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

Abstract

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.

Comments: 19 pages, 14 figures, submitted to Nuclear Physics A