Two photons into π0 π0

We perform a theoretical study based on dispersion relations of the reaction gamma gamma -> pi(0)pi(0) emphasizing the low-energy region. We discuss how the f(0)(980) signal emerges in gamma gamma -> pi(0)pi(0) within the dispersive approach and how this fixes to a large extent the phase of the isoscalar S-wave gamma gamma -> pi pi amplitude above the K ($) over bar threshold. This allows us to make sharper predictions for the cross-section at lower energies and our results could then be used to distinguish between different pi pi isoscalar S-wave parameterizations with the advent of new precise data on gamma gamma -> pi(0)pi(0). We compare our dispersive approach with an updated calculation employing the unitary chiral perturbation theory (U chi PT). We also pay special attention to the role played by the a-resonance in -y-y -4 7r7r and calculate its coupling and width to gamma gamma, for which we obtain Gamma(sigma -> gamma gamma) = (1.68 +/- 0.15) keV.