diff --git a/draft_nico.tex b/draft_nico.tex new file mode 100644 index 0000000..40db64f --- /dev/null +++ b/draft_nico.tex @@ -0,0 +1,571 @@ +% vim: set sts=4 et : +\documentclass[reprint,preprintnumbers,prd,nofootinbib]{revtex4-1} + +\usepackage{amsfonts} +\usepackage{amsmath} +\usepackage{array} +\usepackage{braket} +\usepackage{epstopdf} +\usepackage{graphicx} +\usepackage{hepparticles} +\usepackage{hepnicenames} +\usepackage{hepunits} +\usepackage{hyperref} +\usepackage[% + utf8 +]{inputenc} +\usepackage{slashed} +\usepackage{subfigure} +\usepackage{placeins} +\usepackage[% + normalem +]{ulem} +\usepackage[% + usenames, + svgnames, + dvipsnames +]{xcolor} + + +%% Shortcuts %% +\newcommand{\ie}{\textit{i.e.}} +\newcommand{\nuvec}{\vec{\nu}} +\newcommand{\refapp}[1]{appendix~\ref{app:#1}} +\newcommand{\refeq}[1]{eq.~(\ref{eq:#1})} +\newcommand{\refeqs}[2]{eqs.~(\ref{eq:#1})--(\ref{eq:#2})} +\newcommand{\reffig}[1]{figure~\ref{fig:#1}} +\newcommand{\refsec}[1]{section~\ref{sec:#1}} +\newcommand{\reftab}[1]{table~\ref{tab:#1}} +%\let\oldtheta\theta +%\renewcommand{\theta}{\vartheta} +\newcommand{\eps}{\varepsilon} +\newcommand{\para}{\parallel} +\newcommand{\Gfermi}{G_F} +%\newcommand{\dd}[2][]{{\mathrm{d}^{#1}}#2\,} +\newcommand{\dd}{\ensuremath{\textrm{d}}} +\newcommand{\order}[1]{\ensuremath{\mathcal{O}\left(#1\right)}} +\DeclareMathOperator{\sign}{sgn} +\DeclareMathOperator{\ReNew}{Re} +\DeclareMathOperator{\ImNew}{Im} +\let\Re\ReNew +\let\Im\ImNew +\DeclareMathOperator*{\sumint}{% +\mathchoice% + {\ooalign{$\displaystyle\sum$\cr\hidewidth$\displaystyle\int$\hidewidth\cr}} + {\ooalign{\raisebox{.14\height}{\scalebox{.7}{$\textstyle\sum$}}\cr\hidewidth$\textstyle\int$\hidewidth\cr}} + {\ooalign{\raisebox{.2\height}{\scalebox{.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}} + {\ooalign{\raisebox{.2\height}{\scalebox{.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}} +} +\DeclareMathOperator*{\argmax}{arg\,max} +\newcommand{\wilson}[2][]{\mathcal{C}^\text{#1}_{#2}} +\newcommand{\op}[1]{\mathcal{O}_{#1}} +\newcommand{\la}{\langle} +\newcommand{\ra}{\rangle} +\newcommand{\eqa}[1]{\begin{eqnarray} #1 \end{eqnarray}} + + +\def\deriv {\ensuremath{\mathrm{d}}} +\def\qsq {\ensuremath{q^2}\xspace} + +\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}} +\def\BdToJPsiKst {\decay{\Bd}{\jpsi\Kstarz}} +\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^-}} + +\def\lepton {{\ensuremath{\ell}}\xspace} +\def\ellm {{\ensuremath{\ell^-}}\xspace} +\def\ellp {{\ensuremath{\ell^+}}\xspace} +\def\ellell {\ensuremath{\ell^+ \ell^-}\xspace} +\def\mumu {{\ensuremath{\Pmu^+\Pmu^-}}\xspace} + +\def\lhcb {\mbox{LHCb}\xspace} +\def\belle {\mbox{Belle}\xspace} + +\def\WC {\ensuremath{\mathcal{C}}\xspace} + + + +%% Kinematic Macros %% + +%% Editing %% +%\usepackage[normalem]{ulem} % for \sout{} +%\makeatletter +%\newcommand{\todo}[1]{\textcolor{red}{\textbf{ToDo:} #1}} +%\newcommand{\ok}{\ensuremath{\checkmark}} +%\def\dvd{\@ifstar\@@dvd\@dvd} +%\newcommand{\@dvd}[1]{\textcolor{purple}{[\textbf{DvD:} #1]}} +%\newcommand{\@@dvd}[1]{\textcolor{purple}{#1}} +%\def\rsc{\@ifstar\@@rsc\@rsc} +%\newcommand{\@rsc}[1]{\textcolor{ForestGreen}{[\textbf{RsC:} #1]}} +%\newcommand{\@@rsc}[1]{\textcolor{ForestGreen}{#1}} +%\makeatother + +\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}