Radiative corrections to the charged pion-pair production process π ⁻ γ → π ⁺ π ⁻ π ⁻ at low energies

Abstract

We calculate the one-photon loop radiative corrections to the charged pion-pair production process π ⁻ γ → π ⁺ π ⁻ π ⁻. 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 Λ. 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 _π . The largest contribution comes from the simple one-photon exchange. Radiative corrections to the π ⁺ π ⁻ and π ⁻ π ⁻ 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 π ⁻ π ⁻ scattering are derived additionally.