One Loop Multiphoton Helicity Amplitudes

TL;DR

The paper develops a recursion-based framework using double-off-shell fermion currents to compute one-loop QED helicity amplitudes for n-photon processes and e+e− → γ...γ in the massless limit. By performing all integrals explicitly with dimensional regularization, it yields gauge-invariant, no-cut amplitudes, revealing that all-like-helicity n-photon scattering vanishes for $n \ge 5$ and that the only nonzero case in that class is the $n=4$ box contribution, e.g. $A(1^+,2^+,3^+,4^+) = $ $i e^4/(2\pi^2) [\langle 12\rangle^*\langle 34\rangle^*]/[\langle 12\rangle\langle 34\rangle]$. The method also handles one-loop e+e− → γ...γ via light-by-light and jellyfish diagrams, producing compact, spinor-form results. These findings demonstrate a powerful, largely algebraic approach to complex loop amplitudes with potential extensions to more general helicities and QCD, while avoiding spurious logarithms and discontinuities. $A(1^+,2^+,3^+,4^+) = $ $i e^4/(2\pi^2) [\langle 12\rangle^*\langle 34\rangle^*]/[\langle 12\rangle\langle 34\rangle]$ is a representative finite result for the four-photon case in the all-like-helicity sector.$

Abstract

We use the solutions to the recursion relations for double-off-shell fermion currents to compute helicity amplitudes for $n$-photon scattering and electron-positron annihilation to photons in the massless limit of QED. The form of these solutions is simple enough to allow {\it all}\ of the integrations to be performed explicitly. For $n$-photon scattering, we find that unless $n=4$, the amplitudes for the helicity configurations (+++...+) and (-++...+) vanish to one-loop order.

Figures (1)

• Figure 1: The basic diagram for $n$-photon scattering. The blob represents the sum of all possible tree graphs with $n-1$ photons attached to a fermion line.