$K^-\to\pi^- a$ at Next-to-Leading Order in Chiral Perturbation Theory

The weak decay $K^-\to\pi^- a$ is a powerful probe of axion-like particles (ALPs). In this work, we provide a comprehensive analysis of this process within chiral perturbation theory, extending existing calculations by including complete next-to-leading order (NLO) contributions and isospin-breaking corrections at first order in $(m_u-m_d)$. We show that the consistent incorporation of ALPs in the QCD and weak chiral Lagrangians requires a non-trivial extension of the corresponding operator bases, which we describe in detail. Furthermore, we show that in the presence of an ALP the so-called weak mass term, which is unobservable in the Standard Model, is non-redundant already at leading order. We find that NLO corrections associated with flavor-violating ALP couplings modify the leading-order result by a few percent, with only small uncertainties. On the contrary, the NLO corrections proportional to flavor-conserving ALP couplings lead to an $\mathcal{O}(20\%)$ reduction relative to the leading-order predictions. These corrections are accompanied by a large uncertainties mainly originating from the QCD low-energy constant $L_{4,r}$ as well as from the presence of various unknown weak low-energy constants. We emphasize the importance of a precise determination of these coupling parameters for the successful study of new physics in light meson decays.