diff --git a/draft.tex b/draft.tex index 8513f82..65a92ed 100644 --- a/draft.tex +++ b/draft.tex @@ -30,7 +30,6 @@ %% Shortcuts %% \newcommand{\ie}{\textit{i.e.}} \newcommand{\nuvec}{\vec{\nu}} -\newcommand{\order}[1]{\mathcal{O}\left({#1}\right)} \newcommand{\refapp}[1]{appendix~\ref{app:#1}} \newcommand{\refeq}[1]{eq.~(\ref{eq:#1})} \newcommand{\refeqs}[2]{eqs.~(\ref{eq:#1})--(\ref{eq:#2})} @@ -42,7 +41,9 @@ \newcommand{\eps}{\varepsilon} \newcommand{\para}{\parallel} \newcommand{\Gfermi}{G_F} -\newcommand{\dd}[2][]{{\mathrm{d}^{#1}}#2\,} +%\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} @@ -63,6 +64,41 @@ \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 %% @@ -100,8 +136,155 @@ \maketitle -\section{Introduction} -\label{sec:intro} + +In this work we assume the $\Bz \to \Kstarz \ellell$ 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 correlator parameters for each polarisation 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 +correlators $\mathcal{H}_\lambda(z)$, where $\lambda=\perp, \para,0$ is +the polarization of the \Kstarz, assuming the analytic expansion to be cut at +the order $z^2$. + +In order to test the validity of the adopted parametrizations we repeat +the fit with different configurations: +\begin{itemize} + \item We include the $\mathcal{H}_\lambda(z)$ SM prediction + from~\cite{Danny_2017} as gaussian contraint to the fit. + \item We remove any theoretical assumption on $\mathcal{H}_\lambda(z)$ + and let free-floating all the parameters up to the order $z^2$. + \item We increase the order of the analytical expansion of $\mathcal{H}_\lambda(z)$ + 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 $\Delta\WC_{10}$. + + + + + +\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 Run2. + \label{fig:DeltaC9C10} +} +\end{figure} + + +\begin{figure}[tbh] +\includegraphics[width=.4\textwidth]{plots/B2Kstll_summary.pdf} +\caption{% + . + \label{fig:allComponents} +} +\end{figure} + + + + + \bibliography{references}