What and can tell us about New Physics in transitions?
Abstract
The deviations with respect to the Standard Model that are currently observed in transitions, or anomalies, can be interpreted in terms of different New Physics (NP) scenarios within a modelindependent effective approach. We identify a set of internal tensions of the fit that require further attention and whose theoretical or experimental nature could be determined with more data. In this landscape of NP, we discuss possible ways to discriminate among favoured NP hypotheses in the short term thanks to current and forthcoming observables. While the update of will be an important milestone on the way to disentangle the type of NP we may be observing (LeptonFlavour Universality Violating and/or Lepton Flavour Universal), additional observables, in particular , turn out to be central to determine which NP hypothesis should be preferred. We also analyse the preferences shown by the current global fit concerning various NP hypotheses, using two different tools: the behaviour of the pulls of individual observables under NP scenarios and the directions favoured by approximate quadratic parametrisations of the observables in terms of Wilson coefficients.
LPTOrsay1904
Energies (IFAE), The Barcelona Institute of Science and Technology, Campus UAB, 08193 Bellaterra (Barcelona).
Laboratoire de Physique Théorique, UMR 8627, CNRS, Univ. ParisSud, Université ParisSaclay, 91405 Orsay Cedex, France
1 Introduction and motivation
After the LHCb collaboration announced in April 2017 the measurement of the LeptonFlavour Universality Violating (LFUV) observable , the combined analysis of 175 LFUV and leptonflavour dependent (LFD) observables performed in Ref. Capdevila:2017bsm () using a modelindependent approach showed that the Standard Model (SM) hypothesis is disfavoured compared to various hypotheses of New Physics (NP) contributions in decays, with pulls w.r.t. the SM ranging from 5.3 to 5.8. Similar results were obtained by other groups using different treatments of hadronic uncertainties and sets of observables DescotesGenon:2015uva (); Altmannshofer:2017fio (); Altmannshofer:2017yso (); DAmico:2017mtc (); Geng:2017svp (); Arbey:2018ics (); Ciuchini:2017mik (); Hiller:2017bzc (); Alok:2017sui (). These modelindependent analyses constrain NP scenarios expressed as contributions to the shortdistance Wilson coefficients in the effective Hamiltonian approach for transitions.
The first point to address is obviously whether NP has been discovered, but once this is established, it will prove important to determine the specific pattern of NP discovered. Indeed, even if the amount of data obtained up to now for makes sophisticated global fits to several Wilson coefficients possible Capdevila:2017bsm (); DescotesGenon:2015uva (); Altmannshofer:2017fio (); Altmannshofer:2017yso (); DAmico:2017mtc (); Geng:2017svp (); Arbey:2018ics (); Ciuchini:2017mik (); Hiller:2017bzc (); Alok:2017sui (); Alguero:2018nvb (); Kumar:2019qbv (), the outcome is still not conclusive enough to draw definite conclusions about the actual pattern of NP. Disentangling the realized pattern is an essential guide to build NP models in agreement with these observations. It is therefore usual to limit NP contributions to a few Wilson coefficients, that from now on we will refer as hypotheses, and to build NP models in agreement with these assumptions of the global fits.
Most scenarios discussed in the literature assumed that there is NP in muons only, i.e. the LFUVNP contributions come from allowing the presence of NP in the muon channel and not in the electron one (or it is considered small). Two particularly interesting scenarios have emerged, namely NP in or in , with a larger significance for the first and a smaller one for the latter in Ref. Capdevila:2017bsm (). On the contrary, we also found that a fit restricted to a subset of mainly LFUV observables (15 observables in total) exhibits a marginal preference for the scenario compared to . In addition, the best fit value of the scenario with NP only in is rather different when considering all observables or the LFUV subset.
Recently, several works allowed for NP also in the electron channel, but no particular structure was envisaged from these fits by simply taking some of the electronic Wilson coefficients different from zero Ciuchini:2017mik (); Hurth:2017hxg (); Kumar:2019qbv (). However, in a recent article Alguero:2018nvb (), we allowed the possibility of a specific structure, namely, that the transitions get a common Lepton Flavour Universal (LFU) NP contribution for all charged leptons (electrons, muons and taus). This permitted us to identify new favoured NP hypotheses. This idea was implemented by allowing two NP contributions inside the semileptonic Wilson coefficients:
(1) 
with and where stands for LFUVNP and for LFUNP contributions. We distinguished the two contributions by imposing that . It is important at this point to emphasize the difference between simply allowing the presence of NP also in electrons or allowing the existence of two different kinds of NP contributions (LFU and LFUV). The case of simply allowing NP in the electron channel has been discussed quite extensively in Ref. Hurth:2017hxg () (see also Ref. Ciuchini:2017mik () for a smaller subset of scenarios and without including lowrecoil observables). However, our approach of distinguishing LFU and LFUVNP structures provides new ideas to model building and extends the possible interpretations of the current fits. Performing the fits with this new setting, we obtained our previous results in Ref. Capdevila:2017bsm () but also new scenarios different from Refs. Hurth:2017hxg (); Ciuchini:2017mik (). This can be seen by translating LFU and LFUV contributions into NP contributions to muons and electrons (leaving aside at this stage)
(2) 
This seemingly innocuous redefinition yields interesting consequences, as discussed in Ref. Alguero:2018nvb (). It opens interesting perspectives to explain with different mechanisms the anomalies coming purely from the muon sector (like ) and the ones describing the violation of lepton flavour universality (like ). It may also explain the difference between the best fit point of the scenario in Ref. Capdevila:2017bsm () if one considers all available observables or only LFUV observables, as the former will correspond to and the latter to .
When translated from one language to the other, the most interesting scenarios in Ref. Alguero:2018nvb () become:
(3)  
(4)  
(5) 
This additional NP component in explains the improved significance of the last scenario above (Hyp. V) compared to the favoured scenarios assuming NP in only in Ref. Capdevila:2017bsm (), which can be recast as:
(6)  
(7) 
Let us notice that this approach is also different from all the analyses including NP in electrons Hurth:2017hxg (); Ciuchini:2017mik (); Kumar:2019qbv () where the muonic constrain does not receive any electronic contribution. For completeness, the remaining scenarios considered in Ref. Alguero:2018nvb () were:
(8)  
(9)  
(10) 
We also introduce the additional scenario
(11) 
and the LFUV twodimensional scenario
(12) 
We have named only a subset of scenarios for further reference, with hypotheses I to IV being purely LFUV NP and V to VII allowing both LFUV and LFU NP.
In this situation, it becomes clear that new data will be instrumental to disentangle the different hypotheses. The goal of the present article is to scrutinize the results of the fit from a different perspective to prepare the next step, i.e. to discriminate the most relevant NP scenario among the ones already favoured, complementing our two previous works, Refs. Capdevila:2017bsm () and Capdevila:2017ert (). Currently, the most significant patterns identified exhibit a pull w.r.t the SM very close to each other (within a range of half a ). We explore strategies to disentangle different scenarios and to identify the impact of a precise measurement of . We then combine information on and in order to illustrate that by itself will not be sufficient to disentangle clearly one or a small subset of scenarios, but that a combination of and can be useful, depending on the (future) measured value ^{1}^{1}1Up to now only the Belle experiment has been able to perform a measurement of , leading to Wehle:2016yoi ()..
In section 2 we discuss the inner tensions of the fit in order to point those observables where further experimental or theoretical work would be required. In section 3 we explore how a forthcoming precise measurement of can disentangle or disfavour scenarios assuming that the statistical error is reduced and the central value stays within 2 of its present value. We analyse it considering two different fits, either with all observables or only the LFUV subset. We also discuss the impact of a measurement of in relation with its possible measurement by Belle II and LHCb. We then discuss the structure of the current fits, looking more closely at the deviations of some observables in section 4, focusing on the change in their pulls depending on the NP scenario considered. In section 5 we discuss the structure of the observables in terms of their Wilson coefficients to determine their sensitivities and the directions preferred by each of the anomalies, before drawing our conclusions.
2 Inner tensions of the global fit
In Ref. Capdevila:2017bsm (), we saw that different NP scenarios involving led to a much better description of the data than the SM, with fits reaching pvalues around 6070% (the SM being around 10%) and providing pulls with respect to the SM above 5. The overall agreement is thus already very good within these NP scenarios and from a purely statistical point of view, it should be expected that these fits exhibit slight tensions. It is however interesting to look at these remaining tensions in more detail in order to determine where statistical fluctuations may be reduced with more data or where improved measurements might help to lift the degeneracy among NP scenarios. We focus on three main tensions that we consider particularly relevant in the current global fit.
2.1 in the first bin
A first tension related to occurs in the global fit and it proves interesting to consider both and in order to understand its nature (see Fig. 1).
Let us first consider the second bin (from 1 to 6 GeV) for . Even though the deficit could be consistent with an excess in the electron channel with respect to the muon one, the study of the corresponding bins of points towards a deficit of muons. The mechanism that explains the deviation with respect to the SM in the long second bin of is consistent with all the deviations that have been observed in other channels and different invariant dilepton mass square regions.
The situation is different for the first bin of , where the is clearly compatible with the SM (see Fig. 1). An excess in the electron channel would then be needed in order to explain the observed deficit in . This difference of mechanism between the first and the second bins of can be understood in two ways: i) a specific NP effect Altmannshofer:2017bsz () localised at very low and able to compete with the dominant Wilson coefficient (well determined to be in agreement with the SM expectations from ) Capdevila:2017bsm (); Misiak:2006zs (); Misiak:2006ab (); DescotesGenon:2011yn (); Asner:2010qj (); ii) some experimental issue in measuring dielectron pairs at very small invariant mass, close to the photon pole. It would be very interesting that LHCb keep on their efforts to understand the systematics in this bin.
Another approach to slightly reduce the tension between data and SM in the first bin of through a NP explanation consists in including NP contributions to the channel, in particular, considering righthanded currents affecting electrons, as discussed in Ref. Kumar:2019qbv (). In the scenarios S8S11 (using the notation of Ref. Kumar:2019qbv ()) the prediction of is found to be within range of the current measurement. This could open a new window to explore the existence of righthanded currents and to explain some of the tensions found, even though more data is required in order to be conclusive.
2.2 versus
Another tension in the fit concerns the branching ratio for , in particular when compared with the related decay .
The prediction for the branching ratio involves hadronic form factors to be determined using different theoretical approaches, depending on the dilepton invariant mass region analysed: at large recoil, one can use lightcone sumrules based on lightmeson distribution amplitudes Straub:2015ica (), while lattice form factors are available at low recoil. Due to the difficulty to assess precisely the uncertainties attached to lightcone sum rules, we perform our computation using more conservative results from lightcone sum rules based on meson distribution amplitudes Khodjamirian:2010vf () with conservative error estimates, exploiting QCD factorisation to restore correlations that were not available in Ref. Khodjamirian:2010vf (). We checked that our results are compatible with those obtained in Ref. Straub:2015ica () and that the two approaches yield very similar results for the fits Capdevila:2017bsm (); DescotesGenon:2015uva (); Altmannshofer:2017yso (); Altmannshofer:2017fio ().
A recent update of these form factors is available in Ref. Gubernari:2018wyi () using the same approach as Ref. Khodjamirian:2010vf (), adding corrections from higher twists and providing correlations. We will update our results accordingly in a coming publication, but we do not expect very significant changes for the present article, based on our previous studies Capdevila:2017ert (). For instance, we checked that even if a large reduction of 50% is achieved on the error of the form factors, the resulting uncertainty of key optimized observables like is minor (it would imply a reduction from 10% to 8% for the anomalous bins of ). On the contrary, a large impact is observed in unprotected observables like branching ratios or observables. As our fit is driven by the optimized observables, we expect only minor changes in the outcome of the fits.
Contrary to the case of , there are no computations available using the Bmeson lightcone sum rules of Refs. Khodjamirian:2010vf (); Gubernari:2018wyi () for , and one must rely on the estimates given in Ref. Straub:2015ica (). One can see in Fig. 2 that at low recoil, where lattice form factors are used, the prediction for is expected to be slightly larger than and indeed data (with large error bars) follows the same trend. On the contrary, in the largerecoil region where the lightcone sum rules results of Ref. Straub:2015ica () are used, the SM predictions lead to a larger value for than for . Surprisingly, data shows the opposite trend, which may come from a statistical fluctuation of the data leading to an inversion of the experimental measurements of both modes at large recoil. Alternatively, this issue may signal a problem in the theoretical prediction of the form factors of Ref. Straub:2015ica (). Firstly, these predictions are obtained by combining results in different kinematic regions (lightcone sum rules and lattice QCD) which do not fully agree with each other when they are extrapolated: the fit to a common parametrisation over the whole kinematic space leads to a fit with uncertainties that may be artificially small due to these incompatibilities of the inputs. Moreover, the choice of the parametrisation Khodjamirian:2010vf (); Straub:2015ica () used to describe the form factors over the whole kinematic range has interesting properties of convergence, but it may in some cases lead to potential unitarity violations GonzalezSolis:2018ooo ().
Finally, another issue that specifically affects is the  mixing. As it is well known,  mixing implies that the time evolution of the meson before its decay involves two mass states with different widths that are linear combinations of the flavour states and . The current measurements performed at LHCb are integrated over time, and the neat effect of the evolution between the two mass states is a correction of in the relation between the theoretical computation of the branching ratio and its measurement DescotesGenon:2011pb (); DeBruyn:2012wj (); DeBruyn:2012jp (); DescotesGenon:2015hea (). This effect is taken into account in the global fit DescotesGenon:2015uva () as an additional source of uncertainty for the theoretical estimate of the branching ratios.
The experimental efficiencies should also be corrected for this effect, which depend on the CPasymmetry that can also be affected by NP contributions. It should thus be kept free within a large range in the absence of measurements. Neglecting this effect and assuming a SM value for this asymmetry may lead to an underestimation of some systematics on the efficiencies. For instance, Ref. Dettori:2018bwt () showed that this issue can lead to an additional systematic effect of in the systematics. The impact on efficiencies from NP effects was indeed considered in Ref. Aaij:2015esa () for by varying in the underlying physics model used to compute signal efficiencies, leading to a much smaller effect in this case (of a few percent, in line with backoftheenvelope estimates).
2.3 Tensions between large and low recoil in angular observables
We discuss for the first time here a rather different type of tension, concerning the angular observables at large and low recoil. On the one hand, we observe that branching ratios exhibit the same discrepancy pattern between theory and experiment at low and large recoil ^{2}^{2}2This is true for all modes, apart from the decay , where the experimental errors at low recoil are very large and the normalisation chosen prevents further interpretation Aaij:2015xza (); Aaij:2018gwm ().. On the other hand, the current deviations at LHCb in require NP contributions with opposite sign in the two kinematic regions. Indeed, the pull between the SM value and the LHCb experimental measurement in has the opposite sign (albeit the significance is only 1.2) w.r.t. its largerecoil bins, in particular and . This very slight tension is not there in the case of the Belle data where samesign deviations are observed, even though the error bars are rather large in this case.
For the purposes of illustration, let us consider the NP scenario where there is no LFU contribution and NP occurs only in and . This is illustrated in Fig. 3 where the constraints for these observables (as well as other relevant observables that will be listed below) are shown at 68.3% (left) and 95% (right) CL. One can notice their milder sensitivity to . (blue region) would prefer a negative while (green region) would favour a positive at 68.3% CL.
Black dots indicate the particular solutions and corresponding to the bestfit points of the 1D favoured scenarios in Ref. Capdevila:2017bsm () (hypotheses I and II of the present article). We also indicate the constraints from , , and , since we believe that they are representative of the set of observables driving our global fit ^{3}^{3}3 and observables are known to behave in a more SMlike way than the ones selected here, thus providing weaker constraints.. The former pair of observables (, ) has a large overlap region compatible with the SM while the latter one (, ) overlaps far from the SM point. While and strongly constrain NP solutions, the bins are weakly constraining. Finally, the yellow region in the right panel in Fig. 3 is the overlap of the regions from the five observables obtained after considering the data regions at 95% CL.
In summary, an interesting tension between low and largerecoil regions for is observed at the 2sigma level, favouring contributions of different signs in the two kinematic regions. Although not statistically significant, this inner tension seems to require either different sources of NP or a shift in the data once more statistics is added.
3 Potential of (and ) to disentangle NP hypotheses
In this section we discuss the potential impact of the prospective measurements of and on the global fits in order to distinguish NP hypotheses. We perform the following illustrative exercise: we vary the experimental values of and within suitable ranges, and we perform fits according to these values taken as actual measurements. First only is allowed to vary before we consider the combined impact of and . The ‘pseudodata’ for takes into account the increased statistics available for this observable, and we assume a reduced error by a factor 1.8 w.r.t. the present experimental statistical error following the prospects announced in Ref. miteshtalk (). Specifically for this exercise we take as the upper prospective experimental error for and the lower error .
For each fit (corresponding to a given hypothesis and set of data), both the pull of the hypothesis w.r.t. the SM () and the bestfitpoint (b.f.p) are computed, which we plot as functions of either or . Before discussing the results of our analysis, we first state our assumptions:

We consider two different kind of fits with different subsets of observables Capdevila:2017bsm (): on one side, the global fit (to all available observables) and on the other one, the LFUV fit, where only the observables measuring LFUV are included (plus constraints coming from radiative decays).

Any variation of the experimental value of must manifest itself in a change in the branching ratios and/or . Here we should consider two different cases:

Global Fit: When we consider a fit to all observables, we must add a second assumption. Taking into account the fact that the observed systematic deficit in the muonic branching ratios we assume that NP is purely of LFUV type and affects only muons and not electrons. This means that varying the value of in the global fits should correspond also to a variation of the branching ratio (whereas the electron mode is unaffected). This implies that for the global fits only scenarios measuring LFUV NP (hypotheses I to IV) can be considered in a consistent way.

LFUV Fit: On the contrary, the LFUV fits contain but no LFD branching ratios. We do not need to make any assumption on the changes in and , which opens the possibility of studying hypotheses with both LFU and LFUV NP contributions (hypotheses V to VII).


is freely varied within a range from its current experimental value. It represents a good compromise between a high coverage of the true value and a span compatible with our computational means. is varied within the range in order to ensure that we scan over values corresponding to the most relevant NP scenarios (see Fig. 2 of Ref. Alguero:2018nvb ()).

With the increased statistics available at Run 2, it will be possible for experiments to provide more precise determinations of key observables. Therefore, besides the reduction in the error of and the associated observable (in some of the scenarios as discussed above), we assume a guesstimated uncertainty of order 0.1 for .
The purpose of this analysis is not to provide precise determinations of the pull of the SM and the b.f.p.s for different values of and but rather to gain qualitative knowledge on how experimental measurements of these two observables will drive the analyses. This is particularly true for the b.f.p. plots that will provide only the central value but not the confidence intervals obtained for the NP contributions.
3.1 Global Fits
Figure 4 displays the outcome of the global fit with different experimental central values of under various NP hypotheses varied according to the procedure described above. The shaded vertical band in the plots of Figure 4 highlights the current experimental confidence interval for .
The left panel in Figure 4 illustrates the relevance of on the global fits. For all the NP scenarios considered, except for , we observe that their corresponding undergoes a variation from one end of the range of variation to the other. If we restrict the variation of to only , one can see differences of between the two extremes, as expected from the linearity of on seen in the plots.
The flatness of the under the hypothesis can be easily understood. The theoretical prediction of is insensitive to the value of , so that it remains constant and equal to 1 to a very high accuracy. Therefore the difference between the theoretical and experimental values of does not play any role in the minimisation of the function. As a consequence, the b.f.p. is determined using the other observables of the fit, regardless of the experimental value for (see the righthand side plot of Fig. 4), and the contribution of cancels in the statistic. This explains the observed flat curve for the , up to small variations linked to the numerical minimisation of the function.
The results in Figure 4 show that, for most of the values of scanned, it is not possible to fully disentangle all the NP scenarios, with the exception of , but it is possible to distinguish three regions:.

: The NP scenarios , and show similar pulls (), where the 2D scenario stands out as the scenario with the highest preference over the SM. On the other hand, sits at the level, with a difference with respect to the previous scenarios. Clearly, such small values for enhance the tensions between the SM and the experimental result, as reflected by the pulls and the b.f.p.s. which depart a lot from the SM.

: In this case, the new determination of is nominally close to its current experimental value (or slightly bigger) but with smaller errors. Therefore, the values for the b.f.p.s are numerically similar to the b.f.p.s reported in Ref. Capdevila:2017bsm (). The pulls of the SM are all found between and , and we can only distinguish from all the others except for values around . Among the three NP scenarios without righthanded coefficients, is the one with the highest (all being around of each other for most of this region).

: Here, most of the pulls decrease with respect to their present values, reaching between and . Even when , their significances get down to the range , apart from that remains at . Indeed, if a new measurement of is found in better agreement with its SM prediction, there is still an important number of other tensions (i.e. , and ) that require NP contributions in order to be explained. Interestingly, this case may favour the hypothesis with righthanded currents compared to other hypotheses.
Finally, we study the combined influence of and . The value of is varied as explained above and we repeat the analysis for three different values of : its current experimental value and the ends of its range. In all cases, we work under the hypothesis of LFUV NP affecting only muons, namely, the experimental value of and the associated error are modified as indicated at the beginning of this section. This implies that we restrict ourselves to hypotheses I to IV. The results are presented in Figs. 5 and 6. The first figure shows how the varies with different experimental values of and and the second shows how b.f.p.s are affected.
We observe that prefers small values of . For the hypotheses and ,, the bigger the greater the significance of the (as already observed in Fig. 2 of Ref. Alguero:2018nvb () and Fig. 6 of Ref. Capdevila:2017bsm ()). Except for , increasing the value of amounts to shifting the curves to the right, although the shift size depends on the particular hypothesis. In addition, we extract the following conclusions:

is not able to distinguish between the three NP hypotheses: , and , if the experimental value of is found to be below its current determination. However, it is possible to discriminate between these three scenarios and , since the latter has a lower by .

If stays at its current value and , then the measurement of allows one to disentangle and from . It is always possible to separate from the rest of scenarios except for .

In the case where the central value of is increased by , is clearly able to disentangle from all the other scenarios if .
Regarding the b.f.p.s (Fig. 6) and their behaviour with , we notice that the values of for the scenarios , and tend to cluster together.
In summary, if the average between the new and old measurement of is below its current value, it would marginally increase the preference of the global fit for hypotheses with , while a measurement above its Run1 measurement would favour NP mainly in or . Values of below its Run1 measurement would also clearly disfavour the hypothesis with respect to the other three hypotheses. The observable is an excellent candidate to separate from . In particular, values of bigger than 0.4 would disfavour the hypothesis compared to other hypotheses dominated by .
3.2 LFUV fits
It is also interesting to address the impact of the observables and on the LFUV fits, which currently include and from LHCb, the measurements of by the Belle collaboration, all the observables available, as well as and Iwasaki:2005sy (); Lees:2013nxa (); Aaij:2017vad (). We follow the same guidelines as for the global fits, with the only difference that now hypotheses with LFU NP are also allowed.
Certain features are common in the two series of fits. For instance, shows a constant behaviour and the linearity in the behaviour of the various pulls is still observed. Some remarks of this LFUV fit are in order:

When analysing the impact of (top left panel in Fig. 7), is now the hypothesis with the most significant (though very close to the other ones that cluster together), contrary to the global fits that tend to prefer . This was already observed in the LFUV fits performed in Ref. Capdevila:2017bsm (): scenarios with are quite efficient in explaining and (see Fig. 5 of Ref. Capdevila:2017bsm ()). All hypotheses (apart from ) yield pulls w.r.t. the SM with almost identical significance levels throughout the whole range of variation of , getting closer as deviates more and more from the SM. Hypotheses containing LFU NP are difficult to distinguish: in principle, LFUV observables contain cross terms that are products of LFU and LFUV NP contributions, but they are actually highly suppressed in all current LFUV observables. Let us add that the LFUV fits can yield pulls with very high significances at this level of precision in , moving from to when is varied.

If we assume that the experimental value of the new average for is below its present value and assess the impact of (top right panel in Fig. 7), we observe the same clear separation between the solution and all the other hypotheses already pointed out in the context of global fits. For , it is not possible to identify a predominant hypothesis nor to distinguish between hypotheses (apart from ). Small values of (close to zero or negative) favour as well as hypotheses with .

Keeping the central value of to its current value while scanning over (bottom left panel in Fig. 7), does not help in lifting the degeneracy in significance between the different hypotheses, specially for (). In this region is the preferred hypothesis but all of the pulls fall within less than . However, for , it is indeed possible to classify the hypotheses in two groups: on the one side, preferred by the fit, , and and, on the other side, all the other scenarios with which are disfavoured by the fit. In this case, stands out as the scenario with the highest .

Finally, we analyse the impact of assuming that the new average for lies above its current determination (bottom right panel in Fig. 7). If , there is no clear separation between hypotheses. On the contrary, separate the scenarios in two categories. On the one hand, all the scenarios where is the dominant fit parameter cluster together, with significances between and . On the other hand, all the solutions with show lower significances (by roughly ). In this context, becomes the solution with the highest significance, although rather marginally.
In summary LFUV fits lead us to draw similar conclusions to the ones extracted from the global fits. However they exhibit a stronger clustering of the pulls, especially if only is used to discriminate them. Moreover, if we take a given hypothesis with LFUVNP only, and consider hypotheses obtained by adding further LFUNP contributions, we obtain very similar pulls. Therefore, the only way to distinguish among different LFUNP hypotheses consists in performing the global fits and having access to both muonic and electronic branching ratios. The addition of allows for strategies that enable the discrimination of scenarios with a structure from scenarios with mainly NP in , in a similar way to the case of the global fits.
4 Pulls of individual observables
Discerning among different NP hypotheses using the significance of the proved difficult using the current set of data. In Sec. 3 we have discussed how new data could possibly improve the situation and allow us to disentangle the different scenarios. We discuss here a complementary approach based on analysing the current picture for each hypothesis using the correlated pull of individual observables.
We consider the individual pull of an observable, which can be written for the observable as:
(13) 
where and are the minimal values of the with and without the observables
(14) 
where and are the theoretical prediction and the experimental value for the observable respectively, and corresponds to the covariance matrix element .
This definition, already used in Refs. Lenz:2010gu (); Charles:2011va (); Lenz:2012az (); Charles:2016qtt (), differs from the definition often adopted e.g. in the context of electroweak precision observables :2005ema (): this naive pull is defined as the difference between the experimental value and the theoretical value at the b.f.p., normalised by the uncertainty. As discussed in Ref. demortier (), this definition should be refined. The pull of an observable can be considered as the assessment of the impact of the additional external constraint to the fit given by this observable, which must be assessed using a pull involving the central values and the uncertainties with and without the constraint given by the observable. It can be easily shown that the definition of Ref. demortier () is equivalent to our definition. Moreover, this definition follows the same approach of pulls as for the comparison of hypotheses and it can be expected to follow a normal law with zero mean and unit width. We stress that it includes correlations, so that it might have a different value from the naive pull in the presence of large correlations among observables (from experimental or theoretical nature).
In the following, we will compare the pulls of observables defined in Eq. (13) under the SM hypothesis and under the various NP hypotheses (I to VII) for all observables. However, we will focus mainly on the observables yielding larger than 1.5. These with respect to the SM are expected to be reduced under the various NP hypotheses, but some might remain (while the others disappear), providing interesting insights into the role of the various observables within a given NP hypothesis.
The observables under discussion are visualized in Fig. 8. In this figure, black squares represent the of the observable within the SM computed following Eq. (13), while coloured and empty shapes represent the of the same observable under different NP hypotheses.
4.1 Observables with a large within the Standard Model
Looking at Fig. 8, the first observable considered is : it is very sensitive to LFUV NP, it has a large with respect to the SM and it should be updated very soon. It has also a small in all the NP hypotheses considered except for Hyp. III. Remarkably, it has almost no correlation with any other observable in the fit^{4}^{4}4There is little theoretical correlation between and the rest of the observables as the hadronic form factors, which are the main source of correlated uncertainty among observables, cancel in . The experimental correlation between and the branching ratio is not public and therefore assumed to be zero here. We checked the lack of correlation between and the rest of the observables at the level of our covariance matrix. Capdevila:2018jhy () and, as a consequence, the fit must satisfy both and the rest of the observables in parallel, leading to a potential tension unless the solution for both sectors of the fit is similar. It turns out that for our favoured hypotheses, the b.f.p. for the global fit is well compatible with the current value. Let us consider for instance the hypotheses I and II, i.e. and . We can determine the value of the NP contribution needed to have the theoretical value of match its experimental value exactly. We can then compute the value of the for the global fit for this particular value of the NP contribution. Comparing this value of the with the obtained after fitting all observables (including ), we observe that that the difference is 3 units for Hyp. I and 0.1 units for Hyp. II. This shows a good consistency between the current values of and the rest of the observables. If we perform the same exercise taking an experimental value of that would be larger, the difference between and is in both Hyp. I and II, so that these scenarios could still explain all deviations in a consistent way. Contrarily, if is lower by , Hyp. I would display whereas Hyp. II would remain basically unchanged, indicating that the latter would show a better ability to explain and the rest of the data.
Another observable showing a tension with the SM is . Remarkably, none of the hypotheses considered is able to reduce the tension of this observable (see Fig. 8) so it remains poorly explained either within the SM or under the NP hypotheses considered. Even when we vary the central values for and , this observable cannot be easily accommodated. Under some of our NP hypotheses, its is slightly worse than within the SM. A similar problem is seen in even though for this observable the within the SM is smaller than for . Including a NP contribution to (which affects mainly the first bin) does not improve these two observables as they require contributions of opposite signs. Therefore, the small tensions observed in the first bin for these two observables will require more data to be understood.
The polarization fraction follows a similar trend as . Its current measurement shows little deviation with the SM prediction DescotesGenon:2015uva () in any of its bins due to the large theoretical errors, but its is around under the SM hypothesis due to the correlations with the rest of the observables involved in the fit. Moreover, for all NP hypotheses considered its value for the is larger than in the SM case. Let us add that if we increase by the central values for the observables and branching ratio, the for gets reduced.
In the case of the observables and the tension between the SM prediction and the experimental value is somewhat reduced if one introduces NP contributions. The less efficient hypothesis is Hyp. II, , and the best one is Hyp. I, i.e. a single contribution . As shown in Fig. 9, this hypothesis (grey band) would help accommodating at the same time both the deviations in and