Production of Pion Pairs by a Photon in the Coulomb Field of a Nucleus

We have studied the possibility of measuring the $\pi -\pi$ interaction through electromagnetic pion-pair production in the Coulomb field of a nucleus. The Mandelstam representation was set up for the process $\gamma \left(\mathrm{virtual}\right)+\gamma \to \pi +\pi$, and partial-wave dispersion relations were derived. Solutions for the ensuing integral equations were obtained within the approximation of retaining only the one-pion exchange contribution to the unphysical cut. It is sufficient to consider only the $\pi -\pi$ $S$-wave interaction, for which a current estimate with a virtual $I=0\mathrm{}$ state near threshold is utilized. Our results indicate that the effects of the $\pi -\pi$ final-state interaction on the Coulomb process may be observed in the presence of nuclear pair production, provided that the momentum transfers to the nucleus are sufficiently small, say smaller than 70 Mev/c in the illustrative case of an incident photon of 6 Bev.