CP violation in $K\toμ^+μ^-$ with and without time dependence through a tagged analysis

TL;DR

This work demonstrates that the time-integrated CP asymmetry in $K^0\to\mu^+\mu^-$, when combined with branching ratios ${\cal B}(K^0_L\to\gamma\gamma)$, ${\cal B}(K^0_L\to\mu^+\mu^-)$, and ${\cal B}(K^0_S\to\mu^+\mu^-)$, yields clean access to short-distance physics encoded in the CKM combination $|A^2\lambda^5\bar{\eta}|$ via $K_L$–$K_S$ interference. The authors derive the SM prediction for the time-integrated CP asymmetry $\widetilde{A}_{\rm CP}(t)$ and show how it, together with a measurement of ${\cal B}(K^0_S\to\mu^+\mu^-)$, determines the short-distance amplitude and reduces discrete ambiguities in the $K^0_L\to\mu^+\mu^-$ rate, including the sign of the long-distance contribution from $K^0_L\to\gamma\gamma$. A detailed LHCb-like feasibility study indicates that, in optimistic scenarios with upgraded acceptance and efficient flavor tagging, $|\bar{\eta}|$ could be constrained to about 35% of its SM value, and the sign ambiguity in the SM prediction for ${\cal B}(K^0_L\to\mu^+\mu^-)$ could be resolved at >3$\sigma$ at HL-LHC. These results highlight the kaon sector as a complementary avenue to $B$-physics for precision CKM tests and motivate dedicated experimental efforts at LHCb to exploit tagged kaon decays.

Abstract

We point out that using current knowledge of ${\cal B}(K^0_L\toμ^+μ^-)$ and $ {\cal B}(K^0_L\to γγ)$, one can extract short-distance information from the combined measurement of the time-integrated CP asymmetry, $A_{\rm CP}(K^0\toμ^+μ^-)$, and of ${\cal B}(K^0_S\toμ^+μ^-)$. We discuss the interplay between this set of observables, and demonstrate that determining ${\rm sign}[A_{\rm CP}(K^0\toμ^+μ^-)]$ would eliminate the discrete ambiguity in the Standard Model prediction for ${\cal B}(K^0_L\toμ^+μ^-)$. We then move on to feasibility studies within an LHCb-like setup, using both time-integrated and time-dependent information, employing $K^0$ and $\overline K{}^0$ tagging methods. We find that, within an optimistic scenario, the short-distance amplitude, proportional to the CKM parameter combination $|A^2λ^5\barη|$, could be constrained by LHCb at the level of about $35\%$ of its Standard Model value, and the discrete ambiguity in ${\cal B}(K^0_L\toμ^+μ^-)_{\rm SM}$ could be resolved at more than $3σ$ by the end of the high luminosity LHC.

Figures (7)

• Figure 1: SM prediction for $A_{\rm CP}$ as a function of the end point of the integrals, as in Eq. \ref{['eq:ACP']}. The shaded regions correspond to the $1\sigma$ uncertainties on the predictions.
• Figure 2: Kaon multiplicity per $pp$ simulated collision for $pp\rightarrow K^0 X$ events (black lines) and $pp\rightarrow B_s^0 X$ events. The simulations are done with PYTHIA 8.2.2.4Sjostrand:2014zea using SoftQCD:all settings. It can be seen that $B_s^0$ events have a much higher kaon multiplicity, providing a higher mistag rate than in average $K^0$ events. This explains why we find much better tagging power for an SSK-like algorithm in $pp\rightarrow K^0X$ events compared to $pp\rightarrow B_s^0X$ events.
• Figure 3: Invariant mass distributions from fast simulation before (red) and after (blue) applying an impact parameter cut that eliminates half of the signal. Left: $K_S^0\rightarrow\pi^+(\rightarrow\mu^+\nu)\pi^-(\rightarrow\mu^-\bar{\nu})$ decays. Right: $K_S^0\rightarrow\mu^+\mu^-$ decays. It can be seen that the decays in flight that enter the $K_S^0$ peak get removed almost entirely and, in addition, the mass resolution of the remaining $K_S^0\rightarrow\mu^+\mu^-$ is better than in the unfiltered sample, further improving the signal-background separation.
• Figure 4: Decay-time distributions for $K^0\rightarrow\mu^+\mu^-$ (blue) and $\overline{ K}{}\xspace\xspace^0\rightarrow\mu^+\mu^-$ decays (red), using the decay-time acceptance for the Upgrade II scenario. The functions after the unbinned simultaneous fit are overlaid. The spectra are shown for $Y_{eff} \sim 300$ events.
• Figure 5: Experimental sensitivity to $\vert\bar{\eta}\vert$ (left), and the significance of the sign of $A_{\gamma\gamma}$ assuming the SM (right), as a function of the effective yield, $Y_{eff}$. It can be seen that with $Y_{eff} \sim 300$ events, LHCb can measure $|\bar{\eta}|$ with uncertainty below $40\%$, and resolve ${\rm sign}[A_{\gamma\gamma}]$ at more than three standard deviations. See Tab. \ref{['tab:yeffs']} for various scenarios and the corresponding $Y_{eff}$ values.