γγ→π0π0contribution to the processe+e−→e+e−π0π0

We consider the $O({p}^{4})$ predictions of chiral perturbation theory ($\ensuremath{\chi}\mathrm{PT}$) in the process $\ensuremath{\gamma}\ensuremath{\gamma}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{0}{\ensuremath{\pi}}^{0}$, together with the study of the existing experimental measurements and the planned new ones. At the Frascati $\ensuremath{\Phi}$ factory $\mathrm{DA}\ensuremath{\Phi}\mathrm{NE}$ the rate of the events ${e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}{e}^{+}{e}^{\ensuremath{-}}{\ensuremath{\pi}}^{0}{\ensuremath{\pi}}^{0}$ is large enough to allow for a measurement of the cross section with a good statistical sensitivity. However, a quantitative analysis of the commonly used approximation of considering only real photons for the detection of the $\ensuremath{\gamma}\ensuremath{\gamma}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{0}{\ensuremath{\pi}}^{0}$ signal is still lacking. This motivates us to compute the cross section in $\ensuremath{\chi}\mathrm{PT}$ for the production of ${\ensuremath{\pi}}^{0}$ pairs due to off-shell photons. A measurement of the azimuthal correlations would test the higher order $\ensuremath{\chi}\mathrm{PT}$ corrections independently from the measurement of the cross section. We approximate these corrections by including the contribution of the low-lying vector resonances in the scattering amplitude.