Application of $K \to 3\pi$ amplitudes to semileptonic kaon decays

Abstract

We study dispersive representations of nonlocal form factors in $K^+ \to \pi^+ \ell^+ \ell^-$ and $K^+ \to \pi^+ \nu \bar{\nu}$ decays, with an aim of improving the theoretical description of the spectrum and decay rate of the neutrino mode. Based on unitarity, these representations invoke the $K^+ \pi^- \to \pi^+ \pi^-$ amplitude in $P$-wave and the pion vector form factor. The $P$-wave amplitude can be effectively parameterized within the dispersive Khuri-Treiman framework, and constrained by experimental information on the CP-conserving $K^+ \to \pi^+ \pi^+ \pi^-$ and $K^+ \to \pi^0 \pi^0 \pi^+$ decays. We also emphasize certain relations between charged-lepton and neutrino non-local form factors based on the Operator Product Expansion, which can be used to impose further phenomenological constraints.