On the scalar $\pi K$ form factor beyond the elastic region

TL;DR

This paper develops a comprehensive model for the $K$ form factor across elastic and inelastic regions, aiding precision tests of the Standard Model and analysis of heavy-particle decays involving $K$ final states.

Contribution

It introduces a novel dispersion relation-based representation of the $K$ $S$-wave form factor that extends into the inelastic region using resonance exchange, while preserving unitarity and analytic structure.

Findings

Successfully described the $ au o K_S  ext{pi} u_ au$ spectrum.

Improved estimates of $CP$ asymmetry and forward-backward asymmetry.

Enhanced separation of $S$- and $P$-wave contributions.

Abstract

Pion-kaon ($\pi K$) final states, often appearing in heavy-particle decays at the precision frontier, are important for Standard-Model tests, to describe crossed channels with exotic states, and for spectroscopy of excited kaon resonances. We construct a representation of the $\pi K$ $S$-wave form factor using the elastic $\pi K$ scattering phase shift via dispersion relations in the elastic region and extend this model into the inelastic region using resonance exchange, while maintaining unitarity and the correct analytic structure. As a first application, we successfully described the $\tau \to K_S \pi \nu_\tau$ spectrum to not only achieve a better distinction between $S$- and $P$-wave contributions, but also to provide an improved estimate of the $CP$ asymmetry produced by a tensor operator as well as the forward-backward asymmetry, both of which can be confronted with future data at Belle II.