The $\pi\eta$ interaction and $a_0$ resonances in photon-photon scattering

We revisit the information on the two lightest $a_0$ resonances and $S$-wave $\pi\eta$ scattering that can be extracted from photon-photon scattering experiments. For this purpose we construct a model for the $S$-wave photon-photon amplitudes which satisfies analyticity properties, two-channel unitarity and obeys the soft photon as well as the soft pion constraints. The underlying I=1 hadronic $T$-matrix involves six phenomenological parameters and is able to account for two resonances below 1.5 GeV.We perform a combined fit of the $\gamma\gamma\to \pi\eta$ and $\gamma\gamma\to K_SK_S$ high statistics experimental data from the Belle collaboration. Minimisation of the $\chi^2$ is found to have two distinct solutions with approximately equal $\chi^2$. One of these exhibits a light and narrow excited $a_0$ resonance analogous to the one found in the Belle analysis. This however requires a peculiar coincidence between the $J=0$ and $J=2$ resonance effects which is likely to be unphysical. In both solutions the $a_0(980)$ resonance appears as a pole on the second Riemann sheet. The location of this pole in the physical solution is determined to be $m-i\Gamma/2=1000.7^{+12.9}_{-0.7} -i\,36.6^{+12.7}_{-2.6}$ MeV. The solutions are also compared to experimental data in the kinematical region of the decay $\eta\to\pi^0\gamma\gamma$. In this region an isospin violating contribution associated with $\pi^+\pi^-$ rescattering must be added for which we provide a dispersive evaluation.