Chiral corrections to $\pi^-\gamma\to 3\pi$ processes at low energies

We calculate in chiral perturbation theory the double-pion photoproduction processes $\pi^-\gamma \to \pi^- \pi^0\pi^0$ and $\pi^-\gamma\to \pi^+\pi^-\pi^-$ 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 $\pi^-\gamma \to\pi^-\pi^0\pi^0$ gets enhanced in the region $\sqrt{s}< 7m_\pi$ 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 $\pi^-\gamma\to \pi^+ \pi^-\pi^-$ remains almost unchanged in the region $\sqrt{s}< 6m_\pi$ 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 ($\pi^+\pi^- \to \pi^0\pi^0$ versus $\pi^-\pi^- \to \pi^-\pi^-$). We present also results for the complete set of two-pion invariant mass spectra. The predictions of chiral perturbation theory for the $\pi^-\gamma\to 3\pi$ 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 $\pi^-\gamma$ reactions with various final states.