Different versions of soft-photon theorems exemplified at leading and next-to-leading terms for pion-pion and pion-proton scattering

We investigate the photon emission in pion-pion and pion-proton scattering in the soft-photon limit where the photon energy ω →0 . The expansions of the π^-π⁰→π^-π⁰γ and the π^±p →π^±p γ amplitudes, satisfying the energy-momentum relations, to the orders ω^-1 and ω⁰ are derived. We show that these terms can be expressed completely in terms of the on-shell amplitudes for π^-π⁰→π^-π⁰ and π^±p →π^±p , respectively, and their partial derivatives with respect to s and t . The structure term which is nonsingular for ω →0 is determined to the order ω⁰ from the gauge-invariance constraint using the generalized Ward identities for pions and the proton. For the reaction π^-π⁰→π^-π⁰γ we discuss in detail the soft-photon theorems in the versions of both Low and Weinberg. We show that these two versions are different and must not be confounded. Weinberg's version gives the pole term of a Laurent expansion in ω of the amplitude for π^-π⁰→π^-π⁰γ around the phase-space point of zero radiation. Low's version gives an approximate expression for the above amplitude at a fixed phase-space point, corresponding to nonzero radiation. Clearly, the leading and next-to-leading terms in theses two approaches must be, and are indeed, different. We show their relation. We also discuss the expansions of differential cross sections for π^-π⁰→π^-π⁰γ with respect to ω for ω →0 .