Different versions of soft-photon theorems exemplified at leading and next-to-leading terms for pion-pion and pion-proton scattering - Lebiedowicz, Piotr; Nachtmann, Otto; Szczurek, Antoni (Physical review, 2024)
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−⁢𝜋0⁢𝛾 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−⁢𝜋0⁢𝛾 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−⁢𝜋0⁢𝛾 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−⁢𝜋0⁢𝛾 with respect to 𝜔 for 𝜔 →0.