Quadratic Effective Action for QED in D=2,3 Dimensions

TL;DR

This paper computes the quadratic effective action for QED in 2 and 3 dimensions, revealing the structure of gauge boson propagators, effective potentials, and mass dependence, with results matching bosonized theories and showing screening effects.

Contribution

It provides a nonperturbative analysis of the effective action and propagators in low-dimensional QED, including explicit calculations of the effective potential and boson mass dependence.

Findings

In 2D, the gauge boson has one massless pole for any nonzero fermion mass.

The effective potential between static charges in 2D is linearly increasing with distance.

In 3D, a massive pole appears, leading to screening effects.

Abstract

We calculate the effective action for Quantum Electrodynamics (QED) in D=2,3 dimensions at the quadratic approximation in the gauge fields. We analyse the analytic structure of the corresponding nonlocal boson propagators nonperturbatively in k/m. In two dimensions for any nonzero fermion mass, we end up with one massless pole for the gauge boson . We also calculate in D=2 the effective potential between two static charges separated by a distance L and find it to be a linearly increasing function of L in agreement with the bosonized theory (massive Sine-Gordon model). In three dimensions we find nonperturbatively in k/m one massive pole in the effective bosonic action leading to screening. Fitting the numerical results we derive a simple expression for the functional dependence of the boson mass upon the dimensionless parameter e^{2}/m .