Revisiting $K\rightarrow \pi a$ decays

The theoretical calculation for pseudo--scalars hadronic decays $K \to \pi a$ is reviewed. %with the Axion-Like-Particle escaping the detection, While one-loop penguin contributions are usually considered, tree-level processes have most often been overlooked in literature. Following the Lepage--Brodsky approach the tree-level contribution to the charged and neutral pseudo--scalar decay in ALP is estimated. Assuming generic ALP couplings to SM fermions, the latest NA62/E949 results for the $K^+ \to \pi^+ a$ decay and the present/future KOTO results for the $K^0_L \to \pi^0 a$ decay are used to provide updated bounds on the ALP--fermion Lagrangian sector. Finally, the interplay between the tree-level and one-loop contributions is investigated.