# An empirical model to determine the hadronic resonance contributions to \(\overline{B}{} ^0 \!\rightarrow \overline{K}{} ^{*0} \mu ^+ \mu ^- \) transitions

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## Abstract

A method for analysing the hadronic resonance contributions in \(\overline{B}{} ^0 \!\rightarrow \overline{K}{} ^{*0} \mu ^+ \mu ^- \) decays is presented. This method uses an empirical model that relies on measurements of the branching fractions and polarisation amplitudes of final states involving \(J^{PC}=1^{--}\) resonances, relative to the short-distance component, across the full dimuon mass spectrum of \(\overline{B}{} ^0 \!\rightarrow \overline{K}{} ^{*0} \mu ^+ \mu ^- \) transitions. The model is in good agreement with existing calculations of hadronic non-local effects. The effect of this contribution to the angular observables is presented and it is demonstrated how the narrow resonances in the \(q^2 \) spectrum provide a dramatic enhancement to \(C\!P\)-violating effects in the short-distance amplitude. Finally, a study of the hadronic resonance effects on lepton universality ratios, \(R_{K^{(*)}}\), in the presence of new physics is presented.

## 1 Introduction

Decays with a \(b \!\rightarrow s \,\ell ^+ \ell ^- \) transition receive contributions predominantly from loop-level, flavour changing neutral current transitions. These transitions are mediated by heavy (short-distance) particles and are suppressed in the Standard Model (SM). Over the last few years, discrepancies have emerged when comparing measurements of the properties of \(b \!\rightarrow s \,\ell ^+ \ell ^- \) decays to SM predictions [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. Global analyses of these decays imply that there might be a new vector current which is destructively interfering with the SM contribution to the \(b \!\rightarrow s \,\ell ^+ \ell ^- \) decay, producing inconsistency with the SM at the 4–5\(\sigma \) [11, 12, 13, 14, 15, 16, 17].

In this paper, the possibility that hadronic resonances are interfering with the short-distance amplitude and mimicking physics beyond the SM is considered. This is because in addition to the short-distance contribution to \(b \!\rightarrow s \,\ell ^+ \ell ^- \) decays, the same final state can be obtained through non-local \(b\rightarrow s q\overline{q}\) transitions, where \(q\overline{q}\) denotes a quark-anti-quark pair. An example of such a decay is the decay \(\overline{B}{} ^0 \!\rightarrow {J /\psi } \overline{K}{} ^{*0} \), where the \({J /\psi }\) meson decays into two leptons.^{1} As the decay rate of this process is two orders of magnitude larger than its short-distance counterpart, sizeable interference effects are possible far from the \({J /\psi }\) mass.

The approach presented in this paper models the hadronic contributions originating from charm and light quark resonances as Breit–Wigner amplitudes. This approach is inspired by Refs. [18, 19] and is used to describe the hadronic resonances across the full dimuon mass spectrum of \(B ^0 \!\rightarrow K ^{*0} \mu ^+ \mu ^- \) decays. The LHCb collaboration performed a measurement of the interference between the non-local and short-distance components of \(B ^- \!\rightarrow K ^- \mu ^+\mu ^- \) decays by modelling the hadronic resonance contributions as Breit–Wigner amplitudes [20]. The level of interference was found to be small and the measurement of the short-distance component was found to be compatible with that of previous interpretations.

These non-local contributions are difficult to calculate and to date there is no consensus as to whether the deviations seen in global analyses can be explained by the these intermediate hadronic contributions, or by physics beyond the SM. Differentiating between these two hypotheses is of prime importance for confirming the existence and subsequently characterising phenomena not predicted by the SM. More detailed discussions on this point can be found in Refs. [18, 19, 21, 22, 23, 24, 25, 26, 27, 28].

Due to the more complex amplitude structure of the decay, for each resonant final state there are three relative phases and magnitudes that need to be determined instead of one in the case of the \(B ^- \!\rightarrow K ^- \mu ^+\mu ^- \) decay. Existing measurements of the branching fractions of \(\overline{B}{} ^0 \!\rightarrow {J /\psi } \overline{K}{} ^{*0} \) and \(\overline{B}{} ^0 \!\rightarrow \psi {(2S)} \overline{K}{} ^{*0} \) decays, together with measurements of their polarisation amplitudes [29, 30, 31, 32] can be used to assess the impact of these decays to the observables of the \(\overline{B}{} ^0 \!\rightarrow \overline{K}{} ^{*0} \mu ^+ \mu ^- \) process, up to a single overall phase per resonance that needs to be determined through a simultaneous fit to both the short-distance and non-local components in the \(\overline{K}{} ^{*0} \mu ^+\mu ^-\) final state. In the absence of such a measurement, scanning over all possible values for the global phase for each resonant final state, results in a prediction of the range of hadronic effects that can be compared to more formal calculations. The angular distribution of the decay \(\overline{B}{} ^0 \!\rightarrow \overline{K}{} ^{*0} \mu ^+ \mu ^- \) is sensitive to the strong-phases of non-local contributions, particularly through the observables \(S_{7}\) and \(S_{9}\). This sensitivity allows for a data-driven extraction of the non-local parameters of the proposed model.

The level of \(C\!P\) violation in decays such as \(\overline{B}{} ^0 \!\rightarrow \overline{K}{} ^{*0} \mu ^+ \mu ^- \) depends on weak- and strong-phase differences with interfering processes, such as \(\overline{B}{} ^0 \!\rightarrow {J /\psi } \overline{K}{} ^{*0} \). Therefore, a model for the strong phases of the non-local contributions to \(B ^0 \!\rightarrow K ^{*0} \mu ^+ \mu ^- \) transitions, offers new insight on both the kinematic regions where \(C\!P\) violation might be enhanced, as well as what the level of enhancement could be.

An increasingly large part of the discrepancy in \(b \!\rightarrow s \,\ell ^+ \ell ^- \) transitions is being driven by tests of lepton universality in \(\overline{B}{} \!\rightarrow \overline{K}{} ^{(*)}\ell ^+\ell ^-\) decays [3, 33, 34]. These deviations cannot be explained by hadronic effects (the \({J /\psi }\) meson, for example, decays equally often to electrons and muons). Although a significant deviation from lepton-universality would be a clear indication of physics beyond the SM, the precise characterisation of the new physics model still depends on the treatment of hadronic contributions. The angular distribution of \(\overline{B}{} ^0 \!\rightarrow \overline{K}{} ^{*0} \ell ^+ \ell ^- \) decays is critical in order to both determine the size of the new physics contribution, as well as to distinguish between models with left- or right-handed currents giving rise to new vector and axial-vector couplings.

This paper is organised as follows: Section 2 describes the model of the non-local contributions as well as the experimental inputs; Section 3 presents the comparison of the model to existing calculations; Section 4 shows how current model uncertainties impact both \(C\!P\)-averaged and \(C\!P\)-violating observables of \(\overline{B}{} ^0 \!\rightarrow \overline{K}{} ^{*0} \mu ^+ \mu ^- \) decays, as well as the expected precision of the \(\overline{B}{} ^0 \!\rightarrow \overline{K}{} ^{*0} \mu ^+ \mu ^- \) observables using the data that is expected from the LHCb experiment by the end of Run 2 of the LHC; finally in Section 5 there is a discussion of the impact of the non-local contributions in \(\overline{B}{} ^0 \!\rightarrow \overline{K}{} ^{*0} \ell ^+ \ell ^- \) and \(B ^- \!\rightarrow K ^- \ell ^+\ell ^-\) transitions in the presence of lepton-universality violating physics.

## 2 The model

*B*-meson, \(K^*\)-meson, and lepton respectively, \(q^2\) denotes the mass of the dimuon system squared, \(\lambda =m_{B}^{4}+m_{K^*}^{4}+q^4-2(m_{B}^{2}m_{K^*}^{2}+m_{K^*}^{2}q^{2}+m_{B}^{2}q^{2})\), \(\beta _\ell =\sqrt{1-4m_{\ell }^{2}/q^{2}}\) and

*z*function is given by

*p*is the momentum of the muons in the rest frame of the dimuon system evaluated at

*q*, and \(p_{\mathrm{res}\, j}\) is the momentum evaluated at the mass of the resonance.

Summary of the input values used to model the non-local amplitude components \(\mathcal {G}_{\lambda }\). The input values rely on measurements given in Refs. [20, 29, 30, 31, 32, 40, 41, 42, 43, 44]. The phases are measured relative to \(\theta _{j}^{0}\). As the measurements are given for the decay of the \(B ^0\) meson, in order to convert to the decay of the \(\overline{B}{} ^0\), the phase \(\theta _{j}^{\perp }\) given in the table above must be shifted by \(\pi \)

Mode | \((\eta _{j}^{\parallel },\theta _{j}^{\parallel } \,[rad])\) | \((\eta _{j}^{\perp },\theta _{j}^{\perp }\,[rad])\) | \(\eta _{j}^{0}\) |
---|---|---|---|

\(B^0\rightarrow \rho ^{0}K^{*0}\) | (1.5, 2.6) | (1.9, 2.6) | \(5.1\times 10^{-1}\) |

\(B^0\rightarrow \phi K^{*0}\) | \((2.5\times 10^{+1},2.6)\) | \((3.2\times 10^{+1},2.6)\) | \(1.0\times 10^{+1}\) |

\(B^0\rightarrow {J /\psi } K^{*0}\) | \((4.9\times 10^{+3},-2.9)\) | \((6.5\times 10^{+3},2.9)\) | \(7.1\times 10^{+3}\) |

\(B^0\rightarrow \psi {(2S)} K^{*0}\) | \((5.3\times 10^{+2},-2.8)\) | \((8.1\times 10^{+2},2.8)\) | \(9.6\times 10^{+2}\) |

\(B^0\rightarrow \psi (3770) K^{*0}\) | \((9.3\times 10^{-1},-2.9)\) | (1.5, 2.9) | 1.7 |

\(B^0\rightarrow \psi (4040) K^{*0}\) | \((2.9\times 10^{-1},-2.9)\) | \((5.6\times 10^{-1},2.9)\) | \(6.0\times 10^{-1}\) |

\(B^0\rightarrow \psi (4160) K^{*0}\) | \((8.3\times 10^{-1},-2.9)\) | (2.0, 2.9) | 1.8 |

It is customary that for each helicity amplitude, the expressions of the non-local components \(\mathcal {G}_{\lambda }\) are recast as shifts to the Wilson coefficient \(C_9\), referred to as \(\Delta C_{9\,\,\lambda }^\mathrm{total}\). This convention is particularly useful for comparisons with formal predictions of the non-local contributions.

Measurements of \(\overline{B}{} ^0 \rightarrow V \overline{K}{} ^{*0} \) decays, where *V* denotes any \(J^{PC}=1^{--}\) state, are only sensitive to relative phases of the three transversity amplitudes. Therefore, the convention used in previous measurements of these modes is such that phases \(\theta _\parallel \) and \(\theta _\perp \) are defined relative to \(\theta _0\). Using this convention, the remaining phase difference of each resonant polarisation amplitude relative to the corresponding short-distance one, is given by \(\theta _0\).

### 2.1 Experimental input

In order to assess the impact of the resonances appearing in the dimuon spectrum of \(\overline{B}{} ^0 \!\rightarrow \overline{K}{} ^{*0} \mu ^+ \mu ^- \) decays, knowledge of the resonance parameters \(\eta _{j}\) and \(\theta _{j}\) appearing in Eqs. 7–9 is required. The amplitude analyses of \(\overline{B}{} ^0 \!\rightarrow {J /\psi } \overline{K}{} ^{*0} \) and \(\overline{B}{} ^0 \!\rightarrow \psi {(2S)} \overline{K}{} ^{*0} \) transitions performed by the LHCb, BaBar and Belle collaborations [30, 31, 39] constrain the relative phases and magnitudes of the transversity amplitudes of the resonant decay modes. Combined with the measured branching fractions of these decays by the Belle experiment [30, 32], the parameters \(\eta _{j}^{\parallel ,\perp ,0}\) and \(\theta _{j}^{\parallel ,\perp }\) are determined up to an overall phase, \(\theta _{j}^{0}\), relative to the short-distance amplitude for the \(\overline{B}{} ^0 \!\rightarrow \overline{K}{} ^{*0} \mu ^+ \mu ^- \) decay. Similarly, the amplitude components of \(\overline{B}{} ^0 \!\rightarrow \phi \overline{K}{} ^{*0} \) transitions have been determined up to an overall phase, through the amplitude analyses and branching fraction measurements given in Refs. [40, 41, 42].

For the decay \(\overline{B}{} ^0 \!\rightarrow \rho ^0\overline{K}{} ^{*0} \), the magnitude of the total decay amplitude is set using the world average branching fraction of this transition [38, 43, 44]. As no amplitude analysis of this mode has been performed, the relative phases and magnitudes of the transversity amplitudes are taken to be the same as those of the \(\overline{B}{} ^0 \!\rightarrow \phi \overline{K}{} ^{*0} \) decay. As the overall contribution of the \(\rho ^0\) is expected to be small, this assumption will not impact the main conclusions of this study.

No measurements exist for final states involving the \(\psi (3770)\), \(\psi (4040)\) and \(\psi (4160)\) resonances, denoted as \(\overline{B}{} ^0 \rightarrow V_{\psi } \overline{K}{} ^{*0} \). To estimate the contributions of these final states, the relative phases and magnitudes of the transversity amplitudes are taken from the amplitude analysis of \(\overline{B}{} ^0 \!\rightarrow {J /\psi } \overline{K}{} ^{*0} \) decays. An approximate value of the branching fraction of each of the \(\overline{B}{} ^0 \rightarrow V_{\psi } \overline{K}{} ^{*0} \) modes is obtained by scaling the measured branching fraction of the decay \(\overline{B}{} ^0 \!\rightarrow \psi {(2S)} \overline{K}{} ^{*0} \), with \(\psi {(2S)} \rightarrow \mu ^+ \mu ^- \), by the known ratio of branching fractions between \(B^+\rightarrow \psi {(2S)} K^+\) and \(B^+\rightarrow V_{\psi } K^+\) decays, with \(V_{\psi }\rightarrow \mu ^+ \mu ^- \), given in Ref. [20]. The values used for the relative amplitudes and phases for each resonant contribution are summarised in Table 1.

## 3 Model comparisons

Figure 1 shows the parametrisation of the non-local contributions in the invariant amplitude basis of \(\overline{B}{} ^0 \!\rightarrow \overline{K}{} ^{*0} \mu ^+ \mu ^- \) decays given in Ref. [28]. The relation of this amplitude basis to the helicity basis is also given in Ref. [28]. The predictions using the model described in Sect. 2, where only the contributions from the \({J /\psi }\) and \(\psi {(2S)}\) resonances are considered, are shown for comparison. The free phases \(\theta ^{0}_{{J /\psi }}\) and \(\theta ^{0}_{\psi {(2S)}}\) appearing in Eqs. 7–9 are both set to 0 or \(\pi \). As a consistency check, the model presented in this paper is also shown, with the phases of all transversity amplitudes set to zero. The parameters \(\zeta _{\lambda }\) and \(\omega _{\lambda }\) also appearing in Eqs. 7–9 are chosen such that they are broadly consistent with the values of Ref. [15] and the predictions of Ref. [28], with \(\zeta _{\lambda }\sim 0.08|C_{7}|\) and \(\omega _{\lambda }=\pi \). Ignoring all phases of the transversity amplitudes of \(\overline{B}{} ^0 \!\rightarrow {J /\psi } \overline{K}{} ^{*0} \) and \(\overline{B}{} ^0 \!\rightarrow \psi {(2S)} \overline{K}{} ^{*0} \) decays, the model of \(\Delta C_{9\,\,\lambda }^\mathrm{total}\) described in this analysis is consistent to that of Ref. [28]. However, accounting for the measured relative phases in the resonant decay amplitudes results in large differences between the two models. The level of disagreement depends on the value of the free phases \(\theta ^{0}_{{J /\psi }}\) and \(\theta ^{0}_{\psi {(2S)}}\). The effect of the non-local charm contributions in Ref. [28] are known to move the central value of predictions of angular observables such as \(P_{5}'\) further away from experimental measurements [16]. However, this effect is only true due to the fact that the analysis of Ref. [28] did not account for the phases of the resonant amplitudes. An assessment of the impact of the phases on the angular observables is discussed in Sect. 4.

Building on the ideas of Ref. [28], a recent analysis presented in Ref. [21] provides a prediction of the non-local charm contribution that is valid up to a \(q^2\le m_{ \psi {(2S)}}^{2}\). This prediction also makes use of experimental measurements of \(\overline{B}{} ^0 \!\rightarrow {J /\psi } \overline{K}{} ^{*0} \) and \(\overline{B}{} ^0 \!\rightarrow \psi {(2S)} \overline{K}{} ^{*0} \) decays. In contrast to Ref. [28], the calculations of the non-local contributions are performed at \(q^2<0\) to next-to-leading order in \(\alpha _{s}\). The \(q^2\) parametrisation is given by a *z*-expansion truncated after the second order as in Eq. (5). Figure 2 shows both the real and imaginary parts of the non-local contributions to \(\overline{B}{} ^0 \!\rightarrow \overline{K}{} ^{*0} \mu ^+ \mu ^- \) decays presented in Ref. [21]. As the correlations between the *z*-expansion parameters are not provided, only the central values of the predictions are shown. The phase convention used in Ref. [21] is such that the transversity amplitudes of the \(\overline{B}{} ^0 \!\rightarrow {J /\psi } \overline{K}{} ^{*0} \) and \(\overline{B}{} ^0 \!\rightarrow \psi {(2S)} \overline{K}{} ^{*0} \) decays are related to those presented in this study through \(\eta ^{\parallel }_{j}\rightarrow -\eta ^{\parallel }_{j}\). The model described in Sect. 2, where only the contributions from the \({J /\psi }\) and \(\psi {(2S)}\) resonances are considered, is in qualitative agreement with that of Ref. [21] for the following parameter choice: \(\theta ^{0}_{{J /\psi }}=\pi /8\), \(\theta ^{0}_{\psi {(2S)}}=\pi /8\), \(\zeta _{\lambda }\sim 15\%|C_{7}|\) and \(\omega _{\lambda }=\pi \). The small level of disagreement observed in the imaginary part of the amplitudes at low \(q^2\) is due to the choice of setting \(\omega _{\lambda }=\pi \), with smaller values giving a better agreement.

*D*-meson. This is due to the use of Breit–Wigner functions to approximate the resonant contributions, that experiments can easily adopt.

## 4 Effect on \(\overline{B}{} ^0 \!\rightarrow \overline{K}{} ^{*0} \mu ^+ \mu ^- \) angular observables

Using the model of \(\Delta C_{9\,\,\lambda }^\mathrm{total}\) described in Sect. 2, the effect of the hadronic resonance contributions on the angular observables of \(\overline{B}{} ^0 \!\rightarrow \overline{K}{} ^{*0} \mu ^+ \mu ^- \) decays can be estimated. Figure 3 shows the distribution of the angular observables \(P_{5}^{\prime }\), \(A_\mathrm{FB}\), \(S_7\) and \(F_{L}\) [45, 46] in the SM. The observable \(S_7\) exhibits a particularly large dependence on the strong phases, demonstrating that measurements of the angular distribution of \(\overline{B}{} ^0 \!\rightarrow \overline{K}{} ^{*0} \mu ^+ \mu ^- \) decays can be used to determine the phases of the hadronic resonances. Therefore, this observable can be used to separate short-distance from the non-local contributions, as only the non-local part has a strong-phase difference. The remaining \(C\!P\)-averaged observables can be found in Appendix B. Definitions of these observables can be found for instance in Ref. [47]. As the phase \(\theta _{j}^{0}\) of all the resonant final states appearing in Table 1 are unknown, all possible variations of phases \(\theta _{j}^{0}\) are considered. The uncertainties arising from the combined light-cone sum rules and lattice QCD calculations of \(B\rightarrow K^{*}\) form factors are accounted for using the covariance matrix provided in Ref. [15]. The predictions of these observables using flavio [48] are also shown for comparison. The lack of knowledge of the phase \(\theta _{j}^{0}\) results in a large uncertainty for the prediction of \(P_{5}'\), diluting the sensitivity of this observable to the effects of physics beyond the SM. However, for the choice of \(\theta _{j}^{0}\) that results in a non-local charm contribution that is compatible with the latest prediction presented in Ref. [21] and is shown in Fig. 2), the tension of the prediction with the measured value of \(P_{5}'\) cannot be explained solely through hadronic effects.

### 4.1 Sensitivity to \(C\!P\) violation

### 4.2 Expected experimental precision

The experimental sensitivity to the phases between the short-range and hadronic resonance contributions to \(\overline{B}{} ^0 \!\rightarrow \overline{K}{} ^{*0} \mu ^+ \mu ^- \) is determined using \(\mathcal {O}(10^{6})\) simulated decays that include contributions from both short-distance and non-local components. The size of this sample corresponds to the approximate number of decays expected^{2} to be collected by the LHCb experiment by the end of Run2 of the LHC [49]. The decays are generated with the parameters \(\theta _{j}^{0}\), \(\zeta _\lambda \) and \(\omega _\lambda \) set to zero. The S-wave contribution to \(\overline{B}{} ^0 \!\rightarrow \overline{K}{} ^{*0} \mu ^+ \mu ^- \) decays is accounted for using the angular terms and amplitude expressions as a function of the invariant mass of the \(K\pi \) system given in Refs. [54, 55]. In addition to the S-wave component for the short-range amplitude, S-wave components are introduced with an amplitude and phase (\(\eta _{j}^{S}\), \(\theta _{j}^{S}\)), for the \({J /\psi }\) and the \(\psi {(2S)}\) resonances, based on the measurements given in Refs. [29, 30]. The overall effect of the S-wave contribution to the remaining resonances is considered to be negligible and is therefore ignored. In this study, all Wilson Coefficients are assumed to be real.

In order to ascertain the statistical precision on the non-local contribution, the detector resolution in \(q^2\) needs to be accounted for by smearing the \(q^2\) spectrum of the simulated events. For simplicity, a Gaussian resolution function is used with a width based on the RMS value of the dimuon mass resolution provided in Ref. [20], and converted into a resolution in \(q^2\). As the resolution in the helicity angles are far better than the variations in the angular distributions, any resolution effect in angles can be ignored; the sharp shape of the \(\phi \), \({J /\psi }\) and \(\psi {(2S)}\) resonances mean that a similar argument is not valid for the \(q^2\) distribution.

A four dimensional maximum likelihood fit is performed to the \(q^2\), \(\cos {\theta _l}\), \(\cos {\theta _K}\) and \(\phi \) distributions of the \(\overline{B}{} ^0 \!\rightarrow \overline{K}{} ^{*0} \mu ^+ \mu ^- \) decays in this sample. Both the non-local parameters, including \(\eta _{j}^{S}\) and \(\theta _{j}^{S}\), as well as the Wilson Coefficients \(C_9\) and \(C_{10}\) are left to vary in the fit. The \(B\rightarrow K^*\) form factor parameters however are fixed to their central values given in Ref. [15]. The resulting covariance matrix is used to ascertain the statistical precision on \(\Delta C_{9\,\,\lambda }^\mathrm{total}\). Based on the assessment of the systematic uncertainties in Ref. [20], the dominant source of experimental uncertainty is expected to be statistical in nature. However, the presence of tetra-quark states appearing in \(\overline{B}{} ^0 \!\rightarrow K^-\pi ^+{J /\psi } \) and \(\overline{B}{} ^0 \!\rightarrow K^-\pi ^+\psi {(2S)} \) decays [30, 56] will impact the determination of the non-local parameters. Although the effect is expected to be small, an accurate assessment of the effect is beyond the scope of this study.

The statistical precision on the angular observables is estimated by generating values for the non-local parameters of \(\Delta C_{9\,\,\lambda }^\mathrm{total}\), according to a multivariate Gaussian distribution centred at the values used to simulate the \(\overline{B}{} ^0 \!\rightarrow \overline{K}{} ^{*0} \mu ^+ \mu ^- \) decays, with a covariance matrix obtained from the resulting fit to the simulated data. These values are then propagated to the angular observables in order to obtain their 68% confidence interval as a function of \(q^2\). Figure 5 shows the statistical precision to \(P_{5}'\), \(A_\mathrm{FB}\), \(S_{7}\) and \(F_{L}\) in the SM, where the non-local parameters are given by Table 1 with \(\theta _{j}^{0}=0\). The equivalent plots for the remaining \(\textit{CP}\)-averaged observables can be found in Appendix C.

By the end of Run2 of the LHC, the dominant theoretical uncertainty of the angular observables in the \(q^2\) region \(5<q^2<14\) \({\mathrm {\,GeV^2\!/}c^4}\), will be due to the knowledge of the \(B\rightarrow K^{*}\) form-factors, rather than the non-local components. Future runs of the LHC will result in an even larger number of \(\overline{B}{} ^0 \!\rightarrow \overline{K}{} ^{*0} \mu ^+ \mu ^- \) decays. Therefore, it will, in a fit that combines the experimental data and the form factor uncertainties [57], be possible to use experimental data to further constrain Wilson Coefficients, as well as improve the precision of \(B\rightarrow K^*\) form factors and non-local contributions from charm and light quark resonances.

## 5 Hadronic resonance effects in tests of lepton universality

The model of \(\Delta C_{9}^\mathrm{total}\) discussed in Sect. 2 is used to provide a prediction for \(R_{K^*}\) that accounts for the residual dependence on the unknown phases \(\theta _{j}^{0}\). Figure 6 summarises this prediction in models with values of \(C_{9\,\mu }^\mathrm{NP}\) between -0.5 and -2.0, as suggested by global analyses of \(b\rightarrow s\mu ^+\mu ^-\) transitions. The confidence interval for \(R_{K^*}\) is determined by considering the full variation of the unknown phases \(\theta _{j}^{0}\). The residual form factor uncertainty is found to be subdominant compared to the variation of the phase. A prediction for \(R_K\) is also provided, which uses the long distance contributions measured in Ref. [20] with the 68% confidence interval determined by treating the measured non-local parameters as uncorrelated. It can be seen that when the experimental data is used for measuring the phase of the non-local contribution, the residual uncertainty becomes very small. It is worth noting that for \(C_{9\,\mu }^\mathrm{NP}=0\), there is no dependence on the unknown phase \(\theta _{j}^{0}\). Tabulated values of these predictions can be found in Appendix D. In the presence of new physics entering the Wilson coefficient \(C_{9\,\mu }\), a modest variation of \(R_{K^*}\) with the unknown phase \(\theta _{j}^{0}\) is observed. However, this variation is around 6 times smaller than the estimated uncertainty of \(R_{K^*}\) in the presence of lepton non-universal effects suggested by Ref. [11].

## 6 Conclusions

An empirical model to describe the hadronic resonance contributions in \(\overline{B}{} ^0 \!\rightarrow \overline{K}{} ^{*0} \mu ^+ \mu ^- \) transitions that relies on measurements of the branching fractions and polarisation amplitudes of \(\overline{B}{} ^0 \rightarrow V\overline{K}{} ^{*0} \) decays, is presented. For a particular choice of the relative phases between the short-distance component and the hadronic amplitudes, this model was found to be in good agreement with more formal predictions such as those of Refs. [21, 28]. The approach of this paper can naturally accommodate broad hadronic contributions from \(J^{PC}=1^{--}\) states such as the \(\rho ^0\), the \(\phi \) and charm-resonances above the open charm threshold, which can be inserted into experimental analyses of \(\overline{B}{} ^0 \!\rightarrow \overline{K}{} ^{*0} \mu ^+ \mu ^- \) decays.

The lack of knowledge of the longitudinal phase differences between \(\overline{B}{} ^0 \!\rightarrow \overline{K}{} ^{*0} \mu ^+ \mu ^- \) and \(B ^0 \rightarrow V\overline{K}{} ^{*0} \) decays results in a larger uncertainty on the predictions of the angular observables of \(\overline{B}{} ^0 \!\rightarrow \overline{K}{} ^{*0} \mu ^+ \mu ^- \) decays compared to current approaches. A measurement of these phases is critical as it will reduce the uncertainty in the determination of the Wilson Coefficients.

In addition, the resonant contributions to the decay provide large strong-phase differences that enhance sensitivity to CP violating effects. In this way, there is no need to rely on a time dependent analysis to a \(C\!P\) eigenstate. For the method to be exploited, it is required to have a model of the strong phase differences between short- and non-local contributions to \(\overline{B}{} ^0 \!\rightarrow \overline{K}{} ^{*0} \mu ^+ \mu ^- \) transitions as proposed here.

In the SM, observables such as \(R_{K}\) and \(R_{K^*}\) are independent of hadronic uncertainties. However, in the presence of non-universal effects in \(b \!\rightarrow s \,\ell ^+ \ell ^- \) transitions, these observables receive uncertainties from both the form-factor calculations and the interference between short- and non-local amplitudes. Using the models described in Ref. [20] and in this paper, predictions for \(R_{K}\) and \(R_{K^*}\) are provided for various choices of the Wilson coefficient \(C_{9}^{\mu }\). In order to maximise the potential of observables such as \(R_{K^*}\) as a way of characterising the exact physics model behind potential lepton-universality violating effects, a measurement of the non-local contributions in \(\overline{B}{} ^0 \!\rightarrow \overline{K}{} ^{*0} \mu ^+ \mu ^- \) decays is crucial. The data sample that will be collected by the LHCb experiment by the end of Run2 of the LHC will allow for a simultaneous amplitude analysis of both short-distance and non-local contributions to \(\overline{B}{} ^0 \!\rightarrow \overline{K}{} ^{*0} \mu ^+ \mu ^- \) decays across the full \(q^2\) spectrum of the decay. The model described in this paper, allows for a precise determination of both of these components.

## Footnotes

- 1.
Inclusion of charge conjugate processes is implied throughout this paper unless otherwise noted.

- 2.
The yield of both short-distance and non-local \(\overline{B}{} ^0 \!\rightarrow \overline{K}{} ^{*0} \mu ^+ \mu ^- \) decays is calculated by scaling the number of \(\overline{B}{} ^0 \!\rightarrow {J /\psi } \overline{K}{} ^{*0} \) and short-distance \(\overline{B}{} ^0 \!\rightarrow \overline{K}{} ^{*0} \mu ^+ \mu ^- \) decays given in Ref. [49] by a factor of 4.

## Notes

### Acknowledgements

We would like to thank C. Bobeth, D. van Dyk, and J. Virto for their help in obtaining their predictions of the charm correlator and for explaining in detail their model. We would also like to thank J. Matias, A. Khodjamirian and R. Zwicky for helpful discussions. Many thanks to S. Harnew, C. Langenbruch, S. Maddrell-Mander, J. Rademacker, M.-H. Schune and N. Skidmore for their corrections to the text. GP acknowledges support from the UK Science and Technology Facilities Council (STFC) from the Grant ST/N503952/1, TB acknowledges support from the Royal Society (United Kingdom) and PO acknowledges support from the Swiss National Science Foundation under grant number BSSGI0_155990.

## Supplementary material

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