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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 πœ” β†’0. The expansions of the πœ‹βˆ’β’πœ‹0 β†’πœ‹βˆ’β’πœ‹0⁒𝛾 and the πœ‹Β±β’π‘ β†’πœ‹Β±β’π‘β’π›Ύ amplitudes, satisfying the energy-momentum relations, to the orders πœ”βˆ’1 and πœ”0 are derived. We show that these t...

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Bibliographic Details
Main Authors: Lebiedowicz, Piotr (Author) , Nachtmann, Otto (Author) , Szczurek, Antoni (Author)
Format: Article (Journal)
Language:English
Published: 29 May 2024
In: Physical review
Year: 2024, Volume: 109, Issue: 9, Pages: 094042-1-094042-18
ISSN:2470-0029
DOI:10.1103/PhysRevD.109.094042
Online Access:Verlag, kostenfrei, Volltext: https://doi.org/10.1103/PhysRevD.109.094042
Verlag, kostenfrei, Volltext: https://link.aps.org/doi/10.1103/PhysRevD.109.094042
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Author Notes:Piotr Lebiedowicz, Otto Nachtmann, and Antoni Szczurek
Description
Summary:We investigate the photon emission in pion-pion and pion-proton scattering in the soft-photon limit where the photon energy πœ” β†’0. The expansions of the πœ‹βˆ’β’πœ‹0 β†’πœ‹βˆ’β’πœ‹0⁒𝛾 and the πœ‹Β±β’π‘ β†’πœ‹Β±β’π‘β’π›Ύ amplitudes, satisfying the energy-momentum relations, to the orders πœ”βˆ’1 and πœ”0 are derived. We show that these terms can be expressed completely in terms of the on-shell amplitudes for πœ‹βˆ’β’πœ‹0 β†’πœ‹βˆ’β’πœ‹0 and πœ‹Β±β’π‘ β†’πœ‹Β±β’π‘, respectively, and their partial derivatives with respect to 𝑠 and 𝑑. The structure term which is nonsingular for πœ” β†’0 is determined to the order πœ”0 from the gauge-invariance constraint using the generalized Ward identities for pions and the proton. For the reaction πœ‹βˆ’β’πœ‹0 β†’πœ‹βˆ’β’πœ‹0⁒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 πœ” of the amplitude for πœ‹βˆ’β’πœ‹0 β†’πœ‹βˆ’β’πœ‹0⁒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œ‹βˆ’β’πœ‹0 β†’πœ‹βˆ’β’πœ‹0⁒𝛾 with respect to πœ” for πœ” β†’0.
Item Description:Gesehen am 23.01.2025
Physical Description:Online Resource
ISSN:2470-0029
DOI:10.1103/PhysRevD.109.094042

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