diff --git a/draft.tex b/draft.tex index b43a76a..334d934 100644 --- a/draft.tex +++ b/draft.tex @@ -203,7 +203,7 @@ This relies on the strong correlation {\color{red} between the two decay modes} when examining muons and electrons directly at the level of 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, {\color{blue} most} {\color{red} more} stringent constraints on LFU for a single measurement are expected. +decays is exploited, {\color{red}providing the most stringent constraints on LFU from a single measurement.} {\color{blue} and therefore, most stringent constraints on LFU for a single measurement are expected. } {\color{blue} Let us consider the differential decay rate for $\bar{B}\to \bar{K}^*\ell^+\ell^-$ decays (dominated by the on-shell $\bar{K}^{*0}$ contribution) } @@ -299,10 +299,10 @@ and assumes improvements in the electron mode resolution for LHCb~\cite{Lionetto:XX}. Within the SM setup the Wilson coefficients are set to -$\mathcal{C}^{\rm{NP}}_9 = 4.27$, $\mathcal{C}^{\rm{NP}}_{10} = - 4.17$ and $\mathcal{C}^{\rm{NP}}_7 = -0.34$. +$\mathcal{C}^{\rm{SM}}_9 = 4.27$, $\mathcal{C}^{\rm{SM}}_{10} = - 4.17$ 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.} -$\WC_9^{(e)} = \WC^{\rm{NP}}_9 = \WC^{(\mu)}_9 + 1$ -and $\WC_9^{(\mu)} = -\WC_{10}^{(\mu)} = - 0.7$, + {\color{red} $\WC^{\rm{NP}(\mu)}_9 = - 1$ } +and {\color{red} $\WC_9^{\rm{NP}(\mu)} = -\WC_{10}^{\rm{NP}(\mu)} = - 0.7$}, 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}. @@ -332,7 +332,8 @@ The stability of the model and the convergency to the global minimum is enforced by repeating the fit ${\cal{O}}(500)$ times with randomised starting parameters; the solution with smallest negative log-likelihood is taken as the default. - +{\color{red} We should rephrase in a way that the referee doesn't ask how many times we repeated the fit :) )} + 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. @@ -375,26 +376,30 @@ \end{center} \end{figure*} -The sensitivity to the two NP scenario previously discussed using the proposed observables $\Delta \WC_i$ +The sensitivity to the two NP scenarios previously discussed 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 to the SM as $4.6\,(5.3)\,\sigma$ for 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 NP$_{\WC_9}$ -(NP$_{\WC_9-\WC_{10}}$) scenario respectively. +$9.0(9.6)\,\sigma$ 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_Nev}. +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.} \textbf{TODO: Add also here the plot for the upgrade and also comment on the result itself.} \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 different non-local hadronic parametrisation models - evaluated at (left) \texttt{BMP}$_{\WC_9}$ and (right) \texttt{BMP}$_{\WC_{9,10}}$, - and with the expected statistics after \lhcb Run II. - The contours correspond to $3\,\sigma$ statistical-only uncertainty bands. + {\color{red} 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 \belle II 50~ab$^{-1}$ (dashed) + and \lhcb Upgrade $50\,$fb$^{-1}$ (dotted) and $\,300\,$fb$^{-1}$ (solid) statistics.} \label{fig:DeltaC9C10_Nev} } \end{figure} @@ -413,7 +418,7 @@ 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 -$\mathcal{H}_\lambda[z^2]$ is not a good description of nature. +$\mathcal{H}_\lambda[z^n]$ 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 $\mathcal{H}_\lambda[z^2]$. @@ -423,25 +428,37 @@ $\WC_{10}^{(\mu)}$ at their SM values. 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, -while {\color{red} test bias in C10 and Upgrade} +thanks to the relative cancellation of all the shared parameters between the two channels. +{\color{red} On the contrary, an hypothetical determination of the single WC +$\widetilde{\WC}_9^{(\mu,e)}$ and $\widetilde{\WC}_{10}^{(\mu,e)}$ would result +in a strong bias that mimics the behaviour of NP and makes impossible any claim in this direction.} \textbf{Todo: comment in the conclusion on the use case of the prime WC and also the potential of analysing other channels, in particular for the K*+ in Belle} In conclusion, we propose a clean, robust and model-independent method to combine -all the available information from $\Bz \to \Kstarz \ellell$ decays for a precise -determination of LFU-breaking difference of Wilson coefficients $\Delta\WC_9$ -and $\Delta\WC_{10}$. +all the available information from $B^{0} \to K^{*0} \ellell$ decays for a precise +determination of LFU-breaking differences of WCs $\Delta\WC_9$ +and $\Delta\WC_{10}$, independent on any theoretical uncertainties. Fig.~\ref{fig:allComponents} shows the contribution of all the single constituents of the analysis and how the proposed method takes advantage of the complete description of the decay. -This approach exploits possible differences between the muon and electron channels, -by mean of a shared parametrisation of all the common local (form-factors) and non-local -($\mathcal{H}_\lambda$) hadronic matrix elements. -This results in a clean simultaneous analysis of the two channels, independent on any -theoretical uncertainty; in addition, this method doesn't suffer from the limited -statistics of the electron channel, that would make impossible to perform a complete -angular analysis of the single $\Bz \to \Kstarz e^+ e^-$ decay channel. + +{\color{red} +A remarkable feature of the framework is the possibility to extend the analysis to +include other decay channels involving flavour changing neutral currents. +For instance, the charged decay $B^+ \to K^{*+} \ellell$ underlies the same physics as +the examined neutral mode and is easily accessible at the $B$-factories, while other rare +semileptonic decays as the $B^+ \to K^+ \ellell$ presents a different phenomenology but +guarantees access to the same NP information by mean of the description in term of WCs. +Thus, a global simultaneous fit to all data involving $b \to s \ell^+ \ell^-$ transitions +would provide an historical comprehension of the underling physics. +Secondly, the parameter space of the investigated WCs can naturally be broadened to +incorporate direct measurement of the right-handed $\WC_i'$, currently only weakly +determined~\cite{Capdevila:2017bsm,Altmannshofer:2017yso,Hurth:2017hxg}, +and of possible phases in the WCs - appearing as $\Im \WC_i$ - as predicted by some +models[Ref.]. +} + @@ -462,4 +479,30 @@ \bibliography{references} + +\newpage + + +\section{Supplemental material} + +In this supplemental material we show an extension of the physics case of the proposed method. + +\begin{figure}[tbh] +\includegraphics[width=.4\textwidth]{plots/ellipses_CpMu_Hz.pdf} +\caption{% + . + \label{fig:Cp_Hz} +} +\end{figure} + + +\begin{figure}[tbh] +\includegraphics[width=.4\textwidth]{plots/ellipses_CpMu.pdf} +\caption{% + Figure to be updated.... + \label{fig:Cp} +} +\end{figure} + + \end{document}