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We calculate in chiral perturbation theory the double-pion photoproductionprocesses $\pi^-\gamma \to \pi^- \pi^0\pi^0$ and $\pi^-\gamma\to\pi^+\pi^-\pi^-$ at low energies. At leading order these reactions are governedby the chiral pion-pion interaction. The next-to-leading order correctionsarise from pion-loop diagrams and chiral-invariant counterterms involving thelow-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 fivekinematical variables are given in closed analytical form. We find that thetotal 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 thisbehavior 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 thedynamics of the pion-pair production reactions is much richer, this observedpattern can be understood from the different influence of the chiralcorrections on the pion-pion final state interaction ($\pi^+\pi^- \to\pi^0\pi^0$ versus $\pi^-\pi^- \to \pi^-\pi^-$). We present also results forthe complete set of two-pion invariant mass spectra. The predictions of chiralperturbation theory for the $\pi^-\gamma\to 3\pi$ processes can be tested bythe COMPASS experiment which uses Primakoff scattering of high-energy pions inthe Coulomb field of a heavy nucleus to extract cross sections for$\pi^-\gamma$ reactions with various final states.

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