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

We calculate the one-photon loop radiative corrections to the charged pion-pair production process $π^-γ\toπ^+π^-π^-$. 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 $\widehat k_1$ and $\widehat k_2$ 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.