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

Abstract

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 the muon anomalous magnetic moment. These channels are included in some way in previous evaluations of the ${\pi }^0\omega ,\eta \omega$ and $\eta \varphi$ contributions to $a_{\mu }^{\mathrm{had},\mathrm{LO}}$, where the vector resonances (decaying into ${\pi }^0/\eta +\gamma$) are assumed to be on-shell. Since the separation of resonance and background contributions in a given observable is, in general, a model-dependent procedure, here we use pseudoscalar mesons and the photon as the $in$ 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 the interferences among different contributing to the amplitudes. We find $a_{\mu }^{\mathrm{had},\mathrm{LO}}(P_1^0{P}_2^0\gamma )=(1.13\pm 0.13)\times 10^{-10}$, where uncertainties stem mainly from vector meson dominance model parameters. Improved experimental studies of these exclusive channels in the whole range below 2 GeV would reduce model-dependency.