Renormalization of the 4PI effective action using the functional renormalization group

Techniques based on $n$-particle irreducible effective actions can be used to study systems where perturbation theory does not apply. The main advantage, relative to other nonperturbative continuum methods, is that the hierarchy of integral equations that must be solved truncates at the level of the action, and no additional approximations are needed. The main problem with the method is renormalization, which until now could only be done at the lowest ($n=2$) level. In this paper we show how to obtain renormalized results from an $n$-particle irreducible effective action at any order. We consider a symmetric scalar theory with quartic coupling in four dimensions and show that the 4 loop 4-particle-irreducible calculation can be renormalized using a renormalization group method. The calculation involves one bare mass and one bare coupling constant which are introduced at the level of the Lagrangian, and cannot be done using any known method by introducing counterterms.

Figure 1

The effective action for the symmetric theory to 4 loop order.

Figure 2

Representation of equation (18). The black box on the line in the first figure on the right side represents the insertion ${\partial }_{\kappa }({\mathrm{\Sigma }}_{\kappa }+R_{\kappa })$ and the black box on the vertex in the second figure represents ${\partial }_{\kappa }{V}_{\kappa }$. The vertical dots indicate that there are in total $2m+4n$ legs on the right side of each kernel.

Figure 3

Representation of the integrals in Eqs. (23), (24), (25).

Figure 4

An example of the method used to calculate the flow equations (see text for further details).

Figure 5

The first diagram represents the integral in the second term of Eq. (26), and the second and third diagrams are respectively the first and second integrals in Eq. (27).

Figure 6

The vertex at zero momentum as a function of the number of spatial lattice points $N_s$ with fixed $a_s=1/8$ (or $p_{\max }=25.1$).

Figure 7

The vertex at zero momentum as a function of temperature using two different renormalization points: $T_{\text{renorm}}=2.0$ ($\text{red}+$) and $T_{\text{renorm}}=0.91$ (blue x).

Figure 8

The vertex at zero momentum (red squares) and bare coupling (blue dots) as a function of $p_{\max }$ with $\mathrm{\Delta }p$ held fixed, using $T=1.4$ and $L=14/8$.