$K\to\pi\gamma^*\gamma^*$ transitions at leading order and beyond

The transition amplitude of a kaon to a pion and two off-shell photons is studied. First, it is computed at leading order (one-loop level) of the Chiral Perturbation Theory expansion. Explicit analytical results for the leading-order amplitude are presented, constituting the first complete calculation for the doubly off-shell case. Subsequently, it is reevaluated by employing a refined diagrammatic notation and a generic ansatz incorporating effects beyond leading order. The dependence on the underlying $K\pi PP$ vertex parameters is analyzed. This offers valuable insights into amplitude properties and allows inclusion of unitarity corrections from $K\to3\pi$, yielding the complete $K\to\pi\gamma^*\gamma^*$ amplitude structure. Both the charged and neutral channels are treated in parallel. The presented results provide crucial input for phenomenological studies of related rare decays like $K\to\pi\ell^+\ell^-[\gamma]$ or $K\to\pi\ell_1^+\ell_1^-\ell_2^+\ell_2^-$ and support ongoing precision measurements at experiments like NA62 at CERN. These results may also find application in other related processes, including $\eta^{(\prime)}$ decays.