Revisiting $\gamma^*\to\gamma\pi^0\eta$ near the $\phi(1020)$ using analyticity and the left-cut structure

Amplitudes of the form $\gamma^*(q^2)\to \gamma P_1P_2$ appear as sub-processes in the computation of the muon $g-2$. We test a proposed theoretical modelling against very precise experimental measurements by the KLOE collaboration at $q^2=m^2_\phi$. Starting from an exact parameter-free dispersive representation for the $S$-wave satisfying QCD asymptotic constraints and Low's soft photon theorem we derive, in an effective theory spirit, a two-channel Omn\`es integral representation which involves two subtraction parameters. The discontinuities along the left-hand cuts which, for timelike virtualities, extend both on the real axis and into the complex plane are saturated by the contributions from the light vector mesons. In the case of $P_1P_2=\pi\eta$, we show that a very good fit of the KLOE data can be achieved with two real parameters, using a $T$-matrix previously determined from $\gamma\gamma$ scattering data. This indicates a good compatibility between the two data sets and confirms the validity of the $T$-matrix. The resulting amplitude is also found to be compatible with the chiral soft pion theorem. Applications to the $I=1$ scalar form factors and to the $a_0(980)$ resonance complex pole are presented.