Prospects for Measuring $CP$-Violation in $B_s^0 \rightarrow φμ^+μ^-$ via Time-Dependent Angular Analysis

TL;DR

This study analyzes CP-violation in the rare decay $B_s^0 \rightarrow \phi \mu^+\mu^-$ using a time-dependent angular formalism that incorporates $B_s$ mixing. It introduces new optimised observables $\mathcal{H}_i$, $\mathcal{Z}_i$ and their counterparts $\mathcal{M}_i$, $Q_i$ to reduce hadronic uncertainties and exploits both tagged and untagged time-dependent PDFs. Realistic pseudoexperiments, based on LHCb Upgrade I-like performance, show that a full time-dependent angular analysis can be performed with Run 3–5 data, significantly tightening constraints on Wilson coefficients $\mathcal{C}_i$ and, in particular, their imaginary parts, with optimised observables offering substantial NP sensitivity. The results indicate measurable prospects for $P_5^{t\prime}$ and other observables in central $q^2$ regions, and highlight the complementary strengths of tagged vs untagged analyses for advancing tests of the Standard Model and potential new physics in $b\rightarrow s\ell\ell$ transitions.

Abstract

This work investigates the prospects for performing a time-dependent angular analysis of $B_s^0 \rightarrow φμ^+μ^-$ decays at hadron colliders, and introduces new optimised angular observables associated with $B_s^0$-mixing in the decay rate. The time-dependent normalised decay rate and corresponding probability density function is presented both for when the flavour of the $B_s^0$ meson at production is tagged and when it is untagged. The normalised angular terms linked to $B_s^0$-mixing in the tagged (untagged) case are denoted $\mathcal{Z}_i$ ($\mathcal{H}_i$), and their optimised counterparts as $Q_i$ ($\mathcal{M}_i$). The expected sensitivities of these observables at the end of Run 3, Run 4, and Run 5 of the LHC are determined using pseudoexperiments generated with a decay-time resolution, background level, and signal yield similar to those reported by the LHCb collaboration. It is found that all observables can be extracted with Run 3 statistics, and that the $\mathcal{H}_i$ and $\mathcal{Z}_i$ observables have similar sensitivity, despite the former being suppressed by mixing terms. Moreover, the angular observables only accessible via flavour tagging, such as the equivalent of $P_5^{\prime}$ in the $B^0_s$ system, are found to exhibit sensitivities comparable to current measurements in $B^0 \rightarrow K^{\ast 0} μ^+μ^-$ decays once Run 5 datasets are available. Fits to the observables to extract the Wilson coefficients show a significant increase in precision when either the observables accessible via time-dependent analysis, or flavour tagging, are included. A marked increase in sensitivity to $CP$-violating short-distance effects is observed for a subset of the new optimised $M_i$ and $Q_i$ observables.

Figures (24)

• Figure 1: Visualisation of the helicity angles describing the ${{\mathrm{B}\xspace}\xspace^0_{\mathrm{s}\xspace}\xspace}\xspace\!\rightarrow\xspace \upphi\xspace{\upmu\xspace^+\upmu\xspace^-}\xspace$ decay. $\vartheta_K$ marks the angle between the momentum vectors of the negatively charged kaon and the ${\mathrm{B}\xspace}\xspace^0_{\mathrm{s}\xspace}\xspace$ in the di-kaon rest frame. The angle between the momentum vector of the negatively charged muon and the ${\mathrm{B}\xspace}\xspace^0_{\mathrm{s}\xspace}\xspace$ in the di-muon rest frame is called $\vartheta_\ell$. The angle between the planes spun by the di-kaon and di-muon system is called $\varphi$.
• Figure 2: Example of the generated decay-time distribution for the ${\mathrm{B}\xspace}\xspace^0_{\mathrm{s}\xspace}\xspace$ (left) and ${\overline{ \mathrm{B}\xspace}}\xspace{}^0_{\mathrm{s}\xspace}\xspace$ (right) candidates with an overlay of the corresponding decay-rate. The simulated tagging decision is indicated in the histograms, showing a tag for the ${\mathrm{B}\xspace}\xspace^0_{\mathrm{s}\xspace}\xspace$ (blue) or the ${\overline{ \mathrm{B}\xspace}}\xspace{}^0_{\mathrm{s}\xspace}\xspace$ (red) or no conclusive decision (black). The category tagged ${\mathrm{B}\xspace}\xspace^0_{\mathrm{s}\xspace}\xspace$ is made up from
• Figure 3: Example of a fit to the generated pseudodata in the region where $1.1 < {q^2}\xspace < 6.0\text{\,Ge V}\xspace^2\!/c^4\xspace$ using the model described in the text. In addition to the data, the fit-model and its components are displayed, with the ${\mathrm{B}\xspace}\xspace^0_{\mathrm{s}\xspace}\xspace$ (${\overline{ \mathrm{B}\xspace}}\xspace{}^0_{\mathrm{s}\xspace}\xspace$) shown in blue (red), the untagged component in purple, and the combinatorial background in orange. The characteristic oscillations of the ${\mathrm{B}\xspace}\xspace^0_{\mathrm{s}\xspace}\xspace$-system are well visible in the individual components of the decay-time PDF in the lower left.
• Figure 4: Distribution of the bias and coverage for the ensemble of pseudoexperiments performed for the different number of generated signal decays, $q^2$-regions, and effective tagging-power. Small biases and undercoverage is observed for the smallest sample sizes which vanish in the large sample limit.
• Figure 5: Extrapolated sensitivity to the $C\!P$-averages $S^t_i$, $C\!P$-asymmetries $A^t_i$, $\mathcal{H}_i$, and $\mathcal{Z}_i$ in comparison to the uncertainty of SM predictions evaluated using flavio straub:2018flavio where the local form-factors are taken from Ref. Bharucha:2015bzk, and the non-local form-factors follow the parameterisation of Ref. Altmannshofer:2014rta. The expected range for the sensitivity after the respective luminosity milestones is shown in the shaded area. The observables accessible in the untagged measurement are displayed on the left and the observables which require flavour-tagging are shown on the right. Depending on the assumed tagging-power, the sensitivity to the observables which require flavour-tagging varies.