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We evaluate the contributions of $(\pi^0\pi^0, \pi^0\eta, \eta\eta)\gamma$exclusive channels to the dispersion integral of the leading order HVP of themuon anomalous magnetic moment. These channels are included in some way inprevious evaluations of the $\pi^0\omega, \eta\omega$ and $\eta\phi$contributions to $a_{\mu}^{\rm had, LO}$, where the vector resonances (decayinginto $\pi^0/\eta+ \gamma$) are assumed to be on-shell. Since the separation ofresonance and background contributions in a given observable is, in general, amodel-dependent procedure, here we use pseudoscalar mesons and the photon asthe $in$ and $out$ states of the $e^+e^- \to (\pi^0\pi^0, \pi^0\eta,\eta\eta)\gamma$ $S$-matrix, such that the cross section contains theinterferences among different contributing to the amplitudes. We find $a^{\rmhad, LO}_{\mu}(P^0_1P^0_2\gamma)=(1.13\pm 0.13 ) \times 10^{-10}$, whereuncertainties stem mainly from vector meson dominance model parameters.Improved experimental studies of these exclusive channels in the whole rangebelow 2 GeV would reduce model-dependency.

Contribution of exclusive $(\pi^0\pi^0, \pi^0\eta, \eta\eta)\gamma$ channels to the leading order HVP of the muon $g-2$