diff --git a/draft.tex b/draft.tex index f074f28..c420e54 100644 --- a/draft.tex +++ b/draft.tex @@ -145,13 +145,13 @@ \maketitle -Flavour chang{\color{red}ing} neutral current processes of {\textit{B}} meson decays, dominantly mediated by +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 {\color{blue}loop effects} {\color{red}the decay process} and cause observables to deviate +since as-yet undiscovered particles may contribute 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 {\color{blue}framework to study from differential decay widths to angular observables.} -{\color{red} phenomenology to study, formed by differential decay widths and angular observables.} +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 $B^{0} \to K^{*0} \mu^{+}\mu^{-}$ has been recently reported~\cite{Aaij:2015oid,Wehle:2016yoi,Aaij:2013aln,Aaij:2014pli}, notably in one of the observables with reduced form-factor uncertainties, @@ -172,10 +172,10 @@ The formalism of {\textit{b}} decays is commonly described within an effective field theory~\cite{Altmannshofer:2008dz} - -hereafter only a subset of the Wilson coefficients $C_i$ for the basis of +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 {\color{red} \sout{systematically}} incorporated -by introducing deviations {\color{red} \sout{exclusively}} in the Wilson coefficients (WC)~\cite{Ali:1994bf} +In this framework NP effects are incorporated +by introducing deviations in the WCs~\cite{Ali:1994bf} ({\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 @@ -184,14 +184,14 @@ %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{\color{red}~\cite{Capdevila:2017bsm,Altmannshofer:2017yso,Hurth:2017hxg}}. +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 contributions. -Some promising approaches propose to {\color{blue} \sout{either}} extract these non-local hadronic elements -{\color{red} either} from data-driven analyses~\cite{Blake:2017fyh,Hurth:2017sqw} +Some promising approaches propose to extract these non-local hadronic elements +either from data-driven analyses~\cite{Blake:2017fyh,Hurth:2017sqw} or by using the analytical and dispersive properties of these correlators~\cite{Bobeth:2017vxj}. -However these models still have intrinsic limitations, in particular +However, these models still have intrinsic limitations, in particular in the assumption of the parametrisation of the di-lepton invariant mass. In this Letter we propose a new \textit{model-independent} approach that @@ -200,17 +200,19 @@ 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 -directly at the level of Wilson coefficients. +%This relies on the strong correlation {\color{red} between the two decay modes} when examining muons and electrons +This relies on the strong correlation between the muon and electron modes when examining +directly the difference in Wilson coefficients. Furthermore, in this method the full set of observables available in $\bar{B}\to \bar{K}^*\ell^+\ell^-$ -decays is exploited, {\color{red}providing the most stringent constraints on LFU from a single measurement.} {\color{blue} and therefore, most stringent constraints on LFU for a single measurement are expected. } +decays is exploited, and therefore, most stringent constraints on LFU for a single measurement can be expected. -{\color{blue} Let us consider the differential decay rate for $\bar{B}\to \bar{K}^*\ell^+\ell^-$ -decays (dominated by the on-shell $\bar{K}^{*0}$ contribution) } -{\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} +%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 invariant mass square, $q^2$, and the three angles $\vec{\Omega} = (\cos \theta_\ell, \cos \theta_K, \phi)$~\cite{Altmannshofer:2008dz}. @@ -242,8 +244,8 @@ ${\cal{F}}^{(T)}_{\lambda}(q^{2})$ and $\mathcal{H}_\lambda(q^{2})$ are local and non-local hadronic matrix elements, respectively. While the ${\cal{F}}^{(T)}_{\lambda}(q^{2})$ are form factor parameters set from~\cite{Straub:2015ica} -\footnote{{\color{red} Following a conservative approach, uncertainties on the form factors parameters are -doubled with respect to~\cite{Straub:2015ica}}}, +\footnote{Following a conservative approach, uncertainties on the form factors parameters are +doubled with respect to Ref.~\cite{Straub:2015ica}}, the $\mathcal{H}_\lambda(q^{2})$ are described using two complementary parametrisations~\cite{Bobeth:2017vxj,Hurth:2017sqw} - for brevity only a subset of results is shown for the latter approach. @@ -262,22 +264,21 @@ \begin{equation} \Delta \WC_i = \widetilde{\mathcal{C}}_i^{(\mu)} - \widetilde{\mathcal{C}}_i^{(e)}\,, \end{equation} -where the usual WCs {\color{red} $\mathcal{C}_i^{(\mu,e)}$} are renamed as {\color{red}$\widetilde{\mathcal{C}}_i^{(\mu,e)}$} -in view of {\color{red} the fact} that a precise disentanglement between the physical +where the usual WCs $\mathcal{C}_i^{(\mu,e)}$ are renamed as $\widetilde{\mathcal{C}}_i^{(\mu,e)}$, +since a precise disentanglement between the physical meaning of $\WC_i^{(\mu,e)}$ and the above-mentioned hadronic pollution is impossible at the current stage of the theoretical knowledge. The key feature of this strategy is to realise that all hadronic matrix elements are known to be lepton-flavour universal, and thus are shared among both semileptonic decays. This benefits from the large statistics available for $B^0 \to K^{*0} \mu^+\mu^-$ decays that is sufficient to enable the determination of these multi-space parameters.\footnote{Note -that an amplitude analysis of the {\color{red} single} electron mode has been always previously disregarded, +that an amplitude analysis of the electron mode only has been always previously disregarded, given the limited dataset foreseen in either LHCb or Belle-II experiments.} Therefore, in a common framework these hadronic contributions are treated as nuisance parameters, while only the Wilson coefficients $\widetilde{\WC}_9^{(\mu,e)}$ and $\widetilde{\WC}_{10}^{(\mu,e)}$ are kept separately for the two channels. For consistency the WC $\widetilde{C}_{7}$ is also shared in the fit, given its universal coupling to photons~\cite{Paul:2016urs}. -{\color{red} (Comment: from how is written seems that $C_7$ is floated in the fit, but I like the sentence.)} 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 @@ -291,18 +292,20 @@ 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; -{\color{blue} and the same kinematic regions for both the semileptonic channels in Belle II} -{\color{red} while in Belle II the same kinematic regions is considered for both the semileptonic channels}, namely +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 consistent with published results, +This definition of \qsq ranges are broadly consistent with published results, and assumes improvements in the electron mode resolution for LHCb~\cite{Lionetto:XX}. Within the SM setup the Wilson coefficients are set to $\mathcal{C}^{\rm{SM}}_9 = 4.27$, $\mathcal{C}^{\rm{SM}}_{10} = - 4.17$ and $\mathcal{C}^{\rm{SM}}_7 = -0.34$. -This baseline model is modified in the case of muons for two NP benchmark points (BMP), \textit{i.e.} - {\color{red} $\WC^{\rm{NP}(\mu)}_9 = - 1$ } -and {\color{red} $\WC_9^{\rm{NP}(\mu)} = -\WC_{10}^{\rm{NP}(\mu)} = - 0.7$}, +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}. @@ -330,9 +333,9 @@ \end{enumerate} % The stability of the model and the convergency to the global minimum is enforced by -repeating the fit ${\cal{O}}(500)$ times with randomised starting parameters; +repeating the with randomised starting parameters; the solution with smallest negative log-likelihood is taken as the default. -{\color{red} We should rephrase in a way that the referee doesn't ask how many times we repeated the fit :) )} + 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, @@ -357,7 +360,7 @@ \label{fig:C9ellipse} \end{figure} % -Furthermore, we note that, as commonly stated in the literature (see \textit{i.e.} recent review in~\cite{Capdevila:2017ert}), +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. % @@ -376,30 +379,42 @@ %\end{center} \end{figure} -The sensitivity to the two NP scenarios previously discussed using the proposed observables $\Delta \WC_i$ +The sensitivity to the two benchmark-like NP scenarios using the proposed 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 to the SM as $4.6\,(5.3)\,\sigma$ for -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.} -\textbf{TODO: Add also here the plot for the upgrade and also comment on the result itself.} +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. +Notice that 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. +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. +%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!] \includegraphics[width=.4\textwidth]{plots/ellipses_DeltaC9C10_Nev.pdf} \caption{% - {\color{red} Two-dimensional sensitivity scans for the proposed observables $\Delta\WC_9$ and $\Delta\WC_{10}$ + 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 $3\,\sigma$ statistical-only uncertainty bands expected for the \belle II 50~ab$^{-1}$ (dashed) - and \lhcb Upgrade $50\,$fb$^{-1}$ (dotted) and $\,300\,$fb$^{-1}$ (solid) statistics.} + The contours correspond to $3\,\sigma$ 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} @@ -407,13 +422,17 @@ 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. -A first rudimentary study on the impact of a finite \qsq resolution is preformed +A first rudimentary study on the impact of a finite \qsq resolution is 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~\cite{Aaij:2017vbb}. +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. -\textbf{TODO: Add comment on the S-wave contribution.} +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 (see detailed discussion in Ref.~\cite{Hurth:2017hxg}). +These contributions are in general expected to be small~\cite{Aaij:2015oid,Aaij:2016flj}, +and in the proposed formalism these are introduced in an identical manner for muons and electrons. +Therefore, in this constrained framework these effects are even further suppressed and can then be neglected. 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. @@ -422,29 +441,32 @@ We proceed as follows: we generate toys 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 ``provocative'' set of values that minimize the tension with the $P_5'$ -``anomaly"~\cite{Aaij:2015oid} while keeping $\WC_9^{(\mu)}$ and +We vary the choice of the $\mathcal{H}_\lambda[z^{(3,4)}]$ generated parameters, +including a \textit{provocative} 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 +Despite the mis-modelling of the non-local hadronic effects in the fit results, 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. -{\color{red} On the contrary, an hypothetical determination of the single WC -$\widetilde{\WC}_9^{(\mu,e)}$ and $\widetilde{\WC}_{10}^{(\mu,e)}$ would result -in a strong bias that mimics the behaviour of NP and makes impossible any claim in this direction.} +It is worth emphasizing that a hypothetical determination of the individual +$\widetilde{\WC}_9^{(\mu,e)}$ and $\widetilde{\WC}_{10}^{(\mu,e)}$ WCs would result +in a strong bias that mimics the behaviour of NP and makes impossible any claim in this direction. -\textbf{Todo: comment in the conclusion on the use case of the prime WC and also the potential of analysing other channels, -in particular for the K*+ in Belle} + +% In conclusion, we propose a clean, robust and model-independent method to combine -all the available information from $B^{0} \to K^{*0} \ellell$ decays for a precise +all the available information from $\bar{B}\to \bar{K}^*\ell^+\ell^-$ decays for a precise determination of LFU-breaking differences of WCs $\Delta\WC_9$ and $\Delta\WC_{10}$, independent on any theoretical uncertainties. -Fig.~\ref{fig:allComponents} shows the contribution of all the single constituents of -the analysis and how the proposed method takes advantage of the complete description -of the decay. +Figure~\ref{fig:allComponents} illustrates the usefulness of the newly-proposed observables by combining +the different information from angular analysis to branching ratio measurements. -{\color{red} -A remarkable feature of the framework is the possibility to extend the analysis to +% +%the contribution of all the single constituents of +%the analysis and how the proposed method takes advantage of the complete description +%of the decay. +% +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 $B^+ \to K^{*+} \ellell$ underlies the same physics as the examined neutral mode and is easily accessible at the $B$-factories, while other rare @@ -457,12 +479,8 @@ determined 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}[tbh] +% +\begin{figure}[b] \includegraphics[width=.4\textwidth]{plots/B2Kstll_summary.pdf} \caption{% Sensitivity to the NP$_{\WC_9-\WC_{10}}$ scenario for the expected statistics after the \lhcb RunII. diff --git a/references.bib b/references.bib index 944549a..d3fc2d1 100644 --- a/references.bib +++ b/references.bib @@ -574,3 +574,22 @@ 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;%%" +}