diff --git a/draft.tex b/draft.tex index 627ff81..66b8596 100644 --- a/draft.tex +++ b/draft.tex @@ -36,8 +36,8 @@ \newcommand{\reffig}[1]{figure~\ref{fig:#1}} \newcommand{\refsec}[1]{section~\ref{sec:#1}} \newcommand{\reftab}[1]{table~\ref{tab:#1}} -\let\oldtheta\theta -\renewcommand{\theta}{\vartheta} +%\let\oldtheta\theta +%\renewcommand{\theta}{\vartheta} \newcommand{\eps}{\varepsilon} \newcommand{\para}{\parallel} \newcommand{\Gfermi}{G_F} @@ -61,6 +61,9 @@ \newcommand{\op}[1]{\mathcal{O}_{#1}} \newcommand{\la}{\langle} \newcommand{\ra}{\rangle} +\newcommand{\eqa}[1]{\begin{eqnarray} #1 \end{eqnarray}} + + \def\deriv {\ensuremath{\mathrm{d}}} \def\qsq {\ensuremath{q^2}\xspace} @@ -190,28 +193,54 @@ However these models still have intrinsic limitations, in particular in the assumption of the parametrisation of the di-lepton invariant mass. -In this letter we propose a new \textit{model-independent} approach that +In this Letter we propose a new \textit{model-independent} approach that 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}, +but it is insensitive to the model assumptions of the parametrisation. 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 are expected. +decays is exploited, and therefore, most stringent constraints on LFU for a single measurement are expected. -% Start of proper paper discussion -In this work we assume the $\bar{B}\to \bar{K}^*\ell^+\ell^-$ decay being completely dominated -by the on-shell $\Kstarz$ ($p$-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 $\vec{\Omega} = (\cos \theta_l, \cos \theta_K, \phi)$ [Ref]. -The Probability Density Function ($p.d.f.$) for this decay can be written as +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) +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}. +The probability density function ($p.d.f.$) for this decay can be written as % \begin{equation} p.d.f.^{(i)} = \frac{1}{\Gamma_i} \frac{\dd^4 \Gamma}{\dd q^2 \dd^3 \Omega}, \ - \qquad + \quad \text{with}\quad - \Gamma_i = \int_{q^2} \dd q^2 \frac{\dd\Gamma}{\dd q^2} + \Gamma_i = \int_{q^2} \dd q^2 \frac{\dd\Gamma}{\dd q^2}\,, \end{equation} % -where the \qsq range is defined differently for the two semileptonic channels. +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 +to~\cite{Bobeth:2008ij,Altmannshofer:2008dz} and references therein. + +The $B\to K^*\ell\ell$ decay is conveniently described by the $K^*$ transversity +amplitudes ($\lambda = \perp, \parallel, 0$) +% +\eqa{ + {\cal{A}}_{\lambda}^{L,R} &=& {\cal{N}}_{\lambda}\ \bigg\{ +(C_9 \mp C_{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] +\bigg\}\,,\nonumber +} + +Non-local hadronic matrix elements $\mathcal{H}_\lambda(q^{2})$, +where $\lambda=\perp, \para,0$ is the polarisation of $\bar{K}^{*0}$, +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 $\mathcal{H}_\lambda[z^n]$). + 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 @@ -222,11 +251,6 @@ 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}. -For a definition of $\dd ^4\Gamma/(\dd q^2 \dd ^3\Omega)$ we refer -to~\cite{Bobeth:2008ij,Altmannshofer:2008dz} and references therein. -Concerning the description of the non-local hadronic matrix element we -considered the two recently proposed parametrizations of~\cite{Danny_2017} -and~\cite{Christoph}. In order to study the sensitivity to different NP scenarios, we generated a large number of toys using the following set of parameters: @@ -351,8 +375,9 @@ while {\color{red} test bias in C10 and Upgrade} -\begin{figure}[tbh] -\includegraphics[width=.4\textwidth]{plots/ellipses_DeltaC9C10_a.pdf} \\ +\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 @@ -361,8 +386,8 @@ scenario and the expected statistics after the \lhcb RunII. \label{fig:DeltaC9C10} } -\end{figure} - +\end{center} +\end{figure*} 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 diff --git a/references.bib b/references.bib index 4e00960..82f0efa 100644 --- a/references.bib +++ b/references.bib @@ -456,3 +456,53 @@ reportNumber = "CERN-TH-2017-178, MITP-17-053", SLACcitation = "%%CITATION = ARXIV:1708.04474;%%" } +@article{Bobeth:2008ij, + author = "Bobeth, Christoph and Hiller, Gudrun and Piranishvili, + Giorgi", + title = "{CP Asymmetries in bar $B \to \bar{K}^* (\to \bar{K} \pi) + \bar{\ell} \ell$ and Untagged $\bar{B}_s$, $B_s \to \phi + (\to K^{+} K^-) \bar{\ell} \ell$ Decays at NLO}", + journal = "JHEP", + volume = "07", + year = "2008", + pages = "106", + doi = "10.1088/1126-6708/2008/07/106", + eprint = "0805.2525", + archivePrefix = "arXiv", + primaryClass = "hep-ph", + reportNumber = "DO-TH-08-03", + SLACcitation = "%%CITATION = ARXIV:0805.2525;%%" +} +@article{Boyd:1995cf, + author = "Boyd, C. Glenn and Grinstein, Benjamin and Lebed, Richard + F.", + title = "{Model independent extraction of |V(cb)| using dispersion + relations}", + journal = "Phys. Lett.", + volume = "B353", + year = "1995", + pages = "306-312", + doi = "10.1016/0370-2693(95)00480-9", + eprint = "hep-ph/9504235", + archivePrefix = "arXiv", + primaryClass = "hep-ph", + reportNumber = "UCSD-PTH-95-03", + SLACcitation = "%%CITATION = HEP-PH/9504235;%%" +} +@article{Bourrely:2008za, + author = "Bourrely, Claude and Caprini, Irinel and Lellouch, + Laurent", + title = "{Model-independent description of B ---> pi l nu decays + and a determination of |V(ub)|}", + journal = "Phys. Rev.", + volume = "D79", + year = "2009", + pages = "013008", + doi = "10.1103/PhysRevD.82.099902, 10.1103/PhysRevD.79.013008", + note = "[Erratum: Phys. Rev.D82,099902(2010)]", + eprint = "0807.2722", + archivePrefix = "arXiv", + primaryClass = "hep-ph", + reportNumber = "CPT-P36-2007", + SLACcitation = "%%CITATION = ARXIV:0807.2722;%%" +}