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

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 canceled by those from the vertex correction for small q and {Ω}. 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 {Ω}, but for all ratios of {Ω}/vFq, and we conclude that they can be described by Fermi-liquid forms without a diverging effective mass.