$K\pi$ $I=1/2$ $S$-wave from $\eta_c$ decay data at BaBar and classic Meson-Meson scattering from LASS

A recent analysis of data on the two photon production of the $\eta_c$ and its decay to $K(K\pi)$ has determined the $K\pi$ $S$-wave amplitude in a "model-independent" way assuming primarily that the additional kaon is a spectator in this decay. The purpose of this paper is to fit these results, together with classic $K\pi$ production data from LASS, within a formalism that implements unitarity for the di-meson interaction. This fixes the $I=1/2$ $K\pi\to K\pi$ $S$-wave amplitude up to 2.4 GeV. This resolves the Barrelet ambiguity in the original LASS analysis, and constrains the amount of inelasticity in $K\pi$ scattering, highlighting that this becomes significant beyond 1.8 GeV. This result needs to be checked by experimental information on the many inelastic channels, in particular $K\eta^\prime$ and $K\pi\pi\pi$. Our analysis provides a single representation for the $K\pi$ $S$-wave from threshold, controlled by Chiral Perturbation Theory, through the broad $\kappa$, $K_0^*(1430)$ and $K_0^*(1950)$ resonances. There is no arbitrary sum of Breit-Wigner forms and random backgrounds for real $K\pi$ masses. Rather the form provides a representation that can be translated to other processes with $K\pi$ interactions with their own coupling functions, while automatically maintaining consistency with the chiral dynamics near threshold, with the LASS data and the new results on $\eta_c$ decay.