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\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

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%% Rare decays
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\def\ellell     {\ensuremath{\ell^+ \ell^-}\xspace}
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\def\lhcb {\mbox{LHCb}\xspace}
\def\belle  {\mbox{Belle}\xspace}

\def\WC  {\ensuremath{\mathcal{C}}\xspace}



\begin{document}

\allowdisplaybreaks

\preprint{ZU-TH-15/18}
\title{Supplemental material for ``Towards establishing Lepton Flavour Universality violation in $\bar{B}\to \bar{K}^*\ell^+\ell^-$ 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}


%\appendix

%\section{Supplemental material}


\maketitle

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 $\WC^{\prime \rm{SM}}_{9,10} = 0$.


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 $\WC'^{(\mu)}_9 = \WC'^{(\mu)}_{10} = 0.3$
%
\begin{figure}[h]
\includegraphics[width=.4\textwidth]{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 99\% confidence level statistical-only uncertainty bands evaluated with
	the expected statistics after \lhcb Run II.
    \label{fig:Cp_Hz}
}
\end{figure}
%
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
the LHCb upgrade with $50\,$fb$^{-1}$ and $\,300\,$fb$^{-1}$.
Note that only with the full capability of the LHCb experiment it is possible 
to start disentangling the different NP hypotheses. 


%
\begin{figure}[b]
\includegraphics[width=.4\textwidth]{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 99\% confidence level statistical-only uncertainty bands expected for the LHCb Upgrade (dotted)  
    $50\,$fb$^{-1}$ and (solid) $\,300\,$fb$^{-1}$ statistics.
    \label{fig:Cp}
}
\end{figure}

%\FloatBarrier

\end{document}
