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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.
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 when examining muons and electrons
directly at the level of 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 are expected.
Let us consider the differential decay rate for $\bar{B}\to \bar{K}^*\ell^+\ell^-$
decays (dominated by the on-shell $\bar{K}^{*0}$ contribution)
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{
{\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},
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 are renamed in view of that 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 $B^0 \to K^{*0} \mu^+\mu^-$ decays
that is sufficient to enable the determination of these multi-space parameters.\footnote{Note
that an amplitude analysis of the electron mode 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 \qsq range studied 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 muons and $1.1\,\GeV^2 \leq q^2 \leq 7.0\,\GeV^2$ for electrons in LHCb;
and the same kinematic regions for both the semileptonic channels in Belle II, 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 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
$\mathcal{C}^{\rm{NP}}_9 = 4.07$, $\mathcal{C}^{\rm{NP}}_{10} = - 4.07$ and $\mathcal{C}^{\rm{NP}}_7 = XX$.
This baseline model is modified in the case of muons for two NP benchmark points (BMP), \textit{i.e.}
$\WC_9^{(e)} = \WC^{\rm{NP}}_9 = \WC^{(\mu)}_9 + 1$
and $\WC_9^{(\mu)} = -\WC_{10}^{(\mu)} = - 0.7$,
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}.
An extended unbinned maximum likelihood fit is performed in 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 contraint;
%
\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 ${\cal{O}}(500)$ times 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
$\mathcal{H}_\lambda[z^{n}]$ parametrisations for the \texttt{BMP}$_{\WC_9}$ hypothesis,
with yields corresponding to LHCb Run-II.
We observe that the sensitivity to $\widetilde{\WC}_9^{(\mu,e)}$ is strongly dependent on
the underlying assumption on the modelling of the non-local matrix elements.
On the other hand, Fig.~\ref{fig:C9ellipse} shows a strong correlation between
$\widetilde{\mathcal{C}}_9^{(\mu)}$ and $\widetilde{\mathcal{C}}_9^{(e)}$.
\begin{figure}[tb]
\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}
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.
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*}[t!]
\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{%
$3\,\sigma$ 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{center}
\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