Towards establishing Lepton Flavour Universality violation in decays
Rare semileptonic transitions provide some of the most promising framework to search for new physics effects. Recent analyses of these decays have indicated an anomalous behaviour in measurements of angular distributions of the decay and lepton-flavour-universality observables. Unambiguously establishing if these deviations have a common nature is of paramount importance in order to understand the observed pattern. We propose a novel approach to independently and complementary probe this hypothesis by performing a simultaneous amplitude analysis of and decays. This method enables the direct determination of observables that encode potential non-equal couplings of muons and electrons, and are found to be insensitive to non-perturbative QCD effects. If current hints of new physics are confirmed, our approach could allow an early discovery of physics beyond the Standard Model with LHCb Run-II datasets.
Flavour changing neutral current processes of B meson decays are crucial probes for the Standard Model (SM), since as-yet undiscovered particles may contribute to these transitions and cause observables to deviate from their SM predictions Grossman and Worah (1997); Fleischer (1997); London and Soni (1997); Ciuchini et al. (1997). The decay mode is a prime example (i.e. ), which offers a rich framework to study from differential decay widths to angular observables. An anomalous behaviour in angular and branching fraction analyses of the decay channel has been recently reported Aaij et al. (2016a); Wehle et al. (2016); Aaij et al. (2013a, 2014a), notably in one of the observables with reduced theoretical uncertainties, Aaij et al. (2013b); Descotes-Genon et al. (2016). Several models have been suggested in order to interpret these results as new physics (NP) signatures Gauld et al. (2014); Buras and Girrbach (2013); Altmannshofer and Straub (2013); Crivellin et al. (2015); Hiller and Schmaltz (2014); Biswas et al. (2015); Gripaios et al. (2015). Nonetheless, the vector-like nature of this pattern could be also explained by non-perturbative QCD contributions from operators (i.e. charm loops) that are able to either mimic or camouflage NP effects Jäger and Martin Camalich (2013, 2016). Non-standard measurement in ratios of processes - such as of Aaij et al. (2014b) and Aaij et al. (2017a) - indicate a suppression of the muon channel which is also compatible with the anomaly. In this case an immediate interpretation of lepton flavour universality (LFU) breaking is suggested due to the small theoretical uncertainties in their predictions Hiller and Kruger (2004); Bordone et al. (2016). Whilst the individual level of significance of the present anomalies is still inconclusive, there is an appealing non-trivial consistent pattern shown in global analysis fits Capdevila et al. (2018); Altmannshofer et al. (2017); Hurth et al. (2017a).
The formalism of b decays is commonly described within an effective field theory Altmannshofer et al. (2009), which probes distinct energy scales; with regimes classified into short-distance (high energies) perturbative and non-calculable long-distance effects. These can be parametrised in the weak Lagrangian in terms of effective operators with different Lorentz structures, , with corresponding couplings - referred to as Wilson coefficients (WC). Only a subset of the operators that are most sensitive to NP is examined in this work Bobeth et al. (2017), i.e. (virtual photon exchanges), (vector and axial currents) and corresponding right-handed couplings with flipped helicities. In this framework NP effects are incorporated by introducing deviations in the WCs Ali et al. (1995) from their SM predictions, i.e. . For instance, the anomalous pattern seen in semileptonic decays can be explained by a shift in the coefficient only, or and simultaneously Capdevila et al. (2018); Altmannshofer et al. (2017); Hurth et al. (2017a). A direct experimental determination of the WCs is currently bounded by sizeable uncertainties that arise from non-factorisable hadronic matrix elements that are difficult to assess reliably from first principles. Some promising approaches suggest to extract this contribution from data-driven analyses Blake et al. (2017); Hurth et al. (2017b) and by exploiting analytical properties of its structure Bobeth et al. (2017). However, these models still have intrinsic limitations, in particular in the assumptions that enter in parametrisation of the di-lepton invariant mass distribution.
In this Letter we propose a new model-independent approach that from a simultaneous unbinned amplitude analysis of both and decays can, for the first time, unambiguously determine LFU-breaking from direct measurements of WCs. This work builds on the generalisation of Ref. Bobeth et al. (2017), but it is insensitive to the model assumptions of the parametrisation. This effect relies on the strong correlation between the muon and electron modes imposed by the lepton-flavour universality of the hadronic matrix elements. Furthermore, in this method the full set of observables (e.g , and branching fraction measurements) available in decays is exploited, providing unprecedented precision on LFU in a single analysis.
Consider the differential decay rate for decays (dominated by the on-shell contribution) fully described by four kinematic variables; the di-lepton squared invariant mass, , and the three angles Altmannshofer et al. (2009). The probability density function () for this decay can be written as
with different intervals depending on the lepton flavour under study. For a complete definition of we refer to Bobeth et al. (2008); Altmannshofer et al. (2009) and references therein. It is convenient to explicitly write the WC dependence on the decay width by the transversity amplitudes () as Bobeth et al. (2017)
where is a normalisation factor, and and are referred to “local” and “non-local” hadronic matrix elements, respectively. The are form factors, while encode the aforementioned non-factorisable hadronic contributions and are described using two complementary parametrisations Bobeth et al. (2017); Hurth et al. (2017b) - for brevity only a subset of results is shown for the latter approach. In the following this function is expressed in terms of a “conformal” variable Bobeth et al. (2017); Boyd et al. (1995); Bourrely et al. (2009), with an analytical expansion truncated at a given order (herein referred to as ), after removing singularities related to the and . Some of the drawbacks of this expansion is that a-priori there is no physics argument to justify the order of the polynomial to be curtailed at - which in turn currently limits any claim on NP sensitivity.
In order to overcome these points, we investigate the LFU-breaking hypothesis using direct determinations of the difference of Wilson coefficients between muons and electrons, i.e.
where the usual WCs are renamed as , since an accurate disentanglement between the physical meaning of and the above-mentioned hadronic pollution cannot be achieved at the current stage of the theory Chrzaszcz et al. (2018). The key feature of this strategy is to realise that all hadronic matrix elements are known to be lepton-flavour universal, and thus are shared among both semileptonic decays. This benefits from the large statistics available for decays that is sufficient to enable the determination of these multi-space parameters. Note that an amplitude analysis of the electron mode only has been previously disregarded, given the limited dataset in either LHCb or Belle experiments. In a common framework the hadronic contributions are treated as nuisance parameters, while only the Wilson coefficients and are kept separately for the two channels. For consistency the WC is also shared in the fit and fixed to its SM value, given its universal coupling to photons and the strong constraint from radiative decays Paul and Straub (2017). In the following, all the right-handed WCs are fixed to their SM values, i.e. , while sensitivity studies on the determination of the WCs and are detailed in the appendix.
Signal-only ensembles of pseudo-experiments are generated with sample size corresponding roughly to the yields foreseen in LHCb Run-II [fb] and future upgrades [-fb] Aaij et al. (2017b), and Belle II [ab]. These are extrapolated from Refs. Aaij et al. (2016a, 2017a); Wehle et al. (2016) by scaling respectively with and for LHCb and Belle II, where denotes the designed centre-of-mass energy of the -quark pair. Note that for brevity most of the results are shown for the representative scenario of LHCb Run-II. The studied range corresponds to and for the muon mode and for the electron mode in LHCb; while in Belle II the same kinematic regions are considered for both semileptonic channels, namely and . This definition of ranges are broadly consistent with published results, and assumes improvements in the electron mode resolution for LHCb Lionetto (2018).
Within the SM setup the Wilson coefficients are set to , and (see Bobeth et al. (2017) and references therein), corresponding to a fixed renormalisation scale of GeV. This baseline model is modified for two NP benchmark points (BMP), and , referred respectively to as BMP and BMP, where NP is inserted only in the case of muons, i.e. . These points are favoured by several global fit analyses with similar significance Capdevila et al. (2018); Altmannshofer et al. (2017); Hurth et al. (2017a).
An extended unbinned maximum likelihood fit is performed to these simulated samples, in which multivariate Gaussian terms are added to the likelihood to incorporate prior knowledge on the nuisance parameters. In order to probe the model-independence of the framework, the non-local hadronic parametrisation is modified in several ways, i.e.
baseline SM prediction Bobeth et al. (2017) included as a multivariate Gaussian constraint;
no theoretical assumption on and with free-floating parameters;
higher orders of the analytical expansion of up to and - free floating;
and re-parametrisation of its description as proposed in Ref. Hurth et al. (2017b), i.e. using instead an expansion in terms of GeV.
On the other hand, form factors parameters are taken from Bharucha et al. (2016) and, in order to guarantee a good agreement between Light-Cone Sum Rules Ball and Braun (1998); Khodjamirian et al. (2007) and Lattice results Becirevic et al. (2007); Horgan et al. (2014), their uncertainties are doubled with respect to Ref. Bharucha et al. (2016).
Figure 1 shows the fit results for several alternative parametrisations of the non-local hadronic contribution for the BMP hypothesis, with yields corresponding to LHCb Run-II scenario. We observe that the sensitivity to is strongly dependent on the model assumption used for the non-local matrix elements. Nonetheless, it is noticeable that the high correlation of the and coefficients is sufficient to preserve the true underlying physics at any order of the series expansion and without any parametric theoretical input, i.e. the two-dimensional pull estimator with respect to the LFU hypothesis is unbiased.
We note that, as commonly stated in the literature (see e.g. recent review in Ref. Capdevila et al. (2017)), the determination of is insensitive to the lack of knowledge on the non-local hadronic effects. Nevertheless, its precision is still bounded to the uncertainties on the form factors, that are found to be the limiting factor by the end of Run-II.
The sensitivity to the two benchmark-like NP scenarios using the proposed pseudo observables is shown in Fig. 2. We quantify the maximal expected significance with respect to the SM to be and for BMP and BMP, respectively. Realistic experimental effects are necessary to determine the exact sensitivity achievable. Nevertheless, these results suggest that a first observation (with a single measurement) of LFU breaking appears to be feasible with the expected recorded statistics by the end of LHCb Run II. Furthermore, it is interesting to examine the prospects for confirming this evidence in the upcoming LHCb/Belle upgrades. Figure 3 summarises the two-dimensional statistical-only significances for the designed luminosities. Both LHCb Upgrade and Belle II experiments have comparable sensitivities (within ), while LHCb High-Lumi has an overwhelming significance. These unprecedented datasets will not only yield insights on this phenomena but also enable a deeper understanding of the nature of NP - insensitive to both local and non-local hadronic uncertainties.
Experimental resolution and detector acceptance/efficiency effects are not considered in this work, as these would require further information from current (non-public) or planned B-physics experiments. Nevertheless, preliminary studies on the impact of a finite resolution are performed assuming a -constant asymmetric smearing of the di-lepton invariant mass in the electron mode; the size and asymmetry of such smearing is naively chosen to reproduce the mass fits of Ref. Aaij et al. (2017a). Despite the low asymmetric tail, the determination of and remains unbiased, even if no-correction is applied. Moreover, the differential decay width can receive additional complex amplitudes from signal-like backgrounds, e.g. S-wave from a non-resonant decay and/or a scalar resonance (see detailed discussion in Ref. Hurth et al. (2017a)). These contributions are determined to be small Aaij et al. (2016a, b), and in the proposed formalism they benefit from the same description between the muon and electron mode. Therefore, in this constrained framework these effects are even further suppressed and can then be neglected for the scope of this work.
Another important test to probe the stability of the model consists in analysing potential issues that can rise if the truncation is not a good description of nature. We proceed as follows: we generate ensembles with non-zero coefficients for and , and we perform the fit with . Despite the mis-modelling of the non-local hadronic effects in the fit, we observe that the determination of and is always unbiased, thanks to the relative cancellation of all the shared parameters between the two channels. It is worth mentioning that a hypothetical determination of the individual and WCs can also produce a shift in their central values that mimics the behaviour of NP Chrzaszcz et al. (2018).
In conclusion, we propose a clean and model-independent method to combine all the available information from decays for a precise determination of LFU-breaking differences of WCs, i.e. and . This relies on a shared parametrisation of the local (form-factors) and non-local () hadronic matrix elements between the muonic and electronic channels, that in turn enables the determination of the observables of interest free from any theoretical uncertainty. In addition, this simultaneous analysis is more robust against experimental effects such as mismodeling of the detector resolution, since most parameters are effectively determined from the muon mode. This would be an important benefit for LHCb where the electron resolution is significantly worse than that of muons. Figure 4 illustrates the usefulness of the newly-proposed observables by combining the different information from angular analysis to branching ratio measurements. Due to the inclusiveness of the approach, the expected sensitivity surpasses any of the projections for the foreseen measurements of e.g. or alone - given the benchmark points. Therefore, this novel formalism can be the most immediate method to observe unambiguously NP in decays.
A promising feature of this framework is the possibility to extend the analysis to include other decay channels involving flavour changing neutral currents. For instance, the charged decay undergoes the same physics and is easily accessible at the -factories, while other rare semi-leptonic decays such as and have a different phenomenology but access the same NP information in terms of WC description. Thus, an unbinned global simultaneous fit to all data involving transitions is a natural and appealing extension of this work. Moreover, the parameter space of the investigated WCs can also be broadened to incorporate direct measurement of the right-handed - currently weakly constrained by global fits Capdevila et al. (2018); Altmannshofer et al. (2017); Hurth et al. (2017a).
We acknowledge useful contributions from Gino Isidori, Danny van Dyk and Patrick Owen. This work is supported by the Swiss National Science Foundation (SNF) under contracts 173104 and 174182.
Appendix A Supplemental material
An extension of the physics case of the proposed method is to investigate the sensitivity to the chirality-flipped counterparts of the usual Wilson coefficients, i.e. and . Following the formalism discussed in this letter, the primed WCs are examined by considering in addition to the BMP three different modified NP scenarios for the muon only: ; ; and . Notice that for the electron mode the is set and fixed to the SM value .
Figure 5 shows the fit results for different order of the analytic expansion for the non-local hadronic contribution for a NP scenario with , and yields corresponding to the LHCb Run II expected statistics. The dependency on the determination of and on the order of the expansion clearly saturates after and allows a measurement of the primed Wilson coefficients for the muon decay channel independent on the theoretical hadronic uncertainty. Figure 6 shows the prospects for the sensitivity to the and Wilson coefficients corresponding to the expected statistics at the LHCb upgrade with fb and fb. Note that only with the full capability of the LHCb experiment it is possible to start disentangling the different NP hypotheses.
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