Chiral corrections to Π-γ→3Π processes at low energies

We calculate in chiral perturbation theory the double-pion photoproduction processes Π^-γ→Π^-Π⁰Π⁰ and Π^-γ→Π+Π^-Π^- at low energies. At leading order these reactions are governed by the chiral pion-pion interaction. The next-to-leading order corrections arise from pion-loop diagrams and chiral-invariant counterterms involving the low-energy constants ℓ̄ 1, ℓ̄ 2, ℓ̄ 3 and ℓ̄ 4. The pertinent production amplitudes A1 and A2 depending on five kinematical variables are given in closed analytical form. We find that the total cross section for neutral pion-pair production Π^-γ→Π^-Π⁰Π⁰ gets enhanced in the region s<7mπ by a factor 1.5-1.8 by the next-to-leading order corrections. In contrast to this behavior the total cross section for charged pion-pair production Π^-γ→Π+Π^-Π^- remains almost unchanged in the region s<6mπ in comparison to its tree-level result. Although the dynamics of the pion-pair production reactions is much richer, this observed pattern can be understood from the different influence of the chiral corrections on the pion-pion final state interaction (Π+Π^-→Π⁰Π⁰ versus Π^-Π^-→Π^-Π^-). We present also results for the complete set of two-pion invariant mass spectra. The predictions of chiral perturbation theory for the Π^-γ→3Π processes can be tested by the COMPASS experiment which uses Primakoff scattering of high-energy pions in the Coulomb field of a heavy nucleus to extract cross sections for Π^-γ reactions with various final states.