First measurement of time-dependent $CP$ violation in the decay flavor-changing neutral-current decay $B^{0}\rightarrow K_{S}^{0}μ^{+}μ^{-}$

Abstract: A flavor-tagged time-dependent analysis of $$B^{{0}\rightarrow} K_{S}<sup>{0}μ<sup>{+}μ<sup>{-}$$ decays is performed across the full dimuon mass range excluding the $J/ψ$ and $ψ(2S)$ resonance regions. The analysis uses proton-proton collision data collected by the LHCb experiment in 2011--2018 at center-of-mass energies of 7, 8 and 13TeV, corresponding to an integrated luminosity of 9$fb<sup>{-1}$. The CP violation parameters are determined to be $C=-0.13 \pm 0.32 \pm 0.04$ and $S= +0.82\pm 0.29 \pm 0.05$, where the first uncertainties are statistical and the second are systematic.The results are consistent with the Standard Model prediction. This is the first experimental study of time-dependent CP violation in $b\rightarrow sl<sup>{+}l<sup>{-}$ processes.

• The paper reports the first time-dependent CP violation measurement in B0→KS0μ+μ– decays, providing new constraints on CP-violating phases in b→s transitions.
• It uses LHCb collision data with advanced multivariate classifiers and unbinned likelihood fits to extract CP observables with effective tagging and resolution methods.
• The measured values (C = -0.13±0.32 and S = +0.82±0.29) are consistent with Standard Model predictions, thereby limiting potential new physics scenarios.

Time-Dependent $CP$ Violation in $\mathbf{B^0\rightarrow K_S^0\mu^+\mu^-}$: First Measurement

Motivation and Theoretical Context

The flavor-changing neutral-current (FCNC) transition $b \to s\ell^+\ell^-$ constitutes a powerful probe of physics beyond the Standard Model (SM) due to its suppression at tree level and sensitivity to loop-level contributions. Anomalies have been reported in branching ratios and angular observables for several $B$ meson decays, compatible with non-SM contributions to four-fermion effective couplings [Capdevila:2023yhq, Alguero2023]. However, complementary observables are required to discriminate the underlying dynamics and elucidate possible new sources of $CP$ violation.

Time-dependent analyses of neutral $B$ decays are particularly sensitive to the imaginary components of the Wilson coefficients that parameterize $b \to s\ell^+\ell^-$ transitions [Descotes-Genon2021]. $CP$ violation arising via interference between $B^0$--$\bar{B}^0$ mixing and decay amplitudes provides theoretically robust access to new $CP$-violating phases. The decay $B^0\rightarrow K_S^0\mu^+\mu^-$ enables the first time-dependent $CP$ violation measurement in a $b \to s\ell^+\ell^-$ channel, allowing direct constraints beyond those accessible from time-integrated observables.

The time-dependent $CP$ asymmetry is defined as

$$\mathcal{A}^{CP}(t) = S \sin(\Delta m_d t) - C \cos(\Delta m_d t)$$

with $S$ characterizing mixing-induced $CP$ violation and $C$ direct $CP$ violation. The SM predicts $C \approx 0$ and $S \approx \sin(2\beta)$, where $\beta$ is the well-measured CKM phase [Fleischer2023]. New physics models introducing extra weak phases can substantially alter $C$ and $S$ from their SM values [PRD107075].

Experimental Apparatus and Selection Strategy

The measurement utilizes $pp$ collision data from LHCb spanning 2011--2018, totaling $9\,\mathrm{fb}^{-1}$ at $\sqrt{s}=$ 7, 8, and 13 TeV. The LHCb detector is a forward spectrometer optimized for $b$- and $c$-quark hadron studies, featuring precision tracking, PID, and efficient muon identification [LHCb-DP-2008-001].

Signal candidates are constructed from $K_S^0\to\pi^+\pi^-$ and $\mu^+\mu^-$ pairs. Kinematic, topological, and PID selections—leveraging multivariate classifiers (BDTG)—suppress backgrounds (including misidentified $\Lambda_b$, $D^-$, and combinatorics). PID observables are combined via multivariate techniques and a kinematic fit, with decay topology optimized for maximum signal significance.

The signal is extracted by fitting the $B^0$ mass spectrum with a Crystal Ball function, and combinatoric background is modeled exponentially.

Figure 1: Mass distribution of selected $B^0\rightarrow K_S^0\mu^+\mu^-$ candidates in all categories, superimposed with fit results.

Time-Dependent Analysis: Tagging and Likelihood Fits

The production flavor is determined using both same-side (SS) and opposite-side (OS) tagging algorithms [LHCb-PAPER-2016-039, LHCb-PAPER-2011-027], each yielding a tagging decision and calibrated mistag probability. Combined, these produce effective tagging efficiencies $\varepsilon \langle D^2 \rangle = (2.7\pm1.1)\%$ (Run 1) and $(4.1\pm0.6)\%$ (Run 2). The calibrated mistag probability $\omega$ is crucial for the likelihood modeling.

The decay-time distribution includes convolution with a multi-Gaussian resolution function (effective resolution $\sim60$ fs), and acceptance corrections via a cubic spline parametrization. Systematic effects due to time resolution and acceptance are negligible compared to statistical uncertainties.

$CP$ violation parameters are determined via a weighted unbinned maximum-likelihood fit performed on Run 1 and Run 2 samples, exploiting sWeights for background subtraction [Pivk:2004ty]. Fixed parameters include the $B^0$ lifetime and oscillation frequency, constrained to world averages.

Figure 2: (Left) background-subtracted decay-time distributions for flavor-tagged $B^0$ and $\bar{B}^0$ candidates with fit overlay; (Right) time-dependent $CP$ asymmetry, showing signal yield asymmetry with dilution corrections and maximum-likelihood curve.

Results and Numerical Summary

The total signal yield is $941 \pm 55$. The time-dependent fit in the inclusive $q^2$ range (excluding $J/\psi$ and $\psi(2S)$) yields: $$C = -0.13 \pm 0.32\, (\text{stat}) \pm 0.04\, (\text{syst}) \ S = +0.82 \pm 0.29\, (\text{stat}) \pm 0.05\, (\text{syst})$$ with correlation coefficient $0.50$. These values are consistent with SM predictions ($C\approx0$, $S\approx0.72$) [Fleischer2023, PDG2024]. Subrange fits indicate no statistically significant deviation or $q^2$ dependence.

Profile likelihood scans yield confidence regions for $(C,S)$, compared to the SM expectation.

Figure 3: Two-dimensional confidence regions for $C$ and $S$ as obtained from profile likelihood scans across low-$q^2$, high-$q^2$ and total sample, contrasted with the SM prediction.

Employing the Feldman–Cousins approach, physical boundary $C^2+S^2\leq1$ is enforced, and confidence intervals are derived via constrained likelihood-ratio ordering.

Figure 4: Confidence regions for $C$ and $S$ in the total $q^2$ range under the physical constraint, compared to SM predictions.

Systematics are dominated by flavor tagging calibration and fit biases, but remain sub-leading relative to statistical uncertainties. Alternative fits, splitting by $K_S$ category, magnet polarity, and period, as well as checks in control channels, confirm robustness.

Implications and Outlook

This result constitutes the first measurement of time-dependent $CP$ violation in an exclusive $b \to s\mu^+\mu^-$ decay, representing a new experimental avenue for discriminating new physics contributions to $CP$ violation. The observables $C$ and $S$ exhibit no evidence for deviation from SM expectations, providing constraints on new sources of $CP$-violation in FCNC transitions that are robust against hadronic uncertainties and complementary to existing measurements of branching ratios, angular distributions, and LFU observables.

Given the persistent $b \to s\ell^+\ell^-$ anomalies in time-integrated observables [Capdevila:2023yhq], the absence of sizeable time-dependent $CP$ violation in $B^0\rightarrow K_S^0\mu^+\mu^-$ constrains the parameter space for new physics models, especially those involving complex Wilson coefficients in the effective Hamiltonian (such as $C_9$, $C_{10}$, or $C_7$) [Fleischer:2025ucq, PRD107075, Descotes-Genon2011]. Theoretically clean access to the imaginary parts of these coefficients strengthens global fits and model discrimination [Descotes-Genon2021].

Prospects for the future include increased sensitivity with larger datasets from LHCb upgrades and Belle II, improved flavor tagging, and full angular analyses to disentangle $CP$-even and $CP$-odd eigenstates and probe additional observables and dependencies.

Conclusion

The time-dependent $CP$ violation observables in $B^0\rightarrow K_S^0\mu^+\mu^-$ have been measured for the first time, with results consistent with the Standard Model and no evidence for new sources of $CP$ violation in $b \to s\mu^+\mu^-$ transitions. This methodology augments the arsenal of FCNC probes, facilitating direct constraints on new $CP$-violating dynamics. Future improvements in statistical precision and expanded analysis channels will further advance the discrimination of new physics in the flavor sector.