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

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 $${\tau \rightarrow K_S\pi \nu _\tau }$$ , 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 Pade approximants, $$\sqrt{s_{K_0^*(1430)}}=[1408(48)-i\, 180(48)]\,\text {MeV}$$ and $$\sqrt{s_{K_0^*(1950)}}=[1863(12)-i\,136(20)]\,\text {MeV}$$ , as well as the pole residues. A generalization of the method also allows us to formally define a branching fraction for $${\tau \rightarrow K_0^*(1430)\nu _\tau }$$ in terms of the corresponding residue, leading to the upper limit $${\text {BR}(\tau \rightarrow K_0^*(1430)\nu _\tau )<1.6 \times 10^{-4}}$$ .