Probing CP and flavor violation in neutral kaon decays with ALPs

We analyze the three-body decays of the long-lived neutral kaon $K_L \to \pi\pi a$, where $a$ is an axion-like particle (ALP), and compare them to the two-body decay $K_L \to \pi^0 a$. While the latter requires both flavor violation (FV) and $CP$ violation (CPV), the former can proceed via FV alone, allowing the ratio of decay rates to serve as a probe of CPV of the underlying UV theory. We emphasize the importance of weak-interaction-induced contributions, often neglected in recent calculations. We explore both minimal and non-minimal flavor-violating scenarios, and identify classes of models where ALP production from neutral three-body decays is comparable to - or even dominates over - the two-body decay, despite its reduced phase space. Finally, we discuss the phenomenological implications of our results and show how these decays can provide complementary probes of ALP couplings beyond those accessible via charged kaon channels.