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±𝑝𝛾 amplitudes, satisfying the energy-momentum relations, to the orders 𝜔−1 and 𝜔0 are derived. We show that these t...
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Dokumenttyp: | Article (Journal) |
| Sprache: | Englisch |
| Veröffentlicht: |
29 May 2024
|
| In: |
Physical review
Year: 2024, Jahrgang: 109, Heft: 9, Pages: 094042-1-094042-18 |
| ISSN: | 2470-0029 |
| DOI: | 10.1103/PhysRevD.109.094042 |
| Online-Zugang: | Verlag, kostenfrei, Volltext: https://doi.org/10.1103/PhysRevD.109.094042 Verlag, kostenfrei, Volltext: https://link.aps.org/doi/10.1103/PhysRevD.109.094042 
|
| Verfasserangaben: | Piotr Lebiedowicz, Otto Nachtmann, and Antoni Szczurek |
| Zusammenfassung: | 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±𝑝𝛾 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. |
|---|---|
| Beschreibung: | Gesehen am 23.01.2025 |
| Beschreibung: | Online Resource |
| ISSN: | 2470-0029 |
| DOI: | 10.1103/PhysRevD.109.094042 |