Running coupling constant of a gauge theory in the framework of the Schwinger-Dyson equation: Infrared behavior of three-dimensional quantum electrodynamics

We discuss how to define and obtain the running coupling of a gauge theory in the approach of the Schwinger-Dyson (SD) equation, in order to perform a nonperturbative study of the theory. For this purpose, we introduce the nonlocally generalized gauge fixing into the SD equation, which is used to define the running coupling constant (this method is applicable only to a gauge theory). Some advantages and the validity of this approach are exemplified in QED₃. This confirms the slowing down of the rate of decrease of the running coupling and the existence of the nontrivial infrared fixed point (in the normal phase) of QED₃, claimed recently by Aitchison and Mavromatos, without so many of their approximations. We also argue that the conventional approach is recovered by applying the (inverse) Landau-Khalatnikov transformation to the nonlocal gauge result.