Fermion Propagator in QED - Landau-Khalatnikov-Fradkin Transformations

Abstract

Gauge theories such as quantum electrodynamics (QED) and quantum chromodynamics (QCD) describe the physical world accurately at the level of fundamental particles. They possess gauge symmetry reflected in terms of several identities and transformation laws which impose tight constraints on all conceivable Green functions which define the theory. In this article, we describe and summarize the role played by the Landau-Khalatnikov-Fradkin (LKF) transformations in this context. Within the set of covariant gauges, these transformations tell us how to construct a Green function in an arbitrary gauge, starting from its explicit expression in a particular gauge. In perturbation theory, these transformations are satisfied at every order of approximation. A non-perturbative description of QED and QCD in the continuum is provided by the Schwinger-Dyson Equations (SDEs). These are the fundamental equations of motion encoding the dynamics of Green functions. These equations provide a unified description of weak and strong coupling regimes and are thus increasingly employed to study strongly interacting theories and their transition to the perturbative limit. As these equations are an infinite set of coupled non-linear equations, a truncation is essential to reduce them to a solvable number. LKF transformations provide a stringent constraint on the acceptable truncations which preserve the original symmetries of the gauge theory involved. Most of these truncations consist in cleverly constructing an Anstaz for the electron-photon vertex in QED and the quark-gluon vertex in QCD. In this article, we review the LKF transformations for the fermion propagator. Very importantly, they imply the gauge invariance of the chiral fermion condensate and the pole mass of a fermion. We provide the first demonstration of the latter in this article. Moreover, we also describe how the LKF transformations of the fermion propagator provide gauge-symmetry constraints on a non-perturbative construction of the three-point fermion-boson vertex.