A note on the GG-->PI 0 PI 0 reaction in the 1/N expansion of ChPT

In this work we present the results of a complete calculation of the $\gamma \gamma\rightarrow \pi^0\pi^0$ amplitude to leading order in the large $N$ approximation ($N$ being the number of Goldstone bosons) up to order $m^2_{\pi}/F^2$. The amplitude turns to be proportional to that of $\pi^+\pi^-\rightarrow \pi^0\pi^0$. In spite of the fact that this factorization property cannot hold in general (as it was recently pointed out by Morgan and Penington), it appears here since in the large $N$ limit only the $I=J=0$ channel contributes to the $\gamma \gamma\rightarrow \pi^0\pi^0$ reaction. Moreover it seems to be a reasonable approximation in this case since it is possible to reproduce, as a prediction, the experimental data starting from a one-parameter fit of the $\pi\pi$ scattering data.