In the ABCD model, you have complex flavor fields \(\Phi\) and four color fields \(A\), \(B\), \(C\), and \(D\). The interaction vertex is of the form

That is, two flavor lines and four color lines. Let us look at some few-body processes.

Two-Body

The simplest process is with two distinct bodies. Due to the sextic nature of the interaction vertex, the two-body interaction is 3-loop. You can imagine constructing ladder diagrams with such 3-loop exchanges. With \(n\) exchanges, you have \(4n-1\) loops.

Three-Body

With three distinct bodies, and indeed with any odd number of bodies, you can only form multi-ladder diagrams with two-body three-loop exchanges. The simplest such diagram has two two-body exchanges and is 6-loop.

Four-Body

Besides the multi-ladder diagrams with two-body three-loop exchanges (9-loop), with four distinct bodies you can have three-body interactions by connecting each body with a color digon to make a square polygon. The result is a 5-loop diagram.

Six-Body

With six distinct bodies you can have

• two-body multi-ladders with five exchanges (15-loop)
• three-body hexagon with six exchanges (7-loop)

You can also have four-body interactions that lead to a planar diagram corresponding to the octahedral graph. The octahedron has eight faces, so it corresponds to an 8-loop process. The octahedral graph is the unique 6-quartic graph. Note that the octahedron is also a 3-antiprism.

Eight-Body

With eight distinct bodies you can have

• two-body multi-ladders with seven exchanges (21-loop)
• three-body octagon with eight exchanges (9-loop)

You also have four-body interactions from any of the 6 different 8-quartic graphs. That is, there are 6 different eight-body potentials made with four-body interactions. However, only one of them is planar: the 4-antiprism. Since the 4-antiprism has ten faces, this diagram corresponds to a 10-loop process.

In WolframAlpha, the 4-antiprism graph is "8-quartic graph 3". According to WolframAlpha, "8-quartic graph 1" and "8-quartic graph 4" are toroidal graphs, apparently with genus one. The other three graphs must have genus greater than one.

Ten-Body

With ten distinct bodies you can have

• two-body multi-ladders with nine exchanges (27-loop)
• three-body octagon with ten exchanges (11-loop)

You also have four-body interactions from any of the 59 different 10-quartic graphs. That is, there are 59 different ten-body potentials made with four-body interactions. I have found two of them to be planar: the 5-antiprism and the \(J_{15}\) Johnson solid. Both polyhedra have 12 faces, so these diagrams corresponds to 12-loop processes. According to this site, there are 3 planar regular graphs with ten degree-4 vertices. The third one can be found in WolframAlpha under "10-quartic graph 21".

Twelve-Body

With twelve distinct bodies you can have

• two-body multi-ladders with eleven exchanges (33-loop)
• three-body dodecagon with twelve exchanges (13-loop)

You also have four-body interactions from any of the 1544 different 12-quartic graphs. That is, there are 1544 different twelve-body potentials made with four-body interactions. I have found three of them to be planar: the 6-antiprism, the cuboctahedron, and the \(J_{27}\) Johnson solid. Each polyhedra have 14 faces, so these diagrams corresponds to 14-loop processes. Apparently, there are 13 graphs that are planar. The Chvátal graph is non-planar, and thus yields a non-planar twelve-body potential.

Sixteen-Body

With sixteen distinct bodies you can have

• two-body multi-ladders with fifteen exchanges (45-loop)
• three-body hexadecagon with sixteen exchanges (17-loop)

You also have four-body interactions from any of the 8037418 different 16-quartic graphs. That is, there are 8037418 different sixteen-body potentials made with four-body interactions. I have found three of them to be planar: the 8-antiprism, and the \(J_{28}\) and \(J_{29}\) Johnson solids. Each polyhedra have 18 faces, so these diagrams corresponds to 18-loop processes. Apparently, there are 543 planar graphs. The tesseract graph and the Hoffman graph are non-planar, and thus yield non-planar sixteen-body potentials.