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Subleading soft radiation during scattering of dressed states in QED

Stavros Christodoulou, Nicolaos Toumbas

TL;DR

The paper addresses infrared divergences in QED by employing Faddeev–Kulish dressed states that attach soft-photon clouds up to a scale $E_d$ to charged particles. It extends the dressing to subleading order in the soft momentum and uses the subleading soft photon theorem to compute tree-level amplitudes for three representative processes, showing elastic FK amplitudes are infrared finite and that emission of additional soft photons with $\omega<E_d$ is suppressed. The results establish equivalence between dressed-state amplitudes and the infrared-finite part of traditional amplitudes, effectively reproducing Bloch–Nordsieck inclusive rates without extra soft radiation at this order. This provides a concrete, infrared-safe S-matrix in QED and sets the stage for loop-level analyses and extensions to gravitational theories.

Abstract

We study soft photon emission during scattering of Faddeev-Kulish charged states in QED, at leading order in perturbation theory. The charged asymptotic particles are accompanied by clouds of an infinite number of soft photons of energy less than a characteristic infrared scale $E_d$. When the corresponding ``dressing'' functions are suitably corrected to subleading order in the soft momentum expansion, as advocated in recent work by Choi and Akhoury, we show explicitly that the emission of additional radiative soft photons with energy less than $E_d$ is completely suppressed. Moreover, the dressing renders the elastic amplitudes infrared-finite, order by order in perturbation theory, regulating the infrared divergences due to virtual soft photons, at the energy scale $E_d$. Therefore, the characteristic energy scale of the soft photons in the clouds provides an effective infrared cutoff, allowing for the formulation of an infrared finite S-matrix.

Subleading soft radiation during scattering of dressed states in QED

TL;DR

The paper addresses infrared divergences in QED by employing Faddeev–Kulish dressed states that attach soft-photon clouds up to a scale to charged particles. It extends the dressing to subleading order in the soft momentum and uses the subleading soft photon theorem to compute tree-level amplitudes for three representative processes, showing elastic FK amplitudes are infrared finite and that emission of additional soft photons with is suppressed. The results establish equivalence between dressed-state amplitudes and the infrared-finite part of traditional amplitudes, effectively reproducing Bloch–Nordsieck inclusive rates without extra soft radiation at this order. This provides a concrete, infrared-safe S-matrix in QED and sets the stage for loop-level analyses and extensions to gravitational theories.

Abstract

We study soft photon emission during scattering of Faddeev-Kulish charged states in QED, at leading order in perturbation theory. The charged asymptotic particles are accompanied by clouds of an infinite number of soft photons of energy less than a characteristic infrared scale . When the corresponding ``dressing'' functions are suitably corrected to subleading order in the soft momentum expansion, as advocated in recent work by Choi and Akhoury, we show explicitly that the emission of additional radiative soft photons with energy less than is completely suppressed. Moreover, the dressing renders the elastic amplitudes infrared-finite, order by order in perturbation theory, regulating the infrared divergences due to virtual soft photons, at the energy scale . Therefore, the characteristic energy scale of the soft photons in the clouds provides an effective infrared cutoff, allowing for the formulation of an infrared finite S-matrix.

Paper Structure

This paper contains 15 sections, 100 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Infrared structure of QED and the dressed state formalism
  3. Construction of Faddeev-Kulish states
  4. Formulation of the Faddeev-Kulish S-matrix
  5. Elastic FK amplitudes
  6. Soft radiation and FK amplitudes
  7. Dressed states beyond leading soft order
  8. Application of subleading soft theorem to QED scattering
  9. Subleading corrections to Faddeev-Kulish dressings
  10. Equivalence of dressed and traditional amplitudes
  11. Soft radiation suppression at the tree level
  12. Conclusions
  13. Notation and Conventions
  14. Review of the subleading soft photon theorem
  15. Calculation of TEXT

Figures (3)

  • Figure 1: Tree-level Feynman diagrams for the elastic processes: (a) TEXT, (b) & (c) TEXT, and (d) & (e) TEXT
  • Figure 5: Representatives of the diagrams depicting the elastic TEXT process, in which both interacting particles are accompanied by clouds of soft photons. Diagram (a) corresponds to self-interacting contributions. Diagrams (b), (c) and (d) correspond to non-interacting contributions, while diagrams (e) and (f) illustrate interacting contributions. In all diagrams, a soft photon (blue line) is either emitted and reabsorbed by the same cloud, or exchanged between two clouds, or exchanged between a cloud and an interacting particle.
  • Figure 6: Diagrams depicting soft emission during a generic QED process. The interaction has been denoted with a blob. The first two diagrams describe soft emission from outgoing and incoming fermions, respectively, whereas the last diagram corresponds to soft emission from virtual fermions.