Landau-Khalatnikov-Fradkin Transformations in Quantum Electrodynamics: For Perturbation Theory and Dynamical Mass Generation

We carry out a comprehensive analysis of the Landau-Khalatnikov-Fradkin transformations for a charged fermion propagator at the two-loop level in quantum electrodynamics (QED). Starting with an arbitrary covariant gauge $\xi$ and space-time dimension $d$, we provide its explicit expressions in three and four-dimensional QED. We begin with the tree-level fermion propagator in the Landau gauge and gauge-transform it to obtain an analytical expression for an all order result in an arbitrary covariant gauge. We expand it out to two-loops both for the massless and massive propagators in three and four space-time dimensions. In addition to comparing with all earlier results in the literature wherever possible, we also study constraints of multiplicative renormalizabilty of our results in four-dimensional QED which are logarithmically divergent. Finally, we analyze representative solutions of the fermion propagator which correspond to dynamical chiral symmetry breaking and mass generation in QED. We study the gauge dependence of these emergent solutions, that of the Euclidean pole mass and the chiral fermion condensate.