Soft-photon theorem for pion-proton elastic scattering revisited

Abstract

We discuss the reactions $\pi p\to \pi p$ and $\pi p\to \pi p\gamma$ 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 ${\pi }^{\pm }p\to {\pi }^{\pm }p$ 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 $\omega \to 0$. The Laurent expansion in $\omega$ of the ${\pi }^{\pm }p\to {\pi }^{\pm }p\gamma$ amplitudes is considered and the terms of the orders ${\omega }^{-1}$ and ${\omega }^0$ 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 $\propto {\omega }^{-1}$ in the amplitudes corresponds to Weinberg's soft-photon theorem and is well known from the literature. We derive the next-to-leading term $\propto {\omega }^0$ using only rigorous methods of QFT. We give the relation of the Laurent series for ${\pi }^{0}p\to {\pi }^{0}p\gamma$ and Low's soft-photon theorem. Our formulas for the amplitudes in the limit $\omega \to 0$ are valid for photon momentum $k$ satisfying $k^2⩾0$, $k^0=\omega ⩾0$, that is, for both real and virtual photons. Here we consider a limit where with $\omega \to 0$ we have also $k^2\to 0$. We discuss the behavior of the corresponding cross-sections for ${\pi }^{-}p\to {\pi }^{-}p\gamma$ with respect to $\omega$ for $\omega \to 0$. We consider cross sections for unpolarized as well as polarized protons in the initial and final states.

arXiv comment36 pages, 5 figures, v3 accepted for publication in PRD

References in corpus (9)

• Low's Subleading Soft Theorem as a Symmetry of QED
• Low-Energy Behavior of Gluons and Gravitons from Gauge Invariance
• A Model for Soft High-Energy Scattering: Tensor Pomeron and Vector Odderon
• Soft Photon Theorem for High Energy Amplitudes in Yukawa and Scalar Theories
• The LBK theorem to all orders
• High-energy $\pi \pi$ scattering without and with photon radiation
• Central exclusive diffractive production of a single photon in high-energy proton-proton collisions within the tensor-Pomeron approach
• Different versions of soft-photon theorems exemplified at leading and next-to-leading terms for pion-pion and pion-proton scattering
• Exclusive diffractive bremsstrahlung of one and two photons at forward rapidities: Possibilities for experimental studies in $pp$ collisions at the LHC