The Epstein-Glaser causal approach to the Light-Front QED$_{4}$. II: Vacuum Polarization tensor

TL;DR

This paper constructs the one-loop vacuum polarization tensor in light-front QED$_{4}$ using the perturbative causal approach, resolving the controversy over the instantaneous fermionic propagator term by demonstrating it is unnecessary in this framework.

Contribution

It shows that the fermionic propagator in light-front QED$_{4}$ lacks instantaneous terms within the causal approach, simplifying calculations and resolving existing ambiguities.

Findings

The vacuum polarization tensor matches standard results without extra assumptions.

The fermionic propagator in the causal framework does not contain instantaneous terms.

The approach clarifies why the same results are obtained with or without the instantaneous term.

Abstract

In this work we show how to construct the one-loop vacuum polarization for light-front QED$_{4}$ in the framework of the perturbative causal theory. Usually, in the canonical approach, it is considered for the fermionic propagator the so-called instantaneous term, but it is known in literature that this term is controversial because it can be omitted by computational reasons; for instance, by compensation or vanishing by dimensional regularization. In this work we propose a solution to this paradox. First, in the perturbative causal theory, it is shown that the fermionic propagator does not have instantaneous terms, and with this propagator we calculate the one-loop vacuum polarization, from the calculation it follows the same result as obtained by the standard approach, but without reclaiming any extra assumptions. Moreover, since the perturbative causal theory is defined in the distributional framework, we can also show the reason behind we obtaining the same result whether we consider or not the instantaneous fermionic propagator term.