Simplicity in the Structure of QED and Gravity Amplitudes

TL;DR

This work demonstrates a no-triangle property for one-loop multi-photon amplitudes in massless QED with $n\ge 8$ external photons, showing that all such amplitudes reduce to scalar box integrals. It achieves this via two complementary formalisms: a world-line (string-inspired) reduction that exposes unordered cancellations across external leg orderings, and a unitarity-based approach that connects large-$z$ behavior of tree amplitudes to the loop integral coefficients. The analysis establishes vanishing coefficients for triangles, bubbles, and rational terms beyond certain point counts, and clarifies the universal infrared structure, with the $n$-photon MHV amplitude constrained to particular box configurations. These results highlight deep simplifications in unordered gauge theories and point toward extensions to higher loops and gravity, potentially informing finiteness properties and dual descriptions.

Abstract

We investigate generic properties of one-loop amplitudes in unordered gauge theories in four dimensions. For such theories the organisation of amplitudes in manifestly crossing symmetric expressions poses restrictions on their structure and results in remarkable cancellations. We show that one-loop multi-photon amplitudes in QED with at least eight external photons are given only by scalar box integral functions. This QED `no-triangle' property is true for all helicity configurations and has similarities to the `no-triangle' property found in the case of maximal N = 8 supergravity. Results are derived both via a world-line formalism as well as using on-shell unitarity methods. We show that the simple structure of the loop amplitude originates from the extremely good BCFW scaling behaviour of the QED tree-amplitude.

Figures (7)

• Figure 1: The one-loop $n$-photon amplitude in QED is the sum over all permutations of ordered photon lines attached to a fermion loop.
• Figure 2: Tree-level $e^+e^-+n(\gamma)\to0$ Feynman diagram for the ordering $\sigma\in\mathfrak{S}_n$. All the momenta are assumed to be incoming. The helicity of the fermions is given by $h=\pm\frac{1}{2}$.
• Figure 3: Tree-level scalar Feynman diagram.
• Figure 4: The $e^-e^+\to 4\, \gamma$ tree-amplitude is composed by the Feynman diagrams built from (i) four three-point vertices, (ii) two three-point vertices and one four-point vertex, and (iii) two four-point vertices. The label of the external photon have to be symmetrically distributed over the photon lines.
• Figure 5: Representation of (a) the quadruple cut contributing to the coefficient $c_{4;K_1|K_2|K_3|K_4}$, (b) the triple cut contributing to the coefficient $c_{4;K_1|K_2|K_3}$ and (c) the double cut contributing to the coefficient $c_{2;K_1|K_2}$ of the one-loop amplitude.