diff --git a/draft.tex b/draft.tex index a9bb747..d512e1d 100644 --- a/draft.tex +++ b/draft.tex @@ -283,14 +283,15 @@ 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. -{\color{red} While the form factors ${\cal{F}}^{(T)}_{\lambda}(q^{2})$ parametrise -the fact that the interaction happens inside the hadrons}, +The ${\cal{F}}^{(T)}_{\lambda}(q^{2})$ are form factors, +%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}}, -the $\mathcal{H}_\lambda(q^{2})$ encode the aforementioned non-factorisable +while $\mathcal{H}_\lambda(q^{2})$ encode the aforementioned non-factorisable hadronic contributions and are described using two complementary parametrisations~\cite{Bobeth:2017vxj,Hurth:2017sqw} - for brevity only a subset of results is shown for the latter approach. @@ -329,8 +330,8 @@ and fixed to its SM value, given its universal coupling to photons and the strong constraint from radiative $B$ decays~\cite{Paul:2016urs}. 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.} +while sensitivity studies on the determination of the WCs $\WC_9^{\prime\,(\mu)}$ and $\WC_{10}^{\prime\,(\mu)}$ +are 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 @@ -353,10 +354,10 @@ 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. -{\color{red} This baseline model is modified for two NP benchmark points (BMP), +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.} $\WC_i^{(e)} = \WC_i^{\rm{SM}}$.} +referred respectively 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} $\WC^{\rm{NP}(\mu)}_9 = - 1$ } @@ -391,10 +392,10 @@ % \end{enumerate} -{\color{red}On the other hand, form factors parameters are taken from~\cite{Straub:2015ica} and, in order +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}.} +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; @@ -428,11 +429,11 @@ %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}), +% +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. 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$^{-1}$].} +that are found to be the limiting factor by the end of Run-II. % \begin{figure}[bth!] %\begin{center} @@ -465,9 +466,10 @@ 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, -{\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.} +enable a deeper understanding of the nature of NP - +insensitive to both local and non-local hadronic uncertainties. +%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 @@ -491,13 +493,15 @@ } \end{figure} -{\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} +Experimental resolution and detector acceptance/efficiency effects +are not considered in this work; +further information from current (non-public) or planned \textit{B}-physics experiments are necessary to extend this discussion. +%as they would require additional information from the current \textit{B}-physics experiments. +Nevertheless, preliminary studies %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 +on the impact of a finite \qsq resolution are 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}. @@ -515,9 +519,9 @@ %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 +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.} +$\mathcal{H}_\lambda[z^n]$ 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 $\mathcal{H}_\lambda[z^2]$. @@ -542,7 +546,7 @@ This relies on a shared parametrisation of the local (form-factors) and non-local ($\mathcal{H}_\lambda[z^n]$) hadronic matrix elements between the muonic and electronic channels, that in turn enables the determination of the observables of interest -free from any parametric theoretical uncertainty. +free from any theoretical uncertainty. In addition, this simultaneous analysis is more robust against experimental effects such as mismodeling of the detector resolution, since most parameters are