Soft-photon theorem for pion-proton elastic scattering revisited
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
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2023
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| author | Lebiedowicz, Piotr Nachtmann, Otto Szczurek, Antoni |
| author_facet | Lebiedowicz, Piotr Nachtmann, Otto Szczurek, Antoni |
| contents | We discuss the reactions $πp \to πp$ and $πp \to πp γ$ 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 $π^{\pm} p \to π^{\pm} p$ 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 $ω\to 0$. The Laurent expansion in $ω$ of the $π^{\pm} p \to π^{\pm} p γ$ amplitudes is considered and the terms of the orders $ω^{-1}$ and $ω^{0}$ 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 $\propto ω^{-1}$ in the amplitudes corresponds to Weinberg's soft-photon theorem and is well known from the literature. We derive the next-to-leading term $\propto ω^{0}$ using only rigorous methods of QFT. We give the relation of the Laurent series for $π^{0} p \to π^{0} p γ$ and Low's soft-photon theorem. Our formulas for the amplitudes in the limit $ω\to 0$ are valid for photon momentum $k$ satisfying $k^{2} \geqslant 0$, $k^{0} = ω\geqslant 0$, that is, for both real and virtual photons. Here we consider a limit where with $ω\to 0$ we have also $k^{2} \to 0$. We discuss the behavior of the corresponding cross-sections for $π^{-} p \to π^{-} p γ$ with respect to $ω$ for $ω\to 0$. We consider cross sections for unpolarized as well as polarized protons in the initial and final states. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_12673 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Soft-photon theorem for pion-proton elastic scattering revisited Lebiedowicz, Piotr Nachtmann, Otto Szczurek, Antoni High Energy Physics - Phenomenology High Energy Physics - Theory We discuss the reactions $πp \to πp$ and $πp \to πp γ$ 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 $π^{\pm} p \to π^{\pm} p$ 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 $ω\to 0$. The Laurent expansion in $ω$ of the $π^{\pm} p \to π^{\pm} p γ$ amplitudes is considered and the terms of the orders $ω^{-1}$ and $ω^{0}$ 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 $\propto ω^{-1}$ in the amplitudes corresponds to Weinberg's soft-photon theorem and is well known from the literature. We derive the next-to-leading term $\propto ω^{0}$ using only rigorous methods of QFT. We give the relation of the Laurent series for $π^{0} p \to π^{0} p γ$ and Low's soft-photon theorem. Our formulas for the amplitudes in the limit $ω\to 0$ are valid for photon momentum $k$ satisfying $k^{2} \geqslant 0$, $k^{0} = ω\geqslant 0$, that is, for both real and virtual photons. Here we consider a limit where with $ω\to 0$ we have also $k^{2} \to 0$. We discuss the behavior of the corresponding cross-sections for $π^{-} p \to π^{-} p γ$ with respect to $ω$ for $ω\to 0$. We consider cross sections for unpolarized as well as polarized protons in the initial and final states. |
| title | Soft-photon theorem for pion-proton elastic scattering revisited |
| topic | High Energy Physics - Phenomenology High Energy Physics - Theory |
| url | https://arxiv.org/abs/2307.12673 |