Bethe-Salpeter equations from the 4PI effective action

In this paper we derive a hierarchy of integral equations from the 4PI effective action which have the form of Bethe-Salpeter equations. We show that, together with the equation of motion for the self-consistent 4-vertex, these integral equations are closed, and that their expansions give infinite series of connected diagrams which have the correct symmetry.

Figure 1

The functional ${\Phi }_2$ for the 5-loop 4PI effective action. Bare vertices are denoted by open circles and effective vertices are solid dots.

Figure 2

Equation of motion for the 4-point function from the 5-loop 4PI effective action (different permutations of external indices are indicated with a bracketed numerical factor).

Figure 3

Diagrammatic representation of the BS equation in Eq. (18). Boxes represent the vertex $M$ and circles are the kernel $\Lambda$.

Figure 4

The 4-point kernel from the 5-loop 4PI effective action.

Figure 5

The 6-point kernel from the 5-loop 4PI effective action.

Figure 6

The 8-point kernel from the 5-loop 4PI effective action. The lines with superscript ($-1$) represent inverse propagators that connect indices on either side of the vertex.

Figure 7

The 6-point vertices ${\tilde{M}}_{24}$ and $\widehat{V}$ (represented by the box with the cross in it).

Figure 8

The 8-point vertex ${\tilde{M}}_{44}$.

Figure 9

The BS equation for the vertex $M_{22}$.

Figure 10

The BS equation for the vertex $M_{24}$.

Figure 11

The BS equation for the vertex $M_{42}$.

Figure 12

The BS equation for the vertex $M_{44}$. The lines with superscript ($-1$) represent inverse propagators that connect indices on either side of the vertex.

Figure 13

An example of a diagram which can be produced with different dummy variable assignments.