diff --git a/draft.tex b/draft.tex index 20c2815..a9bb747 100644 --- a/draft.tex +++ b/draft.tex @@ -283,11 +283,13 @@ 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. -While the form factors ${\cal{F}}^{(T)}_{\lambda}(q^{2})$ can be taken from~\cite{Straub:2015ica}\footnote{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}, +{\color{red} While the form factors ${\cal{F}}^{(T)}_{\lambda}(q^{2})$ parametrise +the fact that the interaction happens inside the hadrons}, +%While the form factors ${\cal{F}}^{(T)}_{\lambda}(q^{2})$ can be taken from~\cite{Straub:2015ica}\footnote{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}, %sure full agreement among the LCSRs and Lattice results. -uncertainties on the form factors parameters are doubled with respect to Ref.~\cite{Straub:2015ica}}, +%uncertainties on the form factors parameters are doubled with respect to Ref.~\cite{Straub:2015ica}}, the $\mathcal{H}_\lambda(q^{2})$ encode the aforementioned non-factorisable hadronic contributions and are described using two complementary parametrisations~\cite{Bobeth:2017vxj,Hurth:2017sqw} - @@ -326,7 +328,9 @@ 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}. -Similarly, all the right-handed WCs are fixed to their SM values, \textit{i.e.} $\WC_i^{\prime\,(\mu,e)} = 0$. +In the following, all the right-handed WCs are fixed to their SM values, \textit{i.e.} $\WC_i^{\prime\,(\mu,e)} = 0$, +{\color{red} while a sensitivity study on the determination of the WCs $\WC_9^{\prime\,(\mu)}$ and $\WC_{10}^{\prime\,(\mu)}$ +is 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 @@ -349,13 +353,17 @@ 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 in the case of muons for two NP benchmark points (BMP), -\textit{i.e.} $\WC_9^{(e)} = \WC^{\rm{SM}}_9 = \WC^{(\mu)}_9 + 1$ and +{\color{red} This baseline model is modified for two NP benchmark points (BMP), +$\Delta\WC_9 = - 1$ and $\Delta\WC_9 = - \Delta\WC_{10} = - 0.7$, +respectively referred 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.} $\WC_i^{(e)} = \WC_i^{\rm{SM}}$.} +%This baseline model is modified in the case of muons for two NP benchmark points (BMP), +%\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. +%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}. %but chosen with reduced SM tension in order to examine a more conservative hypothesis. @@ -383,6 +391,10 @@ % \end{enumerate} +{\color{red}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}.} % %The stability of the model and the convergency to the global minimum is enforced by %repeating the fit with randomised starting parameters; @@ -413,9 +425,14 @@ \label{fig:C9ellipse} \end{figure} % -Furthermore, we note that, as commonly stated in the literature (see \textit{e.g.} recent review in Ref.~\cite{Capdevila:2017ert}), +%Furthermore, 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 and thus independent of any model assumption. + +{\color{red}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 and thus independent of any model assumption. +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 time of \lhcb Upgrade [$50\,$fb$^{-1}$].} % \begin{figure}[bth!] %\begin{center} @@ -448,7 +465,9 @@ 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. +enable a deeper understanding of the nature of NP, +{\color{red}in fact, the clear advantage of the proposed pseudo-observables $\Delta\WC_i$ is that they are +insensitive to both local and non-local hadronic uncertainties.} %Note that these unprecedented dataset will enable insight towards the nature of NP. % %the \lhcb Run II, $7.6(8.4)\,\sigma$ for \belle II 50~ab$^{-1}$ dataset and @@ -472,10 +491,13 @@ } \end{figure} -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 performed +{\color{red}As mentioned above, experimental resolution effects and detector acceptance/efficiency +are ignored in this work, as they would require additional information from the current \textit{B}-physics +experiments. Nevertheless, a first preliminary study} +%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. +on the impact of a finite \qsq resolution is 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}. @@ -489,17 +511,20 @@ 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 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 -$\mathcal{H}_\lambda[z^n]$ is not a good description of nature. +%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 +%$\mathcal{H}_\lambda[z^n]$ is not a good description of nature. +{\color{red} 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 $\mathcal{H}_\lambda[z^2]$. -We vary the choice of the $\mathcal{H}_\lambda[z^{(3,4)}]$ generated parameters, -including a set of values that minimises the tension with the $P_5'$ -anomaly~\cite{Aaij:2015oid}, while keeping $\WC_9^{(\mu)}$ and -$\WC_{10}^{(\mu)}$ at their SM values. +%We vary the choice of the $\mathcal{H}_\lambda[z^{(3,4)}]$ generated parameters, +%including a set of values that minimises 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.