Challenges with Internal Photons in Constructive QED

TL;DR

This paper addresses discrepancies in constructive QED calculations involving internal photons, proposing a method using massive photons to reconcile results with Feynman diagrams for processes with massive fermions.

Contribution

It introduces a novel approach using massive photons in constructive QED to resolve inconsistencies with Feynman diagram results for internal photon processes involving massive fermions.

Findings

Constructive techniques do not match Feynman diagrams for massive fermions.

Using massive photons and taking the massless limit yields consistent results.

The approach works for multiple photon-mediated processes, aligning with standard methods.

Abstract

We find the correct spinor amplitude for a simple photon-mediated process and show that, in contrast, the result for the same process using the standard constructive techniques do not agree with Feynman diagrams when the fermions are massive. Along the way, we analyze the $x$ factor used in photon vertices, we work out the spinor shifts for massive particles when the momenta are analytically continued and we consider the large $z$ limit of the amplitudes in this paper and show that the photon-mediated process does not vanish in this limit for any choice of two of its momenta. For comparison with the photon-mediated process, we also describe two processes with external photons that are mediated by massive particles. In both cases, we show that the current techniques are sufficient and that the final results agree with Feynman diagrams. We also demonstrate that by using a massive photon in our calculations and taking the massless limit at the end, we can achieve agreement with Feynman diagrams in all the processes discussed here, including the photon-mediated amplitudes.