10,e}^\mathrm{NP}\,\).
From Fig. 2 we can supplement this picture with the measurement of LFUV ratios. We note how the impact of the \(R_{K}\) measurement can be particularly relevant for the final outcome. Concerning case (i), we see that the interplay of \(R_{K}\) and \(R_{K^*}\) does not favour this scenario any longer within the \(1\sigma \) regions highlighted by the orange bands in the plot. This is in contrast to the previous situation given by the 2014 measurement of \(R_{K}\) and represented in Fig. 2 by the vertical gray band. Most importantly, the tension arising in this NP scenario when accounting for current LFUV ratio measurements is also evident in the case of the PDD approach (right panel in the first row of the figure).
Therefore, we wish to note that – beyond the importance of the present \(R_{K}\) update – the assumptions made in the size of the hadronic contributions when comparing NP scenarios turn out to be crucial. The most evident case of this sort is certainly (iii). Within a conservative approach to QCD power corrections in the \(B \rightarrow K^* \ell ^+ \ell ^\) amplitude, this scenario offers a perfectly viable fit to \(b \rightarrow s \ell ^+ \ell ^\) data. In particular, (iii) provides an optimal description of LFUV ratios according to what depicted in the last row of Fig. 2. However, in terms of model comparison, it remains globally disfavoured with respect to (ii) in virtue of the information arising from the angular measurements of \(B \rightarrow K^* \mu ^+ \mu ^\). Indeed, while NP effects associated to \(O^{LQ}_{2223}\) can actually ameliorate \(b \rightarrow s \ell ^+\ell ^\) anomalies as the ones related to the socalled \(P_{5}'\) observable [180], the phenomenological viability of NP effects encoded in the effective operator \(Q_{10A,e}\) necessarily relies on large hadronic contributions [166], making (iii) a less economic alternative to (ii). This is reflected by the reported \(\varDelta IC\): in the PDD approach the improvement of the SM fit provided by NP effects as in (ii) is several units of IC larger than the one provided by (iii), making (ii) much more favoured by the current experimental dataset.
Let us now turn to the investigation of more complex cases, where BSM dynamics is actually described by a pair of effective operators rather than just a single one. We start focussing on the scenario where the effective couplings of interest turn out to be \((C_{9,\mu }^\mathrm{NP}, C_{9,e}^\mathrm{NP})\). Note from Table 1 that the addition of a NP contribution coming from the electron operator \(Q_{9V,e}\) does not strongly improve the fit obtained with \(Q_{9V,\mu }\): in terms of \(\varDelta IC\), both PMD and PDD approaches slightly penalize this scenario, underlying a marginal improvement in the description of current data in correspondence to the addition of \(C_{9,e}^\mathrm{NP}\). This is also captured by the LFUV ratios in the right panels of Fig. 3, where an improvement is only seen in the value of \(R_K\). Moreover, the prediction for longitudinal and transverse components of \(R_{K^{*},\phi }\) remain essentially the same for the two scenarios 3.
We then move to the inspection of cases where heavy new degrees of freedom can generically couple the lefthanded \(b \rightarrow s\) current to both vectorial and axial leptonic structures or, from the BSM perspective drawn in the SMEFT, to both lefthanded and righthanded leptonic currents. These NP scenarios generalize the specific benchmarks (i), (ii), (iii) discussed at the beginning of the section. Left and central columns in Fig. 4 report the result for the PMD and PDD approach in the case where NP effects lie in the muonic mode only, while the PDD approach for the case of the electronic mode is given in the right column. Comparing with what already illustrated for (i), (ii), (iii), with the help of Table 2 and the second row of Fig. 4 we can easily conclude that NP contributions from \(O^{LQ}_{2223}\)–\(O^{Qe}_{2322}\) are still favoured by data, slightly improving the description of \(R_K\) with respect to the minimal case (i) in the PMD approach, and the minimal case (ii) in the PDD approach. Moreover, the case where NP effects arise from the pair \(O^{LQ}_{1123}\)–\(O^{Qe}_{2311}\) is not favoured over the simpler axial electronic proposal denoted here as (iii). Interestingly, from Table 4 we can also observe that a measurement of the transverse component of the ratios \(R_{K^{*},\phi }\) for these scenarios would be quite indicative. Indeed, the prediction of these LFUV observables from NP effects in \(O^{LQ}_{2223}\)–\(O^{Qe}_{2322}\) points to \( R^{\text {T}}_{K^{*},\phi }[1.1,6]\simeq 0.95\) and \(R^{\text {T}}_{K^{*},\phi }[1.1,6] \simeq 0.85\) in the PMD and PDD approach respectively, compatible among each other only at the 1\(\sigma \) level, and different from the ones obtained for (i) and (ii). On the contrary, the corresponding LFUV prediction from the electronic pair considered here would not be distinguishable from what assessed already in (iii). Finally, from the same Table 4 we also highlight that the study of NP effects from the full set of four operators \(O^{LQ}_{\ell \ell 23}\)–\(O^{Qe}_{23\ell \ell }\) with \(\ell = \{1,2\}\), would not change the important phenomenological interplay found for the pair \(O^{LQ}_{2223}\)–\(O^{Qe}_{2322}\), but would quite distinctively predict transverse ratios \( R^{\text {T}}_{K^{*},\phi }[1.1,6] \simeq 0.75\), independently of the hadronic approach considered. We postpone a thorough discussion on the analysis of these four operators all together to Appendix B, where we study them in the context of the loopgenerated effects reported in Eq. (15), and where we also emphasize the possible connections of \(b \rightarrow s \ell ^+ \ell ^\) anomalies with EW precision physics [199, 200, 207].
4.2 New Physics in both \(b \rightarrow s\) left and righthanded currents
We start by considering NP effects in vectorial and axial muonic currents and described by means of the WET formalism, namely the pairs of NP WCs: \((C_{9,\mu }^\mathrm{NP}\), \( C_{9,\mu }^{\prime , \mathrm{NP}})\), \((C_{9,\mu }^\mathrm{NP}, C_{10,\mu }^{\prime , \mathrm{NP}})\), \((C_{10,\mu }^\mathrm{NP}, C_{9,\mu }^{\prime , \mathrm{NP}})\) and \((C_{10,\mu }^\mathrm{NP}, C_{10,\mu }^{\prime , \mathrm{NP}})\). The two former scenarios are generalizations of the study case (i), and are allowed both in the PMD and PDD approaches, while the latter two can satisfactorily address \(b \rightarrow s \ell ^+ \ell ^\) anomalies only within the PDD approach. Results for all these scenarios can be found in Fig. 5. As first highlighted by the trend in the reported \(\varDelta IC\) and further depicted by \(R_{K}\)–\(R_{K^*}\) plots, the inclusion of righthanded \(b \rightarrow s\) effective couplings allows for an overall better description of data. In particular, from the inspection of the central row in Fig. 5 the scenario involving the operators \(Q_{9V,\mu }\) and \(Q_{10A,\mu }^{\prime }\) provides the best match here to the newly measured \(R_{K}\) together with \(R_{K^*}\) in the \(q^2\)bin [1.1,6] GeV\(^2\). Moreover, it yields an optimal description of \(B_s \rightarrow \mu ^+ \mu ^\) and of the whole angular analysis at the same time – independently of the hadronic approach undertaken – and hence stands out in Table 1 as the study case with the highest \(\varDelta IC\) in both PMD and PDD approaches. This result comes together with the prediction for \( R^{\text {T}}_{K^{*},\phi }[1.1,6] \simeq 1\) in the scenarios with the pairs \(Q_{9V,\mu }\)–\(Q_{9V(10A),\mu }^{\prime }\). We also note that the prediction of the longitudinal ratio unfortunately does not allow to single out within errors the NP case of \(Q_{9V,\mu }\)–\(Q_{9V,\mu }^{\prime }\) with respect to \(Q_{9V,\mu }\)–\(Q_{10,\mu }^{\prime }\).
A different prediction for the transverse and longitudinal LFUV ratios is instead obtained for the pairs \(Q_{10A,\mu }\)–\(Q_{9V(10A),\mu }^{\prime }\), approximately giving \( R^{\text {T}}_{K^{*},\phi }[1.1,6] \simeq R^{\text {L}}_{K^{*},\phi }[1.1,6] \simeq 0.75\). We note that the nontrivial interplay between hadronic physics – addressing here the \(B \rightarrow K^* \mu ^+ \mu ^\) angular analysis – and the experimental weights of the measured LFUV ratios and of \(Br(B_s \rightarrow \mu ^+ \mu ^)\) lead overall to a lower \(\varDelta IC\) value for these two scenarios with respect to the case of \(Q_{9V,\mu }\) and \(Q_{10A,\mu }^{\prime }\) (see Table 1).
We now proceed considering NP effects in lefthanded and righthanded muonic currents employing the gaugeinvariant language of the SMEFT. In particular, we first focus on the scenarios \((C_{2223}^{LQ}, C_{2223}^{ed})\) and \((C_{2223}^{LQ}, C_{2223}^{Ld})\), that are generalizations of the study case ii), therefore viable both in the PMD and in the PDD approach. Their results are shown in Fig. 7. Similarly to what found above for the pairs \(Q_{10A,\mu }\)–\(Q_{9V(10A),\mu }^{\prime }\) and \(Q_{9V(10A),e}\)–\(Q_{9V(10A),e}^{\prime }\), in these scenarios – in spite of the \(R_{K}\) update – the presence of righthanded currents has an overall marginal phenomenological impact. These conclusions are corroborated by the values of \(\varDelta IC\), slightly penalizing these scenarios in comparison with the study case (ii): a marginal improvement in the description of data is indeed obtained at the cost of model complexity in the fit. We eventually point out that the prediction for the longitudinal and transverse LFUV ratios are quite similar within these NP cases, yielding in particular \(R^{\text {T}}_{K^{*},\phi }[1.1,6] \simeq 0.8\).
In the spirit of studying the interplay between lefthanded and righthanded currents in the SMEFT framework, one may investigate also the viability of the above scenarios replacing the role carried out by \(O_{2223}^{LQ}\) with the one of \(O_{2322}^{Qe}\). However, Eq. (19) implies that the 2D scenario \((C_{2322}^{Qe}, C_{2223}^{ed})\) cannot explain the measured value of \(R_K\), since both coefficients contribute to the ratio with upward shifts, in contrast with what is required to account for the experimental data. On the other hand, considering \(C_{2223}^{Ld}\) as the NP term responsible of effects stemming from righthanded currents, positive solutions for this coefficient produce downward shifts in \(R_K\), potentially making the \((C_{2322}^{Qe}, C_{2223}^{Ld})\) scenario a viable solution for this LFUV ratio anomaly, see right panel in Fig. 1. However, as shown e.g. in Ref. [162], downward shifts in \(R_K\) induced by \(C_{2223}^{Ld}\) correspond to upward shifts in \(R_{K^*}\): therefore, since \(C_{2322}^{Qe}\) always contributes positively to this second ratio as well, also this second scenario cannot be considered viable in order to simultaneously address the anomalies in the two LFUV ratios.
Similar results are obtained in the last set of 2D scenarios, involving NP effects in electron channel described by means of the SMEFT formalism, namely \((C_{1123}^{LQ}, C_{1123}^{ed})\), \((C_{1123}^{LQ}, C_{1123}^{Ld})\), \((C_{2311}^{Qe}, C_{1123}^{ed})\) and \((C_{2311}^{Qe}\), \(C_{1123}^{Ld})\). It is interesting to note that, contrarily to what observed for the corresponding muonic case, both scenarios involving the operator \(O_{2311}^{Qe}\) are here allowed, due to the opposite direction of the contribution induced by such operator in the electron sector as shown in Eq. (19). Once again, addressing the information stemming from the angular dataset for the \(B \rightarrow K^*\mu ^+\mu ^\) decay requires these scenarios to be considered only in the PDD approach. The results for these fits, reported in Fig. 8, show a good description of \(R_K\) and \(R_{K^*}\) in all the considered cases. However, once again the \(\varDelta IC\) values reported in Table 2 imply that none of these models is favoured in comparison with the scenarios featuring NP effects in \(O_{2223}^{LQ}\).
We conclude this section briefly discussing the case where all the SMEFT operators are inspected all together. Indeed, the experimental dataset at hand allows us to perform a fit for NP effects present in all the 12 treelevel SMEFT operators, switching on simultaneously the following effective couplings: \(C_{\ell \ell 23}^{LQ}\), \(C_{23\ell \ell }^{Qe}\), \(C_{\ell \ell 23}^{Ld}\), \(C_{\ell \ell 23}^{ed}\), \(C_{\ell \ell 23}^{LedQ}\) and \(C_{\ell \ell 23}^{\prime LedQ}\), with \(\ell =\{1,2\}\). For the sake of completeness, in this scenario we also include scalar operators, particularly constrained by the available experimental information on \(B_{s} \rightarrow \ell ^{+} \ell ^{}\). We report the results of our fit in the PMD and PDD approaches in Fig. 9. Most importantly, we observe that in both approaches \(C_{2223}^{LQ}\) is found to be different from 0: at the \(\sim 6\,\sigma \) level in the PMD approach, at more than \(3\,\sigma \) in the PDD one. For the PMD framework we also note that NP effects in \(O_{2322}^{Qe}\) are singled out at the \(\sim 5\sigma \) level. These findings pretty much reflect what already outlined from Table 2, where the preferred scenario in the PDD approach is indeed the one featuring only \(C_{2223}^{LQ}\), while in the case of a more aggressive approach to QCD power corrections one needs to require also the presence of \(C_{2322}^{Qe}\) in order to accomplish an overall good description of data within the SMEFT. It is finally worth pointing out that the results of key observables as longitudinal and transverse LFUV ratio reported in Table 4 are here compatible with \(R^{\text {T,L}}_{K^{*},\phi }[1.1,6] \simeq 0.7\) within \(1\sigma \) errors.
5 Conclusions

in the considered “WET scenarios”, i.e. the cases where NP contributions do not necessarily stem a priori from \(SU(2)_{L} \otimes U(1)_{Y}\) gaugeinvariant operators at high energies, a preference for NP coupled to both lefthanded quark currents with vector muon coupling and to righthanded quark currents with axial muon coupling stands out regardless of the treatment of hadronic uncertainties;

in the instance of “SMEFT scenarios”, namely when NP effects are explicitly correlated by \(SU(2)_{L} \otimes U(1)_{Y}\) gauge invariance in the UV, several distinct cases are able to address present experimental information depending on the treatment of hadronic effects undertaken; aggressive estimates of hadronic uncertainties point to the simultaneous presence of lefthanded quark and muon couplings and lefthanded quark and righthanded muon couplings; a more conservative analysis leaves room for a broader set of scenarios, including the case of the single purely lefthanded operator with muon coupling;

LFUV effects in the electron sector provide a good description of current \(R_{K^{(*)}}\) measurements, but an overall satisfactory description of experimental results can be obtained only within a conservative approach to QCD effects; within this framework, these NP scenarios are not favoured over ones featuring muon couplings.
Footnotes
 1.
The scale \(\mu _b\) is here set by the scale at which form factors have been computed [178].
 2.
In the present work, while we treat the SM short distance with all available quantum corrections included, for the NP WCs we neglect the running induced by gauge couplings. Consequently, they stay constant from the scale they have been generated, with the notable exception of the SMEFT contributions arsing only at oneloop via RGE, see Sect. 2.2 and Appendix B.
 3.
Here we do not include the negligible contributions to \(C^{(')}_{S,P}\) from the SM for clarity.
 4.
Note that this statement is accurate as long as NP effects do not feed any of the WCs in Eq. (5).
 5.
We are not going to take into account the SMEFT contributions to the CKM parameters recently worked out in [198], since they cannot accommodate for LFUV effects.
 6.
Dimensionsix operators made of Higgs doublets and quark bilinears should also appear [43], but yield a lepton flavour universal contribution. They are severely constrained both by EW and Higgs data, see [199, 200], and by \(\varDelta F= 2\) measurements [197]. They will not be further considered here.
 7.
Within the SMEFT, they cannot simultaneously explain the \(R_{K}\) anomaly as well.
 8.
It is interesting to perform a SM global fit in order to have reference values for the \(IC \) to compare with. The fits yield an \(IC \) of 193 for the PDD approach, and 215 for the PMD one. Recalling that models with smaller values for the IC are preferred, the PDD approach provides a better SM fit compared to the PMD one, since anomalies in the angular analysis of \(B \rightarrow K^* \mu ^+ \mu ^\) can be accommodated through larger longdistance contributions, see Ref. [31].
 9.
 10.
In [166] we were powerexpanding \(h_\lambda \) correlators and enforcing the numerical results obtained from Ref. [23] in the whole largerecoil region as theory weights in the likelihood. Our new procedure for the PMD approach allows now to adopt the outcome of Ref. [23] genuinely as a set of flat priors.
 11.
Nevertheless, we have tested explicitly for the case of the scenario involving \(C_{9, \mu }^{NP}\) that introducing the equivalent of the PDD approach also for the \(B \rightarrow K\) channel does not have a relevant impact on the results of our fit.
 12.
Electron LFUV couplings arising from a \(Z'\) mediator may be also probed by atomicphysics data [222].
Notes
Acknowledgements
We wish to acknowledge Jorge de Blas, Julian Heeck and Marco Nardecchia for insightful discussions. The work of M.F. is supported by the MINECO grant FPA201676005C21P and by Maria de Maetzu program grant MDM20140367 of ICCUB and 2017 SGR 929. The work of M.V. is supported by the NSF Grant no. PHY1620638. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (Grant agreement n\(^o\) 772369). M.C., M.F. and M.V. are grateful to the Mainz Institute for Theoretical Physics (MITP) for its hospitality and its partial support during the completion of this work.
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