diff --git a/draft.tex b/draft.tex index 6c7105e..a40a08d 100644 --- a/draft.tex +++ b/draft.tex @@ -102,29 +102,12 @@ -%% Kinematic Macros %% - -%% Editing %% -%\usepackage[normalem]{ulem} % for \sout{} -%\makeatletter -%\newcommand{\todo}[1]{\textcolor{red}{\textbf{ToDo:} #1}} -%\newcommand{\ok}{\ensuremath{\checkmark}} -%\def\dvd{\@ifstar\@@dvd\@dvd} -%\newcommand{\@dvd}[1]{\textcolor{purple}{[\textbf{DvD:} #1]}} -%\newcommand{\@@dvd}[1]{\textcolor{purple}{#1}} -%\def\rsc{\@ifstar\@@rsc\@rsc} -%\newcommand{\@rsc}[1]{\textcolor{ForestGreen}{[\textbf{RsC:} #1]}} -%\newcommand{\@@rsc}[1]{\textcolor{ForestGreen}{#1}} -%\makeatother - \begin{document} \allowdisplaybreaks \preprint{ZU-TH-15/18} \title{Towards establishing Lepton Flavour Universality violation in $\bar{B}\to \bar{K}^*\ell^+\ell^-$ decays} -%\title{Novel approach for probing Lepton Flavour Universality in $B\to K^*\ell^+\ell^-$ decays} -%\title{Probing Lepton Flavour Universality in $B\to K^*\ell^+\ell^-$ decays} \author{Andrea Mauri} \email{a.mauri@cern.ch} \author{Nicola Serra} @@ -135,8 +118,6 @@ \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 have indicated an anomalous pattern in measurements - %of angulepton-flavour-universality observables. 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. @@ -147,39 +128,27 @@ 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, - %observables that encode potential difference in the coupling for muons and electrons, - %This method enables the direct determination of the difference - %of observables that encode the effects of high energy scales - %of the Wilson Coefficients ${\cal{C}}_{9}$ and ${\cal{C}}_{10}$ - %between electrons and muons, and are found to be insensitive to non-perturbative QCD effects. -%Our method combines the full information in $B^0\to K^*\ell^+\ell^-$ -%decays and, 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. -% We show that considering the current preferred New Physics scenario a first observation of LFU breaking in a single measurement is possible with LHCb Run-II dataset. \end{abstract} \maketitle Flavour changing neutral current processes of {\textit{B}} meson decays -%, dominantly mediated by $b \to s$ amplitudes, 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. -%{\color{red} phenomenology to study, formed by differential decay widths and 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 form-factor uncertainties, 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 -%large hadronic contributions from $b\to s c{\bar{c}}$ operators ({\textit{i.e.}} charm loops) 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}. Non-standard measurement in ratios of $b \to s \ell^+ \ell^-$ processes @@ -195,28 +164,16 @@ an effective field theory~\cite{Altmannshofer:2008dz}, which probes distinct energy scales; with regimes classified into short-distance (high energies) perturbative -%\footnote{At short-distance (high energies) quarks interact weakly, being considered in an asymptotic freedom, which -%allows a pertubative calculation~\cite{PhysRevD.8.3633,Politzer:1974fr,Gross:1998jx}.} 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). -%induced couplings $\mathcal{C}_i$ - referred to as Wilson coefficients (WC) - -%and effective vertex operators with different Lorentz structure, $\mathcal{O}_i$. 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. -% -%hereafter only a subset of the Wilson coefficients (WC) $C_i$ for the basis of -%dimension-six field operators $O_i$ is used for the weak Lagrangian~\cite{Bobeth:2017vxj}. 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, whilst the individual level of significance of the present anomalies is still inconclusive, -%there is an appealing non-trivial consistency shown in -%global analysis fits~\cite{Capdevila:2017bsm,Altmannshofer:2017yso,Hurth:2017hxg}; -%\textit{i.e.} a shift in the coefficient $\mathcal{C}_9$ only, -%or $\mathcal{C}_9$ and $\mathcal{C}_{10}$ simultaneously. 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}. @@ -224,9 +181,7 @@ 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} -%or by using the analytical and dispersive properties of these correlators~\cite{Bobeth:2017vxj}. -%and by exploring properties (\textit{e.g.} analyticity) of the structure of these functions~\cite{Bobeth:2017vxj}. +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. @@ -237,22 +192,14 @@ 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 relies on the strong correlation {\color{red} between the two decay modes} when examining muons and electrons This effect relies on the strong correlation between the muon and electron modes imposed by the lepton-flavour universality of the hadronic matrix elements. -%when examining directly the difference in Wilson coefficients. 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. -%and therefore, most stringent constraints on LFU for a single measurement can be expected. -%Let us consider the differential decay rate for $\bar{B}\to \bar{K}^*\ell^+\ell^-$ Consider the differential decay rate for $\bar{B}\to \bar{K}^*\ell^+\ell^-$ decays (dominated by the on-shell $\bar{K}^{*0}$ contribution) -%{\color{red} (I know that you already changed this to the text above, but "Let us" looks pretty ugly:) ) -%In this work we assume the $\bar{B}\to \bar{K}^*\ell^+\ell^-$ decay -%being completely dominated by the on-shell $\bar{K}^{*0}$ ($p$-wave) contribution. -%The differential decay rate is hence 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}. @@ -266,7 +213,6 @@ \end{equation} % with different \qsq intervals depending on the lepton flavour under study. -%where the \qsq range is defined differently for the two semileptonic channels. For a 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 @@ -284,19 +230,11 @@ ${\cal{F}}^{(T)}_{\lambda}(q^{2})$ 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 the form factors ${\cal{F}}^{(T)}_{\lambda}(q^{2})$ parametrise -%the fact that the interaction happens inside the hadrons}, -%While the form factors ${\cal{F}}^{(T)}_{\lambda}(q^{2})$ can be taken from~\cite{Straub:2015ica}\footnote{In order -%to guarantee a good agreement between Light-Cone Sum Rules~\cite{Ball:1998kk,Khodjamirian:2006st} -%and Lattice results~\cite{Becirevic:2006nm,Horgan:2013hoa}, -%sure full agreement among the LCSRs and Lattice results. -%uncertainties on the form factors parameters are doubled with respect to Ref.~\cite{Straub:2015ica}}, 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 -%~\footnote{After removing below-threshold poles, \textit{i.e.} $J/\psi(1S)$ and $\psi(2S)$, } 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 @@ -358,16 +296,8 @@ $\Delta\WC_9 = - 1$ 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.} $\WC_i^{(e)} = \WC_i^{\rm{SM}}$. -%This baseline model is modified in the case of muons for two NP benchmark points (BMP), -%\textit{i.e.} $\WC_9^{(e)} = \WC^{\rm{SM}}_9 = \WC^{(\mu)}_9 + 1$ and -%{\color{red} $\WC^{\rm{NP}(\mu)}_9 = - 1$ } -%and {\color{red} $\WC_9^{\rm{NP}(\mu)} = -\WC_{10}^{\rm{NP}(\mu)} = - 0.7$}, -%$\WC_{9(10)}^{(e)} = \WC^{\rm{SM}}_{9(10)} = \WC_{9(10)}^{(\mu)} +(-)\,0.7$, -%$\WC_9^{(\mu)} = -\WC_{10}^{(\mu)} = - 0.7$, -%referred to as \texttt{BMP}$_{\WC_9}$ and \texttt{BMP}$_{\WC_{9,10}}$, respectively. These points are favoured by several global fit analyses with similar significance~\cite{Capdevila:2017bsm,Altmannshofer:2017yso,Hurth:2017hxg}. -%but chosen with reduced SM tension in order to examine a more conservative hypothesis. An extended unbinned maximum likelihood fit is performed to these simulated samples, in which multivariate Gaussian terms are added to the likelihood to incorporate prior knowledge @@ -396,10 +326,6 @@ to guarantee a good agreement between Light-Cone Sum Rules~\cite{Ball:1998kk,Khodjamirian:2006st} and Lattice results~\cite{Becirevic:2006nm,Horgan:2013hoa}, their uncertainties are doubled with respect to Ref.~\cite{Straub:2015ica}. -% -%The stability of the model and the convergency to the global minimum is enforced by -%repeating the fit with randomised starting parameters; -%the solution with smallest negative log-likelihood is taken as the default. Figure~\ref{fig:C9ellipse} shows the fit results for several alternative parametrisations @@ -426,10 +352,6 @@ \label{fig:C9ellipse} \end{figure} % -%Furthermore, we note that, as commonly stated in the literature (see \textit{e.g.} recent review in Ref.~\cite{Capdevila:2017ert}), -%the determination of $\WC_{10}^{(\mu,e)}$ is insensitive to the lack of knowledge on the -%non-local hadronic effects and thus independent of any model assumption. -% 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, @@ -452,9 +374,6 @@ The sensitivity to the two benchmark-like NP scenarios using the proposed pseudo observables $\Delta \WC_i$ is shown in Fig.~\ref{fig:DeltaC9C10}. -%Fig.~\ref{fig:DeltaC9C10} shows the sensitivity to the two NP scenarios, NP$_{\WC_9}$ -%and NP$_{\WC_9-\WC_{10}}$ in terms of the two model-independent LFU-breaking -%difference of Wilson coefficients $\Delta\WC_9$ and $\Delta\WC_{10}$. 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. @@ -468,18 +387,6 @@ 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. -%The clear advantage of the proposed pseudo-observables $\Delta\WC_i$ is that they are -%insensitive to both local and non-local hadronic uncertainties.} -%Note that these unprecedented dataset will enable insight towards the nature of NP. -% -%the \lhcb Run II, $7.6(8.4)\,\sigma$ for \belle II 50~ab$^{-1}$ dataset and -%$9.0(9.6)\,\sigma$ for the \lhcb 50~fb$^{-1}$ Upgrade for the \texttt{BMP}$_{\WC_9}$ -%(\texttt{BMP}$_{\WC_9-\WC_{10}}$) scenario respectively. -%{\color{red} The two-dimensional sensitivity pull expected for the \belle II and -%\lhcb Upgrades is shown in Fig.~\ref{fig:DeltaC9C10_Upgrade}. -%We note that the proposed method can provide a first observation of LFU breaking -%in a single measurement with LHCb Run-II dataset, while for a precise determination -%of the nature of NP the expected statistics of future upgrades of \lhcb and \belle II is required.} \begin{figure}[bth!] @@ -496,11 +403,6 @@ 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. -%further information from current (non-public) or planned \textit{B}-physics experiments are necessary to extend this discussion. -%as they would require additional information from the current \textit{B}-physics experiments. -%Modelling detector effects such as \qsq and angles resolution, detector acceptance/efficiency, -%is hardly possible without access to (non-public) information of the current -%\textit{B}-physics experiments. 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 @@ -515,20 +417,12 @@ Therefore, in this constrained framework these effects are even further suppressed and can then be neglected for the scope of this work. -%Another important test to probe the stability of the model consists in changing the -%description of the non-local hadronic effects in the generation of the pseudo-experiments. -%In this way we analyse potential issues that can rise if the truncation -%$\mathcal{H}_\lambda[z^n]$ is not a good description of nature. Another important test to probe the stability of the model consists in analysing potential issues that can rise if the truncation $\mathcal{H}_\lambda[z^n]$ is not a good description of nature. We proceed as follows: we generate ensembles with non-zero coefficients for $\mathcal{H}_\lambda[z^3]$ and $\mathcal{H}_\lambda[z^4]$, and we perform the fit with $\mathcal{H}_\lambda[z^2]$. -%We vary the choice of the $\mathcal{H}_\lambda[z^{(3,4)}]$ generated parameters, -%including a set of values that minimises the tension with the $P_5'$ -%anomaly~\cite{Aaij:2015oid}, while keeping $\WC_9^{(\mu)}$ and -%$\WC_{10}^{(\mu)}$ at their SM values. 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. @@ -557,7 +451,6 @@ 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. -%of projection for the current anomalous measurements 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. @@ -572,8 +465,6 @@ 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}. -%and of possible phases in the WCs - appearing as $\Im \WC_i$ - by mean of a separate -%analysis of the two opposite charge-conjugated modes. % \begin{figure}[t] \includegraphics[width=.4\textwidth]{plots/B2Kstll_summary.pdf}