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.

Comments: 8 pages, 2 figures