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Vertex corrections in gauge theories for two-dimensional condensed matter systems

We calculate the self-energy of two-dimensional fermions that are coupled to transverse gauge fields, taking two-loop corrections into account. Given a bare gauge field propagator that diverges for small momentum transfers q as 1 / q^{eta}, 1 < eta < 2, the fermionic self-energy without vertex corrections vanishes for small frequencies omega as Sigma (omega) propto omega^{gamma with gamma = {frac{2}{1 + eta}} < 1. We show that inclusion of the leading radiative correction to the fermion - gauge field vertex leads to Sigma (omega) propto omega^{gamma} [ 1 - a_{eta} ln (omega_0 / omega) ], where a_η is a positive numerical constant and omega_0 is some finite energy scale. The negative logarithmic correction is consistent with the scenario that higher order vertex corrections push the exponent gamma to larger values.