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

TL;DR

This paper develops dispersive models for nonlocal form factors in rare kaon decays to improve theoretical predictions of decay spectra and rates, utilizing unitarity, experimental data, and the Operator Product Expansion.

Contribution

It introduces a dispersive framework for nonlocal form factors in kaon decays, incorporating experimental constraints and theoretical relations to enhance decay rate predictions.

Findings

Dispersive representations effectively model $K^+ o \pi^+ u ar{ u}$ decay spectra.

Constraints from $K^+ o \pi^+ \pi^+ \pi^-$ decays improve form factor accuracy.

Relations from the Operator Product Expansion link charged-lepton and neutrino form factors.

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.