Gauge-invariant response functions of fermions coupled to a gauge field

TL;DR

This paper investigates gauge-invariant response functions of fermions coupled to a gauge field, demonstrating that certain singularities cancel out and the system behaves like a Fermi liquid without diverging effective mass.

Contribution

It provides a detailed calculation showing the cancellation of singularities in gauge-invariant response functions and confirms the non-renormalization of the gauge field propagator up to two loops.

Findings

Leading singularities cancel between self-energy and vertex corrections.

Gauge field propagator remains unrenormalized up to two-loop order.

Response functions follow Fermi liquid behavior without diverging effective mass.

Abstract

We study a model of fermions interacting with a gauge field and calculate gauge-invariant two-particle Green's functions or response functions. The leading singular contributions from the self-energy correction are found to be cancelled by those from the vertex correction for small $q$ and $\Omega$. As a result, the remaining contributions are not singular enough to change the leading order results of the random phase approximation. It is also shown that the gauge field propagator is not renormalized up to two-loop order. We examine the resulting gauge-invariant two-particle Green's functions for small $q$ and $\Omega$, but for all ratios of $\Omega / v_F q$ and we conclude that they can be described by Fermi liquid forms without a diverging effective mass.