diff --git a/draft.tex b/draft.tex index 66b8596..0ce95b0 100644 --- a/draft.tex +++ b/draft.tex @@ -197,8 +197,10 @@ from a simultaneous amplitude analysis of both $B^0 \to K^{*0} \mu^+\mu^-$ and $B^0 \to K^{*0} e^+e^-$ decays can, for the first time, unambiguously determine LFU-breaking from direct measurements of WCs. -This work builds on the the generalisation of Ref.~\cite{Bobeth:2017vxj}, +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 +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. @@ -206,7 +208,7 @@ decays (dominated by the on-shell $\bar{K}^{*0}$ contribution) fully described by four kinematic variables; the di-lepton invariant mass square, $q^2$, and the three angles -$\vec{\Omega} = (\cos \theta_l, \cos \theta_K, \phi)$~\cite{Altmannshofer:2008dz}. +$\vec{\Omega} = (\cos \theta_\ell, \cos \theta_K, \phi)$~\cite{Altmannshofer:2008dz}. The probability density function ($p.d.f.$) for this decay can be written as % \begin{equation} @@ -218,103 +220,127 @@ % with different \qsq intervals depending on the lepton flavour under study. %where the \qsq range is defined differently for the two semileptonic channels. -For a definition of $\dd ^4\Gamma/(\dd q^2 \dd ^3\Omega)$ we refer +For a complete definition of $\dd ^4\Gamma/(\dd q^2 \dd ^3\Omega)$ we refer to~\cite{Bobeth:2008ij,Altmannshofer:2008dz} and references therein. - -The $B\to K^*\ell\ell$ decay is conveniently described by the $K^*$ transversity -amplitudes ($\lambda = \perp, \parallel, 0$) +It is convenient to explicitly write the WC dependence on the decay width by +the transversity amplitudes ($\lambda=\perp, \para,0$) as~\cite{Bobeth:2017vxj} % \eqa{ - {\cal{A}}_{\lambda}^{L,R} &=& {\cal{N}}_{\lambda}\ \bigg\{ -(C_9 \mp C_{10}) {\cal{F}}_{\lambda}(q^2) \\ + {\cal{A}}_{\lambda}^{(\ell)\,L,R} &=& {\cal{N}}_{\lambda}^{(\ell)}\ \bigg\{ +(C^{(\ell)}_9 \mp C^{(\ell)}_{10}) {\cal{F}}_{\lambda}(q^2) \\ % -&&+\frac{2m_b M_B}{q^2} \bigg[ C_7 {\cal{F}}_{\lambda}^{T}(q^2) - 16\pi^2 \frac{M_B}{m_b} {\cal{H}}_{\lambda}(q^2) \bigg] +&&+\frac{2m_b M_B}{q^2} \bigg[ C^{(\ell)}_7 {\cal{F}}_{\lambda}^{T}(q^2) - 16\pi^2 \frac{M_B}{m_b} {\cal{H}}_{\lambda}(q^2) \bigg] \bigg\}\,,\nonumber } - -Non-local hadronic matrix elements $\mathcal{H}_\lambda(q^{2})$, -where $\lambda=\perp, \para,0$ is the polarisation of $\bar{K}^{*0}$, +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}, +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. In the following this correlator is expressed in terms of a conformal variable $z(q^{2})$~\cite{Bobeth:2017vxj,Boyd:1995cf,Bourrely:2008za}, with an analytical expansion truncated at a given order $z^n$ (herein referred to as $\mathcal{H}_\lambda[z^n]$). +Some of the drawbacks of this expansion is that \textit{a-priori} there is +no physics argument to justify the order of the polynomial to be curtailed at +or even if this series will ever converge - +which in turn currently limits any claim on NP sensitivity. -Prospects for the \lhcb experiment are studied within $1.1\,\GeV^2 -\leq q^2 \leq 8.0\,\GeV^2$ and $11.0\,\GeV^2 \leq q^2 \leq 12.5\,\GeV^2$ -for the muonic mode, and $1.1\,\GeV^2 \leq q^2 \leq 8.0\,\GeV^2$ for -the electron mode. Studies on the \belle~II experiment uses the same -kinematic regions for both the semileptonic channels, namely -$1.1\,\GeV^2 \leq q^2 \leq 9.0\,\GeV^2$ and $10.0\,\GeV^2 -\leq q^2 \leq 13.0\,\GeV^2$. This definition of \qsq ranges -corresponds approximately to what already in use in published work by -\lhcb and \belle~\cite{LHCb-PAPER-2015-051,LHCB-PAPER-2017-013,Belle-Kstll}. - - -In order to study the sensitivity to different NP scenarios, we -generated a large number of toys using the following set of parameters: -the non-local hadronic parameters as in~\cite{Danny_2017}, -the form factor parameters as determined from~\cite{Straub:2015ica}, -but with twice the stated uncertainty, and the CKM Wolfenstein -parameters~\cite{Bona:2006ah}; all the above-mentioned parameters -are shared between the two semileptonic modes and are treated as -nuisance parameters, while only the Wilson coefficients $\WC_9^{(\mu,e)}$ -and $\WC_{10}^{(\mu,e)}$ are kept separately for the two channels. - -We define three benchmark points, depending on the values of the Wilson coefficients -used to generate the ensembles: one ``SM", where the values of the Wilson coefficients -are set to their SM values, and two BSM scenarios, one labelled as ``NP$_{\WC_9}$", -where NP is inserted only in $\WC_9^{(\mu)}$ with a shift with respect to the SM of -$\WC_9^{\text{NP} (\mu)} = - 1$, and the latter -labelled as ``NP$_{\WC_9-\WC_{10}}$", where NP is inserted in $\WC_9^{(\mu)}$ -and $\WC_{10}^{(\mu)}$ with a shift with respect to the SM of -$\WC_9^{\text{NP} (\mu)} = -\WC_{10}^{\text{NP} (\mu)} = - 0.7$. - -The number of events used to generate the pseudo-experiments is -obtained from~\cite{LHCb-PAPER-2015-051,LHCB-PAPER-2017-013,Belle-Kstll} -and extrapolated to the current and future expected statistics to study -the prospects of the \lhcb and \belle~II experiments. - -In all cases under study, we perform an extended unbinned maximum likelihood fit by -including in the likelihood function poissonian terms that take into account the muon -and electron yields obtained in the different kinematic regions. -Multivariate gaussian terms are added to the likelihood to incorporate prior knowledge -on the nuisance parameters as introduced above. -For each generated sample the fit is repeated several times with different initialization -of the fitted parameters. - -The authors of~\cite{Danny_2017} proposed a SM prediction of the non-local -hadronic matrix elements $\mathcal{H}_\lambda(z)$, where $\lambda=\perp, \para,0$ -is the polarization of the \Kstarz, operating an analytic expansion in the ``conformal” -variable $z(q^2)$ and assuming a truncation at the order $z^2$ (in the following we refer -to the analytic expansion of $\mathcal{H}_\lambda$ truncated at the order $z^n$ -as $\mathcal{H}_\lambda[z^n]$). - -In order to test the validity of the adopted parametrizations we repeat -the fit with different configurations: -\begin{itemize} - \item We include the $\mathcal{H}_\lambda[z^2]$ SM prediction - from~\cite{Danny_2017} as gaussian contraint to the fit. - \item We remove any theoretical assumption on $\mathcal{H}_\lambda[z^2]$ - and let free-floating all the parameters. - \item We increase the order of the analytical expansion of $\mathcal{H}_\lambda$ - up to the (free-floating) order of $z^3$ and $z^4$. - \item We re-parametrize the description of the non-local hadronic matrix - element as proposed in~\cite{Christoph}. -\end{itemize} - -We observe that the sensitivity to $\WC_9^{(\mu,e)}$ is strongly dependent on -the assumption underlying the parametrization of the non-local matrix element, -see Fig.~\ref{fig:C9ellipse}. -In this work we renounce to a precise determination of $\WC_9^{(\mu,e)}$, -that will be renamed as $\widetilde{\mathcal{C}}_9^{(\mu,e)}$ in the -following, in view of the fact that a precise disentanglement between the physical -meaning of $\WC_9^{(\mu,e)}$ and the above-mentioned hadronic pollution is +In order to overcome these points, we investigate the LFU-breaking +hypothesis using direct determinations of the difference of Wilson coefficients +between muons and electrons, \textit{i.e.} +\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 +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, +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}. + +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 +[$50\,$-$\,300\,$fb$^{-1}$]~\cite{Aaij:2244311}, and Belle II [$50\,$ab$^{-1}$]. +These are extrapolated from Refs.~\cite{Aaij:2015oid,Aaij:2017vbb,Wehle:2016yoi} +by scaling respectively with $\sigma_{b\bar{b}} \propto \sqrt{s}$ and $\sigma_{b\bar{b}} \propto s$ +for LHCb and Belle II, where $s$ denotes the designed centre-of-mass +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 +$1.1\,\GeV^2 \leq q^2 \leq 8.0\,\GeV^2$ 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 $\mathcal{C}^{\rm{NP}}_7 = XX$. +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 $\WC_9^{(\mu)} = -\WC_{10}^{(\mu)} = - 0.7$, +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}. + +An extended unbinned maximum likelihood fit is performed in 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 +parametrisation is modified in several ways, \textit{i.e.} +% +\begin{enumerate} + % + \item[i.] baseline $\mathcal{H}_\lambda[z^2]$ SM prediction + parametrisation~\cite{Bobeth:2017vxj} as a multivariate gaussian contraint; + % + \item[ii.] no theoretical assumption on $\mathcal{H}_\lambda[z^2]$ + and with free-floating parameters; + % + \item[iii.] higher orders of the analytical expansion of $\mathcal{H}_\lambda[z^{n}]$ + up to $z^3$ and $z^4$ - free floating; + % + \item[iv.] and re-parametrisation of its description as + proposed in~\cite{Hurth:2017sqw}. + % +\end{enumerate} +% +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. + +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. +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] +\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} +} +\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 @@ -325,15 +351,8 @@ a clear sign of LFU-violation. -\begin{figure}[tbh] -\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} -} -\end{figure} + + 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 diff --git a/references.bib b/references.bib index 82f0efa..89ee02b 100644 --- a/references.bib +++ b/references.bib @@ -506,3 +506,55 @@ reportNumber = "CPT-P36-2007", SLACcitation = "%%CITATION = ARXIV:0807.2722;%%" } +@article{Straub:2015ica, + author = "Bharucha, Aoife and Straub, David M. and Zwicky, Roman", + title = "{$B\to V\ell^+\ell^-$ in the Standard Model from + light-cone sum rules}", + journal = "JHEP", + volume = "08", + year = "2016", + pages = "098", + doi = "10.1007/JHEP08(2016)098", + eprint = "1503.05534", + archivePrefix = "arXiv", + primaryClass = "hep-ph", + reportNumber = "TUM-HEP-957-14, CP3-Origins-2015-010-DNRF90, + DIAS-2015-10", + SLACcitation = "%%CITATION = ARXIV:1503.05534;%%" +} +@article{Paul:2016urs, + author = "Paul, Ayan and Straub, David M.", + title = "{Constraints on new physics from radiative $B$ decays}", + journal = "JHEP", + volume = "04", + year = "2017", + pages = "027", + doi = "10.1007/JHEP04(2017)027", + eprint = "1608.02556", + archivePrefix = "arXiv", + primaryClass = "hep-ph", + SLACcitation = "%%CITATION = ARXIV:1608.02556;%%" +} +@techreport{Aaij:2244311, + author = "Aaij, Roel and others", + title = "{Expression of Interest for a Phase-II LHCb Upgrade: + Opportunities in flavour physics, and beyond, in the HL-LHC + era}", + institution = "CERN", + collaboration = "LHCb Collaboration", + address = "Geneva", + number = "CERN-LHCC-2017-003", + month = "Feb", + year = "2017", + reportNumber = "CERN-LHCC-2017-003", + url = "http://cds.cern.ch/record/2244311", +} +@thesis{Lionetto:XX, + author = "Lionetto, Federica", + title = "{Measurement of angular observables of $B^0 \rightarrow K^{*0}e^+e^-$ and $B^0 \rightarrow K^{*0}\mu^+\mu^-$ decays and the upgrade of LHCb}", + month = "Mar", + year = "2018", + reportNumber = "CERN-THESIS-2018-XX", + url = "http://cds.cern.ch/record/2045786", + note = "Presented 22 March 2018", +}