  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.
