Radiative corrections to the charged pion-pair production process {\boldmath${\pi }^{-}\gamma \to {\pi }^{+}{\pi }^{-}{\pi }^{-}$} at low energies

Abstract

We calculate the one-photon loop radiative corrections to the charged pion-pair production process ${\pi }^{-}\gamma \to {\pi }^{+}{\pi }^{-}{\pi }^{-}$. In the low-energy region this reaction is governed by the chiral pion-pion interaction. The pertinent set of 42 irreducible photon-loop diagrams is calculated by using the package FeynCalc. Electromagnetic counterterms with two independent low-energy constants ${\hat{k}}_1$ and ${\hat{k}}_2$ are included in order to remove the ultraviolet divergences generated by the photon-loops. Infrared finiteness of the virtual radiative corrections is achieved by including soft photon radiation below an energy cut-off $\lambda$. The purely electromagnetic interaction of the charged pions mediated by one-photon exchange is also taken into account. The radiative corrections to the total cross section (in the isospin limit) vary between $+10\%$ close to threshold and about $-1\%$ at a center-of-mass energy of $7m_{\pi }$. The largest contribution comes from the simple one-photon exchange. Radiative corrections to the ${\pi }^{+}{\pi }^{-}$ and ${\pi }^{-}{\pi }^{-}$ mass spectra are studied as well. The Coulomb singularity of the final-state interaction produces a kink in the dipion mass spectra. The virtual radiative corrections to elastic ${\pi }^{-}{\pi }^{-}$ scattering are derived additionally.

Comments: 19 pages, 17 figures, accepted for publication in Eur. Phys. J. A