On the Landau-Khalatnikov-Fradkin transformation in Quantum Electrodynamics

The Landau-Khalatnikov-Fradkin (LKF) transformation is a powerful and elegant transformation allowing to study the gauge dependence of the propagator of charged particles interacting with gauge fields. With the help of this transformation, we derive a non-perturbative identity between massless propagators in two different gauges. In the case of four-dimensional Quantum Electrodynamics (QED), from the LKF identity, we find that the corresponding perturbative series can be exactly expressed in terms of a hatted transcendental basis that eliminates all even Euler zeta-functions. This explains the mystery of even zeta-values observed in multi-loop calculations of Euclidean massless correlators for almost three decades now. Our construction further allows us to derive an exact formula relating hatted and standard zeta-functions to all orders of perturbation theory. In the case of quenched three-dimensional QED, from the LKF identity, assuming the finiteness of the perturbative expansion, we state that, exactly in d=3, all odd perturbative coefficients, starting with the third order one, should be zero in any gauge. To check the result, we calculate the three- and four-loop corrections to the massless fermion propagator. The three-loop correction is finite and gauge invariant but, however, the four-loop one has singularities except in the Feynman gauge where it is also finite. These results explicitly show an absence of the finiteness of the perturbative expansion in quenched three-dimensional QED. Moreover, up to four loops, gauge-dependent terms are completely determined by lower order ones in agreement with the LKF transformation.