Soft-photon theorem for pion-proton elastic scattering revisited

We discuss the reactions $\pi p \to \pi p$ and $\pi p \to \pi p \gamma$ 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 $\pi^{\pm} p \to \pi^{\pm} p$ 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 c.m. photon energy $\omega \to 0$. The Laurent expansion in $\omega$ of the $\pi^{\pm} p \to \pi^{\pm} p \gamma$ 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 $\propto \omega^{-1}$ in the amplitudes corresponds to Weinberg's soft-photon theorem and is well known from the literature. We derive the next-to-leading term $\propto \omega^{0}$ using only rigorous methods of QFT. We give the relation of the Laurent series for $\pi^{0} p \to \pi^{0} p \gamma$ and Low's soft-photon theorem. Our formulas for the amplitudes in the limit $\omega \to 0$ are valid for photon momentum $k$ satisfying $k^{2} \geqslant 0$, $k^{0} = \omega \geqslant 0$, that is, for both real and virtual photons. Here we consider a limit where with $\omega \to 0$ we have also $k^{2} \to 0$. We discuss the behavior of the corresponding cross-sections for $\pi^{-} p \to \pi^{-} p \gamma$ with respect to $\omega$ for $\omega \to 0$. We consider cross sections for unpolarized as well as polarized protons in the initial and final states.