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Amplitudes of the form $\gamma^*(q^2)\to \gamma P_1P_2$ appear assub-processes in the computation of the muon $g-2$. We test a proposedtheoretical modelling against very precise experimental measurements by theKLOE collaboration at $q^2=m^2_\phi$. Starting from an exact parameter-freedispersive representation for the $S$-wave satisfying QCD asymptoticconstraints and Low's soft photon theorem we derive, in an effective theoryspirit, a two-channel Omn\`es integral representation which involves twosubtraction parameters. The discontinuities along the left-hand cuts which, fortimelike virtualities, extend both on the real axis and into the complex planeare 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 achievedwith two real parameters, using a $T$-matrix previously determined from$\gamma\gamma$ scattering data. This indicates a good compatibility between thetwo data sets and confirms the validity of the $T$-matrix. The resultingamplitude is also found to be compatible with the chiral soft pion theorem.Applications to the $I=1$ scalar form factors and to the $a_0(980)$ resonancecomplex pole are presented.

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