Soft-photon theorem for pion-proton elastic scattering revisited

We discuss the reactions πp→πp and πp→πpγ from a general quantum field theory (QFT) point of view, describing these reactions in QCD and lowest relevant order of electromagnetism. We consider the pion-proton elastic scattering both off shell and on shell. The on-shell amplitudes for π±p→π±p scattering are described by two invariant amplitudes, while the off-shell amplitudes contain eight invariant amplitudes. We study the photon emission amplitudes in the soft-photon limit where the center-of-mass photon energy ω→0. The Laurent expansion in ω of the π±p→π±pγ amplitudes is considered and the terms of the orders ω−1 and ω0 are derived. These terms can be expressed by the on-shell invariant amplitudes and their partial derivatives with respect to s and t. The pole term ∝ω−1 in the amplitudes corresponds to Weinberg’s soft-photon theorem and is well known from the literature. We derive the next-to-leading term ∝ω0 using only rigorous methods of QFT. We give the relation of the Laurent series for π0p→π0pγ and Low’s soft-photon theorem. Our formulas for the amplitudes in the limit ω→0 are valid for photon momentum k satisfying k2≥0, k0=ω≥0, that is, for both real and virtual photons. Here we consider a limit where with ω→0 we have also k2→0. We discuss the behavior of the corresponding cross sections for π−p→π−pγ with respect to ω for ω→0. We consider cross sections for unpolarized as well as polarized protons in the initial and final states. Published by the American Physical Society 2024