Contribution of exclusive $(π^0π^0, π^0η, ηη)γ$ channels to the leading order HVP of the muon $g-2$

Abstract

We evaluate the contributions of $(π^0π^0, π^0η, ηη)γ$ 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 $π^0ω, ηω$ and $ηϕ$ contributions to $a_μ^{\rm had, LO}$, where the vector resonances (decaying into $π^0/η+ γ$) 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 $e^+e^- \to (π^0π^0, π^0η, ηη)γ$ $S$-matrix, such that the cross section contains the interferences among different contributing to the amplitudes. We find $a^{\rm had, LO}_μ(P^0_1P^0_2γ)=(1.13\pm 0.13 ) \times 10^{-10}$, 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.