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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 Physics effects.
Recent analyses have indicated an anomalous pattern in measurements of lepton-flavour-universality observables.
We propose a novel approach to independently and complementary clarify the nature of these effects
by performing a simultaneous amplitude analysis of $B^0 \to K^{*0} \mu^+\mu^-$ and $B^0 \to K^{*0} e^+e^-$ decays.
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.
We show that considering the current preferred New Physics scenario a first observation of LFU breaking in a single measurement is possible with LHCb Run-II dataset.
\end{abstract}
\maketitle
Flavour change 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 to loop effects 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.
An anomalous behaviour in angular and branching fraction analyses of the decay channel
$B^{0} \to K^{*0} \mu^{+}\mu^{-}$ 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 $C_i$ 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 systematically incorporated
by introducing deviations exclusively in the Wilson coefficients (WC)~\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.
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 either extract these non-local hadronic elements
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 amplitude analysis of both
$B^0 \to K^{*0} \mu^+\mu^-$ and $B^0 \to K^{*0} e^+e^-$ decays
can, for the first time, unambiguously determine LFU-breaking from direct measurements of WCs.
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 are expected.
% Start of proper paper discussion
In this work we assume the $\bar{B}\to \bar{K}^*\ell^+\ell^-$ decay being completely dominated
by the on-shell $\Kstarz$ ($p$-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 $\vec{\Omega} = (\cos \theta_l, \cos \theta_K, \phi)$ [Ref].
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}, \
\qquad
\text{with}\quad
\Gamma_i = \int_{q^2} \dd q^2 \frac{\dd\Gamma}{\dd q^2}
\end{equation}
%
where the \qsq range is defined differently for the two semileptonic channels.
Prospects for the \lhcb experiment are studied within $1.1\,\GeV^2
\leq q^2 \leq 8.0\,\GeV^2$ and $11.0\,\GeV^2 \leq q^2 \leq 12.5\,\GeV^2$
for the muonic mode, and $1.1\,\GeV^2 \leq q^2 \leq 8.0\,\GeV^2$ for
the electron mode. Studies on the \belle~II experiment uses the same
kinematic regions for both the semileptonic channels, namely
$1.1\,\GeV^2 \leq q^2 \leq 9.0\,\GeV^2$ and $10.0\,\GeV^2
\leq q^2 \leq 13.0\,\GeV^2$. This definition of \qsq ranges
corresponds approximately to what already in use in published work by
\lhcb and \belle~\cite{LHCb-PAPER-2015-051,LHCB-PAPER-2017-013,Belle-Kstll}.
For a definition of $\dd ^4\Gamma/(\dd q^2 \dd ^3\Omega)$ we refer
to~\cite{Bobeth:2008ij,Altmannshofer:2008dz} and references therein.
Concerning the description of the non-local hadronic matrix element we
considered the two recently proposed parametrizations of~\cite{Danny_2017}
and~\cite{Christoph}.
In order to study the sensitivity to different NP scenarios, we
generated a large number of toys using the following set of parameters:
the non-local hadronic parameters as in~\cite{Danny_2017},
the form factor parameters as determined from~\cite{Straub:2015ica},
but with twice the stated uncertainty, and the CKM Wolfenstein
parameters~\cite{Bona:2006ah}; all the above-mentioned parameters
are shared between the two semileptonic modes and are treated as
nuisance parameters, while only the Wilson coefficients $\WC_9^{(\mu,e)}$
and $\WC_{10}^{(\mu,e)}$ are kept separately for the two channels.
We define three benchmark points, depending on the values of the Wilson coefficients
used to generate the ensembles: one ``SM", where the values of the Wilson coefficients
are set to their SM values, and two BSM scenarios, one labelled as ``NP$_{\WC_9}$",
where NP is inserted only in $\WC_9^{(\mu)}$ with a shift with respect to the SM of
$\WC_9^{\text{NP} (\mu)} = - 1$, and the latter
labelled as ``NP$_{\WC_9-\WC_{10}}$", where NP is inserted in $\WC_9^{(\mu)}$
and $\WC_{10}^{(\mu)}$ with a shift with respect to the SM of
$\WC_9^{\text{NP} (\mu)} = -\WC_{10}^{\text{NP} (\mu)} = - 0.7$.
The number of events used to generate the pseudo-experiments is
obtained from~\cite{LHCb-PAPER-2015-051,LHCB-PAPER-2017-013,Belle-Kstll}
and extrapolated to the current and future expected statistics to study
the prospects of the \lhcb and \belle~II experiments.
In all cases under study, we perform an extended unbinned maximum likelihood fit by
including in the likelihood function poissonian terms that take into account the muon
and electron yields obtained in the different kinematic regions.
Multivariate gaussian terms are added to the likelihood to incorporate prior knowledge
on the nuisance parameters as introduced above.
For each generated sample the fit is repeated several times with different initialization
of the fitted parameters.
The authors of~\cite{Danny_2017} proposed a SM prediction of the non-local
hadronic matrix elements $\mathcal{H}_\lambda(z)$, where $\lambda=\perp, \para,0$
is the polarization of the \Kstarz, operating an analytic expansion in the ``conformal”
variable $z(q^2)$ and assuming a truncation at the order $z^2$ (in the following we refer
to the analytic expansion of $\mathcal{H}_\lambda$ truncated at the order $z^n$
as $\mathcal{H}_\lambda[z^n]$).
In order to test the validity of the adopted parametrizations we repeat
the fit with different configurations:
\begin{itemize}
\item We include the $\mathcal{H}_\lambda[z^2]$ SM prediction
from~\cite{Danny_2017} as gaussian contraint to the fit.
\item We remove any theoretical assumption on $\mathcal{H}_\lambda[z^2]$
and let free-floating all the parameters.
\item We increase the order of the analytical expansion of $\mathcal{H}_\lambda$
up to the (free-floating) order of $z^3$ and $z^4$.
\item We re-parametrize the description of the non-local hadronic matrix
element as proposed in~\cite{Christoph}.
\end{itemize}
We observe that the sensitivity to $\WC_9^{(\mu,e)}$ is strongly dependent on
the assumption underlying the parametrization of the non-local matrix element,
see Fig.~\ref{fig:C9ellipse}.
In this work we renounce to a precise determination of $\WC_9^{(\mu,e)}$,
that will be renamed as $\widetilde{\mathcal{C}}_9^{(\mu,e)}$ in the
following, in view of the fact that a precise disentanglement between the physical
meaning of $\WC_9^{(\mu,e)}$ and the above-mentioned hadronic pollution is
impossible at the current stage of the theoretical knowledge.
On the other hand, Fig.~\ref{fig:C9ellipse} shows a strong correlation between
$\widetilde{\mathcal{C}}_9^{(\mu)}$ and $\widetilde{\mathcal{C}}_9^{(e)}$.
The method proposed in this letter profits from this correlation to investigate LFU-breaking
directly at the level of Wilson coefficients.
In fact, Fig.~\ref{fig:C9ellipse} also proves that the difference
\begin{equation}
\Delta \WC_9 = \widetilde{\mathcal{C}}_9^{(\mu)} - \widetilde{\mathcal{C}}_9^{(e)}
\end{equation}
is independent on the chosen parametrization and a non-zero $\Delta \WC_9$ would be
a clear sign of LFU-violation.
\begin{figure}[tbh]
\includegraphics[width=.4\textwidth]{plots/ellipses_C9.pdf}
\caption{%
$3\,\sigma$ contours in the $\widetilde{\mathcal{C}}_9^{(\mu)}$ - $\widetilde{\mathcal{C}}_9^{(e)}$
plane obtained for different parametrizations of the non-local hadronic effects from a large number of toys
generated with the NP$_{\WC_9}$ scenario and the expected statistics after the \lhcb Run2.
\label{fig:C9ellipse}
}
\end{figure}
We note that, as commonly stated in the literature~[Refs.], the determination of
$\WC_{10}^{(\mu,e)}$ doesn't suffer from the lack of knowledge on the
non-local hadronic effects and it's hence independent on the tested parametrization.
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 to the SM as $4.6\,(5.3)\,\sigma$ for
the \lhcb RunII, $xx(yy)\,\sigma$ for the \belle II 50~ab$^{-1}$ dataset and
$xx(yy)\,\sigma$ for the \lhcb 50~fb$^{-1}$ Upgrade for the NP$_{\WC_9}$
(NP$_{\WC_9-\WC_{10}}$) scenario respectively.
Modelling detector effects as \qsq and angles resolution or detector acceptance and
efficiency is hardly possible without access to (non-public) information of the current
$B$~physics experiments.
A first rudimentary study on the impact of a finite \qsq resolution is preformed
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~\cite{LHCB-PAPER-2017-013}.
Despite the low \qsq asymmetric tail, the determination of $\Delta\WC_9$ and
$\Delta\WC_{10}$ remains unbiased.
An other 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 the potential issue that can rise if the truncation
$\mathcal{H}_\lambda[z^2]$ 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 ``provocative" set of values that minimize the tension with the $P_5'$
``anomaly"~\cite{LHCb-PAPER-2015-051} 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, 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,
while {\color{red} test bias in C10 and Upgrade}
\begin{figure}[tbh]
\includegraphics[width=.4\textwidth]{plots/ellipses_DeltaC9C10_a.pdf} \\
\includegraphics[width=.4\textwidth]{plots/ellipses_DeltaC9C10_b.pdf}
\caption{%
$3\,\sigma$ contours in the $\Delta\WC_9$ - $\Delta\WC_{10}$ plane obtained for
different parametrizations of the non-local hadronic effects from a large number of toys
generated with the NP$_{\WC_9}$ (top) and NP$_{\WC_9-\WC_{10}}$ (bottom)
scenario and the expected statistics after the \lhcb RunII.
\label{fig:DeltaC9C10}
}
\end{figure}
In conclusion, we propose a clean, robust and model-independent method to combine
all the available information from $\Bz \to \Kstarz \ellell$ decays for a precise
determination of LFU-breaking difference of Wilson coefficients $\Delta\WC_9$
and $\Delta\WC_{10}$.
Fig.~\ref{fig:allComponents} shows the contribution of all the single constituents of
the analysis and how the proposed method takes advantage of the complete description
of the decay.
This approach exploits possible differences between the muon and electron channels,
by mean of a shared parametrization of all the common local (form-factors) and non-local
($\mathcal{H}_\lambda$) hadronic matrix elements.
This results in a clean simultaneous analysis of the two channels, independent on any
theoretical uncertainty; in addition, this method doesn't suffer from the limited
statistics of the electron channel, that would make impossible to perform a complete
angular analysis of the single $\Bz \to \Kstarz e^+ e^-$ decay channel.
\begin{figure}[tbh]
\includegraphics[width=.4\textwidth]{plots/B2Kstll_summary.pdf}
\caption{%
Sensitivity to the NP$_{\WC_9-\WC_{10}}$ scenario for the expected statistics after the \lhcb RunII.
The relative contribution ($1,\,2,\,3\,\sigma$ contours) of each step of the analysis is shown in different colors, together with the
result of full amplitude method proposed in this letter.
\label{fig:allComponents}
}
\end{figure}
\bibliography{references}
\end{document}
Rafael Silva Coutinho