$\pi \pi \rightarrow K \bar{K}$ scattering up to 1.47 GeV with hyperbolic dispersion relations

TL;DR

This paper develops hyperbolic dispersion relations for $ ightarrow Kar{K}$ scattering up to 1.47 GeV, testing data consistency and providing precise parameterizations aligned with crossing symmetry.

Contribution

It introduces a set of hyperbolic dispersion relations extended to 1.47 GeV and uses them to refine data fits, ensuring consistency with fundamental symmetries and providing new parameterizations.

Findings

Identified conflicting data sets for the S-wave.

Provided parameterizations consistent with dispersion relations up to 2 GeV.

Demonstrated the applicability of hyperbolic dispersion relations in this energy range.

Abstract

In this work we provide a dispersive analysis of $\pi\pi \rightarrow K\bar{K}$ scattering. For this purpose we present a set of partial-wave hyperbolic dispersion relations using a family of hyperbolas that maximizes the applicability range of the hyperbolic dispersive representation, which we have extended up to 1.47 GeV. We then use these equations first to test simple fits to different and often conflicting data sets, also showing that some of these data and some popular parameterizations of these waves fail to satisfy the dispersive analysis. Our main result is obtained after imposing these new relations as constraints on the data fits. We thus provide simple and precise parameterizations for the S, P and D waves that describe the experimental data from $K\bar K$ threshold up to 2 GeV, while being consistent with crossing symmetric partial-wave dispersion relations up to their maximum applicability range of 1.47 GeV. For the $S$-wave we have found that two solutions describing two conflicting data sets are possible. The dispersion relations also provide a representation for $S$, $P$ and $D$ waves in the pseudo-physical region.