Classical and Quantum Results on Logarithmic Terms in the Soft Theorem in Four Dimensions

TL;DR

The paper analyzes logarithmic terms in soft theorems for four-dimensional scattering by combining classical scattering data with one-loop quantum amplitudes in a scalar QED–gravity setup. It identifies universal ln ω^{-1} contributions in soft photon and graviton factors, arising from long-range interactions and backreaction, and distinguishes two infrared-sensitive momentum regions that generate these logs. The authors develop a consistent framework for treating infrared divergences and momentum conservation, and they demonstrate a close classical-quantum correspondence, including extensions to general theories and mixed electromagnetic-gravitational interactions. The results generalize known soft-theorem structures and provide robust predictions for the log terms across a broad range of scattering processes.

Abstract

We explore the logarithmic terms in the soft theorem in four dimensions by analyzing classical scattering with generic incoming and outgoing states and one loop quantum scattering amplitudes. The classical and quantum results are consistent with each other. Although most of our analysis in quantum theory is carried out for one loop amplitudes in a theory of (charged) scalars interacting via gravitational and electromagnetic interactions, we expect the results to be valid more generally.

Figures (25)

• Figure 1: One loop contribution to $\Gamma^{(n,1)}$ involving internal photon line connecting two different legs. The thick lines represent scalar particles and the thin lines carrying the symbol $\gamma$ represent photons. There are other diagrams related to this by permutations of the external scalar particles.
• Figure 2: One loop contribution to $\Gamma^{(n,1)}$ involving internal photon line connecting two different points on the same leg. There are other diagrams related to this by permutations of the external scalar particles. In the last term the + on the scalar line represents a counterterm associated with mass renormalization that has to be adjusted to cancel the net contribution proportional to $1/(p_a.k)^2$.
• Figure 3: One loop contribution to $\Gamma^{(n)}$. There are other diagrams related to this by permutations of the external scalar particles.
• Figure 4: Diagrammatic representations of (\ref{['eward1']}) and (\ref{['eward2']}). The arrow on the photon line represents that the polarization of the photon is taken to be equal to the momentum entering the vertex. The circle denotes a simple vertex $-q_c$ with the polarization of the incoming photon stripped off.
• Figure 5: Sum of the first four diagrams in Fig. \ref{['figself']} with $\varepsilon$ replaced by $k$.