diff --git a/prl/B2Kstll_summary.pdf b/prl/B2Kstll_summary.pdf new file mode 100644 index 0000000..58f805a --- /dev/null +++ b/prl/B2Kstll_summary.pdf Binary files differ diff --git a/prl/draft.tex b/prl/draft.tex new file mode 100644 index 0000000..fdad69f --- /dev/null +++ b/prl/draft.tex @@ -0,0 +1,497 @@ +% 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{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} + +\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 of these decays have indicated an anomalous behaviour in measurements + of angular distributions of the decay $B^0\to + K^*\mu^+\mu^-$ and lepton-flavour-universality observables. +Unambiguously establishing if these deviations have a common nature + is of paramount importance in order to understand the observed pattern. + We propose a novel approach to independently and complementary + probe this hypothesis + by performing 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. + This method enables the direct determination of + observables that encode potential non-equal couplings of muons and electrons, + and are found to be insensitive to non-perturbative QCD effects. +If current hints of new physics are confirmed, our approach could +allow an early discovery of physics beyond the Standard +Model with LHCb Run-II datasets. +\end{abstract} + +\maketitle + +Flavour changing neutral current processes of {\textit{B}} meson decays +are crucial probes for the Standard Model (SM), +since as-yet undiscovered particles may contribute to these transitions 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 +$\bar{B}^{0} \to \bar{K}^{*0} \mu^{+}\mu^{-}$ has been recently reported~\cite{Aaij:2015oid,Wehle:2016yoi,Aaij:2013aln,Aaij:2014pli}, +notably in one of the observables with reduced theoretical uncertainties, +$P^{\prime}_{5}$~\cite{Aaij:2013qta,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 +non-perturbative QCD 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,Ciuchini:2015qxb}. +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 consistent pattern shown in +global analysis fits~\cite{Capdevila:2017bsm,Altmannshofer:2017yso,Hurth:2017hxg,Ciuchini:2017mik,Alok:2017sui}. + +The formalism of {\textit{b}} decays is commonly described within +an effective field theory~\cite{Altmannshofer:2008dz}, +which probes distinct energy scales; +with regimes classified into short-distance (high energies) perturbative +and non-calculable long-distance effects. +These can be parametrised in the weak Lagrangian in terms of +effective operators with different Lorentz structures, $\mathcal{O}_i$, +with corresponding couplings $\mathcal{C}_i$ - referred to as Wilson coefficients (WC). +Only a subset of the operators that are most sensitive to NP is examined in this work~\cite{Bobeth:2017vxj}, +\textit{i.e.} $\mathcal{O}_7$ (virtual photon exchanges), $\mathcal{O}_{9,10}$ (vector and axial currents) +and corresponding right-handed couplings with flipped helicities. +In this framework NP effects are incorporated +by introducing deviations in the WCs~\cite{Ali:1994bf} from their SM predictions, +{\textit{i.e.}} $\mathcal{C}_i = \mathcal{C}^{\mathrm{SM}}_i + \mathcal{C}^{\mathrm{NP}}_i$. +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 matrix elements that are difficult to assess reliably from first principles. +Some promising approaches suggest to extract this contribution +from data-driven analyses~\cite{Blake:2017fyh,Hurth:2017sqw} +and by exploiting analytical properties of its structure~\cite{Bobeth:2017vxj}. +However, these models still have intrinsic limitations, in particular +in the assumptions that enter in parametrisation of the di-lepton invariant mass distribution. + +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 effect relies on the strong correlation between the muon and electron modes +imposed by the lepton-flavour universality of the hadronic matrix elements. +Furthermore, in this method the full set of observables +(\textit{e.g} $R_{K^{*}}$, $P^{\prime}_{5}$ and branching fraction measurements) available in $\bar{B}\to \bar{K}^*\ell^+\ell^-$ +decays is exploited, providing unprecedented precision on LFU in a single analysis. + +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 squared invariant mass, $q^2$, 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. = \frac{1}{\Gamma} \frac{\dd^4 \Gamma}{\dd q^2 \dd^3 \Omega}, \ + \quad + \text{with}\quad + \Gamma = \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. +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 referred to ``local'' and ``non-local'' hadronic matrix elements, respectively. +The ${\cal{F}}^{(T)}_{\lambda}(q^{2})$ are form factors, +while $\mathcal{H}_\lambda(q^{2})$ encode the aforementioned non-factorisable +hadronic contributions and +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 function 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 $\mathcal{H}_\lambda[z^n]$), +after removing singularities related to the $J/\psi(1S)$ and $\psi(2S)$. +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 an accurate disentanglement between the physical +meaning of $\WC_i^{(\mu,e)}$ and the above-mentioned hadronic pollution +cannot be achieved at the current stage of the theory~\cite{Chrzaszcz:2018yza}. +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. +Note that an amplitude analysis of the electron mode only has been previously disregarded, +given the limited dataset in either LHCb or Belle experiments. +In a common framework the 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 +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. + +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 luminosity and $\sigma_{b\bar{b}} \propto \sqrt{s}$ +for LHCb, where $s$ denotes the designed centre-of-mass energy of the $b$-quark pair, +and exclusively with luminosity for Belle II. +%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 $\mathcal{C}^{\rm{SM}}_7 = -0.34$ (see~\cite{Bobeth:2017vxj} and references therein), +corresponding to a fixed renormalisation scale of $\mu = 4.2\,$GeV. +This baseline model is modified for two NP benchmark points (BMP), +$\Delta\WC_9 = - 1$ and $\Delta\WC_9 = - \Delta\WC_{10} = - 0.7$, +referred respectively to as \texttt{BMP}$_{\WC_9}$ and \texttt{BMP}$_{\WC_{9,10}}$, +where NP is inserted only in the case of muons, \textit{i.e.} $\WC_i^{(e)} = \WC_i^{\rm{SM}}$. +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 to these simulated samples, +in which multivariate Gaussian terms are added to the log-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~\cite{Bobeth:2017vxj} + included 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 Ref.~\cite{Hurth:2017sqw}, \textit{i.e.} + using instead an expansion in terms of $1 + a_{\lambda} + b_{\lambda}(q^{2}/6\,$GeV$^{2})$. + + % +\end{enumerate} +On the other hand, form factors parameters are taken from~\cite{Straub:2015ica} and, in order +to guarantee a good agreement between Light-Cone Sum Rules~\cite{Ball:1998kk,Khodjamirian:2006st} +and Lattice results~\cite{Becirevic:2006nm,Horgan:2013hoa}, +their uncertainties are doubled with respect to Ref.~\cite{Straub:2015ica}. + + +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 $\mathcal{H}_\lambda[z^n]$ +and without any parametric theoretical input, +\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 99\% confidence level statistical-only uncertainty bands and + the dotted black line indicates the LFU hypothesis. +} +\label{fig:C9ellipse} +\end{figure} +% +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. Nevertheless, its precision is still bounded to the uncertainties on the form factors, +that are found to be the limiting factor by the end of Run-II. +% +\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 99\% confidence level statistical-only uncertainty bands. + \label{fig:DeltaC9C10} +} +%\end{center} +\end{figure} + +The sensitivity to the two benchmark-like NP scenarios using the proposed pseudo observables $\Delta \WC_i$ +is shown in Fig.~\ref{fig:DeltaC9C10}. +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. +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~\cite{Albrecht:2017odf}. +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 - +insensitive to both local and non-local hadronic uncertainties. + + +\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 99\% confidence level 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} + +Experimental resolution and detector acceptance/efficiency effects +are not considered in this work, as these would require further information +from current (non-public) or planned \textit{B}-physics experiments. +Nevertheless, preliminary studies on the impact of a finite \qsq resolution are 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, even if no-correction is +applied. +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 determined to be small~\cite{Aaij:2015oid,Aaij:2016flj}, +and in the proposed formalism they benefit from the same description between the muon and electron mode. +Therefore, in this constrained framework these effects are even further suppressed and can then be neglected +for the scope of this work. + +Another important test to probe the stability of the model consists in +analysing 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 ensembles with non-zero coefficients for +$\mathcal{H}_\lambda[z^3]$ and $\mathcal{H}_\lambda[z^4]$, and we perform the fit +with $\mathcal{H}_\lambda[z^2]$. +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. +It is worth mentioning that a hypothetical determination of the individual +$\widetilde{\WC}_9^{(\mu,e)}$ and $\widetilde{\WC}_{10}^{(\mu,e)}$ WCs +can also produce a shift in their central values +that mimics the behaviour of NP~\cite{Chrzaszcz:2018yza}.% and makes impossible any claim in this direction. + + +% +In conclusion, we propose a clean 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 ($\mathcal{H}_\lambda[z^n]$) +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. +In addition, this simultaneous analysis is more robust against +experimental effects such +as mismodeling of the detector resolution, since most parameters are +effectively determined from the muon mode. This would be an important +benefit for LHCb where the electron resolution is significantly worse +than that of muons. +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 the projections for the foreseen measurements of \textit{e.g.} $R_{K^{*}}$ or $P^{\prime}_{5}$ alone - given the benchmark points. +Therefore, this novel formalism can be the most immediate method to observe unambiguously +NP in $\bar{B}\to \bar{K}^*\ell^+\ell^-$ decays. + +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}. +% +\begin{figure}[t] +\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 (68, 95, 99\% confidence level 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 contracts 173104 and 174182. + + +%\newpage +%\FloatBarrier + + +\FloatBarrier +\bibliography{references} + +\end{document} diff --git a/prl/ellipses_C9.pdf b/prl/ellipses_C9.pdf new file mode 100644 index 0000000..13293f7 --- /dev/null +++ b/prl/ellipses_C9.pdf Binary files differ diff --git a/prl/ellipses_CpMu.pdf b/prl/ellipses_CpMu.pdf new file mode 100644 index 0000000..46ebca3 --- /dev/null +++ b/prl/ellipses_CpMu.pdf Binary files differ diff --git a/prl/ellipses_CpMu_Hz.pdf b/prl/ellipses_CpMu_Hz.pdf new file mode 100644 index 0000000..8a034e5 --- /dev/null +++ b/prl/ellipses_CpMu_Hz.pdf Binary files differ diff --git a/prl/ellipses_DeltaC9C10_Nev.pdf b/prl/ellipses_DeltaC9C10_Nev.pdf new file mode 100644 index 0000000..1ed7ece --- /dev/null +++ b/prl/ellipses_DeltaC9C10_Nev.pdf Binary files differ diff --git a/prl/ellipses_DeltaC9C10_a.pdf b/prl/ellipses_DeltaC9C10_a.pdf new file mode 100644 index 0000000..80501e8 --- /dev/null +++ b/prl/ellipses_DeltaC9C10_a.pdf Binary files differ diff --git a/prl/ellipses_DeltaC9C10_b.pdf b/prl/ellipses_DeltaC9C10_b.pdf new file mode 100644 index 0000000..b20ff0d --- /dev/null +++ b/prl/ellipses_DeltaC9C10_b.pdf Binary files differ diff --git a/prl/references.bib b/prl/references.bib new file mode 100644 index 0000000..b2ef0ea --- /dev/null +++ b/prl/references.bib @@ -0,0 +1,802 @@ +@misc{suppl, + title = "{See Supplemental Material at [URL will be inserted by publisher] + for sensitivity studies on the determination of the Wilson coefficients + $\mathcal{C}_9^{\prime\,(\mu)}$ and $\mathcal{C}{10}^{\prime\,(\mu)}$}" +} + +@article{Chrzaszcz:2018yza, + author = "Chrzaszcz, Marcin and Mauri, Andrea and Serra, Nicola and + Silva Coutinho, Rafael and van Dyk, Danny", + title = "{Prospects for disentangling long- and short-distance + effects in the decays $B\to K^* \mu^+\mu^-$}", + year = "2018", + eprint = "1805.06378", + archivePrefix = "arXiv", + primaryClass = "hep-ph", + reportNumber = "TUM-HEP-1108-17, ZU-TH-25-17", + SLACcitation = "%%CITATION = ARXIV:1805.06378;%%" +} +@article{Alok:2017sui, + author = "Alok, Ashutosh Kumar and Bhattacharya, Bhubanjyoti and + Datta, Alakabha and Kumar, Dinesh and Kumar, Jacky and + London, David", + title = "{New Physics in $b \to s \mu^+ \mu^-$ after the + Measurement of $R_{K^*}$}", + journal = "Phys. Rev.", + volume = "D96", + year = "2017", + number = "9", + pages = "095009", + doi = "10.1103/PhysRevD.96.095009", + eprint = "1704.07397", + archivePrefix = "arXiv", + primaryClass = "hep-ph", + reportNumber = "UDEM-GPP-TH-17-256, WSU-HEP-1708", + SLACcitation = "%%CITATION = ARXIV:1704.07397;%%" +} +@article{Ciuchini:2017mik, + author = "Ciuchini, Marco and Coutinho, Antonio M. and Fedele, + Marco and Franco, Enrico and Paul, Ayan and Silvestrini, + Luca and Valli, Mauro", + title = "{On Flavourful Easter eggs for New Physics hunger and + Lepton Flavour Universality violation}", + journal = "Eur. Phys. J.", + volume = "C77", + year = "2017", + number = "10", + pages = "688", + doi = "10.1140/epjc/s10052-017-5270-2", + eprint = "1704.05447", + archivePrefix = "arXiv", + primaryClass = "hep-ph", + SLACcitation = "%%CITATION = ARXIV:1704.05447;%%" +} +@article{Ciuchini:2015qxb, + author = "Ciuchini, Marco and Fedele, Marco and Franco, Enrico and + Mishima, Satoshi and Paul, Ayan and Silvestrini, Luca and + Valli, Mauro", + title = "{$B\to K^* \ell^+ \ell^-$ decays at large recoil in the + Standard Model: a theoretical reappraisal}", + journal = "JHEP", + volume = "06", + year = "2016", + pages = "116", + doi = "10.1007/JHEP06(2016)116", + eprint = "1512.07157", + archivePrefix = "arXiv", + primaryClass = "hep-ph", + SLACcitation = "%%CITATION = ARXIV:1512.07157;%%" +} +@article{Grossman:1996ke, + author = "Grossman, Yuval and Worah, Mihir P.", + title = "{CP asymmetries in B decays with new physics in decay + amplitudes}", + journal = "Phys.Lett.", + volume = "B395", + pages = "241-249", + doi = "10.1016/S0370-2693(97)00068-3", + year = "1997", + eprint = "hep-ph/9612269", + archivePrefix = "arXiv", + primaryClass = "hep-ph", + reportNumber = "SLAC-PUB-7351", + SLACcitation = "%%CITATION = HEP-PH/9612269;%%", +} +@article{Fleischer:1996bv, + author = "Fleischer, Robert", + title = "{CP violation and the role of electroweak penguins in + nonleptonic $B$ decays}", + journal = "Int. J. Mod. Phys.", + volume = "A12", + year = "1997", + pages = "2459-2522", + doi = "10.1142/S0217751X97001432", + eprint = "hep-ph/9612446", + archivePrefix = "arXiv", + primaryClass = "hep-ph", + reportNumber = "TTP-96-58, TTP96-58", + SLACcitation = "%%CITATION = HEP-PH/9612446;%%" +} +@article{London:1997zk, + author = "London, David and Soni, Amarjit", + title = "{Measuring the CP angle $\beta$ in hadronic $b \to s$ penguin decays}", + journal = "Phys.Lett.", + volume = "B407", + pages = "61-65", + doi = "10.1016/S0370-2693(97)00695-3", + year = "1997", + eprint = "hep-ph/9704277", + archivePrefix = "arXiv", + primaryClass = "hep-ph", + reportNumber = "UDEM-GPP-TH-97-40", + SLACcitation = "%%CITATION = HEP-PH/9704277;%%", +} +@article{Ciuchini:1997zp, + author = "Ciuchini, Marco and Franco, E. and Martinelli, G. and + Masiero, A. and Silvestrini, L.", + title = "{CP violating B decays in the standard model and + supersymmetry}", + journal = "Phys.Rev.Lett.", + volume = "79", + pages = "978-981", + doi = "10.1103/PhysRevLett.79.978", + year = "1997", + eprint = "hep-ph/9704274", + archivePrefix = "arXiv", + primaryClass = "hep-ph", + reportNumber = "CERN-TH-97-047, CERN-TH-97-47, ROM2F-97-10", + SLACcitation = "%%CITATION = HEP-PH/9704274;%%", +} +@article{Altmannshofer:2008dz, + author = "Altmannshofer, Wolfgang and Ball, Patricia and Bharucha, + Aoife and Buras, Andrzej J. and Straub, David M. and Wick, + Michael", + title = "{Symmetries and Asymmetries of $B \to K^{*} \mu^{+} + \mu^{-}$ Decays in the Standard Model and Beyond}", + journal = "JHEP", + volume = "01", + year = "2009", + pages = "019", + doi = "10.1088/1126-6708/2009/01/019", + eprint = "0811.1214", + archivePrefix = "arXiv", + primaryClass = "hep-ph", + reportNumber = "IPPP-08-58, DCPT-08-116, TUM-HEP-696-08", + SLACcitation = "%%CITATION = ARXIV:0811.1214;%%" +} +@article{Ali:1994bf, + author = "Ali, Ahmed and Giudice, G. F. and Mannel, T.", + title = "{Towards a model independent analysis of rare $B$ + decays}", + journal = "Z. Phys.", + volume = "C67", + year = "1995", + pages = "417-432", + doi = "10.1007/BF01624585", + eprint = "hep-ph/9408213", + archivePrefix = "arXiv", + primaryClass = "hep-ph", + reportNumber = "CERN-TH-7346-94", + SLACcitation = "%%CITATION = HEP-PH/9408213;%%" +} +@article{Aaij:2015oid, + author = "Aaij, Roel and others", + title = "{Angular analysis of the $B^{0} \to K^{*0} \mu^{+} + \mu^{-}$ decay using 3 fb$^{-1}$ of integrated + luminosity}", + collaboration = "LHCb", + journal = "JHEP", + volume = "02", + year = "2016", + pages = "104", + doi = "10.1007/JHEP02(2016)104", + eprint = "1512.04442", + archivePrefix = "arXiv", + primaryClass = "hep-ex", + reportNumber = "CERN-PH-EP-2015-314, LHCB-PAPER-2015-051", + SLACcitation = "%%CITATION = ARXIV:1512.04442;%%" +} + +@article{Descotes-Genon:2015uva, + author = "Descotes-Genon, Sébastien and Hofer, Lars and Matias, + Joaquim and Virto, Javier", + title = "{Global analysis of $b\to s\ell\ell$ anomalies}", + journal = "JHEP", + volume = "06", + year = "2016", + pages = "092", + doi = "10.1007/JHEP06(2016)092", + eprint = "1510.04239", + archivePrefix = "arXiv", + primaryClass = "hep-ph", + reportNumber = "LPT-ORSAY-15-68, QFET-2015-29, SI-HEP-2015-19", + SLACcitation = "%%CITATION = ARXIV:1510.04239;%%" +} + +@article{Wehle:2016yoi, + author = "Wehle, S. and others", + title = "{Lepton-Flavor-Dependent Angular Analysis of $B\to K^\ast + \ell^+\ell^-$}", + collaboration = "Belle", + year = "2016", + eprint = "1612.05014", + archivePrefix = "arXiv", + primaryClass = "hep-ex", + reportNumber = "BELLE-PREPRINT-2016-15, KEK-PREPRINT-2016-54", + SLACcitation = "%%CITATION = ARXIV:1612.05014;%%" +} +@article{Gauld:2013qja, + author = "Gauld, Rhorry and Goertz, Florian and Haisch, Ulrich", + title = "{An explicit Z'-boson explanation of the $B \to K^* \mu^+ + \mu^-$ anomaly}", + journal = "JHEP", + volume = "01", + year = "2014", + pages = "069", + doi = "10.1007/JHEP01(2014)069", + eprint = "1310.1082", + archivePrefix = "arXiv", + primaryClass = "hep-ph", + SLACcitation = "%%CITATION = ARXIV:1310.1082;%%" +} +@article{Buras:2013qja, + author = "Buras, Andrzej J. and Girrbach, Jennifer", + title = "{Left-handed $Z'$ and $Z$ FCNC quark couplings facing new + $b \to s \mu^+ \mu^-$ data}", + journal = "JHEP", + volume = "12", + year = "2013", + pages = "009", + doi = "10.1007/JHEP12(2013)009", + eprint = "1309.2466", + archivePrefix = "arXiv", + primaryClass = "hep-ph", + reportNumber = "FLAVOUR(267104)-ERC-50", + SLACcitation = "%%CITATION = ARXIV:1309.2466;%%" +} +@article{Altmannshofer:2013foa, + author = "Altmannshofer, Wolfgang and Straub, David M.", + title = "{New Physics in $B \to K^*\mu\mu$?}", + journal = "Eur. Phys. J.", + volume = "C73", + year = "2013", + pages = "2646", + doi = "10.1140/epjc/s10052-013-2646-9", + eprint = "1308.1501", + archivePrefix = "arXiv", + primaryClass = "hep-ph", + reportNumber = "FERMILAB-PUB-13-310-T, MITP-13-047", + SLACcitation = "%%CITATION = ARXIV:1308.1501;%%" +} +@article{Crivellin:2015era, + author = "Crivellin, Andreas and Hofer, Lars and Matias, Joaquim + and Nierste, Ulrich and Pokorski, Stefan and Rosiek, + Janusz", + title = "{Lepton-flavour violating $B$ decays in generic $Z'$ + models}", + journal = "Phys. Rev.", + volume = "D92", + year = "2015", + number = "5", + pages = "054013", + doi = "10.1103/PhysRevD.92.054013", + eprint = "1504.07928", + archivePrefix = "arXiv", + primaryClass = "hep-ph", + reportNumber = "CERN-PH-TH-2015-091, TTP15-018", + SLACcitation = "%%CITATION = ARXIV:1504.07928;%%" +} +# Leptoquarks +@article{Hiller:2014yaa, + author = "Hiller, Gudrun and Schmaltz, Martin", + title = "{$R_K$ and future $b \to s \ell \ell$ physics beyond the + standard model opportunities}", + journal = "Phys. Rev.", + volume = "D90", + year = "2014", + pages = "054014", + doi = "10.1103/PhysRevD.90.054014", + eprint = "1408.1627", + archivePrefix = "arXiv", + primaryClass = "hep-ph", + reportNumber = "DO-TH-14-17", + SLACcitation = "%%CITATION = ARXIV:1408.1627;%%" +} +@article{Biswas:2014gga, + author = "Biswas, Sanjoy and Chowdhury, Debtosh and Han, Sangeun + and Lee, Seung J.", + title = "{Explaining the lepton non-universality at the LHCb and + CMS within a unified framework}", + journal = "JHEP", + volume = "02", + year = "2015", + pages = "142", + doi = "10.1007/JHEP02(2015)142", + eprint = "1409.0882", + archivePrefix = "arXiv", + primaryClass = "hep-ph", + SLACcitation = "%%CITATION = ARXIV:1409.0882;%%" +} +@article{Gripaios:2014tna, + author = "Gripaios, Ben and Nardecchia, Marco and Renner, S. A.", + title = "{Composite leptoquarks and anomalies in $B$-meson + decays}", + journal = "JHEP", + volume = "05", + year = "2015", + pages = "006", + doi = "10.1007/JHEP05(2015)006", + eprint = "1412.1791", + archivePrefix = "arXiv", + primaryClass = "hep-ph", + SLACcitation = "%%CITATION = ARXIV:1412.1791;%%" +} +# QCD issues +@article{Jager:2012uw, + author = "Jäger, S. and Martin Camalich, J.", + title = "{On $B \to V \ell \ell$ at small dilepton invariant + mass, power corrections, and new physics}", + journal = "JHEP", + volume = "05", + year = "2013", + pages = "043", + doi = "10.1007/JHEP05(2013)043", + eprint = "1212.2263", + archivePrefix = "arXiv", + primaryClass = "hep-ph", + SLACcitation = "%%CITATION = ARXIV:1212.2263;%%" +} +@article{Jager:2014rwa, + author = "Jäger, Sebastian and Martin Camalich, Jorge", + title = "{Reassessing the discovery potential of the $B \to K^{*} + \ell^+\ell^-$ decays in the large-recoil region: SM + challenges and BSM opportunities}", + journal = "Phys. Rev.", + volume = "D93", + year = "2016", + number = "1", + pages = "014028", + doi = "10.1103/PhysRevD.93.014028", + eprint = "1412.3183", + archivePrefix = "arXiv", + primaryClass = "hep-ph", + SLACcitation = "%%CITATION = ARXIV:1412.3183;%%" +} +@article{Aaij:2014ora, + author = "Aaij, Roel and others", + title = "{Test of lepton universality using $B^{+}\rightarrow + K^{+}\ell^{+}\ell^{-}$ decays}", + collaboration = "LHCb", + journal = "Phys. Rev. Lett.", + volume = "113", + year = "2014", + pages = "151601", + doi = "10.1103/PhysRevLett.113.151601", + eprint = "1406.6482", + archivePrefix = "arXiv", + primaryClass = "hep-ex", + reportNumber = "CERN-PH-EP-2014-140, LHCB-PAPER-2014-024", + SLACcitation = "%%CITATION = ARXIV:1406.6482;%%" +} +@article{Aaij:2017vbb, + author = "Aaij, R. and others", + title = "{Test of lepton universality with $B^{0} \rightarrow + K^{*0}\ell^{+}\ell^{-}$ decays}", + collaboration = "LHCb", + journal = "JHEP", + volume = "08", + year = "2017", + pages = "055", + doi = "10.1007/JHEP08(2017)055", + eprint = "1705.05802", + archivePrefix = "arXiv", + primaryClass = "hep-ex", + reportNumber = "LHCB-PAPER-2017-013, CERN-EP-2017-100", + SLACcitation = "%%CITATION = ARXIV:1705.05802;%%" +} +% BR measurements +@article{Aaij:2013aln, + author = "Aaij, R and others", + title = "{Differential branching fraction and angular analysis of + the decay $B_s^0\to\phi\mu^{+}\mu^{-}$}", + collaboration = "LHCb", + journal = "JHEP", + volume = "07", + year = "2013", + pages = "084", + doi = "10.1007/JHEP07(2013)084", + eprint = "1305.2168", + archivePrefix = "arXiv", + primaryClass = "hep-ex", + reportNumber = "CERN-PH-EP-2013-078, LHCB-PAPER-2013-017", + SLACcitation = "%%CITATION = ARXIV:1305.2168;%%" +} +@article{Aaij:2014pli, + author = "Aaij, R. and others", + title = "{Differential branching fractions and isospin asymmetries + of $B \to K^{(*)} \mu^+ \mu^-$ decays}", + collaboration = "LHCb", + journal = "JHEP", + volume = "06", + year = "2014", + pages = "133", + doi = "10.1007/JHEP06(2014)133", + eprint = "1403.8044", + archivePrefix = "arXiv", + primaryClass = "hep-ex", + reportNumber = "LHCB-PAPER-2014-006, CERN-PH-EP-2014-055", + SLACcitation = "%%CITATION = ARXIV:1403.8044;%%" +} +@article{Hiller:2003js, + author = "Hiller, Gudrun and Kruger, Frank", + title = "{More model-independent analysis of $b \to s$ processes}", + journal = "Phys. Rev.", + volume = "D69", + year = "2004", + pages = "074020", + doi = "10.1103/PhysRevD.69.074020", + eprint = "hep-ph/0310219", + archivePrefix = "arXiv", + primaryClass = "hep-ph", + reportNumber = "LMU-18-03, TUM-HEP-519-03", + SLACcitation = "%%CITATION = HEP-PH/0310219;%%" +} +@article{Bordone:2016gaq, + author = "Bordone, Marzia and Isidori, Gino and Pattori, Andrea", + title = "{On the Standard Model predictions for $R_K$ and + $R_{K^*}$}", + journal = "Eur. Phys. 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Rev. Lett.", + volume = "111", + year = "2013", + pages = "191801", + doi = "10.1103/PhysRevLett.111.191801", + eprint = "1308.1707", + archivePrefix = "arXiv", + primaryClass = "hep-ex", + reportNumber = "LHCB-PAPER-2013-037, CERN-PH-EP-2013-146", + SLACcitation = "%%CITATION = ARXIV:1308.1707;%%" +} + diff --git a/prl/supplemental_material.tex b/prl/supplemental_material.tex new file mode 100644 index 0000000..c0fd7e8 --- /dev/null +++ b/prl/supplemental_material.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}