diff --git a/draft.tex b/draft.tex index 78a9d43..7bc39ea 100644 --- a/draft.tex +++ b/draft.tex @@ -137,7 +137,7 @@ Rare semileptonic $b \to s \ell^+ \ell^-$ transitions provide some of the most promising framework to search for New Physics effects. Recent analyses have indicated an anomalous pattern in measurements of lepton-flavour-universality observables. We propose a novel approach to independently and complementary clarify the nature of these effects - by performing a simultaneous amplitude analysis of $B^0 \to K^{*0} \mu^+\mu^-$ and $B^0 \to K^{*0} e^+e^-$ decays. + 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 the difference of the Wilson Coefficients ${\cal{C}}_{9}$ and ${\cal{C}}_{10}$ between electrons and muons, and are found to be insensitive to both local and non-local hadronic contributions. We show that considering the current preferred New Physics scenario a first observation of LFU breaking in a single measurement is possible with LHCb Run-II dataset. @@ -153,7 +153,7 @@ 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 -$B^{0} \to K^{*0} \mu^{+}\mu^{-}$ has been recently reported~\cite{Aaij:2015oid,Wehle:2016yoi,Aaij:2013aln,Aaij:2014pli}, +$\bar{B}^{0} \to \bar{K}^{*0} \mu^{+}\mu^{-}$ 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) @@ -195,14 +195,15 @@ in the assumption of the parametrisation of the di-lepton invariant mass. In this Letter we propose a new \textit{model-independent} approach that -from a simultaneous amplitude analysis of both -$B^0 \to K^{*0} \mu^+\mu^-$ and $B^0 \to K^{*0} e^+e^-$ decays +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 when examining -directly the difference in Wilson coefficients. +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. @@ -270,7 +271,7 @@ impossible at the current stage of the theoretical knowledge. The key feature of this strategy is to realise that all hadronic matrix elements are known to be lepton-flavour universal, and thus are shared among both semileptonic decays. -This benefits from the large statistics available for $B^0 \to K^{*0} \mu^+\mu^-$ decays +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.} @@ -304,7 +305,7 @@ \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} $\WC_9^{\rm{NP}(\mu)} = -\WC_{10}^{\rm{NP}(\mu)} = - 0.7$}, -$\WC_{9(10)}^{(e)} = \WC^{\rm{SM}}_{9(10)} = \WC_{9(10)}^{(\mu)} -(+) 0.7$, +$\WC_{9(10)}^{(e)} = \WC^{\rm{SM}}_{9(10)} = \WC_{9(10)}^{(\mu)} +(-)\,0.7$, %$\WC_9^{(\mu)} = -\WC_{10}^{(\mu)} = - 0.7$, referred to as \texttt{BMP}$_{\WC_9}$ and \texttt{BMP}$_{\WC_{9,10}}$, respectively. These points are favoured by several global fit @@ -334,7 +335,7 @@ \end{enumerate} % The stability of the model and the convergency to the global minimum is enforced by -repeating the with randomised starting parameters; +repeating the fit with randomised starting parameters; the solution with smallest negative log-likelihood is taken as the default. @@ -345,7 +346,7 @@ 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 $\mathcal{H}_\lambda[z^2]$, +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] @@ -471,7 +472,7 @@ 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 $B^+ \to K^{*+} \ellell$ undergoes the same physics +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.