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

Understand

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.