An empirical model to determine the hadronic resonance contributions to $\kern
2.15991pt\overline{\kern-2.15991ptB}{}^{0}\\!\rightarrow\kern
2.39996pt\overline{\kern-2.39996ptK}{}^{*0}\mu^{+}\mu^{-}$ transitions

T. Blake1, U. Egede2, P. Owen3, K.A. Petridis4, G. Pomery4

1University of Warwick, Coventry, United Kingdom
2Imperial College London, London, United Kingdom
3Universität Zürich, Zürich, Switzerland
4Univeristy of Bristol, Bristol, United Kingdom

A method for analysing the hadronic resonance contributions in $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\mu^{+}\mu^{-}$ 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 $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\mu^{+}\mu^{-}$ 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 $b\rightarrow sq\overline{q}$ transitions, where
$q\overline{q}$ denotes a quark-anti-quark pair. An example of such a decay is
the decay $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$, where the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ meson decays into two
leptons.111Inclusion of charge conjugate processes is implied throughout this
paper unless otherwise noted. 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 ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ 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. [21, 22, 23, 24, 19, 25, 18, 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 $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow\psi{(2S)}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ 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 $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\mu^{+}\mu^{-}$ 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
$\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\mu^{+}\mu^{-}$ 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 $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\mu^{+}\mu^{-}$ 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 $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\mu^{+}\mu^{-}$ depends on weak-
and strong-phase differences with interfering processes, such as $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$. 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 $\kern
1.79993pt\overline{\kern-1.79993ptB}{}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{(*)}\ell^{+}\ell^{-}$ decays [33, 3,
34]. These deviations cannot be explained by hadronic effects (the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ 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 $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\ell^{+}\ell^{-}$ 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 $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\mu^{+}\mu^{-}$ decays, as well as
the expected precision of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\mu^{+}\mu^{-}$ 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 $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\ell^{+}\ell^{-}$ and
$B^{-}\\!\rightarrow K^{-}\ell^{+}\ell^{-}$ transitions in the presence of
lepton-universality violating physics.

##  2 The model

The differential decay rate of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\mu^{+}\mu^{-}$ transitions, where
the $\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ is a P-wave state and
ignoring scalar or timelike contributions to the dimuon system, depends on
eight independent observables [35]. Each of these observables is made up of
bilinear combinations of six complex amplitudes representing the three
polarisation states of the $\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$
for both the left- and right-handed chirality of the dilepton system. The
expression for the differential decay rate in terms of the angular observables
and their subsequent definition in terms of amplitudes, can be found in Ref.
[36]. The decay amplitudes are written in terms of the complex valued Wilson
Coefficients $C_{7}$, $C_{9}$ and $C_{10}$, encoding short distance effects,
and the $q^{2}$ dependent form-factors ,
$F_{i}(q^{2})=(V,A_{1},A_{12},T_{1},T_{2},T_{23})$ given in Ref. [15], that
express the $B\rightarrow K^{*}$ matrix elements of the operators involved in
these decays. The coefficient $C_{9}$ corresponds to the coupling strength of
the vector current operator, $C_{10}$ to the axial-vector current operator and
$C_{7}$ to the electromagnetic dipole operator. A detailed review of these
decays, including the operator definitions and the numerical values of the
Wilson Coefficients in the SM, can be found in Ref [36]. The decay amplitudes
in the transversity basis and assuming a narrow $K^{*0}$ can be written as

$\mathcal{A}_{0}^{\rm
L,R}(q^{2})=-8N\frac{m_{B}m_{K^{*}}}{\sqrt{q^{2}}}\left\\{(C_{9}\mp
C_{10})A_{12}(q^{2})+\frac{m_{b}}{m_{B}+m_{K^{*}}}C_{7}T_{23}(q^{2})+\mathcal{G}_{0}(q^{2})\right\\},$
(1)

$\mathcal{A}_{\parallel}^{\rm
L,R}(q^{2})=-N\sqrt{2}(m_{B}^{2}-m_{K^{*}}^{2})\left\\{(C_{9}\mp
C_{10})\frac{A_{1}(q^{2})}{m_{B}-m_{K^{*}}}+\frac{2m_{b}}{q^{2}}C_{7}T_{2}(q^{2})+\mathcal{G}_{\parallel}(q^{2})\right\\},$
(2)

$\mathcal{A}_{\perp}^{\rm L,R}(q^{2})=N\sqrt{2\lambda}\left\\{(C_{9}\mp
C_{10})\frac{V(q^{2})}{m_{B}+m_{K^{*}}}+\frac{2m_{b}}{q^{2}}C_{7}T_{1}(q^{2})+\mathcal{G}_{\perp}(q^{2})\right\\},$
(3)

where $m_{B}$, $m_{K^{*}}$ and $m_{\ell}$ are the masses of the $B$-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

$N=V_{tb}V_{ts}^{*}\sqrt{\frac{G_{F}^{2}\alpha^{2}}{3\times
2^{10}\pi^{5}m_{B}^{3}}q^{2}\lambda^{1/2}\beta_{\mu}}\,.$ (4)

In the above expressions, and for the remainder of this analysis,
contributions from right handed Wilson Coefficients have been omitted. These
are numerically small or zero in the SM and are not currently favoured by
global analyses of $b\\!\rightarrow s\,\ell^{+}\ell^{-}$ processes. Following
Ref. [15], the form factors are written

$F^{i}(q^{2})=\frac{1}{1-q^{2}/m_{R_{i}}^{2}}\sum_{k=0}^{2}\alpha_{k}^{i}[z(q^{2})-z(0)]^{k},$
(5)

where the $z$ function is given by

$z(t)=\frac{\sqrt{t_{+}-t}-\sqrt{t_{+}-t_{0}}}{\sqrt{t_{+}-t}+\sqrt{t_{+}-t_{0}}}\,,$
(6)

with $t_{\pm}=(m_{B}\pm m_{K^{*}})^{2}$ and
$t_{0}=t_{+}(1-\sqrt{1-t_{-}/t_{+}})$. The parameters $m_{R_{i}}$ are taken
from Ref. [15] and the coefficients $\alpha_{k}^{i}$ including their
correlations are taken from a combined fit to light-cone sum rule calculations
and Lattice QCD results given in Refs. [15, 37].

The functions $\mathcal{G}_{\lambda}(q^{2})$ describe the non-local hadronic
contributions to the $B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$ amplitudes and
are given by a simplistic empirical parametrisation inspired by the procedure
of Refs [20, 18, 19]. In particular

$\mathcal{G}_{0}=\frac{m_{b}}{m_{B}+m_{K^{*}}}T_{23}(q^{2})\zeta^{0}e^{i\omega^{0}}+A_{12}(q^{2})\sum_{j}\eta_{j}^{0}e^{i\theta_{j}^{0}}A_{j}^{\rm
res}(q^{2}),$ (7)

$\mathcal{G}_{\parallel}=\frac{2m_{b}}{q^{2}}T_{2}(q^{2})\zeta^{\parallel}e^{i\omega^{\parallel}}+\frac{A_{1}(q^{2})}{m_{B}-m_{K^{*}}}\sum_{j}\eta_{j}^{\parallel}e^{i\theta_{j}^{\parallel}}A_{j}^{\rm
res}(q^{2}),$ (8)

$\mathcal{G}_{\perp}=\frac{2m_{b}}{q^{2}}T_{1}(q^{2})\zeta^{\perp}e^{i\omega^{\perp}}+\frac{V(q^{2})}{m_{B}+m_{K^{*}}}\sum_{j}\eta_{j}^{\perp}e^{i\theta_{j}^{\perp}}A_{j}^{\rm
res}(q^{2}),$ (9)

where the sum in the above expressions represents the coherent sum of vector
meson resonant amplitudes with $\eta_{j}^{\lambda}$ and $\theta_{j}^{\lambda}$
the magnitude and phase of each resonant amplitude relative to $C_{9}$. The
exact normalisation of the $\eta_{j}^{\lambda}$ parameters is shown in
Appendix A and is chosen such the integral of the sum of the squared
magnitudes of the amplitude of a given amplitude produce the correct
experimental branching fraction. Similarly, the parameters $\zeta^{\lambda}$
and $\omega^{\lambda}$ need to be determined from experimental measurements.
In this analysis, the central values of these parameters are set to zero,
unless otherwise specified.

The $q^{2}$ dependence of each resonant amplitude is given by $A_{j}^{\rm
res}(q^{2})$. As indicated from the analysis of the dimuon mass spectrum in
$B^{-}\\!\rightarrow K^{-}\mu^{+}\mu^{-}$ decays from Ref. [20], the
resonances considered in this analysis are the $\rho^{0}$, $\phi$,
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$, $\psi(2S)$, $\psi(3770)$,
$\psi(4040)$ and $\psi(4160)$. Contributions from light-quark resonances are
expected to be either CKM- or loop-suppressed compared to final states
occurring through charmonium resonances. As experiments accumulate more data,
additional broad light-quark states will start becoming statistically
significant and can be easily incorporated in the model. For simplicity,
$A_{j}^{\rm res}$ is modelled by a relativistic Breit–Wigner function given by

$A_{j}^{\rm res}(q^{2})=\frac{m_{{\rm res}\,j}\Gamma_{{\rm res}\,j}}{(m_{{\rm
res}\,j}^{2}-q^{2})-im_{{\rm res}\,j}\Gamma_{j}(q^{2})},$ (10)

where $m_{{\rm res}\,j}$ and $\Gamma_{{\rm res}\,j}$ are the pole mass and
natural width of the $j^{\rm th}$ resonance and their values are taken from
Ref. [38]. The running width $\Gamma_{j}(q^{2})$ is given by

$\Gamma_{j}(q^{2})=\frac{p}{p_{{\rm res}\,j}}\frac{m_{{\rm
res}\,j}}{q}\Gamma_{{\rm res}\,j},$ (11)

where $p$ is the momentum of the muons in the rest frame of the dimuon system
evaluated at $q$, and $p_{{\rm res}\,j}$ is the momentum evaluated at the mass
of the resonance.

This isobar approach, although not rigorous, it provides a model for the
strong phase variation of the amplitude across the full $q^{2}$ spectrum. This
variation can result in sizeable effects even far from the pole of the
resonances as discussed in Sec. 4.

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 $\Delta C_{9\,\,\lambda}^{\rm total}$.
This convention is particularly useful for comparisons with formal predictions
of the non-local contributions.

Measurements of $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
V\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ 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 $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\mu^{+}\mu^{-}$ decays, knowledge
of the resonance parameters $\eta_{j}$ and $\theta_{j}$ appearing in Eqs. 7–9
is required. The amplitude analyses of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow\psi{(2S)}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ transitions performed by the
LHCb, BaBar and Belle collaborations [39, 30, 31] 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 $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\mu^{+}\mu^{-}$ decay. Similarly,
the amplitude components of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow\phi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ 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 $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow\rho^{0}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$, 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 $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow\phi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ 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 $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow V_{\psi}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$. To estimate the contributions of
these final states, the relative phases and magnitudes of the transversity
amplitudes are taken from the amplitude analysis of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ decays. An
approximate value of the branching fraction of each of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow V_{\psi}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ modes is obtained by scaling the
measured branching fraction of the decay $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow\psi{(2S)}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$, 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.

Mode | $(\eta_{j}^{\parallel},\theta_{j}^{\parallel}\,[rad])$ | $(\eta_{j}^{\perp},\theta_{j}^{\perp}\,[rad])$ | $\eta_{j}^{0}$
---|---|---|---
$B^{0}\rightarrow\rho^{0}K^{*0}$ | $(1.5\phantom{\times 10^{++0}},\phantom{+}2.6)$ | $(1.9\phantom{\times 10^{++0}},\phantom{+}2.6)$ | $5.1\times 10^{-1}$
$B^{0}\rightarrow\phi K^{*0}$ | $(2.5\times 10^{+1},\phantom{+}2.6)$ | $(3.2\times 10^{+1},\phantom{+}2.6)$ | $1.0\times 10^{+1}$
$B^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$ | $(4.9\times 10^{+3},-2.9)$ | $(6.5\times 10^{+3},\phantom{+}2.9)$ | $7.1\times 10^{+3}$
$B^{0}\rightarrow\psi{(2S)}K^{*0}$ | $(5.3\times 10^{+2},-2.8)$ | $(8.1\times 10^{+2},\phantom{+}2.8)$ | $9.6\times 10^{+2}$
$B^{0}\rightarrow\psi(3770)K^{*0}$ | $(9.3\times 10^{-1},-2.9)$ | $(1.5\phantom{\times 10^{++0}},\phantom{+}2.9)$ | $1.7\phantom{\times 10^{++0}}$
$B^{0}\rightarrow\psi(4040)K^{*0}$ | $(2.9\times 10^{-1},-2.9)$ | $(5.6\times 10^{-1},\phantom{+}2.9)$ | $6.0\times 10^{-1}$
$B^{0}\rightarrow\psi(4160)K^{*0}$ | $(8.3\times 10^{-1},-2.9)$ | $(2.0\phantom{\times 10^{++0}},\phantom{+}2.9)$ | $1.8\phantom{\times 10^{++0}}$
Table 1: 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 $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$, the phase $\theta_{j}^{\perp}$
given in the table above must be shifted by $\pi$.

##  3 Model comparisons

The study presented in Ref. [28] provides a prediction of the non-local charm
loop contribution to $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\mu^{+}\mu^{-}$ decays. It relies
on QCD light-cone sum rule calculations of $B\rightarrow K^{*}$ matrix
elements for $q^{2}\ll 4m_{c}^{4}$ and extrapolated to larger $q^{2}$ through
a hadronic dispersion relation. The extrapolation uses input from experimental
measurements of the rate and amplitude structure of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow\psi{(2S)}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ decays. As this calculation does
not account for the factorisable next-to-leading order corrections to the
charm loop, all phases of the non-local relative to the short-distance
amplitudes are set to zero.

Figure 1: The non-local contributions to the $B^{0}\\!\rightarrow
K^{*0}\mu^{+}\mu^{-}$ invariant amplitudes as a function of $q^{2}$. The
prediction using the model discussed in Sec. 2 is shown, where only the
contributions from the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and
$\psi{(2S)}$ resonances are considered. The free phases
$\theta^{0}_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ and
$\theta^{0}_{\psi{(2S)}}$ are both set to 0 (cyan solid line) or $\pi$ (cyan
dashed-dotted line). The prediction where all phases of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\psi{(2S)}$ appearing in
Eqs. 7–9 are set to zero is also depicted (black solid line), alongside the
prediction from Ref. [28] (magenta band).

Figure 1 shows the parametrisation of the non-local contributions in the
invariant amplitude basis of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\mu^{+}\mu^{-}$ 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 Sec. 2, where
only the contributions from the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
and $\psi{(2S)}$ resonances are considered, are shown for comparison. The free
phases $\theta^{0}_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ 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
$\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow\psi{(2S)}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ decays, the model of $\Delta
C_{9\,\,\lambda}^{\rm total}$ 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
$\theta^{0}_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ 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 $P_{5}^{\prime}$ 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 Sec. 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 $q^{2}\leq m_{\psi{(2S)}}^{2}$. This prediction also makes use of
experimental measurements of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow\psi{(2S)}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ 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 $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\mu^{+}\mu^{-}$ 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
$\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow\psi{(2S)}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ decays are related to those
presented in this study through
$\eta^{\parallel}_{j}\rightarrow-\eta^{\parallel}_{j}$. The model described in
Sec. 2, where only the contributions from the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\psi{(2S)}$ resonances are
considered, is in qualitative agreement with that of Ref. [21] for the
following parameter choice: $\theta^{0}_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}=\pi/8$, $\theta^{0}_{\psi{(2S)}}=\pi/8$, $\zeta_{\lambda}\sim
15\%|C_{7}|$ 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.

Figure 2: The non-local contributions to the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\mu^{+}\mu^{-}$ transversity
amplitudes as a function of $q^{2}$. The real (left) and imaginary (right)
components are shown separately. The prediction from Ref. [21] is shown
(magenta points). Predictions using the model discussed in Sec. 2, where only
the contributions from the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and
$\psi{(2S)}$ resonances are considered, are overlaid for different choices of
the phases $\theta^{0}_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ and
$\theta^{0}_{\psi{(2S)}}$ (cyan lines). See text for further details.

To conclude, the simplistic model of the non-local contributions to $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\mu^{+}\mu^{-}$ decays presented in
this paper is in good agreement with existing models, provided appropriate
choices of $\theta^{0}_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$,
$\theta^{0}_{\psi{(2S)}}$, $\omega_{\lambda}$ and $\zeta_{\lambda}$. For the
latter, a larger value is required to match the predictions of Ref. [21],
compared to Ref. [28]. The expressions of $\mathcal{G}_{\lambda}(q^{2})$ have
sufficient freedom to capture the $q^{2}$ dependence of formal theory
predictions in the $q^{2}$ range $1<q^{2}<m_{\psi{(2S)}}^{2}$. In addition, in
contrast to current predictions, the model of $\Delta C_{9\,\,\lambda}^{\rm
total}(q^{2})$ can naturally accommodate hadronic contributions from
$J^{PC}=1^{--}$ states composed of light quarks such as the $\phi$ and
$\rho^{0}$, as well as resonances appearing in the region $q^{2}>4m_{D}^{2}$,
where $m_{D}$ denotes the mass of the $D$-meson. This is due to the use of
Breit–Wigner functions to approximate the resonant contributions, that
experiments can easily adopt.

##  4 Effect on $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\mu^{+}\mu^{-}$ angular observables

Using the model of $\Delta C_{9\,\,\lambda}^{\rm total}$ described in Sec. 2,
the effect of the hadronic resonance contributions on the angular observables
of $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\mu^{+}\mu^{-}$ decays can be
estimated. Figure 3 shows the distribution of the angular observables
$P_{5}^{\prime}$, $A_{\rm FB}$, $S_{7}$ 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 $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\mu^{+}\mu^{-}$ 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 $P_{5}^{\prime}$, 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
$P_{5}^{\prime}$ cannot be explained solely through hadronic effects.

Figure 3: Distributions of the angular observables $P_{5}^{\prime}$, $A_{\rm
FB}$ $S_{7}$, and $F_{L}$ as a function of $q^{2}$ for regions below (left)
and above (right) the open charm threshold (cyan). Specific choices are
highlighted for $\theta_{j}^{0}=0$ (hatched band) and $\theta_{j}^{0}=\pi$
(dark band). The measured values of the observables from Ref. [49] are also
shown (black points). The theoretical predictions (magenta band) using flavio
[48] are shown for comparison.

###  4.1 Sensitivity to $C\\!P$ violation

The model of the hadronic resonance contributions to $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\mu^{+}\mu^{-}$ decays described in
this paper provides a prediction for the strong phase differences involved in
these transitions. Direct $C\\!P$ violation will arise when there are
interfering amplitudes that have different weak phases as well as different
strong phases, as discussed within the context of $B^{-}\\!\rightarrow
K^{-}\mu^{+}\mu^{-}$ and $B^{-}\\!\rightarrow\pi^{-}\mu^{+}\mu^{-}$ decays in
Refs. [24, 50]. Therefore, it is interesting to study the effect that
potential weak phases beyond the SM have on angular observables such as the
direct $C\\!P$ asymmetry $A_{C\\!P}$, defined as

$\displaystyle A_{C\\!P}=\dfrac{\dfrac{d\Gamma(B\rightarrow
K^{*}\mu^{+}\mu^{-})}{dq^{2}}-\dfrac{d\overline{\Gamma}(B\rightarrow
K^{*}\mu^{+}\mu^{-})}{dq^{2}}}{\dfrac{d\Gamma(B\rightarrow
K^{*}\mu^{+}\mu^{-})}{dq^{2}}+\dfrac{d\overline{\Gamma}(B\rightarrow
K^{*}\mu^{+}\mu^{-})}{dq^{2}}},$ (12)

where $\Gamma$ and $\overline{\Gamma}$ correspond to the partial widths of the
decays $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\mu^{+}\mu^{-}$ and
$B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$ respectively, as well as the so-
called $C\\!P$-odd angular observables $A_{i}$, defined for instance in Ref.
[36]. The method is similar to what is discusssed in Ref. [51] for
semileptonic charm decays.

Figure 4:  Distribution of observables $A_{C\\!P}$, $A_{3}$ and $A_{9}$ as a
function of $q^{2}$, for $\theta_{j}^{0}$ of all resonances set to $-\pi/2$,
$0$, $\pi/2$ and $\pi$. Two new physics models are considered, one with
$C_{9}^{NP}=-1.0-1.0i$ (left), and one with $C_{7}^{NP}=-0.03i$,
$C_{9}^{NP}=-1.0$ (right).

Figure 4 shows the observables $A_{C\\!P}$, $A_{3}$ and $A_{9}$ for
$\theta_{j}^{0}$ of all resonances set to $-\pi/2$, $0$, $\pi/2$ and $\pi$. To
illustrate the effect that the model of the strong phase differences have in
the presence of new weak phases, two new physics models are considered which
are compatible with existing experimental constraints. One with
$C_{9}^{\rm{NP}}=-1.0-1.0i$, and one with both $C_{7}^{\rm{NP}}=-0.03i$ and
$C_{9}^{\rm{NP}}=-1.0$ [52, 53]. The notation $C_{7,9}^{\rm{NP}}$ denotes the
new physics contribution to the corresponding Wilson Coefficient. In both
these models, all other Wilson Coefficients are set to their SM values. It is
clear that the non-local contribution enhances $C\\!P$-violating effects in
these decays, with the level of this enhancement depending on the value of the
unknown phase $\theta_{j}^{0}$. As it can be seen, there is a huge effect in
the vicinity of the resonances, thus giving sensitivity to an imaginary
component of $C_{9}$ in a way which have not been considered before. The only
other viable way to gain sensitivity would be through a time dependent
analysis of the $B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi$ or the $B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$ with the
$K^{*0}$ decaying to the $C\\!P$ eigenstate $K^{0}_{\rm\scriptscriptstyle
S}\pi^{0}$. In contrast, $C\\!P$-violating effects arising through a weak
phases appearing in the Wilson coefficient $C_{7}$, are best constrained from
measurements of $B\rightarrow K^{*}\gamma$ decays [53].

###  4.2 Expected experimental precision

The experimental sensitivity to the phases between the short-range and
hadronic resonance contributions to $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\mu^{+}\mu^{-}$ 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 expected222The yield of both short-
distance and non-local $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\mu^{+}\mu^{-}$ decays is
calculated by scaling the number of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ and short-distance
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\mu^{+}\mu^{-}$ decays given in
Ref. [49] by a factor of 4. 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 $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\mu^{+}\mu^{-}$ 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 ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ 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 $\phi$,
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ 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 $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\mu^{+}\mu^{-}$ 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 $\Delta
C_{9\,\,\lambda}^{\rm total}$. 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 $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow
K^{-}\pi^{+}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow
K^{-}\pi^{+}\psi{(2S)}$ decays [56, 30] 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.

Figure 5: Predictions of the observables $P_{5}^{\prime}$, $A_{\rm FB}$,
$S_{7}$ and $F_{L}$ in the SM using the expected post-fit precision of the
non-local parameters $\Delta C_{9\,\,\lambda}^{\rm total}$ at the end of Run2
of the LHC. A sample of $\mathcal{O}(10^{6})$ simulated $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\mu^{+}\mu^{-}$ decays that include
contributions from both short-distance and non-local components, is used to
determine the parameters of $\Delta C_{9\,\,\lambda}^{\rm total}$. The decays
are simulated in the SM, with the parameters $\theta_{j}^{0}$,
$\zeta_{\lambda}$ and $\omega_{\lambda}$ set to zero. The 68% confidence
intervals are shown for the statistical uncertainty (cyan band) and the
combination of the statistical uncertainty with the $B\rightarrow K^{*}$ form-
factor uncertainties (magenta band) given in Ref. [15].

The statistical precision on the angular observables is estimated by
generating values for the non-local parameters of $\Delta
C_{9\,\,\lambda}^{\rm total}$, according to a multivariate Gaussian
distribution centred at the values used to simulate the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\mu^{+}\mu^{-}$ 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 $P_{5}^{\prime}$, $A_{\rm FB}$, $S_{7}$ 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 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 $5<q^{2}<14$
${\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$, 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
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\mu^{+}\mu^{-}$ 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

Recent tests of lepton universality in $b\rightarrow s\ell^{+}\ell^{-}$ decays
have revealed hints of non-universal new physics entering in the dimuon Wilson
Coefficient $C_{9}^{\mu}$ [58, 59, 11, 12]. The level of this potential new
physics effect is compatible with the observed anomalies in the amplitude
analyses and branching fraction measurements of $b\rightarrow s\mu^{+}\mu^{-}$
transitions. Lepton universality tests rely on measurements such as the ratios
of branching fractions between decays with muons and electrons in the final
state. The observables $R_{K}$ and $R_{K^{*}}$ are defined as

$\displaystyle
R_{K^{(*)}}=\dfrac{\displaystyle\int_{q^{2}_{min}}^{q^{2}_{max}}\dfrac{d\Gamma(B\rightarrow
K^{(*)}\mu^{+}\mu^{-})}{dq^{2}}dq^{2}}{\displaystyle\int_{q^{2}_{min}}^{q^{2}_{max}}\dfrac{d\Gamma(B\rightarrow
K^{(*)}e^{+}e^{-})}{dq^{2}}dq^{2}}$ (13)

Hadronic effects in $b\rightarrow s\ell^{+}\ell^{-}$ decays are lepton
universal and observables such as $R_{K}$ and $R_{K^{*}}$ can be predicted
precisely in the SM, due to the cancellation of hadronic uncertainties.
Therefore, any significant deviation between measurements and predictions of
these quantities is a clear sign of physics beyond the SM. However, in the
presence of new physics effects that enter through the Wilson Coefficient
$C_{9}^{\mu}$, the cancellation of hadronic uncertainties is no longer exact.
Consequently, in order to determine the exact nature of any potential new
physics model, an accurate determination of the non-local contributions in
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\mu^{+}\mu^{-}$ decays is
essential.

Figure 6: Predictions of $R_{K^{*}}$ at large recoil (hatched magenta) and low
recoil (hatched cyan), and $R_{K}$ at large recoil (solid burgundy) for
different values of $C_{9\,\mu}^{\rm NP}$. The $R_{K}$ values at low recoil
are identical to those at large recoil and thus not shown. The interval for
$R_{K^{*}}$ is determined using the model described in Sec. 2, considering the
full variation of the unknown phases $\theta_{j}^{0}$. In contrast the 68%
confidence interval of the $R_{K}$ prediction is obtained using the measured
non-local contributions in $B^{-}\\!\rightarrow K^{-}\mu^{+}\mu^{-}$ decays
[20].

The model of $\Delta C_{9}^{\rm total}$ discussed in Sec. 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 $C_{9\,\mu}^{\rm NP}$ 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 $C_{9\,\mu}^{\rm NP}=0$, 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 $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\mu^{+}\mu^{-}$ transitions that
relies on measurements of the branching fractions and polarisation amplitudes
of $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow V\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ 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. [28, 21]. 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 $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\mu^{+}\mu^{-}$ decays.

The lack of knowledge of the longitudinal phase differences between $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\mu^{+}\mu^{-}$ and
$B^{0}\rightarrow V\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ decays
results in a larger uncertainty on the predictions of the angular observables
of $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\mu^{+}\mu^{-}$ 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
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\mu^{+}\mu^{-}$ 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 $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\mu^{+}\mu^{-}$ 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 $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\mu^{+}\mu^{-}$ 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.

## 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. Appendix

##  Appendix A Normalisation of the $\eta_{j}^{\lambda}$ parameters

The magnitude of each resonant amplitude $\eta_{j}^{\lambda}$ appearing in Eq.
7, 8 and 9 is given by

$\begin{split}|\eta_{j}^{0}|^{2}&=\displaystyle\dfrac{f_{j}^{0}\mathcal{B}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow V\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0})\times\mathcal{B}(V\rightarrow\mu^{+}\mu^{-})}{\displaystyle\tau_{B}\int\left|8N\frac{m_{B}m_{K^{*}}}{\sqrt{q^{2}}}A_{j}^{\rm
res}(q^{2})A_{12}(q^{2})\right|^{2}dq^{2}},\\\ \\\
|\eta_{j}^{\parallel}|^{2}&=\displaystyle\dfrac{f_{j}^{\parallel}\mathcal{B}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow V\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0})\times\mathcal{B}(V\rightarrow\mu^{+}\mu^{-})}{\displaystyle\tau_{B}\int\left|N\sqrt{2}(m_{B}^{2}-m_{K^{*}}^{2})A_{j}^{\rm
res}(q^{2})\frac{A_{1}(q^{2})}{m_{B}-m_{K^{*}}}\right|^{2}dq^{2}},\\\ \\\
|\eta_{j}^{\perp}|^{2}&=\displaystyle\dfrac{f_{j}^{\perp}\mathcal{B}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow V\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0})\times\mathcal{B}(V\rightarrow\mu^{+}\mu^{-})}{\displaystyle\tau_{B}\int\left|N\sqrt{2\lambda}A_{j}^{\rm
res}(q^{2})\frac{V(q^{2})}{m_{B}+m_{K^{*}}}\right|^{2}dq^{2}},\end{split}$
(14)

where the $f_{j}^{\lambda}$ factors denote the measured polarisation fraction
of $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow V\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ decays and $\tau_{B}$ denotes the
$B$-meson lifetime.

##  Appendix B CP-averaged observables

Figure 7: Distributions of the CP-averaged observables in the SM as a function
of $q^{2}$ below (left) and above (right) the open charm threshold (cyan).
Specific choices are highlighted for $\theta_{j}^{0}=0$ (hatched band) and
$\theta_{j}^{0}=\pi$ (dark band). The measured values of the observables from
Ref. [49] are also shown (black points). The theoretical predictions (magenta
band) using flavio [48] are shown for comparison.

##  Appendix C Experimental sensitivity for CP-averaged observables

Figure 8: Predictions of the remaining $C\\!P$-averaged observables in the SM
using the expected post-fit precision of the long-distance parameters $\Delta
C_{9\,\,\lambda}^{\rm total}$ at the end of Run2 of the LHC. A sample of
$\mathcal{O}(10^{6})$ simulated $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\mu^{+}\mu^{-}$ decays that include
contributions from both short- and long-distance components, is used to
determine the parameters of $\Delta C_{9\,\,\lambda}^{\rm total}$. The decays
are simulated in the SM, with the parameters $\theta_{j}^{0}$,
$\zeta_{\lambda}$ and $\omega_{\lambda}$ set to zero. The 68% confidence
intervals are shown for the statistical uncertainty (cyan band) and the
combination of the statistical uncertainty with the $B\rightarrow K^{*}$ form-
factor uncertainties (magenta band) given in Ref. [15].

##  Appendix D $R_{K^{*}}$ and $R_{K}$ predictions

Table 2: Predictions of $R_{K^{*}}$ and $R_{K}$ at large and low recoil for
different values of $C_{9\,\mu}^{\rm NP}$. The interval for $R_{K^{*}}$ is
determined using the model described in Sec. 2, considering the full variation
of the unknown phases $\theta_{j}^{0}$. The uncertainty due to the residual
form factor dependence is found to be subdominant. In contrast, the 68%
confidence interval of the $R_{K}$ prediction is obtained using the measured
long distance contributions in $B^{-}\\!\rightarrow K^{-}\mu^{+}\mu^{-}$
decays [20].

Observable | $C_{9\,\mu}^{\rm NP}=-0.5$ | $C_{9\,\mu}^{\rm NP}=-1.0$ | $C_{9\,\mu}^{\rm NP}=-1.5$ | $C_{9\,\mu}^{\rm NP}=-2.0$
---|---|---|---|---
|  $1.1<q^{2}<6.0$ ${\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$
$R_{K^{*}}$ | $[0.902,0.912]$ | $[0.827,0.850]$ | $[0.769,0.808]$ | $[0.727,0.784]$
$R_{K}$ | $[0.888,0.889]$ | $[0.792,0.794]$ | $[0.712,0.718]$ | $[0.651,0.658]$
|  $15<q^{2}<19$ ${\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$
$R_{K^{*}}$ | $[0.889,0.894]$ | $[0.796,0.806]$ | $[0.719,0.735]$ | $[0.658,0.680]$
$R_{K}$ | $[0.888,0.889]$ | $[0.792,0.794]$ | $[0.712,0.718]$ | $[0.651,0.658]$

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