Subleading soft dressings of asymptotic states in QED and perturbative quantum gravity

TL;DR

This work addresses infrared divergences in QED and perturbative gravity by constructing subleading Faddeev–Kulish (FK) dressings from asymptotic symmetries, valid to first order in the coupling and subleading in the soft expansion. By enforcing charge conservation for the subleading soft charges associated with superrotation, it builds gravitational and electromagnetic dressings ($W_g$ and $W_e$) that render soft emissions of photons and gravitons trivial in dressed amplitudes, and proves that FK amplitudes reproduce the IR-finite part of the conventional amplitudes, yielding cross sections equivalent to Bloch–Nordsieck results. The paper also demonstrates a gravitational Wilson-line–like equivalence of the dressings and shows that external soft insertions do not contribute at leading or subleading order, reinforcing the correlation between soft and hard sectors in FK states. While loop-corrected subleading soft theorems are not fully incorporated, the authors argue that their impact on dressed amplitudes remains negligible at this order, preserving the IR-finite equivalence and the expected inclusive cross sections. The results illuminate how asymptotic symmetries organize the infrared structure of scattering and provide a controlled framework for analyzing soft radiation in a gauge- and gravity-theoretic setting.

Abstract

We construct Faddeev-Kulish states in QED and perturbative quantum gravity to subleading order in the soft momentum expansion and to first order in the coupling constant, using the charge conservation formula of asymptotic symmetries associated with the tree-level subleading soft theorems. We demonstrate that the emission and absorption of soft photons/gravitons in dressed amplitudes vanish. The fact that no additional soft radiation may be added to a dressed amplitude supports the claim that, in the dressed state formalism, the soft and hard sectors of scattering processes are correlated. We also show that the dressed virtual amplitudes are equivalent to the infrared-finite part of the traditional amplitudes constructed using Fock states. Since there is no real soft radiation in the asymptotic Hilbert space, the dressed state formalism gives the same cross sections as the Bloch-Nordsieck method.

Figures (4)

• Figure 1: Different contributions to the emission amplitude of a soft graviton.
• Figure 2: We consider the simple case of $1\to 1$ gravitational potential scattering, where the incoming and outgoing momenta are $\mathbf p_i$ and $\mathbf p_f$, respectively. The figures illustrate different contributions to the FK amplitude of this process. Blob represents the internal diagram, including the gravitational potential.
• Figure 3: Contributions to the amplitude due to graviton exchange between a dressing and either an external leg or the internal diagram.
• Figure 4: Graviton exchanges that do not involve dressings. The soft graviton loops exponentiate and factor out Weinberg:1965nx.