Direct $CP$ Violation in $K \to \mu^+ \mu^-$

TL;DR

This paper discusses how interference effects between $K_L$ and $K_S$ decays can reveal direct CP violation in $K o \mu^+ \mu^-$ processes, significantly impacting standard model predictions and aiding in new physics investigations.

Contribution

It introduces the impact of interference effects on $K o \mu^+ \mu^-$ decay predictions and proposes a method to determine the sign of $ ext{A}(K_L o \gamma \gamma)$ to reduce theoretical uncertainties.

Findings

Interference can alter $K_S o \mu^+ \mu^-$ predictions by about 60%.

Measurement of interference can determine the sign of $ ext{A}(K_L o \gamma \gamma)$.

Interference effects are significant in analyzing CP violation and new physics models.

Abstract

A rare decay $K_L \to \mu^+ \mu^- $ has been measured precisely, while a rare decay $K_S \to \mu^+ \mu^- $ will be observed by an upgrade of the LHCb experiment. Although both processes are almost CP-conserving decays, we point out that an interference contribution between $K_L$ and $K_S$ in the kaon beam emerges from a genuine direct CP violation. It is found that the interference contribution can change $K_S \to \mu^+ \mu^-$ standard-model predictions at $\mathcal{O}(60\%)$. We also stress that an unknown sign of $\mathcal{A}(K_L \to \gamma \gamma)$ can be determined by a measurement of the interference, which can much reduce a theoretical uncertainty of $\mathcal{B}(K_L \to \mu^+ \mu^-)$. We also investigate the interference in a new physics model, where the $\epsilon'_K / \epsilon_K$ tension is explained by an additional $Z$-penguin contribution.