On the scalar $πK$ form factor beyond the elastic region

Abstract

Pion-kaon ($πK$) pairs occur frequently as final states in heavy-particle decays. A consistent treatment of $πK$ scattering and production amplitudes over a wide energy range is therefore mandatory for multiple applications: in Standard Model tests; to describe crossed channels in the quest for exotic hadronic states; and for an improved spectroscopy of excited kaon resonances. In the elastic region, the phase shifts of $πK$ scattering in a given partial wave are related to the phases of the respective $πK$ form factors by Watson's theorem. Going beyond that, we here construct a representation of the scalar $πK$ form factor that includes inelastic effects via resonance exchange, while fulfilling all constraints from $πK$ scattering and maintaining the correct analytic structure. As a first application, we consider the decay ${τ\to K_Sπν_τ}$, in particular, we study to which extent the $S$-wave $K_0^*(1430)$ and the $P$-wave $K^*(1410)$ resonances can be differentiated and provide an improved estimate of the $CP$ asymmetry produced by a tensor operator. Finally, we extract the pole parameters of the $K_0^*(1430)$ and $K_0^*(1950)$ resonances via Padé approximants, $\sqrt{s_{K_0^*(1430)}}=[1408(48)-i\, 180(48)]$ MeV and $\sqrt{s_{K_0^*(1950)}}=[1863(12)-i\,136(20)]$ MeV, as well as the pole residues. A generalization of the method also allows us to formally define a branching fraction for ${τ\to K_0^*(1430) ν_τ}$ in terms of the corresponding residue, leading to the upper limit ${\text{BR}(τ\to K_0^*(1430) ν_τ)<1.6 \times 10^{-4}}$.