$\eta \to \pi^{0} \gamma \gamma$ and $\gamma \gamma \to \pi^{0} \pi^{0}$ in $O(p^{6})$ chiral perturbation theory

Abstract

$\eta \rightarrow \pi^{0} \gamma \gamma$ and $\ggpipi$ are considered in $O(p^{6})$ chiral perturbation theory. In addition to the usual $\rho,\omega$ contributions, there are two $O(p^{6})$ operators (${\cal L}_{6,m}$) arising from explicit chiral symmetry breaking. Since only one of the two operators contributes to $\etapigg$, the coefficient of this operator ($\equiv d_3$) can be determined in two ways : (i) from the measured decay rate of $\eta \rightarrow \pi^{0} \gamma \gamma$, and (ii) by assuming the resonance saturations of the low energy coefficients in the $O(p^{6})$ chiral lagrangian. We find that two methods lead to vastly different values of $d_3$, which would indicate that either the measured decay rate for $\eta \rightarrow \pi^{0} \gamma \gamma$ is too large by a factor of $2 \sim 3$, or the resonance saturation assumptions do not work for $d_3$.