diff --git a/draft.tex b/draft.tex index 0ce95b0..e972b6f 100644 --- a/draft.tex +++ b/draft.tex @@ -226,6 +226,7 @@ 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) \\ % @@ -305,7 +306,7 @@ \begin{enumerate} % \item[i.] baseline $\mathcal{H}_\lambda[z^2]$ SM prediction - parametrisation~\cite{Bobeth:2017vxj} as a multivariate gaussian contraint; + parametrisation~\cite{Bobeth:2017vxj} as a multivariate Gaussian constraint; % \item[ii.] no theoretical assumption on $\mathcal{H}_\lambda[z^2]$ and with free-floating parameters; @@ -326,88 +327,83 @@ $\mathcal{H}_\lambda[z^{n}]$ parametrisations for the \texttt{BMP}$_{\WC_9}$ hypothesis, with yields corresponding to LHCb Run-II. We observe that the sensitivity to $\widetilde{\WC}_9^{(\mu,e)}$ is strongly dependent on -the underlying assumption on the modelling of the non-local matrix elements. -On the other hand, Fig.~\ref{fig:C9ellipse} shows a strong correlation between -$\widetilde{\mathcal{C}}_9^{(\mu)}$ and $\widetilde{\mathcal{C}}_9^{(e)}$. - -\begin{figure}[tb] +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 $\mathcal{H}_\lambda[z^2]$, +\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{% - $3\,\sigma$ 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} + 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 $3\,\sigma$ statistical-only uncertainty bands and + the dotted black line indicates the LFU hypothesis. } +\label{fig:C9ellipse} \end{figure} +% +Furthermore, we note that, as commonly stated in the literature (see \textit{i.e.} recent review in~\cite{Capdevila:2017ert}), +the determination of $\WC_{10}^{(\mu,e)}$ is insensitive to the lack of knowledge on the +non-local hadronic effects and thus independent of any model assumption. +% +\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 (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. + \label{fig:DeltaC9C10} +} +\end{center} +\end{figure*} - -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. - - - - - -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 $\Delta\WC_{10}$. +The sensitivity to the two NP scenario 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 RunII, $xx(yy)\,\sigma$ for the \belle II 50~ab$^{-1}$ dataset and +the \lhcb Run II, $xx(yy)\,\sigma$ for \belle II 50~ab$^{-1}$ dataset and $xx(yy)\,\sigma$ for the \lhcb 50~fb$^{-1}$ Upgrade for the NP$_{\WC_9}$ (NP$_{\WC_9-\WC_{10}}$) scenario respectively. +\textbf{TODO: Add also here the plot for the upgrade and also comment on the result itself.} - -Modelling detector effects as \qsq and angles resolution or detector acceptance and -efficiency is hardly possible without access to (non-public) information of the current -$B$~physics experiments. +Modelling detector effects such as \qsq and angles resolution, detector acceptance/efficiency, +is hardly possible without access to (non-public) information of the current +\textit{B}-physics experiments. A first rudimentary study on the impact of a finite \qsq resolution is preformed 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~\cite{LHCB-PAPER-2017-013}. +to reproduce the mass fits of~\cite{Aaij:2017vbb}. Despite the low \qsq asymmetric tail, the determination of $\Delta\WC_9$ and $\Delta\WC_{10}$ remains unbiased. -An other important test to probe the stability of the model consists in changing the +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 the potential issue that can rise if the truncation +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. 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 +$\mathcal{H}_\lambda[z^3]$ and $\mathcal{H}_\lambda[z^4]$, and we perform the fit with $\mathcal{H}_\lambda[z^2]$. We vary the choice of the $\mathcal{H}_\lambda[z^{3(4)}]$ generated parameters, -including a ``provocative" set of values that minimize the tension with the $P_5'$ -``anomaly"~\cite{LHCb-PAPER-2015-051} while keeping $\WC_9^{(\mu)}$ and +including a ``provocative'' set of values that minimize the tension with the $P_5'$ +``anomaly"~\cite{Aaij:2015oid} while keeping $\WC_9^{(\mu)}$ and $\WC_{10}^{(\mu)}$ 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} - -\begin{figure*}[t!] -\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{% - $3\,\sigma$ 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 RunII. - \label{fig:DeltaC9C10} -} -\end{center} -\end{figure*} - +\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 $\Delta\WC_9$ @@ -416,7 +412,7 @@ 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 parametrization of all the common local (form-factors) and non-local +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 diff --git a/references.bib b/references.bib index 89ee02b..944549a 100644 --- a/references.bib +++ b/references.bib @@ -558,3 +558,19 @@ url = "http://cds.cern.ch/record/2045786", note = "Presented 22 March 2018", } +@article{Capdevila:2017ert, + author = "Capdevila, Bernat and Descotes-Genon, Sebastien and + Hofer, Lars and Matias, Joaquim", + title = "{Hadronic uncertainties in $B \to K^* \mu^+ \mu^-$: a + state-of-the-art analysis}", + journal = "JHEP", + volume = "04", + year = "2017", + pages = "016", + doi = "10.1007/JHEP04(2017)016", + eprint = "1701.08672", + archivePrefix = "arXiv", + primaryClass = "hep-ph", + reportNumber = "ICCUB-17-003, UAB-FT-990, LPT-ORSAY-16-89", + SLACcitation = "%%CITATION = ARXIV:1701.08672;%%" +}