Renormalization group flow equations for the proper vertices of the background effective average action

We derive a system of coupled flow equations for the proper vertices of the background effective average action and we give an explicit representation of these by means of diagrammatic and momentum space techniques. This explicit representation can be used as a new computational technique that enables the projection of the flow of a large new class of truncations of the background effective average action. In particular, these can be single- or bifield truncations of local or nonlocal character. As an application we study non-Abelian gauge theories. We show how to use this new technique to calculate the beta function of the gauge coupling (without employing the heat kernel expansion) under various approximations. In particular, one of these approximations leads to a derivation of beta functions similar to those proposed as candidates for an “all-orders” beta function. Finally, we discuss some possible phenomenology related to these flows.

Figure 1

Diagrammatic representation of the flow equations for ${\partial }_t{\gamma }_k^{(1;0)}$ and ${\partial }_t{\gamma }_k^{(0;1)}$ as given in (24).

Figure 2

Diagrammatic representation of the flow equations for the vertices ${\partial }_t{\gamma }_k^{(2;0)}$, ${\partial }_t{\gamma }_k^{(1;1)}$, and ${\partial }_t{\gamma }_k^{(0;2)}$ as in Eq. (25).

Figure 3

Diagrammatic representation of the flow equation for the one-vertices ${\partial }_t{\gamma }_k^{(1;0)}$ and ${\partial }_t{\gamma }_k^{(0;1)}$ as given in (33).

Figure 4

Diagrammatic representation of the flow equations for the two-vertices ${\partial }_t{\gamma }_k^{(2;0)}$, ${\partial }_t{\gamma }_k^{(1;1)}$, and ${\partial }_t{\gamma }_k^{(0;2)}$ as in Eq. (34).

Figure 5

Diagrammatic representation of the compact form of the flow equation for the zero-field two-point function ${\overline{\gamma }}_k^{(2)}$ of the gEAA after the field multiplet decomposition within the truncation specified by Eqs. (53) and (54). Thick wavy lines represent the background field ${\overline{A}}_{\mu }$, light wavy lines represent the fluctuation field $a_{\mu }$, while dotted lines represent the ghost fields $\overline{c}$ and $c$.

Figure 6

Diagrammatic representation of the flow equation for the zero-field proper vertex ${\gamma }_k^{(2,0,0;0)}$ of the bEAA used to calculate ${\partial }_t{Z}_{a,k}$.

Figure 7

Graphical representation of the flow equation for the ghost-ghost zero-field proper vertex ${\gamma }_k^{(0,1,1;0)}$ from (122).

Figure 8

The one-loop coefficient $a$ in the cutoff schemes type I (lower curve at $d=2$) and type II (upper curve at $d=2$). For reference we plotted the line $-\frac{11}{3}$.

Figure 9

Single-field beta functions from (144). Asymptotes from the left: type II, RS and type I beta functions. The one-loop beta function (without asymptote) is shown for comparison. Note the similarity between the type II and the RS beta functions.

Figure 10

Bifield beta functions from (154). Asymptotes from the left: Padè, type II, and type I beta functions. Now the type I and type II beta function have the asymptotes in the same place. The one-loop beta function (without asymptote) is shown for comparison.

Figure 11

Implicit plot for the single-field flow of $g_k$ from (157). From top: RS, type II, type I, and one-loop flows.

Figure 12

The mass gap as a function of $g_0$ (with $k_0=91.19\mathrm{GeV}$) for the single-field flows compared to the flow proposed by RS from (159). From top-left the RS, type II, and type I curves.

Figure 13

Implicit plot for the bifield flow of $g_k$ from (160). From the top: RS, type II, type I, and one-loop flows. Note that the type I and II are closer than in Fig. 11.

Figure 14

The mass gap $M$ as a function of $g_0$ (with $k_0=91.19\mathrm{GeV}$) for the (from top-left) RS and bi–field flows.