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\def\PB      {\ensuremath{\mathrm{B}}\xspace}    
\def\B       {{\ensuremath{\PB}}\xspace}
\def\PK      {\ensuremath{\mathrm{K}}\xspace}   
\def\kaon    {{\ensuremath{\PK}}\xspace}
\def\Kstarz  {{\ensuremath{\kaon^{*0}}}\xspace}
\def\Bd      {{\ensuremath{\B^0}}\xspace}
\def\Bz      {{\ensuremath{\B^0}}\xspace}

%% Key decay channels

\def\BdToKstmm    {\decay{\Bd}{\Kstarz\mup\mun}}
\def\BdbToKstmm   {\decay{\Bdb}{\Kstarzb\mup\mun}}

\def\BsToJPsiPhi  {\decay{\Bs}{\jpsi\phi}}
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\def\BdbToJPsiKst {\decay{\Bdb}{\jpsi\Kstarzb}}

%% Rare decays
\def\BdKstee  {\decay{\Bd}{\Kstarz\epem}}
\def\BdbKstee {\decay{\Bdb}{\Kstarzb\epem}}
\def\bsll     {\decay{\bquark}{\squark \ell^+ \ell^-}}

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\def\ellell     {\ensuremath{\ell^+ \ell^-}\xspace}
\def\mumu       {{\ensuremath{\Pmu^+\Pmu^-}}\xspace}

\def\lhcb {\mbox{LHCb}\xspace}
\def\belle  {\mbox{Belle}\xspace}

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\begin{document}

\allowdisplaybreaks

\preprint{ZU-TH-}
\title{Towards establishing Lepton Flavour Universality breaking in $\bar{B}\to \bar{K}^*\ell^+\ell^-$ decays}
%\title{Novel approach for probing Lepton Flavour Universality in $B\to K^*\ell^+\ell^-$ decays}
%\title{Probing Lepton Flavour Universality in $B\to K^*\ell^+\ell^-$ decays}
\author{Andrea Mauri}
\email{a.mauri@cern.ch}
\author{Nicola Serra}
\email{nicola.serra@cern.ch}
\author{Rafael Silva Coutinho}
\email{rafael.silva.coutinho@cern.ch}
\affiliation{Physik-Institut, Universit\"at Z\"urich, Winterthurer Strasse 190, 8057 Z\"urich, Switzerland}

\begin{abstract}
  Rare semileptonic $b \to s \ell^+ \ell^-$ transitions provide some of the most promising framework to search for new physcs effects. 
  Recent analyses of these decays have indicated an anomalous pattern in measurements
  of angular observables in the decay $B^0\to
    K^*\mu^+\mu^-$ and lepton-flavour-universality observables. 
Unambigously establishing a common cause of all these deviations 
is vital in order to clarify the nature of these effects. 
We propose a novel method, based on a simultaneous amplitude analysis
of $\bar{B}^0 \to \bar{K}^{*0} \mu^+\mu^-$  and $\bar{B}^0 \to
\bar{K}^{*0} e^+e^-$ decays, that will allow to access this issue. 
Our method combines the full information in $B^0\to K^*\ell^+\ell^-$
decays and, if current hints of new physics are confirmed, it could lead to a discovery of physics beyond the Standard
Model already with LHCb Run-II datasets.  
 %This method enables the direct determination of the difference of the Wilson Coefficients ${\cal{C}}_{9}$ and ${\cal{C}}_{10}$ between electrons and muons, 
%  and are found to be insensitive to both local and non-local hadronic contributions.  
%%%% Stronger
%This method has is We show that considering current preferred New Physics scenarios a first observation of LFU breaking in a single measurement is possible with LHCb Run-II dataset. 
\end{abstract}

\maketitle



Flavour changing neutral current processes of {\textit{B}} meson decays, dominantly mediated by 
$b \to s$ amplitudes, are crucial probes for the Standard Model (SM), 
since as-yet undiscovered particles may contribute and cause observables to deviate  
from their SM predictions~\cite{Grossman:1996ke,Fleischer:1996bv,London:1997zk,Ciuchini:1997zp}.
The decay mode $\bar{B}\to \bar{K}^*\ell^+\ell^-$ is a prime example (\textit{i.e.} $\ell = \mu, e$), 
which offers a rich framework to study from differential decay widths to angular observables.
%{\color{red} phenomenology to study, formed by differential decay widths and angular observables.}
An anomalous behaviour in angular and branching fraction analyses of the decay channel 
$\bar{B}^{0} \to \bar{K}^{*0} \mu^{+}\mu^{-}$ has been recently reported~\cite{Aaij:2015oid,Wehle:2016yoi,Aaij:2013aln,Aaij:2014pli}, 
notably in one of the observables with reduced form-factor uncertainties, 
$P^{\prime}_{5}$~\cite{Descotes-Genon:2015uva}.
Several models have been suggested in order to interpret these results as New Physics (NP) 
signatures~\cite{Gauld:2013qja,Buras:2013qja,Altmannshofer:2013foa,Crivellin:2015era,Hiller:2014yaa,Biswas:2014gga,Gripaios:2014tna}.  
Nonetheless, the vector-like nature of this pattern could be also explained by 
large hadronic contributions from $b\to s c{\bar{c}}$ operators ({\textit{i.e.}} charm loops) 
that are able to either mimic or camouflage NP effects~\cite{Jager:2012uw,Jager:2014rwa}. 
Non-standard measurement in ratios of $b \to s \ell^+ \ell^-$ processes 
- such as of $R_{K}$~\cite{Aaij:2014ora} and $R_{K^{*}}$~\cite{Aaij:2017vbb} - 
indicate a suppression of the muon channel which is also compatible with the $P^{\prime}_{5}$ anomaly. 
In this case an immediate interpretation of lepton flavour universality (LFU) breaking is 
suggested due to the small theoretical uncertainties in their predictions~\cite{Hiller:2003js,Bordone:2016gaq}. 
Whilst the individual level of significance of the present anomalies is still inconclusive, 
there is an appealing non-trivial consistency shown in 
global analysis fits~\cite{Capdevila:2017bsm,Altmannshofer:2017yso,Hurth:2017hxg}.

The formalism of {\textit{b}} decays is commonly described within 
an effective field theory~\cite{Altmannshofer:2008dz} - 
hereafter only a subset of the Wilson coefficients (WC) $C_i$ for the basis of
dimension-six field operators $O_i$ is used for the weak Lagrangian~\cite{Bobeth:2017vxj}.  
In this framework NP effects are incorporated 
by introducing deviations in the WCs~\cite{Ali:1994bf} 
({\textit{i.e.}} $\mathcal{C}_i = \mathcal{C}^{\mathrm{SM}}_i + \mathcal{C}^{\mathrm{NP}}_i$). 
%For instance, whilst the individual level of significance of the present anomalies is still inconclusive, 
%there is an appealing non-trivial consistency shown in 
%global analysis fits~\cite{Capdevila:2017bsm,Altmannshofer:2017yso,Hurth:2017hxg};
%\textit{i.e.} a shift in the coefficient $\mathcal{C}_9$ only, 
%or $\mathcal{C}_9$ and $\mathcal{C}_{10}$ simultaneously. 
For instance, the anomalous pattern seen in semileptonic decays can be 
explained by a shift in the coefficient $\mathcal{C}_9$ only, 
or $\mathcal{C}_9$ and $\mathcal{C}_{10}$ simultaneously~\cite{Capdevila:2017bsm,Altmannshofer:2017yso,Hurth:2017hxg}. 
A direct experimental determination of the WCs is currently 
bounded by sizeable uncertainties that arise from 
non-factorisable hadronic contributions. 
Some promising approaches propose to  extract these non-local hadronic elements 
either from data-driven analyses~\cite{Blake:2017fyh,Hurth:2017sqw} 
or by using the analytical and dispersive properties of these correlators~\cite{Bobeth:2017vxj}. 
However, these models still have intrinsic limitations, in particular 
in the assumption of the parametrisation of the di-lepton invariant mass. 

In this Letter we propose a new \textit{model-independent} approach that 
from a simultaneous unbinned amplitude analysis of both  
$\bar{B}^0 \to \bar{K}^{*0} \mu^+\mu^-$  and $\bar{B}^0 \to \bar{K}^{*0} e^+e^-$ decays 
can, for the first time, unambiguously determine LFU-breaking from direct measurements of WCs. 
This work builds on the generalisation of Ref.~\cite{Bobeth:2017vxj}, 
but it is insensitive to the model assumptions of the parametrisation. 
%This relies on the strong correlation {\color{red} between the two decay modes} when examining muons and electrons 
This relies on the strong correlation between the muon and electron modes 
imposed by the lepton-flavour universality of the hadronic matrix elements. 
%when examining directly the difference in Wilson coefficients.
Furthermore, in this method the full set of observables available in $\bar{B}\to \bar{K}^*\ell^+\ell^-$  
decays is exploited, and therefore, most stringent constraints on LFU for a single measurement can be expected.  

%Let us consider the differential decay rate for $\bar{B}\to \bar{K}^*\ell^+\ell^-$ 
Consider the differential decay rate for $\bar{B}\to \bar{K}^*\ell^+\ell^-$ 
decays (dominated by the on-shell $\bar{K}^{*0}$ contribution) 
%{\color{red} (I know that you already changed this to the text above, but "Let us" looks pretty ugly:) ) 
%In this work we assume the $\bar{B}\to \bar{K}^*\ell^+\ell^-$ decay 
%being completely dominated by the on-shell $\bar{K}^{*0}$ ($p$-wave) contribution.
%The differential decay rate is hence
fully described by four kinematic variables;  
the di-lepton invariant mass square, $q^2$, and the three angles 
$\vec{\Omega} = (\cos \theta_\ell, \cos \theta_K, \phi)$~\cite{Altmannshofer:2008dz}.
The probability density function ($p.d.f.$) for this decay can be written as
%
\begin{equation}
p.d.f.^{(i)} = \frac{1}{\Gamma_i} \frac{\dd^4 \Gamma}{\dd q^2 \dd^3 \Omega}, \
    \quad
    \text{with}\quad
    \Gamma_i = \int_{q^2} \dd q^2 \frac{\dd\Gamma}{\dd q^2}\,,
\end{equation}
%
with different \qsq intervals depending on the lepton flavour under study. 
%where the \qsq range is defined differently for the two semileptonic channels.
For a complete definition of $\dd ^4\Gamma/(\dd q^2 \dd ^3\Omega)$ we refer 
to~\cite{Bobeth:2008ij,Altmannshofer:2008dz} and references therein.
It is convenient to explicitly write the WC dependence on the decay width by 
the transversity amplitudes ($\lambda=\perp, \para,0$) as~\cite{Bobeth:2017vxj}
%
\eqa{
  \label{eq:amp_dep}
  {\cal{A}}_{\lambda}^{(\ell)\,L,R} &=& {\cal{N}}_{\lambda}^{(\ell)}\ \bigg\{ 
(C^{(\ell)}_9 \mp C^{(\ell)}_{10}) {\cal{F}}_{\lambda}(q^2) \\
%
&&+\frac{2m_b M_B}{q^2} \bigg[ C^{(\ell)}_7 {\cal{F}}_{\lambda}^{T}(q^2) - 16\pi^2 \frac{M_B}{m_b} {\cal{H}}_{\lambda}(q^2) \bigg]
\bigg\}\,,\nonumber 
}
where ${\cal{N}}_{\lambda}^{(\ell)}$ is a normalisation factor, and 
${\cal{F}}^{(T)}_{\lambda}(q^{2})$ and $\mathcal{H}_\lambda(q^{2})$ 
are local and non-local hadronic matrix elements, respectively.  
While the ${\cal{F}}^{(T)}_{\lambda}(q^{2})$ are form factor parameters set from~\cite{Straub:2015ica}
\footnote{Following a conservative approach, uncertainties on the form factors parameters are 
doubled with respect to Ref.~\cite{Straub:2015ica}},
the $\mathcal{H}_\lambda(q^{2})$ 
are described using two complementary parametrisations~\cite{Bobeth:2017vxj,Hurth:2017sqw} - 
for brevity only a subset of results is shown for the latter approach.
In the following this correlator is expressed in terms of a conformal 
variable $z(q^{2})$~\cite{Bobeth:2017vxj,Boyd:1995cf,Bourrely:2008za}, 
with an analytical expansion truncated at a given order  
$z^n$ (herein referred to as $\mathcal{H}_\lambda[z^n]$). 
Some of the drawbacks of this expansion is that \textit{a-priori} there is 
no physics argument to justify the order of the polynomial to be curtailed at 
or even if this series will ever converge - 
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, \textit{i.e.} 
\begin{equation}
\Delta \WC_i = \widetilde{\mathcal{C}}_i^{(\mu)} - \widetilde{\mathcal{C}}_i^{(e)}\,,
\end{equation}
where the usual WCs $\mathcal{C}_i^{(\mu,e)}$ are renamed as $\widetilde{\mathcal{C}}_i^{(\mu,e)}$, 
since a precise disentanglement between the physical 
meaning of $\WC_i^{(\mu,e)}$ and the above-mentioned hadronic pollution is
impossible at the current stage of the theoretical knowledge.
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 $\bar{B}^0 \to \bar{K}^{*0} \mu^+\mu^-$ decays 
that is sufficient to enable the determination of these multi-space parameters.\footnote{Note 
that an amplitude analysis of the electron mode only has been always previously disregarded, 
given the limited dataset foreseen in either LHCb or Belle-II experiments.} 
Therefore, in a common framework these hadronic contributions are treated as 
nuisance parameters, while only the Wilson coefficients $\widetilde{\WC}_9^{(\mu,e)}$ 
and $\widetilde{\WC}_{10}^{(\mu,e)}$ are kept separately for the two channels.
For consistency the WC $\widetilde{C}_{7}$ is also shared in the fit, 
given its universal coupling to photons~\cite{Paul:2016urs}. 

Signal-only ensembles of pseudo-experiments are generated with
sample size corresponding roughly to the yields foreseen in LHCb Run-II [$8\,$fb$^{-1}$] and future upgrades 
[$50\,$-$\,300\,$fb$^{-1}$]~\cite{Aaij:2244311}, and Belle II [$50\,$ab$^{-1}$].
These are extrapolated from Refs.~\cite{Aaij:2015oid,Aaij:2017vbb,Wehle:2016yoi} 
by scaling respectively with $\sigma_{b\bar{b}} \propto \sqrt{s}$ and $\sigma_{b\bar{b}} \propto s$ 
for LHCb and Belle II, where $s$ denotes the designed centre-of-mass
energy of the $b$-quark pair.  
Note that for brevity most of the results are shown for the representative
scenario of LHCb Run-II. 
The studied \qsq range corresponds to 
$1.1\,\GeV^2 \leq q^2 \leq 8.0\,\GeV^2$ and $11.0\,\GeV^2  \leq q^2 \leq 12.5\,\GeV^2$ for 
the muon mode and $1.1\,\GeV^2  \leq q^2 \leq 7.0\,\GeV^2$ for the electron mode in LHCb;
while in Belle II the same kinematic regions are considered for both semileptonic channels, namely 
$1.1\,\GeV^2  \leq q^2 \leq 8.0\,\GeV^2$ and $10.0\,\GeV^2  
\leq q^2 \leq 13.0\,\GeV^2$. 
This definition of \qsq ranges are broadly consistent with published results, 
and assumes improvements in the electron mode resolution for LHCb~\cite{Lionetto:XX}.  

Within the SM setup the Wilson coefficients are set to  
$\mathcal{C}^{\rm{SM}}_9 = 4.27$, $\mathcal{C}^{\rm{SM}}_{10} = - 4.17$ and $\mathcal{C}^{\rm{SM}}_7 = -0.34$.
This baseline model is modified in the case of muons for two NP benchmark points (BMP), 
\textit{i.e.} $\WC_9^{(e)} = \WC^{\rm{SM}}_9 = \WC^{(\mu)}_9 + 1$ and 
%{\color{red} $\WC^{\rm{NP}(\mu)}_9 = - 1$ }
%and  {\color{red} $\WC_9^{\rm{NP}(\mu)} = -\WC_{10}^{\rm{NP}(\mu)} = - 0.7$}, 
$\WC_{9(10)}^{(e)} = \WC^{\rm{SM}}_{9(10)} = \WC_{9(10)}^{(\mu)} +(-)\,0.7$, 
%$\WC_9^{(\mu)} = -\WC_{10}^{(\mu)} = - 0.7$,
referred to as \texttt{BMP}$_{\WC_9}$ and \texttt{BMP}$_{\WC_{9,10}}$, respectively. 
These points are favoured by several global fit 
analyses with similar significance~\cite{Capdevila:2017bsm,Altmannshofer:2017yso,Hurth:2017hxg}, 
but chosen with reduced SM tension in order to examine a more conservative hypothesis. 

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, \textit{i.e.}
%
\begin{enumerate}
    %
    \item[i.] baseline $\mathcal{H}_\lambda[z^2]$ SM prediction 
    parametrisation~\cite{Bobeth:2017vxj} as a multivariate Gaussian constraint;
    %
    \item[ii.] no theoretical assumption on $\mathcal{H}_\lambda[z^2]$ 
    and with free-floating parameters;
    %
  \item[iii.] higher orders of the analytical expansion of $\mathcal{H}_\lambda[z^{n}]$ 
    up to $z^3$ and $z^4$ - free floating;
    %
    \item[iv.] and re-parametrisation of its description as  
      proposed in~\cite{Hurth:2017sqw}.
    %
\end{enumerate}
%
The stability of the model and the convergency to the global minimum is enforced by
repeating the fit with randomised starting parameters; 
the solution with smallest negative log-likelihood is taken as the default. 

 
Figure~\ref{fig:C9ellipse} shows the fit results for several alternative parametrisations 
of the non-local hadronic contribution for the \texttt{BMP}$_{\WC_9}$ hypothesis, 
with yields corresponding to LHCb Run-II scenario.  
We observe that the sensitivity to $\widetilde{\WC}_9^{(\mu,e)}$ is strongly dependent on 
the model assumption used for the non-local matrix elements. 
Nonetheless, it is noticeable that the high correlation of the  
$\widetilde{\mathcal{C}}_9^{(\mu)}$ and $\widetilde{\mathcal{C}}_9^{(e)}$ coefficients 
is sufficient to preserve the true underlying physics at any order of the series expansion $\mathcal{H}_\lambda[z^n]$,  
\textit{i.e.} the two-dimensional pull estimator with respect to the LFU hypothesis is unbiased. 
%
\begin{figure}[t]
\includegraphics[width=.4\textwidth]{plots/ellipses_C9.pdf} 
\caption{%
    Two-dimensional sensitivity scans for the pair of Wilson coefficients 
    $\widetilde{\mathcal{C}}_9^{(\mu)}$ and $\widetilde{\mathcal{C}}_9^{(e)}$ 
    for different non-local hadronic parametrisation models evaluated at \texttt{BMP}$_{\WC_9}$,  
    and with the expected statistics after \lhcb Run II. 
    The contours correspond to $3\,\sigma$ statistical-only uncertainty bands and 
    the dotted black line indicates the LFU hypothesis.   
}
\label{fig:C9ellipse}
\end{figure}
%
Furthermore, we note that, as commonly stated in the literature (see \textit{e.g.} recent review in Ref.~\cite{Capdevila:2017ert}),  
the determination of $\WC_{10}^{(\mu,e)}$ is insensitive to the lack of knowledge on the 
non-local hadronic effects and thus independent of any model assumption. 
%
\begin{figure}[bth!]
%\begin{center}
\includegraphics[width=.4\textwidth]{plots/ellipses_DeltaC9C10_a.pdf}\\%\quad\quad\quad\quad  
\includegraphics[width=.4\textwidth]{plots/ellipses_DeltaC9C10_b.pdf} 
\caption{%
    Two-dimensional sensitivity scans for the proposed observables $\Delta\WC_9$ and $\Delta\WC_{10}$ 
    for different non-local hadronic parametrisation models 
    evaluated at (top) \texttt{BMP}$_{\WC_9}$ and (bottom) \texttt{BMP}$_{\WC_{9,10}}$, 
    and with the expected statistics after \lhcb Run II. 
    The contours correspond to $3\,\sigma$ statistical-only uncertainty bands.
    \label{fig:DeltaC9C10}
}
%\end{center}
\end{figure}

The sensitivity to the two benchmark-like NP scenarios using the proposed observables $\Delta \WC_i$
is shown in Fig.~\ref{fig:DeltaC9C10}. 
%Fig.~\ref{fig:DeltaC9C10} shows the sensitivity to the two NP scenarios, NP$_{\WC_9}$ 
%and NP$_{\WC_9-\WC_{10}}$ in terms of the two model-independent LFU-breaking 
%difference of Wilson coefficients $\Delta\WC_9$ and  $\Delta\WC_{10}$.
We quantify the maximal expected significance with respect to the SM to be $4.6$ and $5.3\,\sigma$ for 
\texttt{BMP}$_{\WC_9}$ and \texttt{BMP}$_{\WC_{9,10}}$, respectively. 
Notice that 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~\ref{fig:DeltaC9C10_Upgrade} summarises the two-dimensional statistical-only significances  
for the designed luminosities. 
Both LHCb Upgrade and Belle II experiments have comparable sensitivities (within $8.0-10\,\sigma$), 
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.  
%Note that these unprecedented dataset will enable insight towards the nature of NP. 
%
%the \lhcb Run II, $7.6(8.4)\,\sigma$ for \belle II 50~ab$^{-1}$ dataset and 
%$9.0(9.6)\,\sigma$ for the \lhcb 50~fb$^{-1}$ Upgrade for the \texttt{BMP}$_{\WC_9}$ 
%(\texttt{BMP}$_{\WC_9-\WC_{10}}$) scenario respectively.
%{\color{red} The two-dimensional sensitivity pull expected for the \belle II and 
%\lhcb Upgrades is shown in Fig.~\ref{fig:DeltaC9C10_Upgrade}. 
%We note that the proposed method can provide a first observation of LFU breaking 
%in a single measurement with LHCb Run-II dataset, while for a precise determination 
%of the nature of NP the expected statistics of future upgrades of \lhcb and \belle II is required.}


\begin{figure}[bth!]
\includegraphics[width=.4\textwidth]{plots/ellipses_DeltaC9C10_Nev.pdf}
\caption{%
    Two-dimensional sensitivity scans for the proposed observables $\Delta\WC_9$ and $\Delta\WC_{10}$ 
    for the two considered NP scenarios: (green) \texttt{BMP}$_{\WC_9}$ and (red) \texttt{BMP}$_{\WC_{9,10}}$.
    The contours correspond to $3\,\sigma$ statistical-only uncertainty bands expected for the (dashed) \belle II 50~ab$^{-1}$ 
    and \lhcb Upgrade (dotted) $50\,$fb$^{-1}$  and (solid) $\,300\,$fb$^{-1}$ statistics.
    \label{fig:DeltaC9C10_Upgrade}
}
\end{figure}

Modelling detector effects such as \qsq and angles resolution, detector acceptance/efficiency, 
is hardly possible without access to (non-public) information of the current 
\textit{B}-physics experiments.
A first rudimentary study on the impact of a finite \qsq resolution is performed 
assuming a \qsq-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.~\cite{Aaij:2017vbb}.
Despite the low \qsq asymmetric tail, the determination of $\Delta\WC_9$ and 
$\Delta\WC_{10}$ remains unbiased.
Moreover, the differential decay width can receive additional complex amplitudes from signal-like backgrounds,  
\textit{e.g.} $K\pi$ S-wave from a non-resonant decay and/or a scalar resonance (see detailed discussion in Ref.~\cite{Hurth:2017hxg}). 
These contributions are in general expected to be small~\cite{Aaij:2015oid,Aaij:2016flj}, 
and in the proposed formalism these are introduced in an identical manner for muons and electrons. 
Therefore, in this constrained framework these effects are even further suppressed and can then be neglected.

Another important test to probe the stability of the model consists in changing the 
description of the non-local hadronic effects in the generation of the pseudo-experiments.
In this way we analyse potential issues that can rise if the truncation 
$\mathcal{H}_\lambda[z^n]$ is not a good description of nature.
We proceed as follows: we generate toys with non-zero coefficients for 
$\mathcal{H}_\lambda[z^3]$ and $\mathcal{H}_\lambda[z^4]$, and we perform the fit 
with $\mathcal{H}_\lambda[z^2]$.
We vary the choice of the $\mathcal{H}_\lambda[z^{(3,4)}]$ generated parameters, 
including a \textit{provocative} set of values that minimises the tension with the $P_5'$ 
anomaly~\cite{Aaij:2015oid}, while keeping $\WC_9^{(\mu)}$ and
$\WC_{10}^{(\mu)}$ at their SM values.
Despite the mis-modelling of the non-local hadronic effects in the fit results, we observe 
that the determination of $\Delta\WC_9$ and $\Delta\WC_{10}$ is always unbiased, 
thanks to the relative cancellation of all the shared parameters between the two channels.
It is worth emphasizing that a hypothetical determination of the individual  
$\widetilde{\WC}_9^{(\mu,e)}$ and $\widetilde{\WC}_{10}^{(\mu,e)}$ WCs would result
in a strong bias that mimics the behaviour of NP and makes impossible any claim in this direction.


%
In conclusion, we propose a clean, robust and model-independent method to combine
all the available information from $\bar{B}\to \bar{K}^*\ell^+\ell^-$ decays for a precise 
determination of LFU-breaking differences of WCs, \textit{i.e.} $\Delta\WC_9$ 
and $\Delta\WC_{10}$.
This relies on a shared parametrisation of the local (form-factors) and non-local ($\mathcal{H}_\lambda[q^2]$)
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.  
Figure~\ref{fig:allComponents} 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 projection for the current anomalous measurements alone given the benchmark points.
Therefore, this novel formalism can be the \textit{holy grail} to observe 
NP in $\bar{B}\to \bar{K}^*\ell^+\ell^-$ decays in near future.

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 $\bar{B}^+ \to \bar{K}^{*+} \ellell$ undergoes the same physics 
and is easily accessible at the $B$-factories, while other rare 
semi-leptonic decays such as $B^+ \to K^+ \ellell$ and $\Lambda_{b} \to \Lambda^{(*)} \ell^+\ell^-$ 
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 $b \to s \ell^+ \ell^-$ 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 $\WC_i^{\prime}$ - 
currently weakly constrained by global fits~\cite{Capdevila:2017bsm,Altmannshofer:2017yso,Hurth:2017hxg}. 
%and of possible phases in the WCs - appearing as $\Im \WC_i$ - by mean of a separate 
%analysis of the two opposite charge-conjugated modes.
%
\begin{figure}[b]
\includegraphics[width=.4\textwidth]{plots/B2Kstll_summary.pdf} 
\caption{%
    Sensitivity to \texttt{BMP}$_{\WC_{9,10}}$ scenario for the expected statistics after the \lhcb Run II.
    The relative contribution ($1,\,2,\,3\,\sigma$ contours) of each step of the analysis is shown in different colours, 
    together with the result of full amplitude method proposed in this letter. 
    \label{fig:allComponents}
}
\end{figure}


%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 contract PZ00P2-174182.   

\bibliography{references}

\FloatBarrier

\appendix

%\newpage


\section{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, 
\textit{i.e.} $\WC^{\prime (\mu)}_9$ and $\WC^{\prime(\mu)}_{10}$.
Following the formalism discussed in this letter, the primed WCs are examined by considering in addition to the 
\texttt{BMP}$_{\WC_{9,10}}$ three different modified NP scenarios for the muon only:
$\WC_{9,10}^{\prime(\mu)} = \WC^{\prime \rm{SM}}_{9,10} = 0$;
$\WC^{\prime (\mu)}_9 = \WC^{\prime (\mu)}_{10} = 0.3$; 
and $\WC^{\prime (\mu)}_9 = - \WC^{\prime (\mu)}_{10} =  0.3$.  
Notice that for the electron mode the $\WC_{9,10}^{\prime(e)}$ is set and fixed to the SM value $\WC^{\prime \rm{SM}}_{9,10} = 0$.

\begin{figure}[b]
\includegraphics[width=.4\textwidth]{plots/ellipses_CpMu_Hz.pdf} 
\caption{%
    Two-dimensional sensitivity scans for the pair of Wilson coefficients $\WC'^{(\mu)}_9$ and 
	$\WC'^{(\mu)}_{10}$ for different non-local hadronic parametrisation models for a NP scenario 
	with $\WC'^{(\mu)}_9 = \WC'^{(\mu)}_{10} = 0.3$.
    The contours correspond to $3\,\sigma$ statistical-only uncertainty bands evaluated with
	the expected statistics after \lhcb Run II.
    \label{fig:Cp_Hz}
}
\end{figure}

Figure~\ref{fig:Cp_Hz} shows the fit results for different order of the analytic expansion for 
the non-local hadronic contribution for a NP scenario with $\WC'^{(\mu)}_9 = \WC'^{(\mu)}_{10} = 0.3$, 
and yields corresponding to the \lhcb Run II expected statistics. 
The dependency on the determination of $\WC'^{(\mu)}_9$ and $\WC'^{(\mu)}_{10}$ on the 
order of the expansion clearly saturates after $\mathcal{H}_\lambda[z^3]$ and allows a measurement
of the primed Wilson coefficients for the muon decay channel $B^{0} \to K^{*0} \mumu$ independent 
on the theoretical hadronic uncertainty.
%
Figure~\ref{fig:Cp} shows the prospects for the sensitivity to the $\WC'^{(\mu)}_9$ and 
$\WC'^{(\mu)}_{10}$ Wilson coefficients corresponding to the expected statistics at 
Note that only with the full capability of the LHCb experiment it is possible 
to start disentangling the different NP hypotheses. 
%
\begin{figure}[t]
\includegraphics[width=.4\textwidth]{plots/ellipses_CpMu.pdf} 
\caption{%
    Two-dimensional sensitivity scans for the pair of Wilson coefficients $\WC'^{(\mu)}_9$ and $\WC'^{(\mu)}_{10}$
    for three NP scenarios: (blue) $\WC'^{(\mu)}_9 = \WC'^{(\mu)}_{10} = 0$, (orange) $\WC'^{(\mu)}_9 = \WC'^{(\mu)}_{10} = 0.3$ 
    and (magenta) $\WC'^{(\mu)}_9 = - \WC'^{(\mu)}_{10} = 0.3$.
    The contours correspond to $3\,\sigma$ statistical-only uncertainty bands expected for the LHCb Upgrade (dotted)  
    $50\,$fb$^{-1}$ and (solid) $\,300\,$fb$^{-1}$ statistics.
    \label{fig:Cp}
}
\end{figure}


\end{document}
