diff --git a/prl/draft.tex b/prl/draft.tex index fdad69f..306b613 100644 --- a/prl/draft.tex +++ b/prl/draft.tex @@ -268,10 +268,8 @@ and fixed to its SM value, given its universal coupling to photons and the strong constraint from radiative $B$ decays~\cite{Paul:2016urs}. In the following, all the right-handed WCs are fixed to their SM values, \textit{i.e.} $\WC_i^{\prime\,(\mu,e)} = 0$, -see Supplemental Material for sensitivity studies on the determination of the -WCs $\WC_9^{\prime\,(\mu)}$ and $\WC_{10}^{\prime\,(\mu)}$~\cite{suppl}. -%while sensitivity studies on the determination of the WCs $\WC_9^{\prime\,(\mu)}$ and $\WC_{10}^{\prime\,(\mu)}$ -%are detailed in the appendix. +while sensitivity studies on the determination of the WCs $\WC_9^{\prime\,(\mu)}$ and $\WC_{10}^{\prime\,(\mu)}$ +are detailed in the supplemental material to this Letter~\cite{appendix}. 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 @@ -345,7 +343,7 @@ \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} +\includegraphics[width=.4\textwidth]{ellipses_C9.pdf} \caption{% Two-dimensional sensitivity scans for the pair of Wilson coefficients $\widetilde{\mathcal{C}}_9^{(\mu)}$ and $\widetilde{\mathcal{C}}_9^{(e)}$ @@ -364,8 +362,8 @@ % \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} +\includegraphics[width=.4\textwidth]{ellipses_DeltaC9C10_a.pdf}\\%\quad\quad\quad\quad +\includegraphics[width=.4\textwidth]{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 @@ -396,7 +394,7 @@ \begin{figure}[bth!] -\includegraphics[width=.4\textwidth]{plots/ellipses_DeltaC9C10_Nev.pdf} +\includegraphics[width=.4\textwidth]{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}$_{\WC_{9,10}}$. @@ -473,7 +471,7 @@ currently weakly constrained by global fits~\cite{Capdevila:2017bsm,Altmannshofer:2017yso,Hurth:2017hxg}. % \begin{figure}[t] -\includegraphics[width=.4\textwidth]{plots/B2Kstll_summary.pdf} +\includegraphics[width=.4\textwidth]{B2Kstll_summary.pdf} \caption{% Sensitivity to \texttt{BMP}$_{\WC_{9,10}}$ scenario for the expected statistics after the \lhcb Run II. The relative contribution (68, 95, 99\% confidence level contours) of each step of the analysis is shown in different colours, diff --git a/prl/suppl.tex b/prl/suppl.tex new file mode 100644 index 0000000..c0fd7e8 --- /dev/null +++ b/prl/suppl.tex @@ -0,0 +1,172 @@ +% 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} + + + +\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} + + +\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 $\WC^{\prime \rm{SM}}_{9,10} = 0$. + +\begin{figure}[t] +\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 99\% confidence level 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 +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}[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 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} + + +\end{document} diff --git a/prl/supplemental_material.tex b/prl/supplemental_material.tex deleted file mode 100644 index c0fd7e8..0000000 --- a/prl/supplemental_material.tex +++ /dev/null @@ -1,172 +0,0 @@ -% 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} - - - -\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} - - -\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 $\WC^{\prime \rm{SM}}_{9,10} = 0$. - -\begin{figure}[t] -\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 99\% confidence level 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 -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}[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 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} - - -\end{document}