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 rightarrow K_Spi 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 Padé 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}}.
Highlights
Ple Breit–Wigner (BW) model, but require the proper consideration of the analytic structure, most conveniently implemented in the framework of dispersion relations
The phase shifts of π K scattering in a given partial wave are related to the phases of the respective π K form factors by Watson’s theorem
In the crossed reaction π π → K K this connection determines scalar meson [16] and nucleon [17] form factors via a coupled-channel T -matrix, while the π K form factors of a given partial wave are directly related to the respective π K scattering amplitudes via Watson’s theorem [18], which states that the phases coincide in the elastic region
Summary
The same principles of unitarity and analyticity upon which modern analyses of π K scattering are based imply a relation to the corresponding form factors. P-wave, inelastic effects are typically included in resonance chiral theory (RChT) [28] via the K ∗(1410), providing an extended parameterization of the phase shift to be used in the Omnès factor or by feeding the corresponding amplitudes into a unitarization scheme such as the N /D method [29,30]. The latter is hard to handle, since it is difficult to prevent its high-order polynomials from generating.
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