Eur. Phys. J. C (2021) 81: 420
https://doi.org/10.1140/epjc/s10052-021-09169-7

Regular Article - Theoretical Physics

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

¹ Forschungszentrum Jülich, Institute for Advanced Simulation, Institut für Kernphysik, and Jülich Center for Hadron Physics, 52425, Jülich, Germany
² Albert Einstein Center for Fundamental Physics, Institute for Theoretical Physics, University of Bern, Sidlerstrasse 5, 3012, Bern, Switzerland
³ Helmholtz-Institut für Strahlen- und Kernphysik and Bethe Center for Theoretical Physics, Universität Bonn, 53115, Bonn, Germany

^b noel@itp.unibe.ch

Received: 4 March 2021
Accepted: 19 April 2021
Published online: 14 May 2021

Abstract

Pion–kaon ($\pi K$) pairs occur frequently as final states in heavy-particle decays. A consistent treatment of $\pi K$ 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 $\pi K$ scattering in a given partial wave are related to the phases of the respective $\pi K$ form factors by Watson’s theorem. Going beyond that, we here construct a representation of the scalar $\pi K$ form factor that includes inelastic effects via resonance exchange, while fulfilling all constraints from $\pi K$ scattering and maintaining the correct analytic structure. As a first application, we consider the decay $\tau \to K_S\pi {\nu }_{\tau }$, in particular, we study to which extent the S-wave $K_0^{\ast }(1430)$ and the P-wave $K^{\ast }(1410)$ 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 $K_0^{\ast }(1430)$ and $K_0^{\ast }(1950)$ resonances via Padé approximants, $\sqrt{s_{K_0^{\ast }(1430)}}=[1408(48)-i180(48)]\text{MeV}$ and $\sqrt{s_{K_0^{\ast }(1950)}}=[1863(12)-i136(20)]\text{MeV}$, as well as the pole residues. A generalization of the method also allows us to formally define a branching fraction for $\tau \to K_0^{\ast }(1430){\nu }_{\tau }$ in terms of the corresponding residue, leading to the upper limit $\text{BR}(\tau \to K_0^{\ast }(1430){\nu }_{\tau })<1.6\times {10}^{-4}$.