N-particle irreducible effective action techniques for gauge theories

Abstract

A loop or coupling expansion of a so-called $n$-particle irreducible ($n\mathrm{P}\mathrm{I}$) generating functional provides a well-defined approximation scheme in terms of self-consistently dressed propagators and $n$-point vertices. A self-consistently complete description determines the functional for arbitrarily high $n$ to a given order in the expansion. We point out an equivalence hierarchy for $n\mathrm{P}\mathrm{I}$ effective actions, which allows one to obtain a self-consistently complete result in practice. The method is applied to a $\mathrm{S}\mathrm{U}(N)$ gauge theory with fermions up to four-loop or $\mathcal{O}({g}^{6})$ corrections. For nonequilibrium we discuss the connection to kinetic theory. The leading-order on-shell results in $g$ can be obtained from the three-loop effective action approximation, which already includes, in particular, all diagrams enhanced by the Landau-Pomeranchuk-Migdal effect. Furthermore, we compare the effective action approach with Schwinger-Dyson (SD) equations. By construction, SD equations are expressed in terms of loop diagrams including both classical and dressed vertices, which lead to ambiguities of whether classical or dressed ones should be employed at a given truncation order. We point out that these problems are absent using effective action techniques. We show that a wide class of truncations of SD equations cannot be obtained from the $n\mathrm{P}\mathrm{I}$ effective action. In turn, our results can be used to resolve SD ambiguities of whether classical or dressed vertices should be employed at a given truncation order.