Six-Photon Amplitudes

TL;DR

The paper addresses the one-loop six-photon (light-by-light) amplitudes in QED, deriving complete analytic results for the two non-vanishing helicity configurations $A_6(--++++)$ and $A_6(-+-+-+)$. It employs two complementary methods—form factor decomposition and unitarity-based multiple cuts—to extract the coefficients of known one-loop master integrals, yielding compact analytic expressions. The authors confirm cut-constructibility, show that all rational parts vanish, and cross-check against independent numerical results (Nagy and Soper), validating the approach. The work demonstrates the applicability of these methods to non-trivial multi-leg amplitudes and lays groundwork for extending to massive particles and other processes relevant to collider phenomenology.

Abstract

We present analytical results for all six-photon helicity amplitudes. For the computation of this loop induced process two recently developed methods, based on form factor decomposition and on multiple cuts, have been used. We obtain compact results, demonstrating the applicability of both methods to one-loop amplitudes relevant to precision collider phenomenology.

Figures (3)

• Figure 1: Quadruple-Cuts, $\hat{d}_1^{(1)}$ (left) and $\hat{d}_{2B}^{(1)}$ (right). Reverse internal-helicity counterparts understood.
• Figure 2: From left to right: quadruple-cuts, $\hat{d}_{2A}^{(1)}$ and $\hat{d}_{1}^{(1)}$, and triple-cut $\hat{c}_{3m}^{(1)}$. Reverse internal-helicity counterparts understood.
• Figure 3: The modulus of the normalised six-photon amplitudes $s\,|{A}_6(++----)|/\alpha^3$ and $s\,|{A}_6(+--++-)|/\alpha^3$ plotted for the kinematics as defined in nagy.