diff --git a/draft.tex b/draft.tex index f45a8a8..8817166 100644 --- a/draft.tex +++ b/draft.tex @@ -145,12 +145,13 @@ \maketitle -Flavour change neutral current processes of {\textit{B}} meson decays, dominantly mediated by +Flavour chang{\color{red}ing} neutral current processes of {\textit{B}} meson decays, dominantly mediated by $b \to s$ amplitudes, are crucial probes for the Standard Model (SM), -since as-yet undiscovered particles may contribute to loop effects and cause observables to deviate +since as-yet undiscovered particles may contribute to {\color{blue}loop effects} {\color{red}the decay process} 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. +which offers a rich {\color{blue}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}, notably in one of the observables with reduced form-factor uncertainties, @@ -173,8 +174,8 @@ an effective field theory~\cite{Altmannshofer:2008dz} - hereafter only a subset of the Wilson coefficients $C_i$ for the basis of dimension-six field operators $O_i$ is used for the weak Lagrangian~\cite{Bobeth:2017vxj}. -In this framework NP effects are systematically incorporated -by introducing deviations exclusively in the Wilson coefficients (WC)~\cite{Ali:1994bf} +In this framework NP effects are {\color{red} \sout{systematically}} incorporated +by introducing deviations {\color{red} \sout{exclusively}} in the Wilson coefficients (WC)~\cite{Ali:1994bf} ({\textit{i.e.}} $\mathcal{C}_i = \mathcal{C}^{\mathrm{SM}}_i + \mathcal{C}^{\mathrm{NP}}_i$). %For instance, whilst the individual level of significance of the present anomalies is still inconclusive, %there is an appealing non-trivial consistency shown in @@ -183,12 +184,12 @@ %or $\mathcal{C}_9$ and $\mathcal{C}_{10}$ simultaneously. 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. +or $\mathcal{C}_9$ and $\mathcal{C}_{10}$ simultaneously{\color{red}~\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 contributions. -Some promising approaches propose to either extract these non-local hadronic elements -from data-driven analyses~\cite{Blake:2017fyh,Hurth:2017sqw} +Some promising approaches propose to {\color{blue} \sout{either}} extract these non-local hadronic elements +{\color{red} either} from data-driven analyses~\cite{Blake:2017fyh,Hurth:2017sqw} or by using the analytical and dispersive properties of these correlators~\cite{Bobeth:2017vxj}. However these models still have intrinsic limitations, in particular in the assumption of the parametrisation of the di-lepton invariant mass. @@ -199,13 +200,17 @@ 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 when examining muons and electrons +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, most stringent constraints on LFU for a single measurement are expected. +decays is exploited, and therefore, {\color{blue} most} {\color{red} more} stringent constraints on LFU for a single measurement are expected. -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) +{\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) } +{\color{red} (I know that you already changed this to the text above, but "Let us" looks pretty ugly:) ) +In this work we assume the $\bar{B}\to \bar{K}^*\ell^+\ell^-$ decay +being completely dominated by the on-shell $\bar{K}^{*0}$ ($p$-wave) contribution. +The differential decay rate is hence} fully described by four kinematic variables; the di-lepton invariant mass square, $q^2$, and the three angles $\vec{\Omega} = (\cos \theta_\ell, \cos \theta_K, \phi)$~\cite{Altmannshofer:2008dz}. @@ -236,7 +241,9 @@ where ${\cal{N}}_{\lambda}^{(\ell)}$ is a normalisation factor, and ${\cal{F}}^{(T)}_{\lambda}(q^{2})$ and $\mathcal{H}_\lambda(q^{2})$ are local and non-local hadronic matrix elements, respectively. -While the ${\cal{F}}^{(T)}_{\lambda}(q^{2})$ are form factor parameters set from~\cite{Straub:2015ica}, +While the ${\cal{F}}^{(T)}_{\lambda}(q^{2})$ are form factor parameters set from~\cite{Straub:2015ica} +\footnote{{\color{red} Following a conservative approach, uncertainties on the form factors parameters are +doubled with respect to~\cite{Straub:2015ica}}}, the $\mathcal{H}_\lambda(q^{2})$ are described using two complementary parametrisations~\cite{Bobeth:2017vxj,Hurth:2017sqw} - for brevity only a subset of results is shown for the latter approach. @@ -255,20 +262,22 @@ \begin{equation} \Delta \WC_i = \widetilde{\mathcal{C}}_i^{(\mu)} - \widetilde{\mathcal{C}}_i^{(e)}\,, \end{equation} -where the usual WCs are renamed in view of that a precise disentanglement between the physical +where the usual WCs {\color{red} $\mathcal{C}_i^{(\mu,e)}$} are renamed as {\color{red}$\widetilde{\mathcal{C}}_i^{(\mu,e)}$} +in view of {\color{red} the fact} that a precise disentanglement between the physical meaning of $\WC_i^{(\mu,e)}$ and the above-mentioned hadronic pollution is 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 that is sufficient to enable the determination of these multi-space parameters.\footnote{Note -that an amplitude analysis of the electron mode has been always previously disregarded, +that an amplitude analysis of the {\color{red} single} electron mode has been always previously disregarded, given the limited dataset foreseen in either LHCb or Belle-II experiments.} Therefore, in a common framework these 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, given its universal coupling to photons~\cite{Paul:2016urs}. +{\color{red} (Comment: from how is written seems that $C_7$ is floated in the fit, but I like the sentence.)} 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 @@ -279,17 +288,18 @@ 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 \qsq range studied 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 muons and $1.1\,\GeV^2 \leq q^2 \leq 7.0\,\GeV^2$ for electrons in LHCb; -and the same kinematic regions for both the semileptonic channels in Belle II, namely +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; +{\color{blue} and the same kinematic regions for both the semileptonic channels in Belle II} +{\color{red} while in Belle II the same kinematic regions is considered for both the 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 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 -$\mathcal{C}^{\rm{NP}}_9 = 4.07$, $\mathcal{C}^{\rm{NP}}_{10} = - 4.07$ and $\mathcal{C}^{\rm{NP}}_7 = XX$. +$\mathcal{C}^{\rm{NP}}_9 = 4.27$, $\mathcal{C}^{\rm{NP}}_{10} = - 4.17$ and $\mathcal{C}^{\rm{NP}}_7 = -0.34$. 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 $\WC_9^{(\mu)} = -\WC_{10}^{(\mu)} = - 0.7$, @@ -297,7 +307,7 @@ 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 in these simulated samples, +An extended unbinned maximum likelihood fit is performed to these simulated samples, in which multivariate Gaussian terms are added to the likelihood to incorporate prior knowledge on the nuisance parameters. In order to probe the model-independence of the framework, the non-local hadronic @@ -323,9 +333,9 @@ repeating the fit ${\cal{O}}(500)$ times with randomised starting parameters; the solution with smallest negative log-likelihood is taken as the default. -Figure~\ref{fig:C9ellipse} shows the fit results for several alternative -$\mathcal{H}_\lambda[z^{n}]$ parametrisations for the \texttt{BMP}$_{\WC_9}$ hypothesis, -with yields corresponding to LHCb Run-II. +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