2PI effective theory at next-to-leading order using the functional renormalization group

We consider a symmetric scalar theory with quartic coupling in four dimensions. We show that the four-loop 2PI calculation can be done using a renormalization group method. The calculation involves one bare coupling constant which is introduced at the level of the Lagrangian and is therefore conceptually simpler than a standard 2PI calculation, which requires multiple counterterms. We explain how our method can be used to do the corresponding calculation at the 4PI level, which cannot be done using any known method by introducing counterterms.

Figure 1

The interacting part of the effective action for the symmetric theory to four-loop order.

Figure 2

Skeleton diagrams in the four-kernel $\mathrm{\Lambda }$ and the six-kernel $\mathrm{\Upsilon }$. The numbers in brackets indicate the number of distinct permutations of external legs, and the numbers that are not bracketed are symmetry factors. The name under each diagram indicates the term in the effective action that produced it (see Fig. 1).

Figure 3

The flow equations in Eqs. (22) and (23).

Figure 4

Example graphs.

Figure 5

The Bethe-Salpeter vertex $M(0)$ and the physical vertex $V(0)$ versus $T$ for $\lambda =2$ and $\lambda =4$ at the two-, three- and four-loop levels in the skeleton expansion of the action.

Figure 6

The physical vertex $V(0)$ versus $T$ for $\lambda =2$ with the renormalization performed at two different temperatures.

Figure 7

The Bethe-Salpeter vertex $M(0)$ and the physical vertex $V(0)$ versus the number of spatial lattice points $N_S$ for $\lambda =T=2$ at the three- and four-loop levels in the skeleton expansion of the action. The curves are shifted so that they cross at $N_s=18$.

Figure 8

The physical vertex $V(0)$ versus $p_{\max }$ with $\lambda =T=2$ and $L=4$ at the four-loop level in the skeleton expansion of the action. For comparison, the results of an incorrect calculation are shown (see the text for further explanation).