Soft-photon theorem for pion-proton elastic scattering revisited

We discuss the reactions pi p -> pi p and pi p -> pi p gamma 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 pi(+/-) p -> pi(+/-) 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 omega -> 0. The Laurent expansion in omega of the pi(+/-) p -> pi(+/-) p gamma amplitudes is considered and the terms of the orders omega(-1) and omega(0) 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 proportional to omega(-1) in the amplitudes corresponds to Weinberg's soft-photon theorem and is well known from the literature. We derive the nextto-leading term proportional to omega(0) using only rigorous methods of QFT. We give the relation of the Laurent series for pi(0) p -> pi(0) p gamma and Low's soft-photon theorem. Our formulas for the amplitudes in the limit omega -> 0 are valid for photon momentum k satisfying k(2) >= 0, k(0) = omega >= 0, that is, for both real and virtual photons. Here we consider a limit where with omega -> 0 we have also k(2) -> 0. We discuss the behavior of the corresponding cross sections for pi(-) p -* pi(-) p gamma with respect to omega for omega -> 0. We consider cross sections for unpolarized as well as polarized protons in the initial and final states.