diff --git a/prd/SupplementalMaterial.zip b/prd/SupplementalMaterial.zip deleted file mode 100644 index a193d1c..0000000 --- a/prd/SupplementalMaterial.zip +++ /dev/null Binary files differ diff --git a/prd/SupplementalMaterial/.DS_Store b/prd/SupplementalMaterial/.DS_Store deleted file mode 100644 index 5008ddf..0000000 --- a/prd/SupplementalMaterial/.DS_Store +++ /dev/null Binary files differ diff --git a/prd/SupplementalMaterial/ellipses_CpMu.pdf b/prd/SupplementalMaterial/ellipses_CpMu.pdf deleted file mode 100644 index 46ebca3..0000000 --- a/prd/SupplementalMaterial/ellipses_CpMu.pdf +++ /dev/null Binary files differ diff --git a/prd/SupplementalMaterial/ellipses_CpMu_Hz.pdf b/prd/SupplementalMaterial/ellipses_CpMu_Hz.pdf deleted file mode 100644 index 8a034e5..0000000 --- a/prd/SupplementalMaterial/ellipses_CpMu_Hz.pdf +++ /dev/null Binary files differ diff --git a/prd/SupplementalMaterial/suppl.tex b/prd/SupplementalMaterial/suppl.tex deleted file mode 100644 index 7d0a4cf..0000000 --- a/prd/SupplementalMaterial/suppl.tex +++ /dev/null @@ -1,181 +0,0 @@ -% vim: set sts=4 et : -\documentclass[reprint,preprintnumbers,prd,nofootinbib]{revtex4-1} - -\usepackage{amsfonts} -\usepackage{amsmath} -\usepackage{array} -\usepackage{braket} -\usepackage{epstopdf} -\usepackage{graphicx} -\usepackage{hepparticles} -\usepackage{hepnicenames} -\usepackage{hepunits} -\usepackage{hyperref} -\usepackage[% - utf8 -]{inputenc} -\usepackage{slashed} -\usepackage{subfigure} -\usepackage{placeins} -\usepackage[% - normalem -]{ulem} -\usepackage[% - usenames, - svgnames, - dvipsnames -]{xcolor} - - -%% Shortcuts %% -\newcommand{\ie}{\textit{i.e.}} -\newcommand{\nuvec}{\vec{\nu}} -\newcommand{\refapp}[1]{appendix~\ref{app:#1}} -\newcommand{\refeq}[1]{eq.~(\ref{eq:#1})} -\newcommand{\refeqs}[2]{eqs.~(\ref{eq:#1})--(\ref{eq:#2})} -\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} -\newcommand{\eps}{\varepsilon} -\newcommand{\para}{\parallel} -\newcommand{\Gfermi}{G_F} -%\newcommand{\dd}[2][]{{\mathrm{d}^{#1}}#2\,} -\newcommand{\dd}{\ensuremath{\textrm{d}}} -\newcommand{\order}[1]{\ensuremath{\mathcal{O}\left(#1\right)}} -\DeclareMathOperator{\sign}{sgn} -\DeclareMathOperator{\ReNew}{Re} -\DeclareMathOperator{\ImNew}{Im} -\let\Re\ReNew -\let\Im\ImNew -\DeclareMathOperator*{\sumint}{% -\mathchoice% - {\ooalign{$\displaystyle\sum$\cr\hidewidth$\displaystyle\int$\hidewidth\cr}} - {\ooalign{\raisebox{.14\height}{\scalebox{.7}{$\textstyle\sum$}}\cr\hidewidth$\textstyle\int$\hidewidth\cr}} - {\ooalign{\raisebox{.2\height}{\scalebox{.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}} - {\ooalign{\raisebox{.2\height}{\scalebox{.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}} -} -\DeclareMathOperator*{\argmax}{arg\,max} -\newcommand{\wilson}[2][]{\mathcal{C}^\text{#1}_{#2}} -\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} - -\def\PB {\ensuremath{\mathrm{B}}\xspace} -\def\B {{\ensuremath{\PB}}\xspace} -\def\PK {\ensuremath{\mathrm{K}}\xspace} -\def\kaon {{\ensuremath{\PK}}\xspace} -\def\Kstarz {{\ensuremath{\kaon^{*0}}}\xspace} -\def\Bd {{\ensuremath{\B^0}}\xspace} -\def\Bz {{\ensuremath{\B^0}}\xspace} - -%% Key decay channels - -\def\BdToKstmm {\decay{\Bd}{\Kstarz\mup\mun}} -\def\BdbToKstmm {\decay{\Bdb}{\Kstarzb\mup\mun}} - -\def\BsToJPsiPhi {\decay{\Bs}{\jpsi\phi}} -\def\BdToJPsiKst {\decay{\Bd}{\jpsi\Kstarz}} -\def\BdbToJPsiKst {\decay{\Bdb}{\jpsi\Kstarzb}} - -%% Rare decays -\def\BdKstee {\decay{\Bd}{\Kstarz\epem}} -\def\BdbKstee {\decay{\Bdb}{\Kstarzb\epem}} -\def\bsll {\decay{\bquark}{\squark \ell^+ \ell^-}} - -\def\lepton {{\ensuremath{\ell}}\xspace} -\def\ellm {{\ensuremath{\ell^-}}\xspace} -\def\ellp {{\ensuremath{\ell^+}}\xspace} -\def\ellell {\ensuremath{\ell^+ \ell^-}\xspace} -\def\mumu {{\ensuremath{\Pmu^+\Pmu^-}}\xspace} - -\def\lhcb {\mbox{LHCb}\xspace} -\def\belle {\mbox{Belle}\xspace} - -\def\WC {\ensuremath{\mathcal{C}}\xspace} - - - -\begin{document} - -\allowdisplaybreaks - -\preprint{ZU-TH-15/18} -\title{Supplemental material for ``Towards establishing Lepton Flavour Universality violation in $\bar{B}\to \bar{K}^*\ell^+\ell^-$ decays''} -\author{Andrea Mauri} -\email{a.mauri@cern.ch} -\author{Nicola Serra} -\email{nicola.serra@cern.ch} -\author{Rafael Silva Coutinho} -\email{rafael.silva.coutinho@cern.ch} -\affiliation{Physik-Institut, Universit\"at Z\"urich, Winterthurer Strasse 190, 8057 Z\"urich, Switzerland} - - -%\appendix - -%\section{Supplemental material} - - -\maketitle - -An extension of the physics case of the proposed method -is to investigate the sensitivity to the chirality-flipped counterparts of the usual Wilson coefficients, -\textit{i.e.} $\WC^{\prime (\mu)}_9$ and $\WC^{\prime(\mu)}_{10}$. -Following the formalism discussed in this letter, the primed WCs are examined by considering in addition to the -\texttt{BMP}$_{\WC_{9,10}}$ three different modified NP scenarios for the muon only: -$\WC_{9,10}^{\prime(\mu)} = \WC^{\prime \rm{SM}}_{9,10} = 0$; -$\WC^{\prime (\mu)}_9 = \WC^{\prime (\mu)}_{10} = 0.3$; -and $\WC^{\prime (\mu)}_9 = - \WC^{\prime (\mu)}_{10} = 0.3$. -Notice that for the electron mode the $\WC_{9,10}^{\prime(e)}$ is set and fixed to the SM value $\WC^{\prime \rm{SM}}_{9,10} = 0$. - - -Figure~\ref{fig:Cp_Hz} shows the fit results for different order of the analytic expansion for -the non-local hadronic contribution for a NP scenario with $\WC'^{(\mu)}_9 = \WC'^{(\mu)}_{10} = 0.3$ -% -\begin{figure}[h] -\includegraphics[width=.4\textwidth]{ellipses_CpMu_Hz.pdf} -\caption{% - Two-dimensional sensitivity scans for the pair of Wilson coefficients $\WC'^{(\mu)}_9$ and - $\WC'^{(\mu)}_{10}$ for different non-local hadronic parametrisation models for a NP scenario - with $\WC'^{(\mu)}_9 = \WC'^{(\mu)}_{10} = 0.3$. - The contours correspond to 99\% confidence level statistical-only uncertainty bands evaluated with - the expected statistics after \lhcb Run II. - \label{fig:Cp_Hz} -} -\end{figure} -% -and yields corresponding to the \lhcb Run II expected statistics. -The dependency on the determination of $\WC'^{(\mu)}_9$ and $\WC'^{(\mu)}_{10}$ on the -order of the expansion clearly saturates after $\mathcal{H}_\lambda[z^3]$ and allows a measurement -of the primed Wilson coefficients for the muon decay channel $B^{0} \to K^{*0} \mumu$ independent -on the theoretical hadronic uncertainty. -% -Figure~\ref{fig:Cp} shows the prospects for the sensitivity to the $\WC'^{(\mu)}_9$ and -$\WC'^{(\mu)}_{10}$ Wilson coefficients corresponding to the expected statistics at -the LHCb upgrade with $50\,$fb$^{-1}$ and $\,300\,$fb$^{-1}$. -Note that only with the full capability of the LHCb experiment it is possible -to start disentangling the different NP hypotheses. - - -% -\begin{figure}[b] -\includegraphics[width=.4\textwidth]{ellipses_CpMu.pdf} -\caption{% - Two-dimensional sensitivity scans for the pair of Wilson coefficients $\WC'^{(\mu)}_9$ and $\WC'^{(\mu)}_{10}$ - for three NP scenarios: (blue) $\WC'^{(\mu)}_9 = \WC'^{(\mu)}_{10} = 0$, (orange) $\WC'^{(\mu)}_9 = \WC'^{(\mu)}_{10} = 0.3$ - and (magenta) $\WC'^{(\mu)}_9 = - \WC'^{(\mu)}_{10} = 0.3$. - The contours correspond to 99\% confidence level statistical-only uncertainty bands expected for the LHCb Upgrade (dotted) - $50\,$fb$^{-1}$ and (solid) $\,300\,$fb$^{-1}$ statistics. - \label{fig:Cp} -} -\end{figure} - -%\FloatBarrier - -\end{document} diff --git a/prd/draft.pdf b/prd/draft.pdf index 4f00004..3d8e2b2 100644 --- a/prd/draft.pdf +++ b/prd/draft.pdf Binary files differ diff --git a/prd/main.zip b/prd/main.zip deleted file mode 100644 index d38b342..0000000 --- a/prd/main.zip +++ /dev/null Binary files differ diff --git a/prd/main/.DS_Store b/prd/main/.DS_Store deleted file mode 100644 index 5008ddf..0000000 --- a/prd/main/.DS_Store +++ /dev/null Binary files differ diff --git a/prd/main/B2Kstll_summary.pdf b/prd/main/B2Kstll_summary.pdf deleted file mode 100644 index 58f805a..0000000 --- a/prd/main/B2Kstll_summary.pdf +++ /dev/null Binary files differ diff --git a/prd/main/draft.bbl b/prd/main/draft.bbl deleted file mode 100644 index cb6db10..0000000 --- a/prd/main/draft.bbl +++ /dev/null @@ -1,680 +0,0 @@ -%merlin.mbs apsrev4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked -%Control: key (0) -%Control: author (8) initials jnrlst -%Control: editor formatted (1) identically to author -%Control: production of article title (-1) disabled -%Control: page (0) single -%Control: year (1) truncated -%Control: production of eprint (0) enabled -\begin{thebibliography}{51}% -\makeatletter -\providecommand \@ifxundefined [1]{% - 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Rev.D82,099902(2010)]},\ \Eprint - {http://arxiv.org/abs/0807.2722} {arXiv:0807.2722 [hep-ph]} \BibitemShut - {NoStop}% -%%CITATION = ARXIV:0807.2722;%% -\bibitem [{\citenamefont {Chrzaszcz}\ \emph {et~al.}(2018)\citenamefont - {Chrzaszcz}, \citenamefont {Mauri}, \citenamefont {Serra}, \citenamefont - {Silva~Coutinho},\ and\ \citenamefont {van Dyk}}]{Chrzaszcz:2018yza}% - \BibitemOpen - \bibfield {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont - {Chrzaszcz}}, \bibinfo {author} {\bibfnamefont {A.}~\bibnamefont {Mauri}}, - \bibinfo {author} {\bibfnamefont {N.}~\bibnamefont {Serra}}, \bibinfo - {author} {\bibfnamefont {R.}~\bibnamefont {Silva~Coutinho}}, \ and\ \bibinfo - {author} {\bibfnamefont {D.}~\bibnamefont {van Dyk}},\ }\href@noop {} {\ - (\bibinfo {year} {2018})},\ \Eprint {http://arxiv.org/abs/1805.06378} - {arXiv:1805.06378 [hep-ph]} \BibitemShut {NoStop}% -%%CITATION = ARXIV:1805.06378;%% -\bibitem [{\citenamefont {Paul}\ and\ \citenamefont - {Straub}(2017)}]{Paul:2016urs}% - \BibitemOpen - \bibfield {author} {\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont - {Paul}}\ and\ \bibinfo {author} {\bibfnamefont {D.~M.}\ \bibnamefont - {Straub}},\ }\href {\doibase 10.1007/JHEP04(2017)027} {\bibfield {journal} - {\bibinfo {journal} {JHEP}\ }\textbf {\bibinfo {volume} {04}},\ \bibinfo - {pages} {027} (\bibinfo {year} {2017})},\ \Eprint - {http://arxiv.org/abs/1608.02556} {arXiv:1608.02556 [hep-ph]} \BibitemShut - {NoStop}% -%%CITATION = ARXIV:1608.02556;%% -\bibitem [{sup()}]{suppl}% - \BibitemOpen - \href@noop {} {\enquote {\bibinfo {title} {{See Supplemental Material at [URL - will be inserted by publisher] for sensitivity studies on the determination - of the Wilson coefficients $\mathcal{C}_9^{\prime\,(\mu)}$ and - $\mathcal{C}{10}^{\prime\,(\mu)}$}},}\ }\BibitemShut {NoStop}% -\bibitem [{\citenamefont {Aaij}\ \emph - {et~al.}(2017{\natexlab{b}})\citenamefont {Aaij} \emph - {et~al.}}]{Aaij:2244311}% - \BibitemOpen - \bibfield {author} {\bibinfo {author} {\bibfnamefont {R.}~\bibnamefont - {Aaij}} \emph {et~al.} (\bibinfo {collaboration} {LHCb Collaboration}),\ - }\href {http://cds.cern.ch/record/2244311} {\emph {\bibinfo {title} - {{Expression of Interest for a Phase-II LHCb Upgrade: Opportunities in - flavour physics, and beyond, in the HL-LHC era}}}},\ \bibinfo {type} {Tech. - Rep.}\ \bibinfo {number} {CERN-LHCC-2017-003}\ (\bibinfo {institution} - {CERN},\ \bibinfo {address} {Geneva},\ \bibinfo {year} {2017})\BibitemShut - {NoStop}% -\bibitem [{\citenamefont {Lionetto}\ \emph {et~al.}(2018)\citenamefont - {Lionetto}, \citenamefont {Serra}, \citenamefont {Steinkamp},\ and\ - \citenamefont {Mueller}}]{Lionetto:2624938}% - \BibitemOpen - \bibfield {author} {\bibinfo {author} {\bibfnamefont {F.}~\bibnamefont - {Lionetto}}, \bibinfo {author} {\bibfnamefont {N.}~\bibnamefont {Serra}}, - \bibinfo {author} {\bibfnamefont {O.}~\bibnamefont {Steinkamp}}, \ and\ - \bibinfo {author} {\bibfnamefont {K.}~\bibnamefont {Mueller}},\ }\href - {https://cds.cern.ch/record/2624938} {\enquote {\bibinfo {title} - {{Measurements of Angular Observables of $B^0\to K^{*0} \mu^+ \mu^-$ and $B^0 - \to K^{*0} e^+ e^-$ Decays and the Upgrade of LHCb}},}\ }\bibinfo - {howpublished} {\url{https://cds.cern.ch/record/2624938}} (\bibinfo {year} - {2018}),\ \bibinfo {note} {presented 22 Mar 2018}\BibitemShut {NoStop}% -\bibitem [{\citenamefont {Bharucha}\ \emph {et~al.}(2016)\citenamefont - {Bharucha}, \citenamefont {Straub},\ and\ \citenamefont - {Zwicky}}]{Straub:2015ica}% - \BibitemOpen - \bibfield {author} {\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont - {Bharucha}}, \bibinfo {author} {\bibfnamefont {D.~M.}\ \bibnamefont - {Straub}}, \ and\ \bibinfo {author} {\bibfnamefont {R.}~\bibnamefont - {Zwicky}},\ }\href {\doibase 10.1007/JHEP08(2016)098} {\bibfield {journal} - {\bibinfo {journal} {JHEP}\ }\textbf {\bibinfo {volume} {08}},\ \bibinfo - {pages} {098} (\bibinfo {year} {2016})},\ \Eprint - {http://arxiv.org/abs/1503.05534} {arXiv:1503.05534 [hep-ph]} \BibitemShut - {NoStop}% -%%CITATION = ARXIV:1503.05534;%% -\bibitem [{\citenamefont {Ball}\ and\ \citenamefont - {Braun}(1998)}]{Ball:1998kk}% - \BibitemOpen - \bibfield {author} {\bibinfo {author} {\bibfnamefont {P.}~\bibnamefont - {Ball}}\ and\ \bibinfo {author} {\bibfnamefont {V.~M.}\ \bibnamefont - {Braun}},\ }\href {\doibase 10.1103/PhysRevD.58.094016} {\bibfield {journal} - {\bibinfo {journal} {Phys. Rev.}\ }\textbf {\bibinfo {volume} {D58}},\ - \bibinfo {pages} {094016} (\bibinfo {year} {1998})},\ \Eprint - {http://arxiv.org/abs/hep-ph/9805422} {arXiv:hep-ph/9805422 [hep-ph]} - \BibitemShut {NoStop}% -%%CITATION = HEP-PH/9805422;%% -\bibitem [{\citenamefont {Khodjamirian}\ \emph {et~al.}(2007)\citenamefont - {Khodjamirian}, \citenamefont {Mannel},\ and\ \citenamefont - {Offen}}]{Khodjamirian:2006st}% - \BibitemOpen - \bibfield {author} {\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont - {Khodjamirian}}, \bibinfo {author} {\bibfnamefont {T.}~\bibnamefont - {Mannel}}, \ and\ \bibinfo {author} {\bibfnamefont {N.}~\bibnamefont - {Offen}},\ }\href {\doibase 10.1103/PhysRevD.75.054013} {\bibfield {journal} - {\bibinfo {journal} {Phys. 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Phys.}\ }\textbf {\bibinfo {volume} {B769}},\ \bibinfo - {pages} {31} (\bibinfo {year} {2007})},\ \Eprint - {http://arxiv.org/abs/hep-ph/0611295} {arXiv:hep-ph/0611295 [hep-ph]} - \BibitemShut {NoStop}% -%%CITATION = HEP-PH/0611295;%% -\bibitem [{\citenamefont {Horgan}\ \emph {et~al.}(2014)\citenamefont {Horgan}, - \citenamefont {Liu}, \citenamefont {Meinel},\ and\ \citenamefont - {Wingate}}]{Horgan:2013hoa}% - \BibitemOpen - \bibfield {author} {\bibinfo {author} {\bibfnamefont {R.~R.}\ \bibnamefont - {Horgan}}, \bibinfo {author} {\bibfnamefont {Z.}~\bibnamefont {Liu}}, - \bibinfo {author} {\bibfnamefont {S.}~\bibnamefont {Meinel}}, \ and\ \bibinfo - {author} {\bibfnamefont {M.}~\bibnamefont {Wingate}},\ }\href {\doibase - 10.1103/PhysRevD.89.094501} {\bibfield {journal} {\bibinfo {journal} {Phys. - Rev.}\ }\textbf {\bibinfo {volume} {D89}},\ \bibinfo {pages} {094501} - (\bibinfo {year} {2014})},\ \Eprint {http://arxiv.org/abs/1310.3722} - {arXiv:1310.3722 [hep-lat]} \BibitemShut {NoStop}% -%%CITATION = ARXIV:1310.3722;%% -\bibitem [{\citenamefont {Capdevila}\ \emph {et~al.}(2017)\citenamefont - {Capdevila}, \citenamefont {Descotes-Genon}, \citenamefont {Hofer},\ and\ - \citenamefont {Matias}}]{Capdevila:2017ert}% - \BibitemOpen - \bibfield {author} {\bibinfo {author} {\bibfnamefont {B.}~\bibnamefont - {Capdevila}}, \bibinfo {author} {\bibfnamefont {S.}~\bibnamefont - {Descotes-Genon}}, \bibinfo {author} {\bibfnamefont {L.}~\bibnamefont - {Hofer}}, \ and\ \bibinfo {author} {\bibfnamefont {J.}~\bibnamefont - {Matias}},\ }\href {\doibase 10.1007/JHEP04(2017)016} {\bibfield {journal} - {\bibinfo {journal} {JHEP}\ }\textbf {\bibinfo {volume} {04}},\ \bibinfo - {pages} {016} (\bibinfo {year} {2017})},\ \Eprint - {http://arxiv.org/abs/1701.08672} {arXiv:1701.08672 [hep-ph]} \BibitemShut - {NoStop}% -%%CITATION = ARXIV:1701.08672;%% -\bibitem [{\citenamefont {Albrecht}\ \emph {et~al.}(2017)\citenamefont - {Albrecht}, \citenamefont {Bernlochner}, \citenamefont {Kenzie}, - \citenamefont {Reichert}, \citenamefont {Straub},\ and\ \citenamefont - {Tully}}]{Albrecht:2017odf}% - \BibitemOpen - \bibfield {author} {\bibinfo {author} {\bibfnamefont {J.}~\bibnamefont - {Albrecht}}, \bibinfo {author} {\bibfnamefont {F.}~\bibnamefont - {Bernlochner}}, \bibinfo {author} {\bibfnamefont {M.}~\bibnamefont {Kenzie}}, - \bibinfo {author} {\bibfnamefont {S.}~\bibnamefont {Reichert}}, \bibinfo - {author} {\bibfnamefont {D.}~\bibnamefont {Straub}}, \ and\ \bibinfo {author} - {\bibfnamefont {A.}~\bibnamefont {Tully}},\ }\href@noop {} {\ (\bibinfo - {year} {2017})},\ \Eprint {http://arxiv.org/abs/1709.10308} {arXiv:1709.10308 - [hep-ph]} \BibitemShut {NoStop}% -%%CITATION = ARXIV:1709.10308;%% -\bibitem [{\citenamefont {Becirevic}\ and\ \citenamefont - {Tayduganov}(2013)}]{Becirevic:2012dp}% - \BibitemOpen - \bibfield {author} {\bibinfo {author} {\bibfnamefont {D.}~\bibnamefont - {Becirevic}}\ and\ \bibinfo {author} {\bibfnamefont {A.}~\bibnamefont - {Tayduganov}},\ }\href {\doibase 10.1016/j.nuclphysb.2012.11.016} {\bibfield - {journal} {\bibinfo {journal} {Nucl. Phys.}\ }\textbf {\bibinfo {volume} - {B868}},\ \bibinfo {pages} {368} (\bibinfo {year} {2013})},\ \Eprint - {http://arxiv.org/abs/1207.4004} {arXiv:1207.4004 [hep-ph]} \BibitemShut - {NoStop}% -%%CITATION = ARXIV:1207.4004;%% -\bibitem [{\citenamefont {Aaij}\ \emph - {et~al.}(2016{\natexlab{b}})\citenamefont {Aaij} \emph - {et~al.}}]{Aaij:2016flj}% - \BibitemOpen - \bibfield {author} {\bibinfo {author} {\bibfnamefont {R.}~\bibnamefont - {Aaij}} \emph {et~al.} (\bibinfo {collaboration} {LHCb}),\ }\href {\doibase - 10.1007/JHEP11(2016)047, 10.1007/JHEP04(2017)142} {\bibfield {journal} - {\bibinfo {journal} {JHEP}\ }\textbf {\bibinfo {volume} {11}},\ \bibinfo - {pages} {047} (\bibinfo {year} {2016}{\natexlab{b}})},\ \bibinfo {note} - {[Erratum: JHEP04,142(2017)]},\ \Eprint {http://arxiv.org/abs/1606.04731} - {arXiv:1606.04731 [hep-ex]} \BibitemShut {NoStop}% -%%CITATION = ARXIV:1606.04731;%% -\end{thebibliography}% diff --git a/prd/main/draft.tex b/prd/main/draft.tex deleted file mode 100644 index c0e48fb..0000000 --- a/prd/main/draft.tex +++ /dev/null @@ -1,495 +0,0 @@ -% vim: set sts=4 et : -\documentclass[reprint,preprintnumbers,prd,nofootinbib]{revtex4-1} - -\usepackage{amsfonts} -\usepackage{amsmath} -\usepackage{array} -\usepackage{braket} -\usepackage{epstopdf} -\usepackage{graphicx} -\usepackage{hepparticles} -\usepackage{hepnicenames} -\usepackage{hepunits} -\usepackage{hyperref} -\usepackage[% - utf8 -]{inputenc} -\usepackage{slashed} -\usepackage{subfigure} -\usepackage{placeins} -\usepackage[% - normalem -]{ulem} -\usepackage[% - usenames, - svgnames, - dvipsnames -]{xcolor} - - -%% Shortcuts %% -\newcommand{\ie}{\textit{i.e.}} -\newcommand{\nuvec}{\vec{\nu}} -\newcommand{\refapp}[1]{appendix~\ref{app:#1}} -\newcommand{\refeq}[1]{eq.~(\ref{eq:#1})} -\newcommand{\refeqs}[2]{eqs.~(\ref{eq:#1})--(\ref{eq:#2})} -\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} -\newcommand{\eps}{\varepsilon} -\newcommand{\para}{\parallel} -\newcommand{\Gfermi}{G_F} -%\newcommand{\dd}[2][]{{\mathrm{d}^{#1}}#2\,} -\newcommand{\dd}{\ensuremath{\textrm{d}}} -\newcommand{\order}[1]{\ensuremath{\mathcal{O}\left(#1\right)}} -\DeclareMathOperator{\sign}{sgn} -\DeclareMathOperator{\ReNew}{Re} -\DeclareMathOperator{\ImNew}{Im} -\let\Re\ReNew -\let\Im\ImNew -\DeclareMathOperator*{\sumint}{% -\mathchoice% - {\ooalign{$\displaystyle\sum$\cr\hidewidth$\displaystyle\int$\hidewidth\cr}} - {\ooalign{\raisebox{.14\height}{\scalebox{.7}{$\textstyle\sum$}}\cr\hidewidth$\textstyle\int$\hidewidth\cr}} - {\ooalign{\raisebox{.2\height}{\scalebox{.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}} - {\ooalign{\raisebox{.2\height}{\scalebox{.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}} -} -\DeclareMathOperator*{\argmax}{arg\,max} -\newcommand{\wilson}[2][]{\mathcal{C}^\text{#1}_{#2}} -\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} - -\def\PB {\ensuremath{\mathrm{B}}\xspace} -\def\B {{\ensuremath{\PB}}\xspace} -\def\PK {\ensuremath{\mathrm{K}}\xspace} -\def\kaon {{\ensuremath{\PK}}\xspace} -\def\Kstarz {{\ensuremath{\kaon^{*0}}}\xspace} -\def\Bd {{\ensuremath{\B^0}}\xspace} -\def\Bz {{\ensuremath{\B^0}}\xspace} - -%% Key decay channels - -\def\BdToKstmm {\decay{\Bd}{\Kstarz\mup\mun}} -\def\BdbToKstmm {\decay{\Bdb}{\Kstarzb\mup\mun}} - -\def\BsToJPsiPhi {\decay{\Bs}{\jpsi\phi}} -\def\BdToJPsiKst {\decay{\Bd}{\jpsi\Kstarz}} -\def\BdbToJPsiKst {\decay{\Bdb}{\jpsi\Kstarzb}} - -%% Rare decays -\def\BdKstee {\decay{\Bd}{\Kstarz\epem}} -\def\BdbKstee {\decay{\Bdb}{\Kstarzb\epem}} -\def\bsll {\decay{\bquark}{\squark \ell^+ \ell^-}} - -\def\lepton {{\ensuremath{\ell}}\xspace} -\def\ellm {{\ensuremath{\ell^-}}\xspace} -\def\ellp {{\ensuremath{\ell^+}}\xspace} -\def\ellell {\ensuremath{\ell^+ \ell^-}\xspace} -\def\mumu {{\ensuremath{\Pmu^+\Pmu^-}}\xspace} - -\def\lhcb {\mbox{LHCb}\xspace} -\def\belle {\mbox{Belle}\xspace} - -\def\WC {\ensuremath{\mathcal{C}}\xspace} - - - -\begin{document} - -\allowdisplaybreaks - -\preprint{ZU-TH-15/18} -\title{Towards establishing Lepton Flavour Universality violation in $\bar{B}\to \bar{K}^*\ell^+\ell^-$ decays} -\author{Andrea Mauri} -\email{a.mauri@cern.ch} -\author{Nicola Serra} -\email{nicola.serra@cern.ch} -\author{Rafael Silva Coutinho} -\email{rafael.silva.coutinho@cern.ch} -\affiliation{Physik-Institut, Universit\"at Z\"urich, Winterthurer Strasse 190, 8057 Z\"urich, Switzerland} - -\begin{abstract} - Rare semileptonic $b \to s \ell^+ \ell^-$ transitions provide some of the most promising framework to search for new physics effects. -Recent analyses of these decays have indicated an anomalous behaviour in measurements - of angular distributions of the decay $B^0\to - K^*\mu^+\mu^-$ and lepton-flavour-universality observables. -Unambiguously establishing if these deviations have a common nature - is of paramount importance in order to understand the observed pattern. - We propose a novel approach to independently and complementary - probe this hypothesis - by performing a simultaneous amplitude analysis of $\bar{B}^0 \to \bar{K}^{*0} \mu^+\mu^-$ and $\bar{B}^0 \to \bar{K}^{*0} e^+e^-$ decays. - This method enables the direct determination of - observables that encode potential non-equal couplings of muons and electrons, - and are found to be insensitive to non-perturbative QCD effects. -If current hints of new physics are confirmed, our approach could -allow an early discovery of physics beyond the Standard -Model with LHCb Run-II datasets. -\end{abstract} - -\maketitle - -Flavour changing neutral current processes of {\textit{B}} meson decays -are crucial probes for the Standard Model (SM), -since as-yet undiscovered particles may contribute to these transitions 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. -An anomalous behaviour in angular and branching fraction analyses of the decay channel -$\bar{B}^{0} \to \bar{K}^{*0} \mu^{+}\mu^{-}$ has been recently reported~\cite{Aaij:2015oid,Wehle:2016yoi,Aaij:2013aln,Aaij:2014pli}, -notably in one of the observables with reduced theoretical uncertainties, -$P^{\prime}_{5}$~\cite{Aaij:2013qta,Descotes-Genon:2015uva}. -Several models have been suggested in order to interpret these results as new physics (NP) -signatures~\cite{Gauld:2013qja,Buras:2013qja,Altmannshofer:2013foa,Crivellin:2015era,Hiller:2014yaa,Biswas:2014gga,Gripaios:2014tna}. -Nonetheless, the vector-like nature of this pattern could be also explained by -non-perturbative QCD contributions from $b\to s c{\bar{c}}$ operators ({\textit{i.e.}} charm loops) -that are able to either mimic or camouflage NP effects~\cite{Jager:2012uw,Jager:2014rwa,Ciuchini:2015qxb}. -Non-standard measurement in ratios of $b \to s \ell^+ \ell^-$ processes -- such as of $R_{K}$~\cite{Aaij:2014ora} and $R_{K^{*}}$~\cite{Aaij:2017vbb} - -indicate a suppression of the muon channel which is also compatible with the $P^{\prime}_{5}$ anomaly. -In this case an immediate interpretation of lepton flavour universality (LFU) breaking is -suggested due to the small theoretical uncertainties in their predictions~\cite{Hiller:2003js,Bordone:2016gaq}. -Whilst the individual level of significance of the present anomalies is still inconclusive, -there is an appealing non-trivial consistent pattern shown in -global analysis fits~\cite{Capdevila:2017bsm,Altmannshofer:2017yso,Hurth:2017hxg,Ciuchini:2017mik,Alok:2017sui}. - -The formalism of {\textit{b}} decays is commonly described within -an effective field theory~\cite{Altmannshofer:2008dz}, -which probes distinct energy scales; -with regimes classified into short-distance (high energies) perturbative -and non-calculable long-distance effects. -These can be parametrised in the weak Lagrangian in terms of -effective operators with different Lorentz structures, $\mathcal{O}_i$, -with corresponding couplings $\mathcal{C}_i$ - referred to as Wilson coefficients (WC). -Only a subset of the operators that are most sensitive to NP is examined in this work~\cite{Bobeth:2017vxj}, -\textit{i.e.} $\mathcal{O}_7$ (virtual photon exchanges), $\mathcal{O}_{9,10}$ (vector and axial currents) -and corresponding right-handed couplings with flipped helicities. -In this framework NP effects are incorporated -by introducing deviations in the WCs~\cite{Ali:1994bf} from their SM predictions, -{\textit{i.e.}} $\mathcal{C}_i = \mathcal{C}^{\mathrm{SM}}_i + \mathcal{C}^{\mathrm{NP}}_i$. -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~\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 matrix elements that are difficult to assess reliably from first principles. -Some promising approaches suggest to extract this contribution -from data-driven analyses~\cite{Blake:2017fyh,Hurth:2017sqw} -and by exploiting analytical properties of its structure~\cite{Bobeth:2017vxj}. -However, these models still have intrinsic limitations, in particular -in the assumptions that enter in parametrisation of the di-lepton invariant mass distribution. - -In this Letter we propose a new \textit{model-independent} approach that -from a simultaneous unbinned amplitude analysis of both -$\bar{B}^0 \to \bar{K}^{*0} \mu^+\mu^-$ and $\bar{B}^0 \to \bar{K}^{*0} e^+e^-$ decays -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 effect relies on the strong correlation between the muon and electron modes -imposed by the lepton-flavour universality of the hadronic matrix elements. -Furthermore, in this method the full set of observables -(\textit{e.g} $R_{K^{*}}$, $P^{\prime}_{5}$ and branching fraction measurements) available in $\bar{B}\to \bar{K}^*\ell^+\ell^-$ -decays is exploited, providing unprecedented precision on LFU in a single analysis. - -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 squared invariant mass, $q^2$, and the three angles -$\vec{\Omega} = (\cos \theta_\ell, \cos \theta_K, \phi)$~\cite{Altmannshofer:2008dz}. -The probability density function ($p.d.f.$) for this decay can be written as -% -\begin{equation} -p.d.f. = \frac{1}{\Gamma} \frac{\dd^4 \Gamma}{\dd q^2 \dd^3 \Omega}, \ - \quad - \text{with}\quad - \Gamma = \int_{q^2} \dd q^2 \frac{\dd\Gamma}{\dd q^2}\,, -\end{equation} -% -with different \qsq intervals depending on the lepton flavour under study. -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. -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{ - \label{eq:amp_dep} - {\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^{(\ell)}_7 {\cal{F}}_{\lambda}^{T}(q^2) - 16\pi^2 \frac{M_B}{m_b} {\cal{H}}_{\lambda}(q^2) \bigg] -\bigg\}\,,\nonumber -} -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. -The ${\cal{F}}^{(T)}_{\lambda}(q^{2})$ are form factors, -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. -In the following -this function 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 $\mathcal{H}_\lambda[z^n]$), -after removing singularities related to the $J/\psi(1S)$ and $\psi(2S)$. -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. - -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 $\mathcal{C}_i^{(\mu,e)}$ are renamed as $\widetilde{\mathcal{C}}_i^{(\mu,e)}$, -since an accurate disentanglement between the physical -meaning of $\WC_i^{(\mu,e)}$ and the above-mentioned hadronic pollution -cannot be achieved at the current stage of the theory~\cite{Chrzaszcz:2018yza}. -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 $\bar{B}^0 \to \bar{K}^{*0} \mu^+\mu^-$ decays -that is sufficient to enable the determination of these multi-space parameters. -Note that an amplitude analysis of the electron mode only has been previously disregarded, -given the limited dataset in either LHCb or Belle experiments. -In a common framework the 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 -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.} $\WC_i^{\prime\,(\mu,e)} = 0$, -while sensitivity studies on the determination of the WCs $\WC_9^{\prime\,(\mu)}$ and $\WC_{10}^{\prime\,(\mu)}$ -are detailed in the supplemental material to this Letter~\cite{suppl}. - -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 luminosity and $\sigma_{b\bar{b}} \propto \sqrt{s}$ -for LHCb, where $s$ denotes the designed centre-of-mass energy of the $b$-quark pair, -and exclusively with luminosity for Belle II. -%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 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; -while in Belle II the same kinematic regions are considered for both 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 broadly consistent with published results, -and assumes improvements in the electron mode resolution for LHCb~\cite{Lionetto:2624938}. - -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 for two NP benchmark points (BMP), -$\Delta\WC_9 = - 1$ and $\Delta\WC_9 = - \Delta\WC_{10} = - 0.7$, -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.} $\WC_i^{(e)} = \WC_i^{\rm{SM}}$. -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 to these simulated samples, -in which multivariate Gaussian terms are added to the log-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~\cite{Bobeth:2017vxj} - included as a multivariate Gaussian constraint; - % - \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 Ref.~\cite{Hurth:2017sqw}, \textit{i.e.} - using instead an expansion in terms of $1 + a_{\lambda} + b_{\lambda}(q^{2}/6\,$GeV$^{2})$. - - % -\end{enumerate} -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}. - - -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 -$\widetilde{\mathcal{C}}_9^{(\mu)}$ and $\widetilde{\mathcal{C}}_9^{(e)}$ coefficients -is sufficient to preserve the true underlying physics at any order of the series expansion $\mathcal{H}_\lambda[z^n]$ -and without any parametric theoretical input, -\textit{i.e.} the two-dimensional pull estimator with respect to the LFU hypothesis is unbiased. -% -\begin{figure}[t] -\includegraphics[width=.4\textwidth]{ellipses_C9.pdf} -\caption{% - Two-dimensional sensitivity scans for the pair of Wilson coefficients - $\widetilde{\mathcal{C}}_9^{(\mu)}$ and $\widetilde{\mathcal{C}}_9^{(e)}$ - for different non-local hadronic parametrisation models evaluated at \texttt{BMP}$_{\WC_9}$, - and with the expected statistics after \lhcb Run II. - The contours correspond to 99\% confidence level statistical-only uncertainty bands and - the dotted black line indicates the LFU hypothesis. -} -\label{fig:C9ellipse} -\end{figure} -% -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 end of Run-II. -% -\begin{figure}[bth!] -%\begin{center} -\includegraphics[width=.4\textwidth]{ellipses_DeltaC9C10_a.pdf}\\%\quad\quad\quad\quad -\includegraphics[width=.4\textwidth]{ellipses_DeltaC9C10_b.pdf} -\caption{% - Two-dimensional sensitivity scans for the proposed observables $\Delta\WC_9$ and $\Delta\WC_{10}$ - for different non-local hadronic parametrisation models - evaluated at (top) \texttt{BMP}$_{\WC_9}$ and (bottom) \texttt{BMP}$_{\WC_{9,10}}$, - and with the expected statistics after \lhcb Run II. - The contours correspond to 99\% confidence level statistical-only uncertainty bands. - \label{fig:DeltaC9C10} -} -%\end{center} -\end{figure} - -The sensitivity to the two benchmark-like NP scenarios using the proposed pseudo observables $\Delta \WC_i$ -is shown in Fig.~\ref{fig:DeltaC9C10}. -We quantify the maximal expected significance with respect to the SM to be $4.6$ and $5.3\,\sigma$ for -\texttt{BMP}$_{\WC_9}$ and \texttt{BMP}$_{\WC_{9,10}}$, respectively. -Realistic experimental effects are necessary to determine the exact sensitivity achievable. -Nevertheless, these results suggest that a first observation (with a single measurement) of LFU breaking -appears to be feasible with the expected recorded statistics by the end of LHCb Run II. -Furthermore, it is interesting to examine the prospects for confirming this evidence in the upcoming LHCb/Belle -upgrades~\cite{Albrecht:2017odf}. -Figure~\ref{fig:DeltaC9C10_Upgrade} summarises the two-dimensional statistical-only significances -for the designed luminosities. -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 - -insensitive to both local and non-local hadronic uncertainties. - - -\begin{figure}[t!] -\includegraphics[width=.4\textwidth]{ellipses_DeltaC9C10_Nev.pdf} -\caption{% - Two-dimensional sensitivity scans for the proposed observables $\Delta\WC_9$ and $\Delta\WC_{10}$ - for the two considered NP scenarios: (green) \texttt{BMP}$_{\WC_9}$ and (red) \texttt{BMP}$_{\WC_{9,10}}$. - The contours correspond to 99\% confidence level statistical-only uncertainty bands expected for the (dashed) \belle II 50~ab$^{-1}$ - and \lhcb Upgrade (dotted) $50\,$fb$^{-1}$ and (solid) $\,300\,$fb$^{-1}$ statistics. - \label{fig:DeltaC9C10_Upgrade} -} -\end{figure} - -Experimental resolution and detector acceptance/efficiency effects -are not considered in this work, as these would require further information -from current (non-public) or planned \textit{B}-physics experiments. -Nevertheless, the precision on this measurement can remain unbiased -either by parametrising these effects in the amplitude model and/or even -re-computing the angles or the $q^2$ variables constraining the $B$ invariant mass~\cite{Lionetto:2624938}. -%Nevertheless, preliminary studies 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}. -%Despite the low \qsq asymmetric tail, the determination of $\Delta\WC_9$ and -%$\Delta\WC_{10}$ remains unbiased, even if no-correction is -%applied. -Moreover, the differential decay width can receive additional complex amplitudes from signal-like backgrounds, -\textit{e.g.} $K\pi$ S-wave from a non-resonant decay and/or a scalar resonance~\cite{Becirevic:2012dp}. -These contributions are determined to be small~\cite{Aaij:2015oid,Aaij:2016flj}, -and in the proposed formalism they benefit from the same description between the muon -and electron mode (see detailed discussion in Ref.~\cite{Chrzaszcz:2018yza}). -Therefore, such contribution does not dilute the expected sensitivity of the measurement. - -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 $\mathcal{H}_\lambda[z^2]$. -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. -It is worth mentioning that a hypothetical determination of the individual -$\widetilde{\WC}_9^{(\mu,e)}$ and $\widetilde{\WC}_{10}^{(\mu,e)}$ WCs -can also produce a shift in their central values -that mimics the behaviour of NP~\cite{Chrzaszcz:2018yza}.% and makes impossible any claim in this direction. - - - -In conclusion, we propose a clean and model-independent method to combine -all the available information from $\bar{B}\to \bar{K}^*\ell^+\ell^-$ decays for a precise -determination of LFU-breaking differences of WCs, \textit{i.e.} $\Delta\WC_9$ -and $\Delta\WC_{10}$. -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 theoretical uncertainty. -In addition, this simultaneous analysis is more robust against -experimental effects such -as mismodeling of the detector resolution, since most parameters are -effectively determined from the muon mode. This would be an important -benefit for LHCb where the electron resolution is significantly worse -than that of muons. -% -\begin{figure}[t!] -\includegraphics[width=.4\textwidth]{B2Kstll_summary.pdf} -\caption{% - Sensitivity to \texttt{BMP}$_{\WC_{9,10}}$ scenario for the expected statistics after the \lhcb Run II. - The relative contribution (68, 95, 99\% confidence level contours) of each step of the analysis is shown in different colours, - together with the result of full amplitude method proposed in this letter. - \label{fig:allComponents} -} -\end{figure} -% -Figure~\ref{fig:allComponents} illustrates the usefulness of the newly-proposed observables by combining -the different information from angular analysis to branching ratio measurements. -Due to the inclusiveness of the approach, the expected sensitivity surpasses any -of the projections for the foreseen measurements of \textit{e.g.} $R_{K^{*}}$ or $P^{\prime}_{5}$ alone - given the benchmark points. -Therefore, this novel formalism can be the most immediate method to observe unambiguously -NP in $\bar{B}\to \bar{K}^*\ell^+\ell^-$ decays. - -A promising feature of this framework is the possibility to extend the analysis to -include other decay channels involving flavour changing neutral currents. -For instance, the charged decay $\bar{B}^+ \to \bar{K}^{*+} \ellell$ undergoes the same physics -and is easily accessible at the $B$-factories, while other rare -semi-leptonic decays such as $B^+ \to K^+ \ellell$ and $\Lambda_{b} \to \Lambda^{(*)} \ell^+\ell^-$ -have a different phenomenology but access the same NP information in terms of WC description. -Thus, an unbinned global simultaneous fit to all data involving $b \to s \ell^+ \ell^-$ transitions -is a natural and appealing extension of this work. -Moreover, the parameter space of the investigated WCs can also be broadened to -incorporate direct measurement of the right-handed $\WC_i^{\prime}$ - -currently weakly constrained by global fits~\cite{Capdevila:2017bsm,Altmannshofer:2017yso,Hurth:2017hxg}. - - -We acknowledge useful contributions from Michele Atzeni, Gino Isidori, Danny van Dyk and Patrick Owen. -This work is supported by the Swiss National Science Foundation (SNF) under contracts 173104 and 174182. - - -%\FloatBarrier -\bibliography{references} - -\end{document} diff --git a/prd/main/ellipses_C9.pdf b/prd/main/ellipses_C9.pdf deleted file mode 100644 index 13293f7..0000000 --- a/prd/main/ellipses_C9.pdf +++ /dev/null Binary files differ diff --git a/prd/main/ellipses_DeltaC9C10_Nev.pdf b/prd/main/ellipses_DeltaC9C10_Nev.pdf deleted file mode 100644 index 1ed7ece..0000000 --- a/prd/main/ellipses_DeltaC9C10_Nev.pdf +++ /dev/null Binary files differ diff --git a/prd/main/ellipses_DeltaC9C10_a.pdf b/prd/main/ellipses_DeltaC9C10_a.pdf deleted file mode 100644 index 80501e8..0000000 --- a/prd/main/ellipses_DeltaC9C10_a.pdf +++ /dev/null Binary files differ diff --git a/prd/main/ellipses_DeltaC9C10_b.pdf b/prd/main/ellipses_DeltaC9C10_b.pdf deleted file mode 100644 index b20ff0d..0000000 --- a/prd/main/ellipses_DeltaC9C10_b.pdf +++ /dev/null Binary files differ diff --git a/prd/main/references.bib b/prd/main/references.bib deleted file mode 100644 index 09be6a3..0000000 --- a/prd/main/references.bib +++ /dev/null @@ -1,820 +0,0 @@ -@misc{suppl, - title = "{See Supplemental Material at [URL will be inserted by publisher] - for sensitivity studies on the determination of the Wilson coefficients - $\mathcal{C}_9^{\prime\,(\mu)}$ and $\mathcal{C}{10}^{\prime\,(\mu)}$}" -} -@article{Becirevic:2012dp, - author = "Becirevic, Damir and Tayduganov, Andrey", - title = "{Impact of $B\to K^\ast_0 \ell^+\ell^-$ on the New - Physics search in $B\to K^\ast \ell^+\ell^-$ decay}", - journal = "Nucl. 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Phys.", - volume = "C67", - year = "1995", - pages = "417-432", - doi = "10.1007/BF01624585", - eprint = "hep-ph/9408213", - archivePrefix = "arXiv", - primaryClass = "hep-ph", - reportNumber = "CERN-TH-7346-94", - SLACcitation = "%%CITATION = HEP-PH/9408213;%%" -} -@article{Aaij:2015oid, - author = "Aaij, Roel and others", - title = "{Angular analysis of the $B^{0} \to K^{*0} \mu^{+} - \mu^{-}$ decay using 3 fb$^{-1}$ of integrated - luminosity}", - collaboration = "LHCb", - journal = "JHEP", - volume = "02", - year = "2016", - pages = "104", - doi = "10.1007/JHEP02(2016)104", - eprint = "1512.04442", - archivePrefix = "arXiv", - primaryClass = "hep-ex", - reportNumber = "CERN-PH-EP-2015-314, LHCB-PAPER-2015-051", - SLACcitation = "%%CITATION = ARXIV:1512.04442;%%" -} - -@article{Descotes-Genon:2015uva, - author = "Descotes-Genon, Sébastien and Hofer, Lars and Matias, - Joaquim and Virto, Javier", - title = "{Global analysis of $b\to s\ell\ell$ anomalies}", - journal = "JHEP", - volume = "06", - year = "2016", - pages = "092", - doi = "10.1007/JHEP06(2016)092", - eprint = "1510.04239", - archivePrefix = "arXiv", - primaryClass = "hep-ph", - reportNumber = "LPT-ORSAY-15-68, QFET-2015-29, SI-HEP-2015-19", - SLACcitation = "%%CITATION = ARXIV:1510.04239;%%" -} - -@article{Wehle:2016yoi, - author = "Wehle, S. and others", - title = "{Lepton-Flavor-Dependent Angular Analysis of $B\to K^\ast - \ell^+\ell^-$}", - collaboration = "Belle", - year = "2016", - eprint = "1612.05014", - archivePrefix = "arXiv", - primaryClass = "hep-ex", - reportNumber = "BELLE-PREPRINT-2016-15, KEK-PREPRINT-2016-54", - SLACcitation = "%%CITATION = ARXIV:1612.05014;%%" -} -@article{Gauld:2013qja, - author = "Gauld, Rhorry and Goertz, Florian and Haisch, Ulrich", - title = "{An explicit Z'-boson explanation of the $B \to K^* \mu^+ - \mu^-$ anomaly}", - journal = "JHEP", - volume = "01", - year = "2014", - pages = "069", - doi = "10.1007/JHEP01(2014)069", - eprint = "1310.1082", - archivePrefix = "arXiv", - primaryClass = "hep-ph", - SLACcitation = "%%CITATION = ARXIV:1310.1082;%%" -} -@article{Buras:2013qja, - author = "Buras, Andrzej J. and Girrbach, Jennifer", - title = "{Left-handed $Z'$ and $Z$ FCNC quark couplings facing new - $b \to s \mu^+ \mu^-$ data}", - journal = "JHEP", - volume = "12", - year = "2013", - pages = "009", - doi = "10.1007/JHEP12(2013)009", - eprint = "1309.2466", - archivePrefix = "arXiv", - primaryClass = "hep-ph", - reportNumber = "FLAVOUR(267104)-ERC-50", - SLACcitation = "%%CITATION = ARXIV:1309.2466;%%" -} -@article{Altmannshofer:2013foa, - author = "Altmannshofer, Wolfgang and Straub, David M.", - title = "{New Physics in $B \to K^*\mu\mu$?}", - journal = "Eur. Phys. J.", - volume = "C73", - year = "2013", - pages = "2646", - doi = "10.1140/epjc/s10052-013-2646-9", - eprint = "1308.1501", - archivePrefix = "arXiv", - primaryClass = "hep-ph", - reportNumber = "FERMILAB-PUB-13-310-T, MITP-13-047", - SLACcitation = "%%CITATION = ARXIV:1308.1501;%%" -} -@article{Crivellin:2015era, - author = "Crivellin, Andreas and Hofer, Lars and Matias, Joaquim - and Nierste, Ulrich and Pokorski, Stefan and Rosiek, - Janusz", - title = "{Lepton-flavour violating $B$ decays in generic $Z'$ - models}", - journal = "Phys. Rev.", - volume = "D92", - year = "2015", - number = "5", - pages = "054013", - doi = "10.1103/PhysRevD.92.054013", - eprint = "1504.07928", - archivePrefix = "arXiv", - primaryClass = "hep-ph", - reportNumber = "CERN-PH-TH-2015-091, TTP15-018", - SLACcitation = "%%CITATION = ARXIV:1504.07928;%%" -} -# Leptoquarks -@article{Hiller:2014yaa, - author = "Hiller, Gudrun and Schmaltz, Martin", - title = "{$R_K$ and future $b \to s \ell \ell$ physics beyond the - standard model opportunities}", - journal = "Phys. Rev.", - volume = "D90", - year = "2014", - pages = "054014", - doi = "10.1103/PhysRevD.90.054014", - eprint = "1408.1627", - archivePrefix = "arXiv", - primaryClass = "hep-ph", - reportNumber = "DO-TH-14-17", - SLACcitation = "%%CITATION = ARXIV:1408.1627;%%" -} -@article{Biswas:2014gga, - author = "Biswas, Sanjoy and Chowdhury, Debtosh and Han, Sangeun - and Lee, Seung J.", - title = "{Explaining the lepton non-universality at the LHCb and - CMS within a unified framework}", - journal = "JHEP", - volume = "02", - year = "2015", - pages = "142", - doi = "10.1007/JHEP02(2015)142", - eprint = "1409.0882", - archivePrefix = "arXiv", - primaryClass = "hep-ph", - SLACcitation = "%%CITATION = ARXIV:1409.0882;%%" -} -@article{Gripaios:2014tna, - author = "Gripaios, Ben and Nardecchia, Marco and Renner, S. A.", - title = "{Composite leptoquarks and anomalies in $B$-meson - decays}", - journal = "JHEP", - volume = "05", - year = "2015", - pages = "006", - doi = "10.1007/JHEP05(2015)006", - eprint = "1412.1791", - archivePrefix = "arXiv", - primaryClass = "hep-ph", - SLACcitation = "%%CITATION = ARXIV:1412.1791;%%" -} -# QCD issues -@article{Jager:2012uw, - author = "Jäger, S. and Martin Camalich, J.", - title = "{On $B \to V \ell \ell$ at small dilepton invariant - mass, power corrections, and new physics}", - journal = "JHEP", - volume = "05", - year = "2013", - pages = "043", - doi = "10.1007/JHEP05(2013)043", - eprint = "1212.2263", - archivePrefix = "arXiv", - primaryClass = "hep-ph", - SLACcitation = "%%CITATION = ARXIV:1212.2263;%%" -} -@article{Jager:2014rwa, - author = "Jäger, Sebastian and Martin Camalich, Jorge", - title = "{Reassessing the discovery potential of the $B \to K^{*} - \ell^+\ell^-$ decays in the large-recoil region: SM - challenges and BSM opportunities}", - journal = "Phys. Rev.", - volume = "D93", - year = "2016", - number = "1", - pages = "014028", - doi = "10.1103/PhysRevD.93.014028", - eprint = "1412.3183", - archivePrefix = "arXiv", - primaryClass = "hep-ph", - SLACcitation = "%%CITATION = ARXIV:1412.3183;%%" -} -@article{Aaij:2014ora, - author = "Aaij, Roel and others", - title = "{Test of lepton universality using $B^{+}\rightarrow - K^{+}\ell^{+}\ell^{-}$ decays}", - collaboration = "LHCb", - journal = "Phys. Rev. Lett.", - volume = "113", - year = "2014", - pages = "151601", - doi = "10.1103/PhysRevLett.113.151601", - eprint = "1406.6482", - archivePrefix = "arXiv", - primaryClass = "hep-ex", - reportNumber = "CERN-PH-EP-2014-140, LHCB-PAPER-2014-024", - SLACcitation = "%%CITATION = ARXIV:1406.6482;%%" -} -@article{Aaij:2017vbb, - author = "Aaij, R. and others", - title = "{Test of lepton universality with $B^{0} \rightarrow - K^{*0}\ell^{+}\ell^{-}$ decays}", - collaboration = "LHCb", - journal = "JHEP", - volume = "08", - year = "2017", - pages = "055", - doi = "10.1007/JHEP08(2017)055", - eprint = "1705.05802", - archivePrefix = "arXiv", - primaryClass = "hep-ex", - reportNumber = "LHCB-PAPER-2017-013, CERN-EP-2017-100", - SLACcitation = "%%CITATION = ARXIV:1705.05802;%%" -} -% BR measurements -@article{Aaij:2013aln, - author = "Aaij, R and others", - title = "{Differential branching fraction and angular analysis of - the decay $B_s^0\to\phi\mu^{+}\mu^{-}$}", - collaboration = "LHCb", - journal = "JHEP", - volume = "07", - year = "2013", - pages = "084", - doi = "10.1007/JHEP07(2013)084", - eprint = "1305.2168", - archivePrefix = "arXiv", - primaryClass = "hep-ex", - reportNumber = "CERN-PH-EP-2013-078, LHCB-PAPER-2013-017", - SLACcitation = "%%CITATION = ARXIV:1305.2168;%%" -} -@article{Aaij:2014pli, - author = "Aaij, R. and others", - title = "{Differential branching fractions and isospin asymmetries - of $B \to K^{(*)} \mu^+ \mu^-$ decays}", - collaboration = "LHCb", - journal = "JHEP", - volume = "06", - year = "2014", - pages = "133", - doi = "10.1007/JHEP06(2014)133", - eprint = "1403.8044", - archivePrefix = "arXiv", - primaryClass = "hep-ex", - reportNumber = "LHCB-PAPER-2014-006, CERN-PH-EP-2014-055", - SLACcitation = "%%CITATION = ARXIV:1403.8044;%%" -} -@article{Hiller:2003js, - author = "Hiller, Gudrun and Kruger, Frank", - title = "{More model-independent analysis of $b \to s$ processes}", - journal = "Phys. Rev.", - volume = "D69", - year = "2004", - pages = "074020", - doi = "10.1103/PhysRevD.69.074020", - eprint = "hep-ph/0310219", - archivePrefix = "arXiv", - primaryClass = "hep-ph", - reportNumber = "LMU-18-03, TUM-HEP-519-03", - SLACcitation = "%%CITATION = HEP-PH/0310219;%%" -} -@article{Bordone:2016gaq, - author = "Bordone, Marzia and Isidori, Gino and Pattori, Andrea", - title = "{On the Standard Model predictions for $R_K$ and - $R_{K^*}$}", - journal = "Eur. Phys. J.", - volume = "C76", - year = "2016", - number = "8", - pages = "440", - doi = "10.1140/epjc/s10052-016-4274-7", - eprint = "1605.07633", - archivePrefix = "arXiv", - primaryClass = "hep-ph", - SLACcitation = "%%CITATION = ARXIV:1605.07633;%%" -} -@article{Capdevila:2017bsm, - author = "Capdevila, Bernat and Crivellin, Andreas and - Descotes-Genon, Sébastien and Matias, Joaquim and Virto, - Javier", - title = "{Patterns of New Physics in $b\to s\ell^+\ell^-$ - transitions in the light of recent data}", - journal = "JHEP", - volume = "01", - year = "2018", - pages = "093", - doi = "10.1007/JHEP01(2018)093", - eprint = "1704.05340", - archivePrefix = "arXiv", - primaryClass = "hep-ph", - reportNumber = "PSI-PR-17-05, LPT-ORSAY-17-19", - SLACcitation = "%%CITATION = ARXIV:1704.05340;%%" -} -@article{Altmannshofer:2017yso, - author = "Altmannshofer, Wolfgang and Stangl, Peter and Straub, - David M.", - title = "{Interpreting Hints for Lepton Flavor Universality - Violation}", - journal = "Phys. Rev.", - volume = "D96", - year = "2017", - number = "5", - pages = "055008", - doi = "10.1103/PhysRevD.96.055008", - eprint = "1704.05435", - archivePrefix = "arXiv", - primaryClass = "hep-ph", - SLACcitation = "%%CITATION = ARXIV:1704.05435;%%" -} -@article{Hurth:2017hxg, - author = "Hurth, T. and Mahmoudi, F. and Martinez Santos, D. and - Neshatpour, S.", - title = "{Lepton nonuniversality in exclusive - $b{\rightarrow}s{\ell}{\ell}$ decays}", - journal = "Phys. Rev.", - volume = "D96", - year = "2017", - number = "9", - pages = "095034", - doi = "10.1103/PhysRevD.96.095034", - eprint = "1705.06274", - archivePrefix = "arXiv", - primaryClass = "hep-ph", - reportNumber = "CERN-TH-2017-110, IPM-P.A-466, MITP-17-035", - SLACcitation = "%%CITATION = ARXIV:1705.06274;%%" -} -@article{Bobeth:2017vxj, - author = "Bobeth, Christoph and Chrzaszcz, Marcin and van Dyk, - Danny and Virto, Javier", - title = "{Long-distance effects in $B\to K^*\ell\ell$ from - Analyticity}", - year = "2017", - eprint = "1707.07305", - archivePrefix = "arXiv", - primaryClass = "hep-ph", - reportNumber = "EOS-2017-01, MIT-CTP-4918, TUM-HEP-1087-17, ZU-TH-17-17", - SLACcitation = "%%CITATION = ARXIV:1707.07305;%%" -} -@article{Blake:2017fyh, - author = "Blake, Thomas and Egede, Ulrik and Owen, Patrick and - Pomery, Gabriela and Petridis, Konstantinos Alexandros", - title = "{An empirical model of the long-distance contributions to - $\bar{B}^{0} \rightarrow \bar{K}^{*0}\mu^{+}\mu^{-}$ - transitions}", - year = "2017", - eprint = "1709.03921", - archivePrefix = "arXiv", - primaryClass = "hep-ph", - SLACcitation = "%%CITATION = ARXIV:1709.03921;%%" -} -@article{Hurth:2017sqw, - author = "Hurth, T. and Langenbruch, C. and Mahmoudi, F.", - title = "{Direct determination of Wilson coefficients using - $B^0\to K^{*0}\mu^+\mu^-$ decays}", - journal = "JHEP", - volume = "11", - year = "2017", - pages = "176", - doi = "10.1007/JHEP11(2017)176", - eprint = "1708.04474", - archivePrefix = "arXiv", - primaryClass = "hep-ph", - 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. 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Rev.", - volume = "D89", - year = "2014", - number = "9", - pages = "094501", - doi = "10.1103/PhysRevD.89.094501", - eprint = "1310.3722", - archivePrefix = "arXiv", - primaryClass = "hep-lat", - reportNumber = "DAMTP-2013-45, MIT-CTP-4507", - SLACcitation = "%%CITATION = ARXIV:1310.3722;%%" -} -@inproceedings{Mauri:2018, - author = "Chrzaszcz, Marcin and Mauri, Andrea and Serra, Nicola and Silva Coutinho, Rafael and van Dyk, Danny", - title = "{Prospects for data-driven analyses of the decays $B \to K^* \ell^+ \ell^-$}", - booktitle = "{$b \to s \ell \ell$ 2018: 6th Workshop on Rare Semileptonic B Decays, February 20-22}", - url = "https://indico.ph.tum.de/event/3782/sessions/558/attachments/2681/3011/NonLocalMEs2.pdf", - year = "2018", - eprint = "[In preparation]", -} -@article{Aaij:2013qta, - author = "Aaij, R and others", - title = "{Measurement of Form-Factor-Independent Observables in - the Decay $B^{0} \to K^{*0} \mu^+ \mu^-$}", - collaboration = "LHCb", - journal = "Phys. Rev. Lett.", - volume = "111", - year = "2013", - pages = "191801", - doi = "10.1103/PhysRevLett.111.191801", - eprint = "1308.1707", - archivePrefix = "arXiv", - primaryClass = "hep-ex", - reportNumber = "LHCB-PAPER-2013-037, CERN-PH-EP-2013-146", - SLACcitation = "%%CITATION = ARXIV:1308.1707;%%" -} - diff --git a/prd/old/SupplementalMaterial.zip b/prd/old/SupplementalMaterial.zip new file mode 100644 index 0000000..a193d1c --- /dev/null +++ b/prd/old/SupplementalMaterial.zip Binary files differ diff --git a/prd/old/SupplementalMaterial/.DS_Store b/prd/old/SupplementalMaterial/.DS_Store new file mode 100644 index 0000000..5008ddf --- /dev/null +++ b/prd/old/SupplementalMaterial/.DS_Store Binary files differ diff --git a/prd/old/SupplementalMaterial/ellipses_CpMu.pdf b/prd/old/SupplementalMaterial/ellipses_CpMu.pdf new file mode 100644 index 0000000..46ebca3 --- /dev/null +++ b/prd/old/SupplementalMaterial/ellipses_CpMu.pdf Binary files differ diff --git a/prd/old/SupplementalMaterial/ellipses_CpMu_Hz.pdf b/prd/old/SupplementalMaterial/ellipses_CpMu_Hz.pdf new file mode 100644 index 0000000..8a034e5 --- /dev/null +++ b/prd/old/SupplementalMaterial/ellipses_CpMu_Hz.pdf Binary files differ diff --git a/prd/old/SupplementalMaterial/suppl.tex b/prd/old/SupplementalMaterial/suppl.tex new file mode 100644 index 0000000..7d0a4cf --- /dev/null +++ b/prd/old/SupplementalMaterial/suppl.tex @@ -0,0 +1,181 @@ +% vim: set sts=4 et : +\documentclass[reprint,preprintnumbers,prd,nofootinbib]{revtex4-1} + +\usepackage{amsfonts} +\usepackage{amsmath} +\usepackage{array} +\usepackage{braket} +\usepackage{epstopdf} +\usepackage{graphicx} +\usepackage{hepparticles} +\usepackage{hepnicenames} +\usepackage{hepunits} +\usepackage{hyperref} +\usepackage[% + utf8 +]{inputenc} +\usepackage{slashed} +\usepackage{subfigure} +\usepackage{placeins} +\usepackage[% + normalem +]{ulem} +\usepackage[% + usenames, + svgnames, + dvipsnames +]{xcolor} + + +%% Shortcuts %% +\newcommand{\ie}{\textit{i.e.}} +\newcommand{\nuvec}{\vec{\nu}} +\newcommand{\refapp}[1]{appendix~\ref{app:#1}} +\newcommand{\refeq}[1]{eq.~(\ref{eq:#1})} +\newcommand{\refeqs}[2]{eqs.~(\ref{eq:#1})--(\ref{eq:#2})} +\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} +\newcommand{\eps}{\varepsilon} +\newcommand{\para}{\parallel} +\newcommand{\Gfermi}{G_F} +%\newcommand{\dd}[2][]{{\mathrm{d}^{#1}}#2\,} +\newcommand{\dd}{\ensuremath{\textrm{d}}} +\newcommand{\order}[1]{\ensuremath{\mathcal{O}\left(#1\right)}} +\DeclareMathOperator{\sign}{sgn} +\DeclareMathOperator{\ReNew}{Re} +\DeclareMathOperator{\ImNew}{Im} +\let\Re\ReNew +\let\Im\ImNew +\DeclareMathOperator*{\sumint}{% +\mathchoice% + {\ooalign{$\displaystyle\sum$\cr\hidewidth$\displaystyle\int$\hidewidth\cr}} + {\ooalign{\raisebox{.14\height}{\scalebox{.7}{$\textstyle\sum$}}\cr\hidewidth$\textstyle\int$\hidewidth\cr}} + {\ooalign{\raisebox{.2\height}{\scalebox{.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}} + {\ooalign{\raisebox{.2\height}{\scalebox{.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}} +} +\DeclareMathOperator*{\argmax}{arg\,max} +\newcommand{\wilson}[2][]{\mathcal{C}^\text{#1}_{#2}} +\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} + +\def\PB {\ensuremath{\mathrm{B}}\xspace} +\def\B {{\ensuremath{\PB}}\xspace} +\def\PK {\ensuremath{\mathrm{K}}\xspace} +\def\kaon {{\ensuremath{\PK}}\xspace} +\def\Kstarz {{\ensuremath{\kaon^{*0}}}\xspace} +\def\Bd {{\ensuremath{\B^0}}\xspace} +\def\Bz {{\ensuremath{\B^0}}\xspace} + +%% Key decay channels + +\def\BdToKstmm {\decay{\Bd}{\Kstarz\mup\mun}} +\def\BdbToKstmm {\decay{\Bdb}{\Kstarzb\mup\mun}} + +\def\BsToJPsiPhi {\decay{\Bs}{\jpsi\phi}} +\def\BdToJPsiKst {\decay{\Bd}{\jpsi\Kstarz}} +\def\BdbToJPsiKst {\decay{\Bdb}{\jpsi\Kstarzb}} + +%% Rare decays +\def\BdKstee {\decay{\Bd}{\Kstarz\epem}} +\def\BdbKstee {\decay{\Bdb}{\Kstarzb\epem}} +\def\bsll {\decay{\bquark}{\squark \ell^+ \ell^-}} + +\def\lepton {{\ensuremath{\ell}}\xspace} +\def\ellm {{\ensuremath{\ell^-}}\xspace} +\def\ellp {{\ensuremath{\ell^+}}\xspace} +\def\ellell {\ensuremath{\ell^+ \ell^-}\xspace} +\def\mumu {{\ensuremath{\Pmu^+\Pmu^-}}\xspace} + +\def\lhcb {\mbox{LHCb}\xspace} +\def\belle {\mbox{Belle}\xspace} + +\def\WC {\ensuremath{\mathcal{C}}\xspace} + + + +\begin{document} + +\allowdisplaybreaks + +\preprint{ZU-TH-15/18} +\title{Supplemental material for ``Towards establishing Lepton Flavour Universality violation in $\bar{B}\to \bar{K}^*\ell^+\ell^-$ decays''} +\author{Andrea Mauri} +\email{a.mauri@cern.ch} +\author{Nicola Serra} +\email{nicola.serra@cern.ch} +\author{Rafael Silva Coutinho} +\email{rafael.silva.coutinho@cern.ch} +\affiliation{Physik-Institut, Universit\"at Z\"urich, Winterthurer Strasse 190, 8057 Z\"urich, Switzerland} + + +%\appendix + +%\section{Supplemental material} + + +\maketitle + +An extension of the physics case of the proposed method +is to investigate the sensitivity to the chirality-flipped counterparts of the usual Wilson coefficients, +\textit{i.e.} $\WC^{\prime (\mu)}_9$ and $\WC^{\prime(\mu)}_{10}$. +Following the formalism discussed in this letter, the primed WCs are examined by considering in addition to the +\texttt{BMP}$_{\WC_{9,10}}$ three different modified NP scenarios for the muon only: +$\WC_{9,10}^{\prime(\mu)} = \WC^{\prime \rm{SM}}_{9,10} = 0$; +$\WC^{\prime (\mu)}_9 = \WC^{\prime (\mu)}_{10} = 0.3$; +and $\WC^{\prime (\mu)}_9 = - \WC^{\prime (\mu)}_{10} = 0.3$. +Notice that for the electron mode the $\WC_{9,10}^{\prime(e)}$ is set and fixed to the SM value $\WC^{\prime \rm{SM}}_{9,10} = 0$. + + +Figure~\ref{fig:Cp_Hz} shows the fit results for different order of the analytic expansion for +the non-local hadronic contribution for a NP scenario with $\WC'^{(\mu)}_9 = \WC'^{(\mu)}_{10} = 0.3$ +% +\begin{figure}[h] +\includegraphics[width=.4\textwidth]{ellipses_CpMu_Hz.pdf} +\caption{% + Two-dimensional sensitivity scans for the pair of Wilson coefficients $\WC'^{(\mu)}_9$ and + $\WC'^{(\mu)}_{10}$ for different non-local hadronic parametrisation models for a NP scenario + with $\WC'^{(\mu)}_9 = \WC'^{(\mu)}_{10} = 0.3$. + The contours correspond to 99\% confidence level statistical-only uncertainty bands evaluated with + the expected statistics after \lhcb Run II. + \label{fig:Cp_Hz} +} +\end{figure} +% +and yields corresponding to the \lhcb Run II expected statistics. +The dependency on the determination of $\WC'^{(\mu)}_9$ and $\WC'^{(\mu)}_{10}$ on the +order of the expansion clearly saturates after $\mathcal{H}_\lambda[z^3]$ and allows a measurement +of the primed Wilson coefficients for the muon decay channel $B^{0} \to K^{*0} \mumu$ independent +on the theoretical hadronic uncertainty. +% +Figure~\ref{fig:Cp} shows the prospects for the sensitivity to the $\WC'^{(\mu)}_9$ and +$\WC'^{(\mu)}_{10}$ Wilson coefficients corresponding to the expected statistics at +the LHCb upgrade with $50\,$fb$^{-1}$ and $\,300\,$fb$^{-1}$. +Note that only with the full capability of the LHCb experiment it is possible +to start disentangling the different NP hypotheses. + + +% +\begin{figure}[b] +\includegraphics[width=.4\textwidth]{ellipses_CpMu.pdf} +\caption{% + Two-dimensional sensitivity scans for the pair of Wilson coefficients $\WC'^{(\mu)}_9$ and $\WC'^{(\mu)}_{10}$ + for three NP scenarios: (blue) $\WC'^{(\mu)}_9 = \WC'^{(\mu)}_{10} = 0$, (orange) $\WC'^{(\mu)}_9 = \WC'^{(\mu)}_{10} = 0.3$ + and (magenta) $\WC'^{(\mu)}_9 = - \WC'^{(\mu)}_{10} = 0.3$. + The contours correspond to 99\% confidence level statistical-only uncertainty bands expected for the LHCb Upgrade (dotted) + $50\,$fb$^{-1}$ and (solid) $\,300\,$fb$^{-1}$ statistics. + \label{fig:Cp} +} +\end{figure} + +%\FloatBarrier + +\end{document} diff --git a/prd/old/draft.pdf b/prd/old/draft.pdf new file mode 100644 index 0000000..4f00004 --- /dev/null +++ b/prd/old/draft.pdf Binary files differ diff --git a/prd/old/main.zip b/prd/old/main.zip new file mode 100644 index 0000000..d38b342 --- /dev/null +++ b/prd/old/main.zip Binary files differ diff --git a/prd/old/main/.DS_Store b/prd/old/main/.DS_Store new file mode 100644 index 0000000..5008ddf --- /dev/null +++ b/prd/old/main/.DS_Store Binary files differ diff --git a/prd/old/main/B2Kstll_summary.pdf b/prd/old/main/B2Kstll_summary.pdf new file mode 100644 index 0000000..58f805a --- /dev/null +++ b/prd/old/main/B2Kstll_summary.pdf Binary files differ diff --git a/prd/old/main/draft.bbl b/prd/old/main/draft.bbl new file mode 100644 index 0000000..cb6db10 --- /dev/null +++ b/prd/old/main/draft.bbl @@ -0,0 +1,680 @@ +%merlin.mbs apsrev4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked +%Control: key (0) +%Control: author (8) initials jnrlst +%Control: editor formatted (1) identically to author +%Control: production of article title (-1) disabled +%Control: page (0) single +%Control: year (1) truncated +%Control: production of eprint (0) enabled +\begin{thebibliography}{51}% +\makeatletter +\providecommand \@ifxundefined [1]{% + \@ifx{#1\undefined} +}% +\providecommand \@ifnum [1]{% + \ifnum #1\expandafter \@firstoftwo + \else \expandafter \@secondoftwo + \fi +}% +\providecommand \@ifx [1]{% + \ifx #1\expandafter \@firstoftwo + \else \expandafter \@secondoftwo + \fi +}% +\providecommand \natexlab [1]{#1}% +\providecommand \enquote [1]{``#1''}% +\providecommand \bibnamefont [1]{#1}% +\providecommand \bibfnamefont [1]{#1}% +\providecommand \citenamefont [1]{#1}% +\providecommand \href@noop [0]{\@secondoftwo}% +\providecommand \href [0]{\begingroup \@sanitize@url 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{arXiv:1708.04474 [hep-ph]} \BibitemShut + {NoStop}% +%%CITATION = ARXIV:1708.04474;%% +\bibitem [{\citenamefont {Bobeth}\ \emph {et~al.}(2008)\citenamefont {Bobeth}, + \citenamefont {Hiller},\ and\ \citenamefont {Piranishvili}}]{Bobeth:2008ij}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {C.}~\bibnamefont + {Bobeth}}, \bibinfo {author} {\bibfnamefont {G.}~\bibnamefont {Hiller}}, \ + and\ \bibinfo {author} {\bibfnamefont {G.}~\bibnamefont {Piranishvili}},\ + }\href {\doibase 10.1088/1126-6708/2008/07/106} {\bibfield {journal} + {\bibinfo {journal} {JHEP}\ }\textbf {\bibinfo {volume} {07}},\ \bibinfo + {pages} {106} (\bibinfo {year} {2008})},\ \Eprint + {http://arxiv.org/abs/0805.2525} {arXiv:0805.2525 [hep-ph]} \BibitemShut + {NoStop}% +%%CITATION = ARXIV:0805.2525;%% +\bibitem [{\citenamefont {Boyd}\ \emph {et~al.}(1995)\citenamefont {Boyd}, + \citenamefont {Grinstein},\ and\ \citenamefont {Lebed}}]{Boyd:1995cf}% + \BibitemOpen + \bibfield {author} {\bibinfo 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Lett.}\ }\textbf {\bibinfo {volume} {B353}},\ \bibinfo + {pages} {306} (\bibinfo {year} {1995})},\ \Eprint + {http://arxiv.org/abs/hep-ph/9504235} {arXiv:hep-ph/9504235 [hep-ph]} + \BibitemShut {NoStop}% +%%CITATION = HEP-PH/9504235;%% +\bibitem [{\citenamefont {Bourrely}\ \emph {et~al.}(2009)\citenamefont + {Bourrely}, \citenamefont {Caprini},\ and\ \citenamefont + {Lellouch}}]{Bourrely:2008za}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {C.}~\bibnamefont + {Bourrely}}, \bibinfo {author} {\bibfnamefont {I.}~\bibnamefont {Caprini}}, \ + and\ \bibinfo {author} {\bibfnamefont {L.}~\bibnamefont {Lellouch}},\ }\href + {\doibase 10.1103/PhysRevD.82.099902, 10.1103/PhysRevD.79.013008} {\bibfield + {journal} {\bibinfo {journal} {Phys. Rev.}\ }\textbf {\bibinfo {volume} + {D79}},\ \bibinfo {pages} {013008} (\bibinfo {year} {2009})},\ \bibinfo + {note} {[Erratum: Phys. Rev.D82,099902(2010)]},\ \Eprint + {http://arxiv.org/abs/0807.2722} {arXiv:0807.2722 [hep-ph]} \BibitemShut + {NoStop}% +%%CITATION = ARXIV:0807.2722;%% +\bibitem [{\citenamefont {Chrzaszcz}\ \emph {et~al.}(2018)\citenamefont + {Chrzaszcz}, \citenamefont {Mauri}, \citenamefont {Serra}, \citenamefont + {Silva~Coutinho},\ and\ \citenamefont {van Dyk}}]{Chrzaszcz:2018yza}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont + {Chrzaszcz}}, \bibinfo {author} {\bibfnamefont {A.}~\bibnamefont {Mauri}}, + \bibinfo {author} {\bibfnamefont {N.}~\bibnamefont {Serra}}, \bibinfo + {author} {\bibfnamefont {R.}~\bibnamefont {Silva~Coutinho}}, \ and\ \bibinfo + {author} {\bibfnamefont {D.}~\bibnamefont {van Dyk}},\ }\href@noop {} {\ + (\bibinfo {year} {2018})},\ \Eprint {http://arxiv.org/abs/1805.06378} + {arXiv:1805.06378 [hep-ph]} \BibitemShut {NoStop}% +%%CITATION = ARXIV:1805.06378;%% +\bibitem [{\citenamefont {Paul}\ and\ \citenamefont + {Straub}(2017)}]{Paul:2016urs}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont + {Paul}}\ and\ \bibinfo {author} {\bibfnamefont {D.~M.}\ \bibnamefont + {Straub}},\ }\href {\doibase 10.1007/JHEP04(2017)027} {\bibfield {journal} + {\bibinfo {journal} {JHEP}\ }\textbf {\bibinfo {volume} {04}},\ \bibinfo + {pages} {027} (\bibinfo {year} {2017})},\ \Eprint + {http://arxiv.org/abs/1608.02556} {arXiv:1608.02556 [hep-ph]} \BibitemShut + {NoStop}% +%%CITATION = ARXIV:1608.02556;%% +\bibitem [{sup()}]{suppl}% + \BibitemOpen + \href@noop {} {\enquote {\bibinfo {title} {{See Supplemental Material at [URL + will be inserted by publisher] for sensitivity studies on the determination + of the Wilson coefficients $\mathcal{C}_9^{\prime\,(\mu)}$ and + $\mathcal{C}{10}^{\prime\,(\mu)}$}},}\ }\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Aaij}\ \emph + {et~al.}(2017{\natexlab{b}})\citenamefont {Aaij} \emph + {et~al.}}]{Aaij:2244311}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {R.}~\bibnamefont + {Aaij}} \emph {et~al.} (\bibinfo {collaboration} {LHCb Collaboration}),\ + }\href {http://cds.cern.ch/record/2244311} {\emph {\bibinfo {title} + {{Expression of Interest for a Phase-II LHCb Upgrade: Opportunities in + flavour physics, and beyond, in the HL-LHC era}}}},\ \bibinfo {type} {Tech. + Rep.}\ \bibinfo {number} {CERN-LHCC-2017-003}\ (\bibinfo {institution} + {CERN},\ \bibinfo {address} {Geneva},\ \bibinfo {year} {2017})\BibitemShut + {NoStop}% +\bibitem [{\citenamefont {Lionetto}\ \emph {et~al.}(2018)\citenamefont + {Lionetto}, \citenamefont {Serra}, \citenamefont {Steinkamp},\ and\ + \citenamefont {Mueller}}]{Lionetto:2624938}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {F.}~\bibnamefont + {Lionetto}}, \bibinfo {author} {\bibfnamefont {N.}~\bibnamefont {Serra}}, + \bibinfo {author} {\bibfnamefont {O.}~\bibnamefont {Steinkamp}}, \ and\ + \bibinfo {author} {\bibfnamefont {K.}~\bibnamefont {Mueller}},\ }\href + {https://cds.cern.ch/record/2624938} {\enquote {\bibinfo {title} + {{Measurements of Angular Observables of $B^0\to K^{*0} \mu^+ \mu^-$ and $B^0 + \to K^{*0} e^+ e^-$ Decays and the Upgrade of LHCb}},}\ }\bibinfo + {howpublished} {\url{https://cds.cern.ch/record/2624938}} (\bibinfo {year} + {2018}),\ \bibinfo {note} {presented 22 Mar 2018}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Bharucha}\ \emph {et~al.}(2016)\citenamefont + {Bharucha}, \citenamefont {Straub},\ and\ \citenamefont + {Zwicky}}]{Straub:2015ica}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont + {Bharucha}}, \bibinfo {author} {\bibfnamefont {D.~M.}\ \bibnamefont + {Straub}}, \ and\ \bibinfo {author} {\bibfnamefont {R.}~\bibnamefont + {Zwicky}},\ }\href {\doibase 10.1007/JHEP08(2016)098} {\bibfield {journal} + {\bibinfo {journal} {JHEP}\ }\textbf {\bibinfo {volume} {08}},\ \bibinfo + {pages} {098} (\bibinfo {year} {2016})},\ \Eprint + {http://arxiv.org/abs/1503.05534} {arXiv:1503.05534 [hep-ph]} \BibitemShut + {NoStop}% +%%CITATION = ARXIV:1503.05534;%% +\bibitem [{\citenamefont {Ball}\ and\ \citenamefont + {Braun}(1998)}]{Ball:1998kk}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {P.}~\bibnamefont + {Ball}}\ and\ \bibinfo {author} {\bibfnamefont {V.~M.}\ \bibnamefont + {Braun}},\ }\href {\doibase 10.1103/PhysRevD.58.094016} {\bibfield {journal} + {\bibinfo {journal} {Phys. Rev.}\ }\textbf {\bibinfo {volume} {D58}},\ + \bibinfo {pages} {094016} (\bibinfo {year} {1998})},\ \Eprint + {http://arxiv.org/abs/hep-ph/9805422} {arXiv:hep-ph/9805422 [hep-ph]} + \BibitemShut {NoStop}% +%%CITATION = HEP-PH/9805422;%% +\bibitem [{\citenamefont {Khodjamirian}\ \emph {et~al.}(2007)\citenamefont + {Khodjamirian}, \citenamefont {Mannel},\ and\ \citenamefont + {Offen}}]{Khodjamirian:2006st}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont + {Khodjamirian}}, \bibinfo {author} {\bibfnamefont {T.}~\bibnamefont + {Mannel}}, \ and\ \bibinfo {author} {\bibfnamefont {N.}~\bibnamefont + {Offen}},\ }\href {\doibase 10.1103/PhysRevD.75.054013} {\bibfield {journal} + {\bibinfo {journal} {Phys. Rev.}\ }\textbf {\bibinfo {volume} {D75}},\ + \bibinfo {pages} {054013} (\bibinfo {year} {2007})},\ \Eprint + {http://arxiv.org/abs/hep-ph/0611193} {arXiv:hep-ph/0611193 [hep-ph]} + \BibitemShut {NoStop}% +%%CITATION = HEP-PH/0611193;%% +\bibitem [{\citenamefont {Becirevic}\ \emph {et~al.}(2007)\citenamefont + {Becirevic}, \citenamefont {Lubicz},\ and\ \citenamefont + {Mescia}}]{Becirevic:2006nm}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {D.}~\bibnamefont + {Becirevic}}, \bibinfo {author} {\bibfnamefont {V.}~\bibnamefont {Lubicz}}, \ + and\ \bibinfo {author} {\bibfnamefont {F.}~\bibnamefont {Mescia}},\ }\href + {\doibase 10.1016/j.nuclphysb.2007.01.032} {\bibfield {journal} {\bibinfo + {journal} {Nucl. Phys.}\ }\textbf {\bibinfo {volume} {B769}},\ \bibinfo + {pages} {31} (\bibinfo {year} {2007})},\ \Eprint + {http://arxiv.org/abs/hep-ph/0611295} {arXiv:hep-ph/0611295 [hep-ph]} + \BibitemShut {NoStop}% +%%CITATION = HEP-PH/0611295;%% +\bibitem [{\citenamefont {Horgan}\ \emph {et~al.}(2014)\citenamefont {Horgan}, + \citenamefont {Liu}, \citenamefont {Meinel},\ and\ \citenamefont + {Wingate}}]{Horgan:2013hoa}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {R.~R.}\ \bibnamefont + {Horgan}}, \bibinfo {author} {\bibfnamefont {Z.}~\bibnamefont {Liu}}, + \bibinfo {author} {\bibfnamefont {S.}~\bibnamefont {Meinel}}, \ and\ \bibinfo + {author} {\bibfnamefont {M.}~\bibnamefont {Wingate}},\ }\href {\doibase + 10.1103/PhysRevD.89.094501} {\bibfield {journal} {\bibinfo {journal} {Phys. + Rev.}\ }\textbf {\bibinfo {volume} {D89}},\ \bibinfo {pages} {094501} + (\bibinfo {year} {2014})},\ \Eprint {http://arxiv.org/abs/1310.3722} + {arXiv:1310.3722 [hep-lat]} \BibitemShut {NoStop}% +%%CITATION = ARXIV:1310.3722;%% +\bibitem [{\citenamefont {Capdevila}\ \emph {et~al.}(2017)\citenamefont + {Capdevila}, \citenamefont {Descotes-Genon}, \citenamefont {Hofer},\ and\ + \citenamefont {Matias}}]{Capdevila:2017ert}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {B.}~\bibnamefont + {Capdevila}}, \bibinfo {author} {\bibfnamefont {S.}~\bibnamefont + {Descotes-Genon}}, \bibinfo {author} {\bibfnamefont {L.}~\bibnamefont + {Hofer}}, \ and\ \bibinfo {author} {\bibfnamefont {J.}~\bibnamefont + {Matias}},\ }\href {\doibase 10.1007/JHEP04(2017)016} {\bibfield {journal} + {\bibinfo {journal} {JHEP}\ }\textbf {\bibinfo {volume} {04}},\ \bibinfo + {pages} {016} (\bibinfo {year} {2017})},\ \Eprint + {http://arxiv.org/abs/1701.08672} {arXiv:1701.08672 [hep-ph]} \BibitemShut + {NoStop}% +%%CITATION = ARXIV:1701.08672;%% +\bibitem [{\citenamefont {Albrecht}\ \emph {et~al.}(2017)\citenamefont + {Albrecht}, \citenamefont {Bernlochner}, \citenamefont {Kenzie}, + \citenamefont {Reichert}, \citenamefont {Straub},\ and\ \citenamefont + {Tully}}]{Albrecht:2017odf}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {J.}~\bibnamefont + {Albrecht}}, \bibinfo {author} {\bibfnamefont {F.}~\bibnamefont + {Bernlochner}}, \bibinfo {author} {\bibfnamefont {M.}~\bibnamefont {Kenzie}}, + \bibinfo {author} {\bibfnamefont {S.}~\bibnamefont {Reichert}}, \bibinfo + {author} {\bibfnamefont {D.}~\bibnamefont {Straub}}, \ and\ \bibinfo {author} + {\bibfnamefont {A.}~\bibnamefont {Tully}},\ }\href@noop {} {\ (\bibinfo + {year} {2017})},\ \Eprint {http://arxiv.org/abs/1709.10308} {arXiv:1709.10308 + [hep-ph]} \BibitemShut {NoStop}% +%%CITATION = ARXIV:1709.10308;%% +\bibitem [{\citenamefont {Becirevic}\ and\ \citenamefont + {Tayduganov}(2013)}]{Becirevic:2012dp}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {D.}~\bibnamefont + {Becirevic}}\ and\ \bibinfo {author} {\bibfnamefont {A.}~\bibnamefont + {Tayduganov}},\ }\href {\doibase 10.1016/j.nuclphysb.2012.11.016} {\bibfield + {journal} {\bibinfo {journal} {Nucl. Phys.}\ }\textbf {\bibinfo {volume} + {B868}},\ \bibinfo {pages} {368} (\bibinfo {year} {2013})},\ \Eprint + {http://arxiv.org/abs/1207.4004} {arXiv:1207.4004 [hep-ph]} \BibitemShut + {NoStop}% +%%CITATION = ARXIV:1207.4004;%% +\bibitem [{\citenamefont {Aaij}\ \emph + {et~al.}(2016{\natexlab{b}})\citenamefont {Aaij} \emph + {et~al.}}]{Aaij:2016flj}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {R.}~\bibnamefont + {Aaij}} \emph {et~al.} (\bibinfo {collaboration} {LHCb}),\ }\href {\doibase + 10.1007/JHEP11(2016)047, 10.1007/JHEP04(2017)142} {\bibfield {journal} + {\bibinfo {journal} {JHEP}\ }\textbf {\bibinfo {volume} {11}},\ \bibinfo + {pages} {047} (\bibinfo {year} {2016}{\natexlab{b}})},\ \bibinfo {note} + {[Erratum: JHEP04,142(2017)]},\ \Eprint {http://arxiv.org/abs/1606.04731} + {arXiv:1606.04731 [hep-ex]} \BibitemShut {NoStop}% +%%CITATION = ARXIV:1606.04731;%% +\end{thebibliography}% diff --git a/prd/old/main/draft.tex b/prd/old/main/draft.tex new file mode 100644 index 0000000..c0e48fb --- /dev/null +++ b/prd/old/main/draft.tex @@ -0,0 +1,495 @@ +% vim: set sts=4 et : +\documentclass[reprint,preprintnumbers,prd,nofootinbib]{revtex4-1} + +\usepackage{amsfonts} +\usepackage{amsmath} +\usepackage{array} +\usepackage{braket} +\usepackage{epstopdf} +\usepackage{graphicx} +\usepackage{hepparticles} +\usepackage{hepnicenames} +\usepackage{hepunits} +\usepackage{hyperref} +\usepackage[% + utf8 +]{inputenc} +\usepackage{slashed} +\usepackage{subfigure} +\usepackage{placeins} +\usepackage[% + normalem +]{ulem} +\usepackage[% + usenames, + svgnames, + dvipsnames +]{xcolor} + + +%% Shortcuts %% +\newcommand{\ie}{\textit{i.e.}} +\newcommand{\nuvec}{\vec{\nu}} +\newcommand{\refapp}[1]{appendix~\ref{app:#1}} +\newcommand{\refeq}[1]{eq.~(\ref{eq:#1})} +\newcommand{\refeqs}[2]{eqs.~(\ref{eq:#1})--(\ref{eq:#2})} +\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} +\newcommand{\eps}{\varepsilon} +\newcommand{\para}{\parallel} +\newcommand{\Gfermi}{G_F} +%\newcommand{\dd}[2][]{{\mathrm{d}^{#1}}#2\,} +\newcommand{\dd}{\ensuremath{\textrm{d}}} +\newcommand{\order}[1]{\ensuremath{\mathcal{O}\left(#1\right)}} +\DeclareMathOperator{\sign}{sgn} +\DeclareMathOperator{\ReNew}{Re} +\DeclareMathOperator{\ImNew}{Im} +\let\Re\ReNew +\let\Im\ImNew +\DeclareMathOperator*{\sumint}{% +\mathchoice% + {\ooalign{$\displaystyle\sum$\cr\hidewidth$\displaystyle\int$\hidewidth\cr}} + {\ooalign{\raisebox{.14\height}{\scalebox{.7}{$\textstyle\sum$}}\cr\hidewidth$\textstyle\int$\hidewidth\cr}} + {\ooalign{\raisebox{.2\height}{\scalebox{.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}} + {\ooalign{\raisebox{.2\height}{\scalebox{.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}} +} +\DeclareMathOperator*{\argmax}{arg\,max} +\newcommand{\wilson}[2][]{\mathcal{C}^\text{#1}_{#2}} +\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} + +\def\PB {\ensuremath{\mathrm{B}}\xspace} +\def\B {{\ensuremath{\PB}}\xspace} +\def\PK {\ensuremath{\mathrm{K}}\xspace} +\def\kaon {{\ensuremath{\PK}}\xspace} +\def\Kstarz {{\ensuremath{\kaon^{*0}}}\xspace} +\def\Bd {{\ensuremath{\B^0}}\xspace} +\def\Bz {{\ensuremath{\B^0}}\xspace} + +%% Key decay channels + +\def\BdToKstmm {\decay{\Bd}{\Kstarz\mup\mun}} +\def\BdbToKstmm {\decay{\Bdb}{\Kstarzb\mup\mun}} + +\def\BsToJPsiPhi {\decay{\Bs}{\jpsi\phi}} +\def\BdToJPsiKst {\decay{\Bd}{\jpsi\Kstarz}} +\def\BdbToJPsiKst {\decay{\Bdb}{\jpsi\Kstarzb}} + +%% Rare decays +\def\BdKstee {\decay{\Bd}{\Kstarz\epem}} +\def\BdbKstee {\decay{\Bdb}{\Kstarzb\epem}} +\def\bsll {\decay{\bquark}{\squark \ell^+ \ell^-}} + +\def\lepton {{\ensuremath{\ell}}\xspace} +\def\ellm {{\ensuremath{\ell^-}}\xspace} +\def\ellp {{\ensuremath{\ell^+}}\xspace} +\def\ellell {\ensuremath{\ell^+ \ell^-}\xspace} +\def\mumu {{\ensuremath{\Pmu^+\Pmu^-}}\xspace} + +\def\lhcb {\mbox{LHCb}\xspace} +\def\belle {\mbox{Belle}\xspace} + +\def\WC {\ensuremath{\mathcal{C}}\xspace} + + + +\begin{document} + +\allowdisplaybreaks + +\preprint{ZU-TH-15/18} +\title{Towards establishing Lepton Flavour Universality violation in $\bar{B}\to \bar{K}^*\ell^+\ell^-$ decays} +\author{Andrea Mauri} +\email{a.mauri@cern.ch} +\author{Nicola Serra} +\email{nicola.serra@cern.ch} +\author{Rafael Silva Coutinho} +\email{rafael.silva.coutinho@cern.ch} +\affiliation{Physik-Institut, Universit\"at Z\"urich, Winterthurer Strasse 190, 8057 Z\"urich, Switzerland} + +\begin{abstract} + Rare semileptonic $b \to s \ell^+ \ell^-$ transitions provide some of the most promising framework to search for new physics effects. +Recent analyses of these decays have indicated an anomalous behaviour in measurements + of angular distributions of the decay $B^0\to + K^*\mu^+\mu^-$ and lepton-flavour-universality observables. +Unambiguously establishing if these deviations have a common nature + is of paramount importance in order to understand the observed pattern. + We propose a novel approach to independently and complementary + probe this hypothesis + by performing a simultaneous amplitude analysis of $\bar{B}^0 \to \bar{K}^{*0} \mu^+\mu^-$ and $\bar{B}^0 \to \bar{K}^{*0} e^+e^-$ decays. + This method enables the direct determination of + observables that encode potential non-equal couplings of muons and electrons, + and are found to be insensitive to non-perturbative QCD effects. +If current hints of new physics are confirmed, our approach could +allow an early discovery of physics beyond the Standard +Model with LHCb Run-II datasets. +\end{abstract} + +\maketitle + +Flavour changing neutral current processes of {\textit{B}} meson decays +are crucial probes for the Standard Model (SM), +since as-yet undiscovered particles may contribute to these transitions 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. +An anomalous behaviour in angular and branching fraction analyses of the decay channel +$\bar{B}^{0} \to \bar{K}^{*0} \mu^{+}\mu^{-}$ has been recently reported~\cite{Aaij:2015oid,Wehle:2016yoi,Aaij:2013aln,Aaij:2014pli}, +notably in one of the observables with reduced theoretical uncertainties, +$P^{\prime}_{5}$~\cite{Aaij:2013qta,Descotes-Genon:2015uva}. +Several models have been suggested in order to interpret these results as new physics (NP) +signatures~\cite{Gauld:2013qja,Buras:2013qja,Altmannshofer:2013foa,Crivellin:2015era,Hiller:2014yaa,Biswas:2014gga,Gripaios:2014tna}. +Nonetheless, the vector-like nature of this pattern could be also explained by +non-perturbative QCD contributions from $b\to s c{\bar{c}}$ operators ({\textit{i.e.}} charm loops) +that are able to either mimic or camouflage NP effects~\cite{Jager:2012uw,Jager:2014rwa,Ciuchini:2015qxb}. +Non-standard measurement in ratios of $b \to s \ell^+ \ell^-$ processes +- such as of $R_{K}$~\cite{Aaij:2014ora} and $R_{K^{*}}$~\cite{Aaij:2017vbb} - +indicate a suppression of the muon channel which is also compatible with the $P^{\prime}_{5}$ anomaly. +In this case an immediate interpretation of lepton flavour universality (LFU) breaking is +suggested due to the small theoretical uncertainties in their predictions~\cite{Hiller:2003js,Bordone:2016gaq}. +Whilst the individual level of significance of the present anomalies is still inconclusive, +there is an appealing non-trivial consistent pattern shown in +global analysis fits~\cite{Capdevila:2017bsm,Altmannshofer:2017yso,Hurth:2017hxg,Ciuchini:2017mik,Alok:2017sui}. + +The formalism of {\textit{b}} decays is commonly described within +an effective field theory~\cite{Altmannshofer:2008dz}, +which probes distinct energy scales; +with regimes classified into short-distance (high energies) perturbative +and non-calculable long-distance effects. +These can be parametrised in the weak Lagrangian in terms of +effective operators with different Lorentz structures, $\mathcal{O}_i$, +with corresponding couplings $\mathcal{C}_i$ - referred to as Wilson coefficients (WC). +Only a subset of the operators that are most sensitive to NP is examined in this work~\cite{Bobeth:2017vxj}, +\textit{i.e.} $\mathcal{O}_7$ (virtual photon exchanges), $\mathcal{O}_{9,10}$ (vector and axial currents) +and corresponding right-handed couplings with flipped helicities. +In this framework NP effects are incorporated +by introducing deviations in the WCs~\cite{Ali:1994bf} from their SM predictions, +{\textit{i.e.}} $\mathcal{C}_i = \mathcal{C}^{\mathrm{SM}}_i + \mathcal{C}^{\mathrm{NP}}_i$. +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~\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 matrix elements that are difficult to assess reliably from first principles. +Some promising approaches suggest to extract this contribution +from data-driven analyses~\cite{Blake:2017fyh,Hurth:2017sqw} +and by exploiting analytical properties of its structure~\cite{Bobeth:2017vxj}. +However, these models still have intrinsic limitations, in particular +in the assumptions that enter in parametrisation of the di-lepton invariant mass distribution. + +In this Letter we propose a new \textit{model-independent} approach that +from a simultaneous unbinned amplitude analysis of both +$\bar{B}^0 \to \bar{K}^{*0} \mu^+\mu^-$ and $\bar{B}^0 \to \bar{K}^{*0} e^+e^-$ decays +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 effect relies on the strong correlation between the muon and electron modes +imposed by the lepton-flavour universality of the hadronic matrix elements. +Furthermore, in this method the full set of observables +(\textit{e.g} $R_{K^{*}}$, $P^{\prime}_{5}$ and branching fraction measurements) available in $\bar{B}\to \bar{K}^*\ell^+\ell^-$ +decays is exploited, providing unprecedented precision on LFU in a single analysis. + +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 squared invariant mass, $q^2$, and the three angles +$\vec{\Omega} = (\cos \theta_\ell, \cos \theta_K, \phi)$~\cite{Altmannshofer:2008dz}. +The probability density function ($p.d.f.$) for this decay can be written as +% +\begin{equation} +p.d.f. = \frac{1}{\Gamma} \frac{\dd^4 \Gamma}{\dd q^2 \dd^3 \Omega}, \ + \quad + \text{with}\quad + \Gamma = \int_{q^2} \dd q^2 \frac{\dd\Gamma}{\dd q^2}\,, +\end{equation} +% +with different \qsq intervals depending on the lepton flavour under study. +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. +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{ + \label{eq:amp_dep} + {\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^{(\ell)}_7 {\cal{F}}_{\lambda}^{T}(q^2) - 16\pi^2 \frac{M_B}{m_b} {\cal{H}}_{\lambda}(q^2) \bigg] +\bigg\}\,,\nonumber +} +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. +The ${\cal{F}}^{(T)}_{\lambda}(q^{2})$ are form factors, +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. +In the following +this function 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 $\mathcal{H}_\lambda[z^n]$), +after removing singularities related to the $J/\psi(1S)$ and $\psi(2S)$. +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. + +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 $\mathcal{C}_i^{(\mu,e)}$ are renamed as $\widetilde{\mathcal{C}}_i^{(\mu,e)}$, +since an accurate disentanglement between the physical +meaning of $\WC_i^{(\mu,e)}$ and the above-mentioned hadronic pollution +cannot be achieved at the current stage of the theory~\cite{Chrzaszcz:2018yza}. +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 $\bar{B}^0 \to \bar{K}^{*0} \mu^+\mu^-$ decays +that is sufficient to enable the determination of these multi-space parameters. +Note that an amplitude analysis of the electron mode only has been previously disregarded, +given the limited dataset in either LHCb or Belle experiments. +In a common framework the 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 +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.} $\WC_i^{\prime\,(\mu,e)} = 0$, +while sensitivity studies on the determination of the WCs $\WC_9^{\prime\,(\mu)}$ and $\WC_{10}^{\prime\,(\mu)}$ +are detailed in the supplemental material to this Letter~\cite{suppl}. + +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 luminosity and $\sigma_{b\bar{b}} \propto \sqrt{s}$ +for LHCb, where $s$ denotes the designed centre-of-mass energy of the $b$-quark pair, +and exclusively with luminosity for Belle II. +%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 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; +while in Belle II the same kinematic regions are considered for both 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 broadly consistent with published results, +and assumes improvements in the electron mode resolution for LHCb~\cite{Lionetto:2624938}. + +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 for two NP benchmark points (BMP), +$\Delta\WC_9 = - 1$ and $\Delta\WC_9 = - \Delta\WC_{10} = - 0.7$, +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.} $\WC_i^{(e)} = \WC_i^{\rm{SM}}$. +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 to these simulated samples, +in which multivariate Gaussian terms are added to the log-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~\cite{Bobeth:2017vxj} + included as a multivariate Gaussian constraint; + % + \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 Ref.~\cite{Hurth:2017sqw}, \textit{i.e.} + using instead an expansion in terms of $1 + a_{\lambda} + b_{\lambda}(q^{2}/6\,$GeV$^{2})$. + + % +\end{enumerate} +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}. + + +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 +$\widetilde{\mathcal{C}}_9^{(\mu)}$ and $\widetilde{\mathcal{C}}_9^{(e)}$ coefficients +is sufficient to preserve the true underlying physics at any order of the series expansion $\mathcal{H}_\lambda[z^n]$ +and without any parametric theoretical input, +\textit{i.e.} the two-dimensional pull estimator with respect to the LFU hypothesis is unbiased. +% +\begin{figure}[t] +\includegraphics[width=.4\textwidth]{ellipses_C9.pdf} +\caption{% + Two-dimensional sensitivity scans for the pair of Wilson coefficients + $\widetilde{\mathcal{C}}_9^{(\mu)}$ and $\widetilde{\mathcal{C}}_9^{(e)}$ + for different non-local hadronic parametrisation models evaluated at \texttt{BMP}$_{\WC_9}$, + and with the expected statistics after \lhcb Run II. + The contours correspond to 99\% confidence level statistical-only uncertainty bands and + the dotted black line indicates the LFU hypothesis. +} +\label{fig:C9ellipse} +\end{figure} +% +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 end of Run-II. +% +\begin{figure}[bth!] +%\begin{center} +\includegraphics[width=.4\textwidth]{ellipses_DeltaC9C10_a.pdf}\\%\quad\quad\quad\quad +\includegraphics[width=.4\textwidth]{ellipses_DeltaC9C10_b.pdf} +\caption{% + Two-dimensional sensitivity scans for the proposed observables $\Delta\WC_9$ and $\Delta\WC_{10}$ + for different non-local hadronic parametrisation models + evaluated at (top) \texttt{BMP}$_{\WC_9}$ and (bottom) \texttt{BMP}$_{\WC_{9,10}}$, + and with the expected statistics after \lhcb Run II. + The contours correspond to 99\% confidence level statistical-only uncertainty bands. + \label{fig:DeltaC9C10} +} +%\end{center} +\end{figure} + +The sensitivity to the two benchmark-like NP scenarios using the proposed pseudo observables $\Delta \WC_i$ +is shown in Fig.~\ref{fig:DeltaC9C10}. +We quantify the maximal expected significance with respect to the SM to be $4.6$ and $5.3\,\sigma$ for +\texttt{BMP}$_{\WC_9}$ and \texttt{BMP}$_{\WC_{9,10}}$, respectively. +Realistic experimental effects are necessary to determine the exact sensitivity achievable. +Nevertheless, these results suggest that a first observation (with a single measurement) of LFU breaking +appears to be feasible with the expected recorded statistics by the end of LHCb Run II. +Furthermore, it is interesting to examine the prospects for confirming this evidence in the upcoming LHCb/Belle +upgrades~\cite{Albrecht:2017odf}. +Figure~\ref{fig:DeltaC9C10_Upgrade} summarises the two-dimensional statistical-only significances +for the designed luminosities. +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 - +insensitive to both local and non-local hadronic uncertainties. + + +\begin{figure}[t!] +\includegraphics[width=.4\textwidth]{ellipses_DeltaC9C10_Nev.pdf} +\caption{% + Two-dimensional sensitivity scans for the proposed observables $\Delta\WC_9$ and $\Delta\WC_{10}$ + for the two considered NP scenarios: (green) \texttt{BMP}$_{\WC_9}$ and (red) \texttt{BMP}$_{\WC_{9,10}}$. + The contours correspond to 99\% confidence level statistical-only uncertainty bands expected for the (dashed) \belle II 50~ab$^{-1}$ + and \lhcb Upgrade (dotted) $50\,$fb$^{-1}$ and (solid) $\,300\,$fb$^{-1}$ statistics. + \label{fig:DeltaC9C10_Upgrade} +} +\end{figure} + +Experimental resolution and detector acceptance/efficiency effects +are not considered in this work, as these would require further information +from current (non-public) or planned \textit{B}-physics experiments. +Nevertheless, the precision on this measurement can remain unbiased +either by parametrising these effects in the amplitude model and/or even +re-computing the angles or the $q^2$ variables constraining the $B$ invariant mass~\cite{Lionetto:2624938}. +%Nevertheless, preliminary studies 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}. +%Despite the low \qsq asymmetric tail, the determination of $\Delta\WC_9$ and +%$\Delta\WC_{10}$ remains unbiased, even if no-correction is +%applied. +Moreover, the differential decay width can receive additional complex amplitudes from signal-like backgrounds, +\textit{e.g.} $K\pi$ S-wave from a non-resonant decay and/or a scalar resonance~\cite{Becirevic:2012dp}. +These contributions are determined to be small~\cite{Aaij:2015oid,Aaij:2016flj}, +and in the proposed formalism they benefit from the same description between the muon +and electron mode (see detailed discussion in Ref.~\cite{Chrzaszcz:2018yza}). +Therefore, such contribution does not dilute the expected sensitivity of the measurement. + +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 $\mathcal{H}_\lambda[z^2]$. +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. +It is worth mentioning that a hypothetical determination of the individual +$\widetilde{\WC}_9^{(\mu,e)}$ and $\widetilde{\WC}_{10}^{(\mu,e)}$ WCs +can also produce a shift in their central values +that mimics the behaviour of NP~\cite{Chrzaszcz:2018yza}.% and makes impossible any claim in this direction. + + + +In conclusion, we propose a clean and model-independent method to combine +all the available information from $\bar{B}\to \bar{K}^*\ell^+\ell^-$ decays for a precise +determination of LFU-breaking differences of WCs, \textit{i.e.} $\Delta\WC_9$ +and $\Delta\WC_{10}$. +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 theoretical uncertainty. +In addition, this simultaneous analysis is more robust against +experimental effects such +as mismodeling of the detector resolution, since most parameters are +effectively determined from the muon mode. This would be an important +benefit for LHCb where the electron resolution is significantly worse +than that of muons. +% +\begin{figure}[t!] +\includegraphics[width=.4\textwidth]{B2Kstll_summary.pdf} +\caption{% + Sensitivity to \texttt{BMP}$_{\WC_{9,10}}$ scenario for the expected statistics after the \lhcb Run II. + The relative contribution (68, 95, 99\% confidence level contours) of each step of the analysis is shown in different colours, + together with the result of full amplitude method proposed in this letter. + \label{fig:allComponents} +} +\end{figure} +% +Figure~\ref{fig:allComponents} illustrates the usefulness of the newly-proposed observables by combining +the different information from angular analysis to branching ratio measurements. +Due to the inclusiveness of the approach, the expected sensitivity surpasses any +of the projections for the foreseen measurements of \textit{e.g.} $R_{K^{*}}$ or $P^{\prime}_{5}$ alone - given the benchmark points. +Therefore, this novel formalism can be the most immediate method to observe unambiguously +NP in $\bar{B}\to \bar{K}^*\ell^+\ell^-$ decays. + +A promising feature of this framework is the possibility to extend the analysis to +include other decay channels involving flavour changing neutral currents. +For instance, the charged decay $\bar{B}^+ \to \bar{K}^{*+} \ellell$ undergoes the same physics +and is easily accessible at the $B$-factories, while other rare +semi-leptonic decays such as $B^+ \to K^+ \ellell$ and $\Lambda_{b} \to \Lambda^{(*)} \ell^+\ell^-$ +have a different phenomenology but access the same NP information in terms of WC description. +Thus, an unbinned global simultaneous fit to all data involving $b \to s \ell^+ \ell^-$ transitions +is a natural and appealing extension of this work. +Moreover, the parameter space of the investigated WCs can also be broadened to +incorporate direct measurement of the right-handed $\WC_i^{\prime}$ - +currently weakly constrained by global fits~\cite{Capdevila:2017bsm,Altmannshofer:2017yso,Hurth:2017hxg}. + + +We acknowledge useful contributions from Michele Atzeni, Gino Isidori, Danny van Dyk and Patrick Owen. +This work is supported by the Swiss National Science Foundation (SNF) under contracts 173104 and 174182. + + +%\FloatBarrier +\bibliography{references} + +\end{document} diff --git a/prd/old/main/ellipses_C9.pdf b/prd/old/main/ellipses_C9.pdf new file mode 100644 index 0000000..13293f7 --- /dev/null +++ b/prd/old/main/ellipses_C9.pdf Binary files differ diff --git a/prd/old/main/ellipses_DeltaC9C10_Nev.pdf b/prd/old/main/ellipses_DeltaC9C10_Nev.pdf new file mode 100644 index 0000000..1ed7ece --- /dev/null +++ b/prd/old/main/ellipses_DeltaC9C10_Nev.pdf Binary files differ diff --git a/prd/old/main/ellipses_DeltaC9C10_a.pdf b/prd/old/main/ellipses_DeltaC9C10_a.pdf new file mode 100644 index 0000000..80501e8 --- /dev/null +++ b/prd/old/main/ellipses_DeltaC9C10_a.pdf Binary files differ diff --git a/prd/old/main/ellipses_DeltaC9C10_b.pdf b/prd/old/main/ellipses_DeltaC9C10_b.pdf new file mode 100644 index 0000000..b20ff0d --- /dev/null +++ b/prd/old/main/ellipses_DeltaC9C10_b.pdf Binary files differ diff --git a/prd/old/main/references.bib b/prd/old/main/references.bib new file mode 100644 index 0000000..09be6a3 --- /dev/null +++ b/prd/old/main/references.bib @@ -0,0 +1,820 @@ +@misc{suppl, + title = "{See Supplemental Material at [URL will be inserted by publisher] + for sensitivity studies on the determination of the Wilson coefficients + $\mathcal{C}_9^{\prime\,(\mu)}$ and $\mathcal{C}{10}^{\prime\,(\mu)}$}" +} +@article{Becirevic:2012dp, + author = "Becirevic, Damir and Tayduganov, Andrey", + title = "{Impact of $B\to K^\ast_0 \ell^+\ell^-$ on the New + Physics search in $B\to K^\ast \ell^+\ell^-$ decay}", + journal = "Nucl. 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Phys.", + volume = "C67", + year = "1995", + pages = "417-432", + doi = "10.1007/BF01624585", + eprint = "hep-ph/9408213", + archivePrefix = "arXiv", + primaryClass = "hep-ph", + reportNumber = "CERN-TH-7346-94", + SLACcitation = "%%CITATION = HEP-PH/9408213;%%" +} +@article{Aaij:2015oid, + author = "Aaij, Roel and others", + title = "{Angular analysis of the $B^{0} \to K^{*0} \mu^{+} + \mu^{-}$ decay using 3 fb$^{-1}$ of integrated + luminosity}", + collaboration = "LHCb", + journal = "JHEP", + volume = "02", + year = "2016", + pages = "104", + doi = "10.1007/JHEP02(2016)104", + eprint = "1512.04442", + archivePrefix = "arXiv", + primaryClass = "hep-ex", + reportNumber = "CERN-PH-EP-2015-314, LHCB-PAPER-2015-051", + SLACcitation = "%%CITATION = ARXIV:1512.04442;%%" +} + +@article{Descotes-Genon:2015uva, + author = "Descotes-Genon, Sébastien and Hofer, Lars and Matias, + Joaquim and Virto, Javier", + title = "{Global analysis of $b\to s\ell\ell$ anomalies}", + journal = "JHEP", + volume = "06", + year = "2016", + pages = "092", + doi = "10.1007/JHEP06(2016)092", + eprint = "1510.04239", + archivePrefix = "arXiv", + primaryClass = "hep-ph", + reportNumber = "LPT-ORSAY-15-68, QFET-2015-29, SI-HEP-2015-19", + SLACcitation = "%%CITATION = ARXIV:1510.04239;%%" +} + +@article{Wehle:2016yoi, + author = "Wehle, S. and others", + title = "{Lepton-Flavor-Dependent Angular Analysis of $B\to K^\ast + \ell^+\ell^-$}", + collaboration = "Belle", + year = "2016", + eprint = "1612.05014", + archivePrefix = "arXiv", + primaryClass = "hep-ex", + reportNumber = "BELLE-PREPRINT-2016-15, KEK-PREPRINT-2016-54", + SLACcitation = "%%CITATION = ARXIV:1612.05014;%%" +} +@article{Gauld:2013qja, + author = "Gauld, Rhorry and Goertz, Florian and Haisch, Ulrich", + title = "{An explicit Z'-boson explanation of the $B \to K^* \mu^+ + \mu^-$ anomaly}", + journal = "JHEP", + volume = "01", + year = "2014", + pages = "069", + doi = "10.1007/JHEP01(2014)069", + eprint = "1310.1082", + archivePrefix = "arXiv", + primaryClass = "hep-ph", + SLACcitation = "%%CITATION = ARXIV:1310.1082;%%" +} +@article{Buras:2013qja, + author = "Buras, Andrzej J. and Girrbach, Jennifer", + title = "{Left-handed $Z'$ and $Z$ FCNC quark couplings facing new + $b \to s \mu^+ \mu^-$ data}", + journal = "JHEP", + volume = "12", + year = "2013", + pages = "009", + doi = "10.1007/JHEP12(2013)009", + eprint = "1309.2466", + archivePrefix = "arXiv", + primaryClass = "hep-ph", + reportNumber = "FLAVOUR(267104)-ERC-50", + SLACcitation = "%%CITATION = ARXIV:1309.2466;%%" +} +@article{Altmannshofer:2013foa, + author = "Altmannshofer, Wolfgang and Straub, David M.", + title = "{New Physics in $B \to K^*\mu\mu$?}", + journal = "Eur. Phys. J.", + volume = "C73", + year = "2013", + pages = "2646", + doi = "10.1140/epjc/s10052-013-2646-9", + eprint = "1308.1501", + archivePrefix = "arXiv", + primaryClass = "hep-ph", + reportNumber = "FERMILAB-PUB-13-310-T, MITP-13-047", + SLACcitation = "%%CITATION = ARXIV:1308.1501;%%" +} +@article{Crivellin:2015era, + author = "Crivellin, Andreas and Hofer, Lars and Matias, Joaquim + and Nierste, Ulrich and Pokorski, Stefan and Rosiek, + Janusz", + title = "{Lepton-flavour violating $B$ decays in generic $Z'$ + models}", + journal = "Phys. Rev.", + volume = "D92", + year = "2015", + number = "5", + pages = "054013", + doi = "10.1103/PhysRevD.92.054013", + eprint = "1504.07928", + archivePrefix = "arXiv", + primaryClass = "hep-ph", + reportNumber = "CERN-PH-TH-2015-091, TTP15-018", + SLACcitation = "%%CITATION = ARXIV:1504.07928;%%" +} +# Leptoquarks +@article{Hiller:2014yaa, + author = "Hiller, Gudrun and Schmaltz, Martin", + title = "{$R_K$ and future $b \to s \ell \ell$ physics beyond the + standard model opportunities}", + journal = "Phys. Rev.", + volume = "D90", + year = "2014", + pages = "054014", + doi = "10.1103/PhysRevD.90.054014", + eprint = "1408.1627", + archivePrefix = "arXiv", + primaryClass = "hep-ph", + reportNumber = "DO-TH-14-17", + SLACcitation = "%%CITATION = ARXIV:1408.1627;%%" +} +@article{Biswas:2014gga, + author = "Biswas, Sanjoy and Chowdhury, Debtosh and Han, Sangeun + and Lee, Seung J.", + title = "{Explaining the lepton non-universality at the LHCb and + CMS within a unified framework}", + journal = "JHEP", + volume = "02", + year = "2015", + pages = "142", + doi = "10.1007/JHEP02(2015)142", + eprint = "1409.0882", + archivePrefix = "arXiv", + primaryClass = "hep-ph", + SLACcitation = "%%CITATION = ARXIV:1409.0882;%%" +} +@article{Gripaios:2014tna, + author = "Gripaios, Ben and Nardecchia, Marco and Renner, S. A.", + title = "{Composite leptoquarks and anomalies in $B$-meson + decays}", + journal = "JHEP", + volume = "05", + year = "2015", + pages = "006", + doi = "10.1007/JHEP05(2015)006", + eprint = "1412.1791", + archivePrefix = "arXiv", + primaryClass = "hep-ph", + SLACcitation = "%%CITATION = ARXIV:1412.1791;%%" +} +# QCD issues +@article{Jager:2012uw, + author = "Jäger, S. and Martin Camalich, J.", + title = "{On $B \to V \ell \ell$ at small dilepton invariant + mass, power corrections, and new physics}", + journal = "JHEP", + volume = "05", + year = "2013", + pages = "043", + doi = "10.1007/JHEP05(2013)043", + eprint = "1212.2263", + archivePrefix = "arXiv", + primaryClass = "hep-ph", + SLACcitation = "%%CITATION = ARXIV:1212.2263;%%" +} +@article{Jager:2014rwa, + author = "Jäger, Sebastian and Martin Camalich, Jorge", + title = "{Reassessing the discovery potential of the $B \to K^{*} + \ell^+\ell^-$ decays in the large-recoil region: SM + challenges and BSM opportunities}", + journal = "Phys. Rev.", + volume = "D93", + year = "2016", + number = "1", + pages = "014028", + doi = "10.1103/PhysRevD.93.014028", + eprint = "1412.3183", + archivePrefix = "arXiv", + primaryClass = "hep-ph", + SLACcitation = "%%CITATION = ARXIV:1412.3183;%%" +} +@article{Aaij:2014ora, + author = "Aaij, Roel and others", + title = "{Test of lepton universality using $B^{+}\rightarrow + K^{+}\ell^{+}\ell^{-}$ decays}", + collaboration = "LHCb", + journal = "Phys. Rev. Lett.", + volume = "113", + year = "2014", + pages = "151601", + doi = "10.1103/PhysRevLett.113.151601", + eprint = "1406.6482", + archivePrefix = "arXiv", + primaryClass = "hep-ex", + reportNumber = "CERN-PH-EP-2014-140, LHCB-PAPER-2014-024", + SLACcitation = "%%CITATION = ARXIV:1406.6482;%%" +} +@article{Aaij:2017vbb, + author = "Aaij, R. and others", + title = "{Test of lepton universality with $B^{0} \rightarrow + K^{*0}\ell^{+}\ell^{-}$ decays}", + collaboration = "LHCb", + journal = "JHEP", + volume = "08", + year = "2017", + pages = "055", + doi = "10.1007/JHEP08(2017)055", + eprint = "1705.05802", + archivePrefix = "arXiv", + primaryClass = "hep-ex", + reportNumber = "LHCB-PAPER-2017-013, CERN-EP-2017-100", + SLACcitation = "%%CITATION = ARXIV:1705.05802;%%" +} +% BR measurements +@article{Aaij:2013aln, + author = "Aaij, R and others", + title = "{Differential branching fraction and angular analysis of + the decay $B_s^0\to\phi\mu^{+}\mu^{-}$}", + collaboration = "LHCb", + journal = "JHEP", + volume = "07", + year = "2013", + pages = "084", + doi = "10.1007/JHEP07(2013)084", + eprint = "1305.2168", + archivePrefix = "arXiv", + primaryClass = "hep-ex", + reportNumber = "CERN-PH-EP-2013-078, LHCB-PAPER-2013-017", + SLACcitation = "%%CITATION = ARXIV:1305.2168;%%" +} +@article{Aaij:2014pli, + author = "Aaij, R. and others", + title = "{Differential branching fractions and isospin asymmetries + of $B \to K^{(*)} \mu^+ \mu^-$ decays}", + collaboration = "LHCb", + journal = "JHEP", + volume = "06", + year = "2014", + pages = "133", + doi = "10.1007/JHEP06(2014)133", + eprint = "1403.8044", + archivePrefix = "arXiv", + primaryClass = "hep-ex", + reportNumber = "LHCB-PAPER-2014-006, CERN-PH-EP-2014-055", + SLACcitation = "%%CITATION = ARXIV:1403.8044;%%" +} +@article{Hiller:2003js, + author = "Hiller, Gudrun and Kruger, Frank", + title = "{More model-independent analysis of $b \to s$ processes}", + journal = "Phys. Rev.", + volume = "D69", + year = "2004", + pages = "074020", + doi = "10.1103/PhysRevD.69.074020", + eprint = "hep-ph/0310219", + archivePrefix = "arXiv", + primaryClass = "hep-ph", + reportNumber = "LMU-18-03, TUM-HEP-519-03", + SLACcitation = "%%CITATION = HEP-PH/0310219;%%" +} +@article{Bordone:2016gaq, + author = "Bordone, Marzia and Isidori, Gino and Pattori, Andrea", + title = "{On the Standard Model predictions for $R_K$ and + $R_{K^*}$}", + journal = "Eur. Phys. J.", + volume = "C76", + year = "2016", + number = "8", + pages = "440", + doi = "10.1140/epjc/s10052-016-4274-7", + eprint = "1605.07633", + archivePrefix = "arXiv", + primaryClass = "hep-ph", + SLACcitation = "%%CITATION = ARXIV:1605.07633;%%" +} +@article{Capdevila:2017bsm, + author = "Capdevila, Bernat and Crivellin, Andreas and + Descotes-Genon, Sébastien and Matias, Joaquim and Virto, + Javier", + title = "{Patterns of New Physics in $b\to s\ell^+\ell^-$ + transitions in the light of recent data}", + journal = "JHEP", + volume = "01", + year = "2018", + pages = "093", + doi = "10.1007/JHEP01(2018)093", + eprint = "1704.05340", + archivePrefix = "arXiv", + primaryClass = "hep-ph", + reportNumber = "PSI-PR-17-05, LPT-ORSAY-17-19", + SLACcitation = "%%CITATION = ARXIV:1704.05340;%%" +} +@article{Altmannshofer:2017yso, + author = "Altmannshofer, Wolfgang and Stangl, Peter and Straub, + David M.", + title = "{Interpreting Hints for Lepton Flavor Universality + Violation}", + journal = "Phys. Rev.", + volume = "D96", + year = "2017", + number = "5", + pages = "055008", + doi = "10.1103/PhysRevD.96.055008", + eprint = "1704.05435", + archivePrefix = "arXiv", + primaryClass = "hep-ph", + SLACcitation = "%%CITATION = ARXIV:1704.05435;%%" +} +@article{Hurth:2017hxg, + author = "Hurth, T. and Mahmoudi, F. and Martinez Santos, D. and + Neshatpour, S.", + title = "{Lepton nonuniversality in exclusive + $b{\rightarrow}s{\ell}{\ell}$ decays}", + journal = "Phys. Rev.", + volume = "D96", + year = "2017", + number = "9", + pages = "095034", + doi = "10.1103/PhysRevD.96.095034", + eprint = "1705.06274", + archivePrefix = "arXiv", + primaryClass = "hep-ph", + reportNumber = "CERN-TH-2017-110, IPM-P.A-466, MITP-17-035", + SLACcitation = "%%CITATION = ARXIV:1705.06274;%%" +} +@article{Bobeth:2017vxj, + author = "Bobeth, Christoph and Chrzaszcz, Marcin and van Dyk, + Danny and Virto, Javier", + title = "{Long-distance effects in $B\to K^*\ell\ell$ from + Analyticity}", + year = "2017", + eprint = "1707.07305", + archivePrefix = "arXiv", + primaryClass = "hep-ph", + reportNumber = "EOS-2017-01, MIT-CTP-4918, TUM-HEP-1087-17, ZU-TH-17-17", + SLACcitation = "%%CITATION = ARXIV:1707.07305;%%" +} +@article{Blake:2017fyh, + author = "Blake, Thomas and Egede, Ulrik and Owen, Patrick and + Pomery, Gabriela and Petridis, Konstantinos Alexandros", + title = "{An empirical model of the long-distance contributions to + $\bar{B}^{0} \rightarrow \bar{K}^{*0}\mu^{+}\mu^{-}$ + transitions}", + year = "2017", + eprint = "1709.03921", + archivePrefix = "arXiv", + primaryClass = "hep-ph", + SLACcitation = "%%CITATION = ARXIV:1709.03921;%%" +} +@article{Hurth:2017sqw, + author = "Hurth, T. and Langenbruch, C. and Mahmoudi, F.", + title = "{Direct determination of Wilson coefficients using + $B^0\to K^{*0}\mu^+\mu^-$ decays}", + journal = "JHEP", + volume = "11", + year = "2017", + pages = "176", + doi = "10.1007/JHEP11(2017)176", + eprint = "1708.04474", + archivePrefix = "arXiv", + primaryClass = "hep-ph", + 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;%%" +} +@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:2624938, + author = "Lionetto, Federica and Serra, Nicola and Steinkamp, Olaf + and Mueller, Katharina", + title = "{Measurements of Angular Observables of $B^0\to K^{*0} + \mu^+ \mu^-$ and $B^0 \to K^{*0} e^+ e^-$ Decays and the + Upgrade of LHCb}", + month = "Feb", + year = "2018", + reportNumber = "CERN-THESIS-2018-074", + url = "https://cds.cern.ch/record/2624938", + note = "Presented 22 Mar 2018", + howpublished = "\url{https://cds.cern.ch/record/2624938}" +} +@article{Capdevila:2017ert, + author = "Capdevila, Bernat and Descotes-Genon, Sebastien and + Hofer, Lars and Matias, Joaquim", + title = "{Hadronic uncertainties in $B \to K^* \mu^+ \mu^-$: a + state-of-the-art analysis}", + journal = "JHEP", + volume = "04", + year = "2017", + pages = "016", + doi = "10.1007/JHEP04(2017)016", + eprint = "1701.08672", + archivePrefix = "arXiv", + primaryClass = "hep-ph", + reportNumber = "ICCUB-17-003, UAB-FT-990, LPT-ORSAY-16-89", + SLACcitation = "%%CITATION = ARXIV:1701.08672;%%" +} +@article{Aaij:2016flj, + author = "Aaij, Roel and others", + title = "{Measurements of the S-wave fraction in $B^{0}\rightarrow + K^{+}\pi^{-}\mu^{+}\mu^{-}$ decays and the + $B^{0}\rightarrow K^{\ast}(892)^{0}\mu^{+}\mu^{-}$ + differential branching fraction}", + collaboration = "LHCb", + journal = "JHEP", + volume = "11", + year = "2016", + pages = "047", + doi = "10.1007/JHEP11(2016)047, 10.1007/JHEP04(2017)142", + note = "[Erratum: JHEP04,142(2017)]", + eprint = "1606.04731", + archivePrefix = "arXiv", + primaryClass = "hep-ex", + reportNumber = "CERN-EP-2016-141, LHCB-PAPER-2016-012", + SLACcitation = "%%CITATION = ARXIV:1606.04731;%%" +} +@article{Albrecht:2017odf, + author = "Albrecht, Johannes and Bernlochner, Florian and Kenzie, + Matthew and Reichert, Stefanie and Straub, David and + Tully, Alison", + title = "{Future prospects for exploring present day anomalies in + flavour physics measurements with Belle II and LHCb}", + year = "2017", + eprint = "1709.10308", + archivePrefix = "arXiv", + primaryClass = "hep-ph", + SLACcitation = "%%CITATION = ARXIV:1709.10308;%%" +} +@article{PhysRevD.8.3633, + title = {Asymptotically free gauge theories. 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David", + title = "{Asymptotic freedom: an approach to strong interactions}", + journal = "Phys.Rept.", + volume = "14", + pages = "129-180", + doi = "10.1016/0370-1573(74)90014-3", + year = "1974", + reportNumber = "PRINT-74-1223 (HARVARD)", + SLACcitation = "%%CITATION = PRPLC,14,129;%%", +} +@article{Gross:1998jx, + author = "Gross, David J.", + title = "{Twenty five years of asymptotic freedom}", + journal = "Nucl.Phys.Proc.Suppl.", + volume = "74", + pages = "426-446", + doi = "10.1016/S0920-5632(99)00208-X", + year = "1999", + eprint = "hep-th/9809060", + archivePrefix = "arXiv", + primaryClass = "hep-th", + SLACcitation = "%%CITATION = HEP-TH/9809060;%%", +} +@article{Ball:1998kk, + author = "Ball, Patricia and Braun, Vladimir M.", + title = "{Exclusive semileptonic and rare B meson decays in QCD}", + journal = "Phys. Rev.", + volume = "D58", + year = "1998", + pages = "094016", + doi = "10.1103/PhysRevD.58.094016", + eprint = "hep-ph/9805422", + archivePrefix = "arXiv", + primaryClass = "hep-ph", + reportNumber = "CERN-TH-98-162, NORDITA-98-41-HE", + SLACcitation = "%%CITATION = HEP-PH/9805422;%%" +} +@article{Khodjamirian:2006st, + author = "Khodjamirian, Alexander and Mannel, Thomas and Offen, + Nils", + title = "{Form-factors from light-cone sum rules with B-meson + distribution amplitudes}", + journal = "Phys. Rev.", + volume = "D75", + year = "2007", + pages = "054013", + doi = "10.1103/PhysRevD.75.054013", + eprint = "hep-ph/0611193", + archivePrefix = "arXiv", + primaryClass = "hep-ph", + reportNumber = "SI-HEP-2006-03", + SLACcitation = "%%CITATION = HEP-PH/0611193;%%" +} +@article{Becirevic:2006nm, + author = "Becirevic, Damir and Lubicz, Vittorio and Mescia, + Federico", + title = "{An Estimate of the B ---> K* gamma form factor}", + journal = "Nucl. Phys.", + volume = "B769", + year = "2007", + pages = "31-43", + doi = "10.1016/j.nuclphysb.2007.01.032", + eprint = "hep-ph/0611295", + archivePrefix = "arXiv", + primaryClass = "hep-ph", + reportNumber = "LPT-ORSAY-06-73, RM3-TH-06-22", + SLACcitation = "%%CITATION = HEP-PH/0611295;%%" +} +@article{Horgan:2013hoa, + author = "Horgan, Ronald R. and Liu, Zhaofeng and Meinel, Stefan + and Wingate, Matthew", + title = "{Lattice QCD calculation of form factors describing the + rare decays $B \to K^* \ell^+ \ell^-$ and $B_s \to \phi + \ell^+ \ell^-$}", + journal = "Phys. Rev.", + volume = "D89", + year = "2014", + number = "9", + pages = "094501", + doi = "10.1103/PhysRevD.89.094501", + eprint = "1310.3722", + archivePrefix = "arXiv", + primaryClass = "hep-lat", + reportNumber = "DAMTP-2013-45, MIT-CTP-4507", + SLACcitation = "%%CITATION = ARXIV:1310.3722;%%" +} +@inproceedings{Mauri:2018, + author = "Chrzaszcz, Marcin and Mauri, Andrea and Serra, Nicola and Silva Coutinho, Rafael and van Dyk, Danny", + title = "{Prospects for data-driven analyses of the decays $B \to K^* \ell^+ \ell^-$}", + booktitle = "{$b \to s \ell \ell$ 2018: 6th Workshop on Rare Semileptonic B Decays, February 20-22}", + url = "https://indico.ph.tum.de/event/3782/sessions/558/attachments/2681/3011/NonLocalMEs2.pdf", + year = "2018", + eprint = "[In preparation]", +} +@article{Aaij:2013qta, + author = "Aaij, R and others", + title = "{Measurement of Form-Factor-Independent Observables in + the Decay $B^{0} \to K^{*0} \mu^+ \mu^-$}", + collaboration = "LHCb", + journal = "Phys. Rev. Lett.", + volume = "111", + year = "2013", + pages = "191801", + doi = "10.1103/PhysRevLett.111.191801", + eprint = "1308.1707", + archivePrefix = "arXiv", + primaryClass = "hep-ex", + reportNumber = "LHCB-PAPER-2013-037, CERN-PH-EP-2013-146", + SLACcitation = "%%CITATION = ARXIV:1308.1707;%%" +} + diff --git a/prd/old/suppl.pdf b/prd/old/suppl.pdf new file mode 100644 index 0000000..78fbf6b --- /dev/null +++ b/prd/old/suppl.pdf Binary files differ diff --git a/prd/prd.zip b/prd/prd.zip new file mode 100644 index 0000000..f072b5e --- /dev/null +++ b/prd/prd.zip Binary files differ diff --git a/prd/prd/B2Kstll_summary.pdf b/prd/prd/B2Kstll_summary.pdf new file mode 100644 index 0000000..58f805a --- /dev/null +++ b/prd/prd/B2Kstll_summary.pdf Binary files differ diff --git a/prd/prd/draft.bbl b/prd/prd/draft.bbl new file mode 100644 index 0000000..25e5373 --- /dev/null +++ b/prd/prd/draft.bbl @@ -0,0 +1,674 @@ +%merlin.mbs apsrev4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked +%Control: key (0) +%Control: author (8) initials jnrlst +%Control: editor formatted (1) identically to author +%Control: production of article title (-1) disabled +%Control: page (0) single +%Control: year (1) truncated +%Control: production of eprint (0) enabled +\begin{thebibliography}{50}% +\makeatletter +\providecommand \@ifxundefined [1]{% + \@ifx{#1\undefined} +}% +\providecommand \@ifnum [1]{% + \ifnum #1\expandafter \@firstoftwo + \else \expandafter \@secondoftwo + \fi +}% +\providecommand \@ifx [1]{% + \ifx #1\expandafter \@firstoftwo + \else \expandafter \@secondoftwo + \fi +}% +\providecommand \natexlab [1]{#1}% +\providecommand \enquote [1]{``#1''}% +\providecommand \bibnamefont [1]{#1}% +\providecommand \bibfnamefont [1]{#1}% +\providecommand \citenamefont [1]{#1}% +\providecommand \href@noop [0]{\@secondoftwo}% +\providecommand \href [0]{\begingroup \@sanitize@url \@href}% +\providecommand \@href[1]{\@@startlink{#1}\@@href}% +\providecommand \@@href[1]{\endgroup#1\@@endlink}% +\providecommand \@sanitize@url [0]{\catcode `\\12\catcode `\$12\catcode + `\&12\catcode `\#12\catcode `\^12\catcode `\_12\catcode `\%12\relax}% +\providecommand \@@startlink[1]{}% +\providecommand \@@endlink[0]{}% +\providecommand \url [0]{\begingroup\@sanitize@url \@url }% +\providecommand \@url [1]{\endgroup\@href {#1}{\urlprefix }}% +\providecommand \urlprefix [0]{URL }% +\providecommand \Eprint [0]{\href }% +\providecommand \doibase [0]{http://dx.doi.org/}% +\providecommand \selectlanguage [0]{\@gobble}% +\providecommand \bibinfo [0]{\@secondoftwo}% +\providecommand \bibfield [0]{\@secondoftwo}% +\providecommand \translation [1]{[#1]}% +\providecommand \BibitemOpen [0]{}% +\providecommand \bibitemStop [0]{}% +\providecommand \bibitemNoStop [0]{.\EOS\space}% +\providecommand \EOS [0]{\spacefactor3000\relax}% +\providecommand \BibitemShut [1]{\csname bibitem#1\endcsname}% +\let\auto@bib@innerbib\@empty +% +\bibitem [{\citenamefont {Grossman}\ and\ \citenamefont + {Worah}(1997)}]{Grossman:1996ke}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Y.}~\bibnamefont + {Grossman}}\ and\ \bibinfo {author} {\bibfnamefont {M.~P.}\ \bibnamefont + {Worah}},\ }\href {\doibase 10.1016/S0370-2693(97)00068-3} {\bibfield + {journal} {\bibinfo {journal} {Phys.Lett.}\ }\textbf {\bibinfo {volume} + {B395}},\ \bibinfo {pages} {241} (\bibinfo {year} {1997})},\ \Eprint + {http://arxiv.org/abs/hep-ph/9612269} {arXiv:hep-ph/9612269 [hep-ph]} + \BibitemShut {NoStop}% +%%CITATION = HEP-PH/9612269;%% +\bibitem [{\citenamefont {Fleischer}(1997)}]{Fleischer:1996bv}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {R.}~\bibnamefont + {Fleischer}},\ }\href {\doibase 10.1142/S0217751X97001432} {\bibfield + {journal} {\bibinfo {journal} {Int. 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Phys.}\ }\textbf {\bibinfo {volume} + {B868}},\ \bibinfo {pages} {368} (\bibinfo {year} {2013})},\ \Eprint + {http://arxiv.org/abs/1207.4004} {arXiv:1207.4004 [hep-ph]} \BibitemShut + {NoStop}% +%%CITATION = ARXIV:1207.4004;%% +\bibitem [{\citenamefont {Aaij}\ \emph + {et~al.}(2016{\natexlab{b}})\citenamefont {Aaij} \emph + {et~al.}}]{Aaij:2016flj}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {R.}~\bibnamefont + {Aaij}} \emph {et~al.} (\bibinfo {collaboration} {LHCb}),\ }\href {\doibase + 10.1007/JHEP11(2016)047, 10.1007/JHEP04(2017)142} {\bibfield {journal} + {\bibinfo {journal} {JHEP}\ }\textbf {\bibinfo {volume} {11}},\ \bibinfo + {pages} {047} (\bibinfo {year} {2016}{\natexlab{b}})},\ \bibinfo {note} + {[Erratum: JHEP04,142(2017)]},\ \Eprint {http://arxiv.org/abs/1606.04731} + {arXiv:1606.04731 [hep-ex]} \BibitemShut {NoStop}% +%%CITATION = ARXIV:1606.04731;%% +\end{thebibliography}% diff --git a/prd/prd/draft.tex b/prd/prd/draft.tex new file mode 100644 index 0000000..7f35a19 --- /dev/null +++ b/prd/prd/draft.tex @@ -0,0 +1,592 @@ +% vim: set sts=4 et : +\documentclass[reprint,preprintnumbers,prd,nofootinbib]{revtex4-1} + +\usepackage{amsfonts} +\usepackage{amsmath} +\usepackage{array} +\usepackage{braket} +\usepackage{epstopdf} +\usepackage{graphicx} +\usepackage{hepparticles} +\usepackage{hepnicenames} +\usepackage{hepunits} +\usepackage{hyperref} +\usepackage[% + utf8 +]{inputenc} +\usepackage{slashed} +\usepackage{subfigure} +\usepackage{placeins} +\usepackage[% + normalem +]{ulem} +\usepackage[% + usenames, + svgnames, + dvipsnames +]{xcolor} + + +%% Shortcuts %% +\newcommand{\ie}{\textit{i.e.}} +\newcommand{\nuvec}{\vec{\nu}} +\newcommand{\refapp}[1]{appendix~\ref{app:#1}} +\newcommand{\refeq}[1]{eq.~(\ref{eq:#1})} +\newcommand{\refeqs}[2]{eqs.~(\ref{eq:#1})--(\ref{eq:#2})} +\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} +\newcommand{\eps}{\varepsilon} +\newcommand{\para}{\parallel} +\newcommand{\Gfermi}{G_F} +%\newcommand{\dd}[2][]{{\mathrm{d}^{#1}}#2\,} +\newcommand{\dd}{\ensuremath{\textrm{d}}} +\newcommand{\order}[1]{\ensuremath{\mathcal{O}\left(#1\right)}} +\DeclareMathOperator{\sign}{sgn} +\DeclareMathOperator{\ReNew}{Re} +\DeclareMathOperator{\ImNew}{Im} +\let\Re\ReNew +\let\Im\ImNew +\DeclareMathOperator*{\sumint}{% +\mathchoice% + {\ooalign{$\displaystyle\sum$\cr\hidewidth$\displaystyle\int$\hidewidth\cr}} + {\ooalign{\raisebox{.14\height}{\scalebox{.7}{$\textstyle\sum$}}\cr\hidewidth$\textstyle\int$\hidewidth\cr}} + {\ooalign{\raisebox{.2\height}{\scalebox{.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}} + {\ooalign{\raisebox{.2\height}{\scalebox{.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}} +} +\DeclareMathOperator*{\argmax}{arg\,max} +\newcommand{\wilson}[2][]{\mathcal{C}^\text{#1}_{#2}} +\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} + +\def\PB {\ensuremath{\mathrm{B}}\xspace} +\def\B {{\ensuremath{\PB}}\xspace} +\def\PK {\ensuremath{\mathrm{K}}\xspace} +\def\kaon {{\ensuremath{\PK}}\xspace} +\def\Kstarz {{\ensuremath{\kaon^{*0}}}\xspace} +\def\Bd {{\ensuremath{\B^0}}\xspace} +\def\Bz {{\ensuremath{\B^0}}\xspace} + +%% Key decay channels + +\def\BdToKstmm {\decay{\Bd}{\Kstarz\mup\mun}} +\def\BdbToKstmm {\decay{\Bdb}{\Kstarzb\mup\mun}} + +\def\BsToJPsiPhi {\decay{\Bs}{\jpsi\phi}} +\def\BdToJPsiKst {\decay{\Bd}{\jpsi\Kstarz}} +\def\BdbToJPsiKst {\decay{\Bdb}{\jpsi\Kstarzb}} + +%% Rare decays +\def\BdKstee {\decay{\Bd}{\Kstarz\epem}} +\def\BdbKstee {\decay{\Bdb}{\Kstarzb\epem}} +\def\bsll {\decay{\bquark}{\squark \ell^+ \ell^-}} + +\def\lepton {{\ensuremath{\ell}}\xspace} +\def\ellm {{\ensuremath{\ell^-}}\xspace} +\def\ellp {{\ensuremath{\ell^+}}\xspace} +\def\ellell {\ensuremath{\ell^+ \ell^-}\xspace} +\def\mumu {{\ensuremath{\Pmu^+\Pmu^-}}\xspace} + +\def\lhcb {\mbox{LHCb}\xspace} +\def\belle {\mbox{Belle}\xspace} + +\def\WC {\ensuremath{\mathcal{C}}\xspace} + + + +\begin{document} + +\allowdisplaybreaks + +\preprint{ZU-TH-15/18} +\title{Towards establishing Lepton Flavour Universality violation in $\bar{B}\to \bar{K}^*\ell^+\ell^-$ decays} +\author{Andrea Mauri} +\email{a.mauri@cern.ch} +\author{Nicola Serra} +\email{nicola.serra@cern.ch} +\author{Rafael Silva Coutinho} +\email{rafael.silva.coutinho@cern.ch} +\affiliation{Physik-Institut, Universit\"at Z\"urich, Winterthurer Strasse 190, 8057 Z\"urich, Switzerland} + +\begin{abstract} + Rare semileptonic $b \to s \ell^+ \ell^-$ transitions provide some of the most promising framework to search for new physics effects. +Recent analyses of these decays have indicated an anomalous behaviour in measurements + of angular distributions of the decay $B^0\to + K^*\mu^+\mu^-$ and lepton-flavour-universality observables. +Unambiguously establishing if these deviations have a common nature + is of paramount importance in order to understand the observed pattern. + We propose a novel approach to independently and complementary + probe this hypothesis + by performing a simultaneous amplitude analysis of $\bar{B}^0 \to \bar{K}^{*0} \mu^+\mu^-$ and $\bar{B}^0 \to \bar{K}^{*0} e^+e^-$ decays. + This method enables the direct determination of + observables that encode potential non-equal couplings of muons and electrons, + and are found to be insensitive to non-perturbative QCD effects. +If current hints of new physics are confirmed, our approach could +allow an early discovery of physics beyond the Standard +Model with LHCb Run-II datasets. +\end{abstract} + +\maketitle + +Flavour changing neutral current processes of {\textit{B}} meson decays +are crucial probes for the Standard Model (SM), +since as-yet undiscovered particles may contribute to these transitions 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. +An anomalous behaviour in angular and branching fraction analyses of the decay channel +$\bar{B}^{0} \to \bar{K}^{*0} \mu^{+}\mu^{-}$ has been recently reported~\cite{Aaij:2015oid,Wehle:2016yoi,Aaij:2013aln,Aaij:2014pli}, +notably in one of the observables with reduced theoretical uncertainties, +$P^{\prime}_{5}$~\cite{Aaij:2013qta,Descotes-Genon:2015uva}. +Several models have been suggested in order to interpret these results as new physics (NP) +signatures~\cite{Gauld:2013qja,Buras:2013qja,Altmannshofer:2013foa,Crivellin:2015era,Hiller:2014yaa,Biswas:2014gga,Gripaios:2014tna}. +Nonetheless, the vector-like nature of this pattern could be also explained by +non-perturbative QCD contributions from $b\to s c{\bar{c}}$ operators ({\textit{i.e.}} charm loops) +that are able to either mimic or camouflage NP effects~\cite{Jager:2012uw,Jager:2014rwa,Ciuchini:2015qxb}. +Non-standard measurement in ratios of $b \to s \ell^+ \ell^-$ processes +- such as of $R_{K}$~\cite{Aaij:2014ora} and $R_{K^{*}}$~\cite{Aaij:2017vbb} - +indicate a suppression of the muon channel which is also compatible with the $P^{\prime}_{5}$ anomaly. +In this case an immediate interpretation of lepton flavour universality (LFU) breaking is +suggested due to the small theoretical uncertainties in their predictions~\cite{Hiller:2003js,Bordone:2016gaq}. +Whilst the individual level of significance of the present anomalies is still inconclusive, +there is an appealing non-trivial consistent pattern shown in +global analysis fits~\cite{Capdevila:2017bsm,Altmannshofer:2017yso,Hurth:2017hxg,Ciuchini:2017mik,Alok:2017sui}. + +The formalism of {\textit{b}} decays is commonly described within +an effective field theory~\cite{Altmannshofer:2008dz}, +which probes distinct energy scales; +with regimes classified into short-distance (high energies) perturbative +and non-calculable long-distance effects. +These can be parametrised in the weak Lagrangian in terms of +effective operators with different Lorentz structures, $\mathcal{O}_i$, +with corresponding couplings $\mathcal{C}_i$ - referred to as Wilson coefficients (WC). +Only a subset of the operators that are most sensitive to NP is examined in this work~\cite{Bobeth:2017vxj}, +\textit{i.e.} $\mathcal{O}_7$ (virtual photon exchanges), $\mathcal{O}_{9,10}$ (vector and axial currents) +and corresponding right-handed couplings with flipped helicities. +In this framework NP effects are incorporated +by introducing deviations in the WCs~\cite{Ali:1994bf} from their SM predictions, +{\textit{i.e.}} $\mathcal{C}_i = \mathcal{C}^{\mathrm{SM}}_i + \mathcal{C}^{\mathrm{NP}}_i$. +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~\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 matrix elements that are difficult to assess reliably from first principles. +Some promising approaches suggest to extract this contribution +from data-driven analyses~\cite{Blake:2017fyh,Hurth:2017sqw} +and by exploiting analytical properties of its structure~\cite{Bobeth:2017vxj}. +However, these models still have intrinsic limitations, in particular +in the assumptions that enter in parametrisation of the di-lepton invariant mass distribution. + +In this Letter we propose a new \textit{model-independent} approach that +from a simultaneous unbinned amplitude analysis of both +$\bar{B}^0 \to \bar{K}^{*0} \mu^+\mu^-$ and $\bar{B}^0 \to \bar{K}^{*0} e^+e^-$ decays +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 effect relies on the strong correlation between the muon and electron modes +imposed by the lepton-flavour universality of the hadronic matrix elements. +Furthermore, in this method the full set of observables +(\textit{e.g} $R_{K^{*}}$, $P^{\prime}_{5}$ and branching fraction measurements) available in $\bar{B}\to \bar{K}^*\ell^+\ell^-$ +decays is exploited, providing unprecedented precision on LFU in a single analysis. + +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 squared invariant mass, $q^2$, and the three angles +$\vec{\Omega} = (\cos \theta_\ell, \cos \theta_K, \phi)$~\cite{Altmannshofer:2008dz}. +The probability density function ($p.d.f.$) for this decay can be written as +% +\begin{equation} +p.d.f. = \frac{1}{\Gamma} \frac{\dd^4 \Gamma}{\dd q^2 \dd^3 \Omega}, \ + \quad + \text{with}\quad + \Gamma = \int_{q^2} \dd q^2 \frac{\dd\Gamma}{\dd q^2}\,, +\end{equation} +% +with different \qsq intervals depending on the lepton flavour under study. +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. +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{ + \label{eq:amp_dep} + {\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^{(\ell)}_7 {\cal{F}}_{\lambda}^{T}(q^2) - 16\pi^2 \frac{M_B}{m_b} {\cal{H}}_{\lambda}(q^2) \bigg] +\bigg\}\,,\nonumber +} +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. +The ${\cal{F}}^{(T)}_{\lambda}(q^{2})$ are form factors, +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. +In the following +this function 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 $\mathcal{H}_\lambda[z^n]$), +after removing singularities related to the $J/\psi(1S)$ and $\psi(2S)$. +Further information about the formalism is given in appendix~\ref{app:formalism}. +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. + +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 $\mathcal{C}_i^{(\mu,e)}$ are renamed as $\widetilde{\mathcal{C}}_i^{(\mu,e)}$, +since an accurate disentanglement between the physical +meaning of $\WC_i^{(\mu,e)}$ and the above-mentioned hadronic pollution +cannot be achieved at the current stage of the theory~\cite{Chrzaszcz:2018yza}. +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 $\bar{B}^0 \to \bar{K}^{*0} \mu^+\mu^-$ decays +that is sufficient to enable the determination of these multi-space parameters. +Note that an amplitude analysis of the electron mode only has been previously disregarded, +given the limited dataset in either LHCb or Belle experiments. +In a common framework the 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 +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.} $\WC_i^{\prime\,(\mu,e)} = 0$, +while sensitivity studies on the determination of the WCs $\WC_9^{\prime\,(\mu)}$ and $\WC_{10}^{\prime\,(\mu)}$ +are detailed in appendix~\ref{app:WC_primed}. + +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 luminosity and $\sigma_{b\bar{b}} \propto \sqrt{s}$ +for LHCb, where $s$ denotes the designed centre-of-mass energy of the $b$-quark pair, +and exclusively with luminosity for Belle II. +%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 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; +while in Belle II the same kinematic regions are considered for both 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 broadly consistent with published results, +and assumes improvements in the electron mode resolution for LHCb~\cite{Lionetto:2624938}. + +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 for two NP benchmark points (BMP), +$\Delta\WC_9 = - 1$ and $\Delta\WC_9 = - \Delta\WC_{10} = - 0.7$, +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.} $\WC_i^{(e)} = \WC_i^{\rm{SM}}$. +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 to these simulated samples, +in which multivariate Gaussian terms are added to the log-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 +(see appendix~\ref{app:formalism} for a detailed discussion), \textit{i.e.} +% +\begin{enumerate} + % + \item[i.] baseline $\mathcal{H}_\lambda[z^2]$ SM prediction~\cite{Bobeth:2017vxj} + included as a multivariate Gaussian constraint; + % + \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 the non-local hadronic matrix elements as + proposed in Ref.~\cite{Hurth:2017sqw}, \textit{i.e.} + %using instead an expansion in terms of $1 + a_{\lambda} + b_{\lambda}(q^{2}/6\,$GeV$^{2})$. + including them as multiplicative factors to the corresponding leading hadronic terms. + % +\end{enumerate} +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}. + + +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 +$\widetilde{\mathcal{C}}_9^{(\mu)}$ and $\widetilde{\mathcal{C}}_9^{(e)}$ coefficients +is sufficient to preserve the true underlying physics at any order of the series expansion $\mathcal{H}_\lambda[z^n]$ +and without any parametric theoretical input, +\textit{i.e.} the two-dimensional pull estimator with respect to the LFU hypothesis is unbiased. +% +\begin{figure}[t] +\includegraphics[width=.4\textwidth]{ellipses_C9.pdf} +\caption{% + Two-dimensional sensitivity scans for the pair of Wilson coefficients + $\widetilde{\mathcal{C}}_9^{(\mu)}$ and $\widetilde{\mathcal{C}}_9^{(e)}$ + for different non-local hadronic parametrisation models evaluated at \texttt{BMP}$_{\WC_9}$, + and with the expected statistics after \lhcb Run II. + The contours correspond to 99\% confidence level statistical-only uncertainty bands and + the dotted black line indicates the LFU hypothesis. +} +\label{fig:C9ellipse} +\end{figure} +% +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 end of Run-II. +% +\begin{figure}[bth!] +%\begin{center} +\includegraphics[width=.4\textwidth]{ellipses_DeltaC9C10_a.pdf}\\%\quad\quad\quad\quad +\includegraphics[width=.4\textwidth]{ellipses_DeltaC9C10_b.pdf} +\caption{% + Two-dimensional sensitivity scans for the proposed observables $\Delta\WC_9$ and $\Delta\WC_{10}$ + for different non-local hadronic parametrisation models + evaluated at (top) \texttt{BMP}$_{\WC_9}$ and (bottom) \texttt{BMP}$_{\WC_{9,10}}$, + and with the expected statistics after \lhcb Run II. + The contours correspond to 99\% confidence level statistical-only uncertainty bands. + \label{fig:DeltaC9C10} +} +%\end{center} +\end{figure} + +The sensitivity to the two benchmark-like NP scenarios using the proposed pseudo observables $\Delta \WC_i$ +is shown in Fig.~\ref{fig:DeltaC9C10}. +We quantify the maximal expected significance with respect to the SM to be $4.6$ and $5.3\,\sigma$ for +\texttt{BMP}$_{\WC_9}$ and \texttt{BMP}$_{\WC_{9,10}}$, respectively. +Realistic experimental effects are necessary to determine the exact sensitivity achievable. +Nevertheless, these results suggest that a first observation (with a single measurement) of LFU breaking +appears to be feasible with the expected recorded statistics by the end of LHCb Run II. +Furthermore, it is interesting to examine the prospects for confirming this evidence in the upcoming LHCb/Belle +upgrades~\cite{Albrecht:2017odf}. +Figure~\ref{fig:DeltaC9C10_Upgrade} summarises the two-dimensional statistical-only significances +for the designed luminosities. +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 - +insensitive to both local and non-local hadronic uncertainties. + + +\begin{figure}[bth!] +\includegraphics[width=.4\textwidth]{ellipses_DeltaC9C10_Nev.pdf} +\caption{% + Two-dimensional sensitivity scans for the proposed observables $\Delta\WC_9$ and $\Delta\WC_{10}$ + for the two considered NP scenarios: (green) \texttt{BMP}$_{\WC_9}$ and (red) \texttt{BMP}$_{\WC_{9,10}}$. + The contours correspond to 99\% confidence level statistical-only uncertainty bands expected for the (dashed) \belle II 50~ab$^{-1}$ + and \lhcb Upgrade (dotted) $50\,$fb$^{-1}$ and (solid) $\,300\,$fb$^{-1}$ statistics. + \label{fig:DeltaC9C10_Upgrade} +} +\end{figure} + +Experimental resolution and detector acceptance/efficiency effects +are not considered in this work, as these would require further information +from current (non-public) or planned \textit{B}-physics experiments. +Nevertheless, the precision on this measurement can remain unbiased +either by parametrising these effects in the amplitude model and/or even +re-computing the angles or the $q^2$ variables constraining the $B$ invariant mass~\cite{Lionetto:2624938}. +%Nevertheless, preliminary studies 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}. +%Despite the low \qsq asymmetric tail, the determination of $\Delta\WC_9$ and +%$\Delta\WC_{10}$ remains unbiased, even if no-correction is +%applied. +Moreover, the differential decay width can receive additional complex amplitudes from signal-like backgrounds, +\textit{e.g.} $K\pi$ S-wave from a non-resonant decay and/or a scalar resonance~\cite{Becirevic:2012dp}. +These contributions are determined to be small~\cite{Aaij:2015oid,Aaij:2016flj}, +and in the proposed formalism they benefit from the same description between the muon +and electron mode (see detailed discussion in Ref.~\cite{Chrzaszcz:2018yza}). +Therefore, such contribution does not dilute the expected sensitivity of the measurement. + +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 $\mathcal{H}_\lambda[z^2]$. +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. +It is worth mentioning that a hypothetical determination of the individual +$\widetilde{\WC}_9^{(\mu,e)}$ and $\widetilde{\WC}_{10}^{(\mu,e)}$ WCs +can also produce a shift in their central values +that mimics the behaviour of NP~\cite{Chrzaszcz:2018yza}.% and makes impossible any claim in this direction. + + +% +In conclusion, we propose a clean and model-independent method to combine +all the available information from $\bar{B}\to \bar{K}^*\ell^+\ell^-$ decays for a precise +determination of LFU-breaking differences of WCs, \textit{i.e.} $\Delta\WC_9$ +and $\Delta\WC_{10}$. +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 theoretical uncertainty. +In addition, this simultaneous analysis is more robust against +experimental effects such +as mismodeling of the detector resolution, since most parameters are +effectively determined from the muon mode. This would be an important +benefit for LHCb where the electron resolution is significantly worse +than that of muons. +Figure~\ref{fig:allComponents} illustrates the usefulness of the newly-proposed observables by combining +the different information from angular analysis to branching ratio measurements. +Due to the inclusiveness of the approach, the expected sensitivity surpasses any +of the projections for the foreseen measurements of \textit{e.g.} $R_{K^{*}}$ or $P^{\prime}_{5}$ alone - given the benchmark points. +Therefore, this novel formalism can be the most immediate method to observe unambiguously +NP in $\bar{B}\to \bar{K}^*\ell^+\ell^-$ decays. + +A promising feature of this framework is the possibility to extend the analysis to +include other decay channels involving flavour changing neutral currents. +For instance, the charged decay $\bar{B}^+ \to \bar{K}^{*+} \ellell$ undergoes the same physics +and is easily accessible at the $B$-factories, while other rare +semi-leptonic decays such as $B^+ \to K^+ \ellell$ and $\Lambda_{b} \to \Lambda^{(*)} \ell^+\ell^-$ +have a different phenomenology but access the same NP information in terms of WC description. +Thus, an unbinned global simultaneous fit to all data involving $b \to s \ell^+ \ell^-$ transitions +is a natural and appealing extension of this work. +Moreover, the parameter space of the investigated WCs can also be broadened to +incorporate direct measurement of the right-handed $\WC_i^{\prime}$ - +currently weakly constrained by global fits~\cite{Capdevila:2017bsm,Altmannshofer:2017yso,Hurth:2017hxg}. +% +\begin{figure}[t] +\includegraphics[width=.4\textwidth]{B2Kstll_summary.pdf} +\caption{% + Sensitivity to \texttt{BMP}$_{\WC_{9,10}}$ scenario for the expected statistics after the \lhcb Run II. + The relative contribution (68, 95, 99\% confidence level contours) of each step of the analysis is shown in different colours, + together with the result of full amplitude method proposed in this letter. + \label{fig:allComponents} +} +\end{figure} + + +\section*{Acknowledgments} + +We acknowledge useful contributions from Michele Atzeni, Gino Isidori, Danny van Dyk and Patrick Owen. +This work is supported by the Swiss National Science Foundation (SNF) under contracts 173104 and 174182. + + +%\newpage +%\FloatBarrier + +\appendix + +\section{Formalism}\label{app:formalism} + +The non-local hadronic matrix elements $\mathcal{H}_\lambda(q^2)$ are investigated +using two complementary parametrisations~\cite{Bobeth:2017vxj,Hurth:2017sqw}. + +The nominal parametrisation~\cite{Bobeth:2017vxj} is obtained through the mapping +\begin{equation} +q^{2} \mapsto z(q^{2}) \equiv \frac{\sqrt{t_+ - q^2} - \sqrt{t_+ - t_0}}{\sqrt{t_+ - q^2} + \sqrt{t_+ - t_0}} \, , +\end{equation} +where $t_+ = 4 M_D^2$ and $t_0 = t_+ - \sqrt{t_+ ( t_+ - M^2_{\psi(2S)})}$, +which leads to the functions $\mathcal{H}_\lambda(z)$ that are characterised +by two singularities at $z_{J/\psi}$ and $z_{\psi(2S)}$. +These can be expressed as +\begin{equation} +\mathcal{H}_\lambda(z) = \frac{1 - z z^*_{J/\psi}}{z - z_{J/\psi}} \frac{1 - z z^*_{\psi(2S)}}{z - z_{\psi(2S)}} \hat{\mathcal{H}}_\lambda(z) \, , +\end{equation} +where the functions $ \hat{\mathcal{H}}_\lambda(z)$ are analytical and can be Taylor-expanded around $z = 0$ as +\begin{equation} +\hat{\mathcal{H}}_\lambda(z) = \Big[ \sum^n_{k=0} \alpha_k^{(\lambda)} z^k \Big] \mathcal{F}_\lambda(z) \, . +\end{equation} +Several orders of the polynomials are studied in the text. +Note that any additional order $k$ introduces a complex parameter, $\alpha_k^{(\lambda)}$, for each of the polarisations $\lambda = \perp, \parallel, 0$. +These nuisance parameters can be either free floated in the fit (nominal configuration labelled as $\mathcal{H}_\lambda[z^2,...,z^4]$) +or Gaussian constrained to their SM prediction (labelled as $\mathcal{H}_\lambda[z^2]$ \textit{with theo. priors} in the plots). + +Finally, the non-local hadronic matrix elements are reparametrised following Ref.~\cite{Hurth:2017sqw}, in which +these non-local hadronic contributions are included as multiplicative factors, leading to a reformulation +of the amplitudes of Eq.~\ref{eq:amp_dep} as +\eqa{ + {\cal{A}}_{\lambda}^{(\ell)\,L,R} = {\cal{N}}_{\lambda}^{(\ell)}\ \bigg\{ && +(C^{(\ell)}_9 \mp C^{(\ell)}_{10}) {\cal{F}}_{\lambda}(q^2) \big[ 1 + a_{\lambda} + b_{\lambda} \frac{q^{2}}{\mathrm{\scalebox{0.8}{6 GeV$^2$}}} \big] \nonumber \\ +% +&& +\frac{2m_b M_B}{q^2} C^{(\ell)}_7 {\cal{F}}_{\lambda}^{T}(q^2) +\bigg\}\,, +} +% +where $a_{\lambda}$ and $b_{\lambda}$ are complex coefficients Gaussian constrained around zero. + + +\section{Right-handed Wilson coefficients}\label{app:WC_primed} + +An extension of the physics case of the proposed method +is to investigate the sensitivity to the chirality-flipped counterparts of the usual Wilson coefficients, +\textit{i.e.} $\WC^{\prime (\mu)}_9$ and $\WC^{\prime(\mu)}_{10}$. +Following the formalism discussed in this letter, the primed WCs are examined by considering in addition to the +\texttt{BMP}$_{\WC_{9,10}}$ three different modified NP scenarios for the muon only: +$\WC_{9,10}^{\prime(\mu)} = \WC^{\prime \rm{SM}}_{9,10} = 0$; +$\WC^{\prime (\mu)}_9 = \WC^{\prime (\mu)}_{10} = 0.3$; +and $\WC^{\prime (\mu)}_9 = - \WC^{\prime (\mu)}_{10} = 0.3$. +Notice that for the electron mode the $\WC_{9,10}^{\prime(e)}$ is set and fixed to the SM value $\WC^{\prime \rm{SM}}_{9,10} = 0$. + +\begin{figure}[t] +\includegraphics[width=.4\textwidth]{ellipses_CpMu_Hz.pdf} +\caption{% + Two-dimensional sensitivity scans for the pair of Wilson coefficients $\WC'^{(\mu)}_9$ and + $\WC'^{(\mu)}_{10}$ for different non-local hadronic parametrisation models for a NP scenario + with $\WC'^{(\mu)}_9 = \WC'^{(\mu)}_{10} = 0.3$. + The contours correspond to 99\% confidence level statistical-only uncertainty bands evaluated with + the expected statistics after \lhcb Run II. + \label{fig:Cp_Hz} +} +\end{figure} + +Figure~\ref{fig:Cp_Hz} shows the fit results for different order of the analytic expansion for +the non-local hadronic contribution for a NP scenario with $\WC'^{(\mu)}_9 = \WC'^{(\mu)}_{10} = 0.3$, +and yields corresponding to the \lhcb Run II expected statistics. +The dependency on the determination of $\WC'^{(\mu)}_9$ and $\WC'^{(\mu)}_{10}$ on the +order of the expansion clearly saturates after $\mathcal{H}_\lambda[z^3]$ and allows a measurement +of the primed Wilson coefficients for the muon decay channel $B^{0} \to K^{*0} \mumu$ independent +on the theoretical hadronic uncertainty. +% +Figure~\ref{fig:Cp} shows the prospects for the sensitivity to the $\WC'^{(\mu)}_9$ and +$\WC'^{(\mu)}_{10}$ Wilson coefficients corresponding to the expected statistics at +the LHCb upgrade with $50\,$fb$^{-1}$ and $\,300\,$fb$^{-1}$. +Note that only with the full capability of the LHCb experiment it is possible +to start disentangling the different NP hypotheses. +% +\begin{figure}[t] +\includegraphics[width=.4\textwidth]{ellipses_CpMu.pdf} +\caption{% + Two-dimensional sensitivity scans for the pair of Wilson coefficients $\WC'^{(\mu)}_9$ and $\WC'^{(\mu)}_{10}$ + for three NP scenarios: (blue) $\WC'^{(\mu)}_9 = \WC'^{(\mu)}_{10} = 0$, (orange) $\WC'^{(\mu)}_9 = \WC'^{(\mu)}_{10} = 0.3$ + and (magenta) $\WC'^{(\mu)}_9 = - \WC'^{(\mu)}_{10} = 0.3$. + The contours correspond to 99\% confidence level statistical-only uncertainty bands expected for the LHCb Upgrade (dotted) + $50\,$fb$^{-1}$ and (solid) $\,300\,$fb$^{-1}$ statistics. + \label{fig:Cp} +} +\end{figure} + +\FloatBarrier +\bibliography{references} + +\end{document} diff --git a/prd/prd/ellipses_C9.pdf b/prd/prd/ellipses_C9.pdf new file mode 100644 index 0000000..13293f7 --- /dev/null +++ b/prd/prd/ellipses_C9.pdf Binary files differ diff --git a/prd/prd/ellipses_CpMu.pdf b/prd/prd/ellipses_CpMu.pdf new file mode 100644 index 0000000..46ebca3 --- /dev/null +++ b/prd/prd/ellipses_CpMu.pdf Binary files differ diff --git a/prd/prd/ellipses_CpMu_Hz.pdf b/prd/prd/ellipses_CpMu_Hz.pdf new file mode 100644 index 0000000..8a034e5 --- /dev/null +++ b/prd/prd/ellipses_CpMu_Hz.pdf Binary files differ diff --git a/prd/prd/ellipses_DeltaC9C10_Nev.pdf b/prd/prd/ellipses_DeltaC9C10_Nev.pdf new file mode 100644 index 0000000..1ed7ece --- /dev/null +++ b/prd/prd/ellipses_DeltaC9C10_Nev.pdf Binary files differ diff --git a/prd/prd/ellipses_DeltaC9C10_a.pdf b/prd/prd/ellipses_DeltaC9C10_a.pdf new file mode 100644 index 0000000..80501e8 --- /dev/null +++ b/prd/prd/ellipses_DeltaC9C10_a.pdf Binary files differ diff --git a/prd/prd/ellipses_DeltaC9C10_b.pdf b/prd/prd/ellipses_DeltaC9C10_b.pdf new file mode 100644 index 0000000..b20ff0d --- /dev/null +++ b/prd/prd/ellipses_DeltaC9C10_b.pdf Binary files differ diff --git a/prd/prd/references.bib b/prd/prd/references.bib new file mode 100644 index 0000000..c36ca1d --- /dev/null +++ b/prd/prd/references.bib @@ -0,0 +1,815 @@ +@article{Becirevic:2012dp, + author = "Becirevic, Damir and Tayduganov, Andrey", + title = "{Impact of $B\to K^\ast_0 \ell^+\ell^-$ on the New + Physics search in $B\to K^\ast \ell^+\ell^-$ decay}", + journal = "Nucl. 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Phys. J.", + volume = "C73", + year = "2013", + pages = "2646", + doi = "10.1140/epjc/s10052-013-2646-9", + eprint = "1308.1501", + archivePrefix = "arXiv", + primaryClass = "hep-ph", + reportNumber = "FERMILAB-PUB-13-310-T, MITP-13-047", + SLACcitation = "%%CITATION = ARXIV:1308.1501;%%" +} +@article{Crivellin:2015era, + author = "Crivellin, Andreas and Hofer, Lars and Matias, Joaquim + and Nierste, Ulrich and Pokorski, Stefan and Rosiek, + Janusz", + title = "{Lepton-flavour violating $B$ decays in generic $Z'$ + models}", + journal = "Phys. Rev.", + volume = "D92", + year = "2015", + number = "5", + pages = "054013", + doi = "10.1103/PhysRevD.92.054013", + eprint = "1504.07928", + archivePrefix = "arXiv", + primaryClass = "hep-ph", + reportNumber = "CERN-PH-TH-2015-091, TTP15-018", + SLACcitation = "%%CITATION = ARXIV:1504.07928;%%" +} +# Leptoquarks +@article{Hiller:2014yaa, + author = "Hiller, Gudrun and Schmaltz, Martin", + title = "{$R_K$ and future $b \to s \ell \ell$ physics beyond the + standard model opportunities}", + journal = "Phys. Rev.", + volume = "D90", + year = "2014", + pages = "054014", + doi = "10.1103/PhysRevD.90.054014", + eprint = "1408.1627", + archivePrefix = "arXiv", + primaryClass = "hep-ph", + reportNumber = "DO-TH-14-17", + SLACcitation = "%%CITATION = ARXIV:1408.1627;%%" +} +@article{Biswas:2014gga, + author = "Biswas, Sanjoy and Chowdhury, Debtosh and Han, Sangeun + and Lee, Seung J.", + title = "{Explaining the lepton non-universality at the LHCb and + CMS within a unified framework}", + journal = "JHEP", + volume = "02", + year = "2015", + pages = "142", + doi = "10.1007/JHEP02(2015)142", + eprint = "1409.0882", + archivePrefix = "arXiv", + primaryClass = "hep-ph", + SLACcitation = "%%CITATION = ARXIV:1409.0882;%%" +} +@article{Gripaios:2014tna, + author = "Gripaios, Ben and Nardecchia, Marco and Renner, S. A.", + title = "{Composite leptoquarks and anomalies in $B$-meson + decays}", + journal = "JHEP", + volume = "05", + year = "2015", + pages = "006", + doi = "10.1007/JHEP05(2015)006", + eprint = "1412.1791", + archivePrefix = "arXiv", + primaryClass = "hep-ph", + SLACcitation = "%%CITATION = ARXIV:1412.1791;%%" +} +# QCD issues +@article{Jager:2012uw, + author = "Jäger, S. and Martin Camalich, J.", + title = "{On $B \to V \ell \ell$ at small dilepton invariant + mass, power corrections, and new physics}", + journal = "JHEP", + volume = "05", + year = "2013", + pages = "043", + doi = "10.1007/JHEP05(2013)043", + eprint = "1212.2263", + archivePrefix = "arXiv", + primaryClass = "hep-ph", + SLACcitation = "%%CITATION = ARXIV:1212.2263;%%" +} +@article{Jager:2014rwa, + author = "Jäger, Sebastian and Martin Camalich, Jorge", + title = "{Reassessing the discovery potential of the $B \to K^{*} + \ell^+\ell^-$ decays in the large-recoil region: SM + challenges and BSM opportunities}", + journal = "Phys. Rev.", + volume = "D93", + year = "2016", + number = "1", + pages = "014028", + doi = "10.1103/PhysRevD.93.014028", + eprint = "1412.3183", + archivePrefix = "arXiv", + primaryClass = "hep-ph", + SLACcitation = "%%CITATION = ARXIV:1412.3183;%%" +} +@article{Aaij:2014ora, + author = "Aaij, Roel and others", + title = "{Test of lepton universality using $B^{+}\rightarrow + K^{+}\ell^{+}\ell^{-}$ decays}", + collaboration = "LHCb", + journal = "Phys. Rev. Lett.", + volume = "113", + year = "2014", + pages = "151601", + doi = "10.1103/PhysRevLett.113.151601", + eprint = "1406.6482", + archivePrefix = "arXiv", + primaryClass = "hep-ex", + reportNumber = "CERN-PH-EP-2014-140, LHCB-PAPER-2014-024", + SLACcitation = "%%CITATION = ARXIV:1406.6482;%%" +} +@article{Aaij:2017vbb, + author = "Aaij, R. and others", + title = "{Test of lepton universality with $B^{0} \rightarrow + K^{*0}\ell^{+}\ell^{-}$ decays}", + collaboration = "LHCb", + journal = "JHEP", + volume = "08", + year = "2017", + pages = "055", + doi = "10.1007/JHEP08(2017)055", + eprint = "1705.05802", + archivePrefix = "arXiv", + primaryClass = "hep-ex", + reportNumber = "LHCB-PAPER-2017-013, CERN-EP-2017-100", + SLACcitation = "%%CITATION = ARXIV:1705.05802;%%" +} +% BR measurements +@article{Aaij:2013aln, + author = "Aaij, R and others", + title = "{Differential branching fraction and angular analysis of + the decay $B_s^0\to\phi\mu^{+}\mu^{-}$}", + collaboration = "LHCb", + journal = "JHEP", + volume = "07", + year = "2013", + pages = "084", + doi = "10.1007/JHEP07(2013)084", + eprint = "1305.2168", + archivePrefix = "arXiv", + primaryClass = "hep-ex", + reportNumber = "CERN-PH-EP-2013-078, LHCB-PAPER-2013-017", + SLACcitation = "%%CITATION = ARXIV:1305.2168;%%" +} +@article{Aaij:2014pli, + author = "Aaij, R. and others", + title = "{Differential branching fractions and isospin asymmetries + of $B \to K^{(*)} \mu^+ \mu^-$ decays}", + collaboration = "LHCb", + journal = "JHEP", + volume = "06", + year = "2014", + pages = "133", + doi = "10.1007/JHEP06(2014)133", + eprint = "1403.8044", + archivePrefix = "arXiv", + primaryClass = "hep-ex", + reportNumber = "LHCB-PAPER-2014-006, CERN-PH-EP-2014-055", + SLACcitation = "%%CITATION = ARXIV:1403.8044;%%" +} +@article{Hiller:2003js, + author = "Hiller, Gudrun and Kruger, Frank", + title = "{More model-independent analysis of $b \to s$ processes}", + journal = "Phys. Rev.", + volume = "D69", + year = "2004", + pages = "074020", + doi = "10.1103/PhysRevD.69.074020", + eprint = "hep-ph/0310219", + archivePrefix = "arXiv", + primaryClass = "hep-ph", + reportNumber = "LMU-18-03, TUM-HEP-519-03", + SLACcitation = "%%CITATION = HEP-PH/0310219;%%" +} +@article{Bordone:2016gaq, + author = "Bordone, Marzia and Isidori, Gino and Pattori, Andrea", + title = "{On the Standard Model predictions for $R_K$ and + $R_{K^*}$}", + journal = "Eur. Phys. J.", + volume = "C76", + year = "2016", + number = "8", + pages = "440", + doi = "10.1140/epjc/s10052-016-4274-7", + eprint = "1605.07633", + archivePrefix = "arXiv", + primaryClass = "hep-ph", + SLACcitation = "%%CITATION = ARXIV:1605.07633;%%" +} +@article{Capdevila:2017bsm, + author = "Capdevila, Bernat and Crivellin, Andreas and + Descotes-Genon, Sébastien and Matias, Joaquim and Virto, + Javier", + title = "{Patterns of New Physics in $b\to s\ell^+\ell^-$ + transitions in the light of recent data}", + journal = "JHEP", + volume = "01", + year = "2018", + pages = "093", + doi = "10.1007/JHEP01(2018)093", + eprint = "1704.05340", + archivePrefix = "arXiv", + primaryClass = "hep-ph", + reportNumber = "PSI-PR-17-05, LPT-ORSAY-17-19", + SLACcitation = "%%CITATION = ARXIV:1704.05340;%%" +} +@article{Altmannshofer:2017yso, + author = "Altmannshofer, Wolfgang and Stangl, Peter and Straub, + David M.", + title = "{Interpreting Hints for Lepton Flavor Universality + Violation}", + journal = "Phys. Rev.", + volume = "D96", + year = "2017", + number = "5", + pages = "055008", + doi = "10.1103/PhysRevD.96.055008", + eprint = "1704.05435", + archivePrefix = "arXiv", + primaryClass = "hep-ph", + SLACcitation = "%%CITATION = ARXIV:1704.05435;%%" +} +@article{Hurth:2017hxg, + author = "Hurth, T. and Mahmoudi, F. and Martinez Santos, D. and + Neshatpour, S.", + title = "{Lepton nonuniversality in exclusive + $b{\rightarrow}s{\ell}{\ell}$ decays}", + journal = "Phys. Rev.", + volume = "D96", + year = "2017", + number = "9", + pages = "095034", + doi = "10.1103/PhysRevD.96.095034", + eprint = "1705.06274", + archivePrefix = "arXiv", + primaryClass = "hep-ph", + reportNumber = "CERN-TH-2017-110, IPM-P.A-466, MITP-17-035", + SLACcitation = "%%CITATION = ARXIV:1705.06274;%%" +} +@article{Bobeth:2017vxj, + author = "Bobeth, Christoph and Chrzaszcz, Marcin and van Dyk, + Danny and Virto, Javier", + title = "{Long-distance effects in $B\to K^*\ell\ell$ from + Analyticity}", + year = "2017", + eprint = "1707.07305", + archivePrefix = "arXiv", + primaryClass = "hep-ph", + reportNumber = "EOS-2017-01, MIT-CTP-4918, TUM-HEP-1087-17, ZU-TH-17-17", + SLACcitation = "%%CITATION = ARXIV:1707.07305;%%" +} +@article{Blake:2017fyh, + author = "Blake, Thomas and Egede, Ulrik and Owen, Patrick and + Pomery, Gabriela and Petridis, Konstantinos Alexandros", + title = "{An empirical model of the long-distance contributions to + $\bar{B}^{0} \rightarrow \bar{K}^{*0}\mu^{+}\mu^{-}$ + transitions}", + year = "2017", + eprint = "1709.03921", + archivePrefix = "arXiv", + primaryClass = "hep-ph", + SLACcitation = "%%CITATION = ARXIV:1709.03921;%%" +} +@article{Hurth:2017sqw, + author = "Hurth, T. and Langenbruch, C. and Mahmoudi, F.", + title = "{Direct determination of Wilson coefficients using + $B^0\to K^{*0}\mu^+\mu^-$ decays}", + journal = "JHEP", + volume = "11", + year = "2017", + pages = "176", + doi = "10.1007/JHEP11(2017)176", + eprint = "1708.04474", + archivePrefix = "arXiv", + primaryClass = "hep-ph", + 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;%%" +} +@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:2624938, + author = "Lionetto, Federica and Serra, Nicola and Steinkamp, Olaf + and Mueller, Katharina", + title = "{Measurements of Angular Observables of $B^0\to K^{*0} + \mu^+ \mu^-$ and $B^0 \to K^{*0} e^+ e^-$ Decays and the + Upgrade of LHCb}", + month = "Feb", + year = "2018", + reportNumber = "CERN-THESIS-2018-074", + url = "https://cds.cern.ch/record/2624938", + note = "Presented 22 Mar 2018", + howpublished = "\url{https://cds.cern.ch/record/2624938}" +} +@article{Capdevila:2017ert, + author = "Capdevila, Bernat and Descotes-Genon, Sebastien and + Hofer, Lars and Matias, Joaquim", + title = "{Hadronic uncertainties in $B \to K^* \mu^+ \mu^-$: a + state-of-the-art analysis}", + journal = "JHEP", + volume = "04", + year = "2017", + pages = "016", + doi = "10.1007/JHEP04(2017)016", + eprint = "1701.08672", + archivePrefix = "arXiv", + primaryClass = "hep-ph", + reportNumber = "ICCUB-17-003, UAB-FT-990, LPT-ORSAY-16-89", + SLACcitation = "%%CITATION = ARXIV:1701.08672;%%" +} +@article{Aaij:2016flj, + author = "Aaij, Roel and others", + title = "{Measurements of the S-wave fraction in $B^{0}\rightarrow + K^{+}\pi^{-}\mu^{+}\mu^{-}$ decays and the + $B^{0}\rightarrow K^{\ast}(892)^{0}\mu^{+}\mu^{-}$ + differential branching fraction}", + collaboration = "LHCb", + journal = "JHEP", + volume = "11", + year = "2016", + pages = "047", + doi = "10.1007/JHEP11(2016)047, 10.1007/JHEP04(2017)142", + note = "[Erratum: JHEP04,142(2017)]", + eprint = "1606.04731", + archivePrefix = "arXiv", + primaryClass = "hep-ex", + reportNumber = "CERN-EP-2016-141, LHCB-PAPER-2016-012", + SLACcitation = "%%CITATION = ARXIV:1606.04731;%%" +} +@article{Albrecht:2017odf, + author = "Albrecht, Johannes and Bernlochner, Florian and Kenzie, + Matthew and Reichert, Stefanie and Straub, David and + Tully, Alison", + title = "{Future prospects for exploring present day anomalies in + flavour physics measurements with Belle II and LHCb}", + year = "2017", + eprint = "1709.10308", + archivePrefix = "arXiv", + primaryClass = "hep-ph", + SLACcitation = "%%CITATION = ARXIV:1709.10308;%%" +} +@article{PhysRevD.8.3633, + title = {Asymptotically free gauge theories. I}, + author = {Gross, David J. and Wilczek, Frank}, + journal = {Phys. Rev. D}, + volume = {8}, + issue = {10}, + pages = {3633--3652}, + numpages = {0}, + year = {1973}, + month = {Nov}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevD.8.3633}, + url = {http://link.aps.org/doi/10.1103/PhysRevD.8.3633} +} +@article{Politzer:1974fr, + author = "Politzer, H. David", + title = "{Asymptotic freedom: an approach to strong interactions}", + journal = "Phys.Rept.", + volume = "14", + pages = "129-180", + doi = "10.1016/0370-1573(74)90014-3", + year = "1974", + reportNumber = "PRINT-74-1223 (HARVARD)", + SLACcitation = "%%CITATION = PRPLC,14,129;%%", +} +@article{Gross:1998jx, + author = "Gross, David J.", + title = "{Twenty five years of asymptotic freedom}", + journal = "Nucl.Phys.Proc.Suppl.", + volume = "74", + pages = "426-446", + doi = "10.1016/S0920-5632(99)00208-X", + year = "1999", + eprint = "hep-th/9809060", + archivePrefix = "arXiv", + primaryClass = "hep-th", + SLACcitation = "%%CITATION = HEP-TH/9809060;%%", +} +@article{Ball:1998kk, + author = "Ball, Patricia and Braun, Vladimir M.", + title = "{Exclusive semileptonic and rare B meson decays in QCD}", + journal = "Phys. Rev.", + volume = "D58", + year = "1998", + pages = "094016", + doi = "10.1103/PhysRevD.58.094016", + eprint = "hep-ph/9805422", + archivePrefix = "arXiv", + primaryClass = "hep-ph", + reportNumber = "CERN-TH-98-162, NORDITA-98-41-HE", + SLACcitation = "%%CITATION = HEP-PH/9805422;%%" +} +@article{Khodjamirian:2006st, + author = "Khodjamirian, Alexander and Mannel, Thomas and Offen, + Nils", + title = "{Form-factors from light-cone sum rules with B-meson + distribution amplitudes}", + journal = "Phys. Rev.", + volume = "D75", + year = "2007", + pages = "054013", + doi = "10.1103/PhysRevD.75.054013", + eprint = "hep-ph/0611193", + archivePrefix = "arXiv", + primaryClass = "hep-ph", + reportNumber = "SI-HEP-2006-03", + SLACcitation = "%%CITATION = HEP-PH/0611193;%%" +} +@article{Becirevic:2006nm, + author = "Becirevic, Damir and Lubicz, Vittorio and Mescia, + Federico", + title = "{An Estimate of the B ---> K* gamma form factor}", + journal = "Nucl. Phys.", + volume = "B769", + year = "2007", + pages = "31-43", + doi = "10.1016/j.nuclphysb.2007.01.032", + eprint = "hep-ph/0611295", + archivePrefix = "arXiv", + primaryClass = "hep-ph", + reportNumber = "LPT-ORSAY-06-73, RM3-TH-06-22", + SLACcitation = "%%CITATION = HEP-PH/0611295;%%" +} +@article{Horgan:2013hoa, + author = "Horgan, Ronald R. and Liu, Zhaofeng and Meinel, Stefan + and Wingate, Matthew", + title = "{Lattice QCD calculation of form factors describing the + rare decays $B \to K^* \ell^+ \ell^-$ and $B_s \to \phi + \ell^+ \ell^-$}", + journal = "Phys. Rev.", + volume = "D89", + year = "2014", + number = "9", + pages = "094501", + doi = "10.1103/PhysRevD.89.094501", + eprint = "1310.3722", + archivePrefix = "arXiv", + primaryClass = "hep-lat", + reportNumber = "DAMTP-2013-45, MIT-CTP-4507", + SLACcitation = "%%CITATION = ARXIV:1310.3722;%%" +} +@inproceedings{Mauri:2018, + author = "Chrzaszcz, Marcin and Mauri, Andrea and Serra, Nicola and Silva Coutinho, Rafael and van Dyk, Danny", + title = "{Prospects for data-driven analyses of the decays $B \to K^* \ell^+ \ell^-$}", + booktitle = "{$b \to s \ell \ell$ 2018: 6th Workshop on Rare Semileptonic B Decays, February 20-22}", + url = "https://indico.ph.tum.de/event/3782/sessions/558/attachments/2681/3011/NonLocalMEs2.pdf", + year = "2018", + eprint = "[In preparation]", +} +@article{Aaij:2013qta, + author = "Aaij, R and others", + title = "{Measurement of Form-Factor-Independent Observables in + the Decay $B^{0} \to K^{*0} \mu^+ \mu^-$}", + collaboration = "LHCb", + journal = "Phys. Rev. Lett.", + volume = "111", + year = "2013", + pages = "191801", + doi = "10.1103/PhysRevLett.111.191801", + eprint = "1308.1707", + archivePrefix = "arXiv", + primaryClass = "hep-ex", + reportNumber = "LHCB-PAPER-2013-037, CERN-PH-EP-2013-146", + SLACcitation = "%%CITATION = ARXIV:1308.1707;%%" +} + diff --git a/prd/suppl.pdf b/prd/suppl.pdf deleted file mode 100644 index 78fbf6b..0000000 --- a/prd/suppl.pdf +++ /dev/null Binary files differ