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

We revisit the information on the two lightest $$a_0$$ a 0 resonances and S -wave $$\pi \eta $$ π η 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 $$\gamma \gamma \rightarrow \pi \eta $$ γ γ → π η and $$\gamma \gamma \rightarrow K_SK_S$$ γ γ → K S K S high statistics experimental data from the Belle collaboration. Minimisation of the $$\chi ^2$$ χ 2 is found to have two distinct solutions with approximately equal $$\chi ^2$$ χ 2 . One of these exhibits a light and narrow excited $$a_0$$ a 0 resonance analogous to the one found in the Belle analysis. This however requires a peculiar coincidence between the $$J=0$$ J = 0 and $$J=2$$ J = 2 resonance effects which is likely to be unphysical. In both solutions the $$a_0(980)$$ 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 $$m-i\varGamma /2=1000.7^{+12.9}_{-0.7} -i\,36.6^{+12.7}_{-2.6}$$ m - i Γ / 2 = 1000 . 7 - 0.7 + 12.9 - i 36 . 6 - 2.6 + 12.7 MeV. The solutions are also compared to experimental data in the kinematical region of the decay $$\eta \rightarrow \pi ^0\gamma \gamma $$ η → π 0 γ γ . In this region an isospin violating contribution associated with $${\pi ^+}{\pi ^-}$$ π + π - rescattering must be added for which we provide a dispersive evaluation.