$n$-particle irreducible effective action techniques for gauge theories

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.

Figure 1

The figure shows together with Fig. 2 the diagrammatic representation of ${\Gamma }_2^0[D,G,V_3,V_3^{(\mathrm{g}\mathrm{h})},V_4]$ as given in Eq. (3.19). Here the wiggled lines denote the gauge field propagator $D$ and the unwiggled lines the ghost propagator $G$. The thick circles denote the dressed and the small ones the classical vertices. This functional contains all terms of ${\Gamma }_2$ that depend on the classical vertices $g{V}_{03}$, $g{V}_{03}^{(\mathrm{g}\mathrm{h})}$, and $g^2{V}_{04}$ for an $\mathrm{S}\mathrm{U}(N)$ gauge theory. There are no further contributions to ${\Gamma }_2^0$ appearing at higher order in the expansion. For the gauge theory with fermions there is in addition the same contribution as in Fig. 2 with the unwiggled propagator lines representing the fermion propagator $\Delta$ and the ghost vertices replaced by the corresponding fermion vertices $V_{03}^{(\mathrm{f})}$ and $V_3^{(\mathrm{f})}$ [cf. Eq. (3.12)].

Figure 2

Ghost/fermion part of ${\Gamma }_2^0$.

Figure 3

The figure shows together with Fig. 4 the diagrammatic representation of ${\Gamma }_2^{\mathrm{i}\mathrm{n}\mathrm{t}}[D,G,V_3,V_3^{(\mathrm{g}\mathrm{h})},V_4]$ to three-loop order as given in Eq. (3.20). For the gauge theory with fermions, to this order there is in addition the same contribution as in Fig. 4 with the unwiggled propagator lines representing the fermion propagator $\Delta$ and the ghost vertex replaced by the corresponding fermion vertex $V_3^{(\mathrm{f})}$. This functional contains no explicit dependence on the classical vertices independent of the order of approximation.

Figure 4

Ghost/fermion part of ${\Gamma }_2^{\mathrm{i}\mathrm{n}\mathrm{t}}$ to three-loop order.

Figure 5

The self-energy for the gauge field ($\Pi$) and the ghost/fermion ($\Sigma$) propagators as obtained from the self-consistently complete two-loop approximation of the effective action. Note that at this order all vertices correspond to the classical ones.

Figure 6

The self-energy for the gauge field ($\Pi$) and the ghost/fermion ($\Sigma$) propagators as obtained from the self-consistently complete three-loop approximation of the effective action (cf. Fig. 7 for the vertices).

Figure 7

The gauge field three-vertex as well as the ghost (fermion) vertex as obtained from the self-consistently complete three-loop approximation of the effective action. Note that apart from the isolated classical three-vertex, all vertices in the equations correspond to dressed ones since at this order the four-vertex equals the classical vertex.

Figure 8

Standard Schwinger-Dyson equation for the proper three-vertex $V_3$. We have not displayed additional diagrams involving ghost or fermion vertices for brevity. We show it for comparison with the three-loop effective action result displayed in Fig. 7. One observes that a naive truncation of the Schwinger-Dyson equation at the one-loop level does not agree with the latter, since the second and third diagrams contain a classical three-vertex instead of a dressed one as in Fig. 7. (Note that the four-vertex equals the classical one at this order in the self-consistently complete loop expansion.)

Figure 9

Infinite series of self-energy contributions with dressed propagator lines and classical vertices.