Constraining ALP-Meson overlaps from $K\pi$ form factors

We present the first constraints on the overlaps between an Axion-like particle (ALP) and the $\pi^0$ and $\eta$ mesons from the analysis of the distortions to the $\langle K|\overline{s}\gamma^\mu u | \pi\rangle$ form factors. We demonstrate that these distortions can be tightly constrained by combining data from $\tau^-\to \pi^0 K^-\nu$ and $K^+\to \pi^0\ell^+\nu$ decays, and go on to map the constraints to the ALP-meson overlaps. We establish that, in general, the ALP-meson and meson-ALP overlaps are different due to the presence of ALP-quark derivative couplings in the UV Lagrangian, and need to be treated separately. Using lattice results and BaBar, Belle, and NA48/2 data, we obtain exclusion limits on the overlaps and give projections for Belle II. Our techniques are independent of the branching ratios of the ALP, hence, robust against ALP decay channel assumptions. For masses of the ALP below 1 GeV, the bounds on the effective scale of the ALP physics extend to $\mathcal{O}$(TeV) for restricted regions of the parameter space for the ALP-$\pi$ and $\pi$-ALP overlaps. On the other hand, these bounds persist for extended regions of the parameter space for ALP-$\eta$ and $\eta$-ALP overlaps.