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Chiral corrections to $π^-γ\to 3π$ processes at low energies

We calculate in chiral perturbation theory the double-pion photoproduction processes $π^-γ\to π^- π^0π^0$ and $π^-γ\to π^+π^-π^-$ 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 $\bar\ell_1,\, \bar\ell_2,\, \bar\ell_3$ and $\bar\ell_4$. The pertinent production amplitudes $A_1$ and $A_2$ depending on five kinematical variables are given in closed analytical form. We find that the total cross section for neutral pion-pair production $π^-γ\toπ^-π^0π^0$ gets enhanced in the region $\sqrt{s}< 7m_π$ 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 $π^-γ\to π^+ π^-π^-$ remains almost unchanged in the region $\sqrt{s}< 6m_π$ 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 ($π^+π^- \to π^0π^0$ versus $π^-π^- \to π^-π^-$). We present also results for the complete set of two-pion invariant mass spectra. The predictions of chiral perturbation theory for the $π^-γ\to 3π$ 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.