More Indications for Lepton Nonuniversality in $b \to s \ell^+ \ell^-$

Recently the LHCb collaboration has confirmed the evidence for lepton flavour nonuniversality at the $3.1\sigma$ level via an updated measurement of $R_K$. In this work we analyse this evidence within a model-independent approach. We make projections for future measurements which indicate that LHCb will be in the position to discover lepton nonuniversality with the Run 3 data in a single observable. We analyse other ratios based on our analysis of the present measurements of the ratios $R_{K^{(*)}}$ and analyse if they are able to differentiate between various new physics options within the effective field theory at present or in the near future. We also compare the present deviations in the ratios with NP indications in the angular observables of exclusive $b \to s \ell\ell$ transitions. Finally, we update our global analysis considering all $b \to s \ell\ell$ observables altogether, including a 20-parameter fit in connection of a Wilks' test.


Introduction
Ever since the measurement of the full angular observables of the exclusive B → K * µ + µ − decay with 1 fb −1 data by LHCb [1] which indicated New Physics (NP) in C µ 9 [2][3][4][5][6], rare b → s observables have been showing the strongest hints for NP. Updated measurements of the B → K * µ + µ − angular observables by the LHCb experiment with 3 and 4.7 fb −1 data [7,8] as well as measurements in other exclusive modes such as B s → φµ + µ − [9] have indicated signs of NP (with deviations of more than 2σ for some observables/bins). Although the SM predictions of some of the angular observables of the aforementioned modes have rather small uncertainties, in general the observables of the exclusive decays suffer from hadronic uncertainties, which often do not allow us to confidently separate possible NP effects from hadronic effects.
Another group of rare decays which have shown signs of NP are lepton flavour universality violating (LFUV) observables R K ( * ) ≡ BR(B +(0 * ) → K +(0 * ) µ + µ − )/BR(B +(0 * ) → K +(0 * ) e + e − ) [10] where the ratios of the branching fractions of muons compared to electrons are considered. The LFUV observables are theoretically very clean with SM uncertainties less than one percent 1 . The first tension in LFUV observables was measured for R K in the [1.1, 6.0] GeV 2 bin with the LHCb Run-1 data with 2.6σ significance [12]. This tension was confirmed with a signficance of 2.5σ when combining the Run 2 and the re-optimised Run 1 result [13] which had smaller uncertainty although the central value was measured to be closer to the SM prediction. LHCb found similar tensions at the level of 2.3 and 2.5σ for R K * in the two low-q 2 bins [0.045, 1.1] and [1.1, 6.0] GeV 2 , respectively [14]. These tensions within the theoretically clean ratios were shown to be rather consistent with the previously found tensions in the angular observables [15][16][17][18].
Among the non-LFUV observables, the BR(B s → µ + µ − ) is one of the cleanest observables giving a very good handle on the muon sector without involving the electron sector. Moreover, assuming no NP contributions due to scalar and pseudo-scalar operators (which is indicated by b → s + − global fits), the short-distance contribution to this decay is only via C µ( ) 10 . Recently LHCb has updated two of the clean observables, namely R K and BR(B s → µ + µ − ) using the complete dataset collected so far [19,20]. The LHCb experiment measures 3.1σ tension with the SM prediction for R K which compared to the previous result with 5 fb −1 data [13] has exactly the same central value but now due to smaller experimental uncertainties has an increased tension with the SM.
The new LHCb measurement of BR(B s → µ + µ − ) gives [19] In our fits, for the experimental value of BR(B s → µ + µ − ) we combine the recent LHCb update [19] with the ATLAS [21] and CMS [22] results considering a joint 2D likelihood as shown in Fig. 1. For the combined experimental measurement of the B s → µ + µ − decay we have In the next section we discuss the impact of the recent LHCb measurements on the fit to clean observables and investigate in detail the role of BR(B s → µ + µ − ) in the two-dimensional fit to LFUV observables. We also analyse the consistency of the clean observables and the rest of the b → s data regarding new physics. In section 3 we update our global b → s analysis in a multidimensional New Physics fit and consider the Wilks' test. In section 4, we present the future prospects of the clean observables, and make predictions for further ratios. The conclusions are given in section 5.

New physics analysis
In this section we consider the new physics analysis of R K ( * ) and BR(B s,d → µ + µ − ) and compare with other b → s data. The nonfactorisable power corrections in exclusive b → s decays are still not under control and have to be guesstimated, but promising approaches like the one in Ref. [23] may solve this problem in the near future (see Ref. [24] for recent progress). Thus, it is still reasonable to make separate analyses of the theoretically very clean ratios and the other b → s observables to crosscheck the consistency of the two data sets.
In Table 1 the results of the one operator fits to new physics using only the data on R K , R K * and B s,d → µµ are shown where for our analysis we have used the SuperIso public program [25]. Compared to our 2019 fits in Ref. [26] we see increased significances for the NP fits to the theoretically clean ratios R K and R K * . In general, the SM pull of the one operator fits are changed by more than 1σ. This is clearly due to the increased tension of the recent R K measurement with the SM. Our findings are in agreement with the recent model-independent analyses in Refs. [27][28][29][30]. For the rest of the b → s + − observables (except R K , R K * and B s,d → µµ), we also find larger SM pulls of the one operator fits compared to the analysis in Ref [26]. As shown in Table 2, there is a 1 − 2σ increase of the SM pull which is due to a new measurement of the angular observables of the B 0 → K * 0 µ + µ − decay (see Ref [31] for more details) and due to the inclusion of further observables such as the angular observables of the charged B + → K * + µ + µ − decay (see Ref. [32] ) and the branching ratio and angular observables of the Λ b → Λµ + µ − baryonic decays 2 . We emphasize again that the large SM-pulls beyond 5σ are based on guesstimates of the 10% nonfactorisable power corrections within the angular observables.
We also present two operator fits and analyse the role of the modes B s,d → µ + µ − : The one-dimensional fits are coherent in the preferred NP scenario whether the clean observables are considered for the fit, or the rest of the b → s data, indicating either a negative δC 9 or a positive δC 10 for both sets of observables. This coherence is however not trivial in the two operator fits. In the fit to the clean observables it is crucial to also consider BR(B s,d → µ + µ − ) in order to get the correct sign for {C 9 , C 10 } as shown in Fig. 2 where the best fit value of the fit to only LFUV ratios (the colored region of the right plot) indicates positive δC 10 as well as a positive δC 9 , and it is only after including BR(B s,d → µ + µ − ) (the black contour of the right plot) that similar best fit signs are obtained for the clean and the rest of the observables. This feature is due to the degeneracy that the ratios R K ( * ) have in C µ 9 and C µ 10 as can be seen by the circular contours of Fig. 3 where a positive δC µ 9 explains the data when simultaneously having a rather large value of δC µ 10 (not consistent with other b → s observables). The best fit value of the fit to only R K ( * ) is {C µ 9 , C µ 10 } = {1.3 ± 0.1, 4.0 ± 4.0} as indicated with the yellow diamond in Fig. 3      BR(B s,d → µ + µ − ) is also included the best fit value is {C µ 9 , C µ 10 } = {−0.2 ± 0.3, 0.5 ± 0.2} as indicated with a green cross.
It should be noted that the LHCb measured central value of R K * in the very low q 2 bin cannot be reached with any combination of NP in δC µ 9,10 since this bin is dominated by the photon contribution via the radiative Wilson coefficient C 7 .
The comparison of the fits to the two separate sets of observables confirm our observations in previous analyses. Within the one operator fits the C 10 -like Wilson coefficients play a significant role in the set of clean observables, but not in the complementary set. And the two operator fits in Figure 2 indicate that there is consistency between the two sets of observables at the 2σ level only.

Global fit
In the next step we show the global one and two operator fits in Table 3 and in Fig. 4 respectively using the b → s data altogether. In the one operator fit, the hierarchy of the preferred NP scenarios have remained the same as is in 2019, with the most prominent scenario indicating beyond the SM contributions to the muon Wilson coefficient δC µ 9 followed by δC µ LL and the universal (not lepton flavour-dependent) Wilson coefficient δC 9 . The significance of these scenarios have increased by more than 2σ compared to 2019 using the same 10% assumption for the power corrections. This increase is mainly due to the updated measurement of the B 0 → K * 0 µ + µ − angular observables as well as the measurement of B + → K * + µ + µ − and finally the recent LHCb measurements of R K and B s → µ + µ − where interestingly at each step the data has indicated the same preferred NP scenario (see also Refs. [31,32]  For the two operator fit the most prominent scenario still involves a universal NP contribution to δC 9 together with δC µ LL very slightly preferred over δC e LL followed by NP in the {δC e 9 , δC µ 9 } and {δC µ 10 , δC µ 9 } as given in the caption of Fig. 4. All the four mentioned scenarios have very similar Pull SM and all involve a NP contribution to δC (µ) 9 . Our results are in part consistent with, but also in part different from the results in the recent model-independent analyses in Refs. [27][28][29]33] (see also [34]). There are two obvious reasons which are responsible for larger discrepancies in the SM-pulls, namely different guesstimates of the nonfactorisable power corrections and different choices of the set of observables used in the fit.
In general NP contributions do not necessarily contribute to only one or two Wilson coefficients and could simultaneously involve several operator structures. In these cases the one and two operator fits lead to unnaturally large SM-pulls. Indeed, beyond simplified models, general NP scenarios contain a variety of new particles and new couplings. Therefore, taking a more agnostic approach to the behaviour of NP contributions, as first proposed in Refs. [35][36][37] we make a 20-dimensional fit, varying all the relevant b → s Wilson coefficients, thus, considering the most general description of NP effects in the b → s channel. We also established criteria to identify possible insensitive parameters and flat directions regarding NP. We note that our method avoids any look-elsewhere effect by starting with the most general description of possible NP effects and by eliminating insensitive parameters and flat directions based on the fit and not based on data. For an alternative approach to include the look-elsewhere effect see Ref. [38].
The results in Table 4 show that compared to our previous analyses [26,35] we find now that the fit constrains all four parameters C e Q1 , C e Q2 .C e Q1 , C e Q2 which previously were shown to be undetermined due to their large uncertainties in the previous analyses and negligible impact on the fit (resulting in the number of effective degrees of freedom to be 16). In Ref. [35] a criterium was presented to single out such   undetermined parameters 3 . But also in the present fit the parameters C e 10 and C e 10 have larger uncertainties and can be shown to have a small impact on the fit. In addition one finds that there is degeneracy in the sense that the fit constrains the difference of these two WCs only, so for both WCs large values are possible. Removing one of the two WCs from the fit one finds the other one is well-constrained and does not effectively change the χ 2 . Therefore we have 19 effective degrees of freedom.  Table 5: PullSM of 1, 2, 6, 10 and 20 dimensional fit. The "All non-primed WC" includes in addition to the previous row, the scalar and pseudoscalar Wilson coefficients. The last row also includes the chirality-flipped counterparts of the Wilson coefficients. In the last column the significance of improvement of the fit compared to the scenario of the previous row is given. The number in parentheses corresponds to the effective degrees of freedom (see the text for further details).
The Pull SM of the 20-parameter fit has increased by more than 2σ compared to the 2019 results. If we consider the fit with the 19 effective parameters we currently find a SM pull of 5.6σ.
The Wilks' test allows us to estimate the impact of the various parameters even further. The likelihood ratio test via Wilks' theorem enables us to estimate the significance of adding Wilson coefficients into a fit when one goes from one nested scenario to a more general one. In our previous analysis in Ref. [35] we found that adding Wilson coefficients to the "C µ 9 only" scenario was improving the fit only marginally. The Wilks' test with the present data shows that adding C µ 10 and C e 9 , C e 10 and also C 7 and C 8 improves the fit significantly and establishes the importance of these fit parameters (see Table 5). This can be explained by the fact that to a great degree, the tension and its increase compared to our previous analysis in Ref. [35] is due to the updated LFUV ratio R K which can be described equally well by NP contributions to the electron and muon sectors. Furthermore, there is now more data on observables with electrons in the final state.

Future prospects and predictions for other ratios
Upgrades of the LHCb experiment are planned. The first upgrade will lead to a total integrated luminosity of 50 fb −1 . A second upgrade at a high-luminosity LHC will lead to an integrated luminosity of 300 fb −1 . We also use a third intermediate benchmark with an integrated luminosity of 18 fb −1 to analyse the future prospects of the clean observables R K , R K * and BR(B s → µ + µ − ).
Our estimates of the future systematical uncertainties are based on the following considerations. From Table 2 in [20], the efficiency ratio between the electron mode and the muon mode is approximately onethird. The LHCb Upgrade will replace the hardware trigger by a software trigger which is expected to yield electron efficiencies closer to muon efficiencies (see Table 2 in [39]). We assume the efficiency ratio in the LHCb Upgrade grows from one-third to ∼ 60%. The ultimate systematic uncertainty for R K is expected to be ∼ 1% [40], and we assume that a similar ultimate systematic could hence be achieved for R K * (also in line with [41]). For the B 0 s → µ + µ − branching fraction, two important systematic sources depend on external information: the value of the b-quark hadronisation fractions f d /f s , and the branching fraction of the B + → J/ψ(→ µ + µ − )K + decay. Hence, an irreducible systematic of ∼ 4% is assumed in our approach. We also assume that ATLAS, CMS, and LHCb will keep relative weights in the B 0 s → µ + µ − branching fraction world average similar to the ones they currently have. Considering the decrease in experimental uncertainties as described above we investigate the NP fits to the clean observables. Keeping the experimental central values as what they are currently do not give acceptable fits which is partly due to the different preferred NP scenarios for R K and R K * in the [1.1, 6] GeV 2 bin and partly due to the rather small value of R K * in the [0.045, 1.1] GeV 2 bin which cannot be reached with NP in the preferred NP scenarios. Instead we make a different but similarly strong assumption that future experimental results are in agreement with one of the current NP scenarios from the fit to clean observables.
However, it should be noted that the significance is rather strongly dependent on the presumed systematic uncertainties as well as the considered scenario. This can be seen in Fig. 5 where Pull SM Pull SM with R K ( * ) and BR(B s → µ + µ − ) prospects LHCb lum. 18 Table 6: Predictions of PullSM for the fit to δC µ 9 , δC µ 10 and δC µ LL (as given in the right panel of Table 1) for the LHCb upgrade scenarios with 18, 50 and 300 fb −1 luminosity collected.  Table 1) remains unchanged. The red, green and blue band correspond to R for each LFUV ratios is individually shown for the C µ 9 and C µ 10 scenarios with different assumptions on the systematic uncertainties. For the C µ 9 case, R K can individually reach 5σ significance at ∼ 16 fb −1 luminosity. On the one hand for the C µ 10 scenario, with the ultimate systematic uncertainty for R K * ([1.1, 6]) it gives 5σ Pull SM at ∼ 13 fb −1 , however, assuming the current systematic uncertainty remains, it does not reach 3σ significance. On the other hand for the same C µ 10 scenario, with R K , there is 5σ significance with ∼ 20 fb −1 luminosity and it is much less dependent on the assumption on the systematic error. In both scenarios R K * ([0.045, 1.1]) does not give a large Pull SM as it is mostly dominated by C 7 and with the considered NP scenarios the predicted value for this bin (as given in the next paragraph) are very close to the SM prediction, 0.906 ± 0.028.
Nonetheless, a single LFUV observable cannot individually pinpoint the correct NP scenario. For example, the predicted 68% confidence interval with 9 fb −1 for the (very) low bin of R K * within the C µ 9 scenario is given by ([0.897, 0 Table 1 this problem becomes even more pronounced. The analytical dependence of the ratios on the NP-WCs given in Ref. [42] explain this feature. So it is expected that this feature stays valid also in future scenarios. In Table 7 we give the 68% confidence interval predictions for other LFUV ratios of muons in the final state over electrons, assuming the various NP fits to R K ( * ) and BR(B s → µ + µ − ) as given in the right panel of Table 1. There are a number of the ratios which are able to discern among the various scenarios (see also [43]). From the first row of Table 7, R F L ([1.1, 6]) is predicted to have distinct intervals whether the considered scenario is NP in C µ 9 or C µ 10 , however there are still overlaps with other cases e.g. C e 9 which can be disentangled to some extent by considering further observables such as R S5 ([1.1, 6]).
In Table 8 and 9 we give the 1σ range predictions of these LFUV observables for the 18 and 50 fb −1 luminosity benchmarks, respectively where several of the observables give more distinct predictions for the various NP scenarios.
In Table 7, some of the observables such as R AFB ([1.1, 6]) have a rather large uncertainty which is due to zero-crossings. In such cases, it is more suitable to consider observable differences [44]. Similar observables are also defined for the optimised P ( ) i observables in [45]. In Appendix A, we give the prediction for the alternative set of observables in Tables 10, 11 and 12 Table 7 for more details.

Conclusions
The current experimental data on b → s transitions show deviations in several observables with respect to the Standard Model predictions. The latest LHCb update of the leptonic decay BR(B s → µ + µ − ) and the lepton flavour violating ratio R K have further strengthened the New Physics description of the so-called B-anomalies which when taken together with the previously measured anomalies in the two bins of R K * results in more than 4σ significance.  Considering all available observables of b → s processes, the significance of the improved description of the data by New Physics contributions becomes even higher, this is however reliant on the assumptions made on the size of the not well-known power corrections in a number of observables of the exclusive B → K ( * ) and B s → φ decays. In order to have an unbiased determination of the structure of New Physics contributions, we considered a 20-dimensional fit where all the relevant b → s operators are taken in account, still finding a large Pull SM . Interestingly, while in our previous analysis there was no significant indication of preference for going beyond one or two operator fits, we now see such an indication for considering simultaneously electron and muon contributions.
Assuming any of the favoured NP descriptions of the lepton-flavour universality violating observables remain, we show that New Physics can be established with more than 5σ significance already with 18 fb −1 of integrated luminosity. However, the preferred scenario, in general cannot be determined by only considering the R K ( * ) observables. To disentangle the preferred New Physics scenario we also give predictions for further ratios with data already gathered by LHCb as well as the projected data with 18 and 50 fb −1 luminosity.

A Predictions for further LFUV observables
Predictions with best fit values of "clean" observables Obs. C