Landau-Khalatnikov-Fradkin Gauge Transformations for the Propagator and Vertex in QED and QED₂

Abstract

The non-perturbative Landau-Khalatnikov-Fradkin (LKF) transformations describe how Green functions in quantum field theory transform under a change in the photon field’s linear covariant gauge parameter (denoted ξ). The transformations are framed most simply in coordinate space where they are multiplicative. They imply that information on gaugedependent contributions from higher order diagrams in the perturbative series is contained in lower order contributions, which is useful in multi-loop calculations. We study the LKF transformations for the propagator and the vertex in both scalar and spinor QED, in some particular dimensions. A novelty of our work is to derive momentum-space integral representations of these transformations; our expressions are also applicable to the longitudinal and transverse parts of the vertex. Applying these transformations to the tree-level Green functions, we show that the one-loop terms obtained from the LKF transformation agree with the gauge dependent parts obtained from perturbation theory. Our results will be presented in more comprehensive form elsewhere.