The Reaction $\gamma\gamma\to\pi^0\pi^0$ in Generalized Chiral Perturbation Theory

The cross section for $\gamma\gamma\to\pi^0\pi^0$ and the pion polarizabilities are computed, within generalized chiral perturbation theory, in the full one loop approximation, {\it i.e.} up to and including order $O({p^5})$. The result depends on the parameter ${\alpha}_{\pi\pi}$ defining the tree level $\pi - \pi$ scattering amplitude and on an additional low energy constant. The latter is shown to be related by an exact sum-rule to the $e^+ e^-$ data. The parameter ${\alpha}_{\pi\pi}$ is related to the quark mass ratio $r = m_s /{\hat m}$ via the expansion of pseudoscalar meson masses. The generalized one loop $\gamma\gamma\to\pi^0\pi^0$ amplitude agrees with the experimental data in the threshold region provided $r= {m_s}/{\hat m}\lapprox 10$. Higher order corrections are estimated comparing our calculation with the dispersive approach.