# Counting higher-order corrections in “ABC theory”

I am trying to understand how to enumerate higher-order Feynman diagrams.

In his book on Elementary Particle Physics, Griffiths considers a simple "ABC toy theory" which has:

• three (scalar, massive) particles A, B, and C (which are their own antiparticles), and
• only one allowed interaction vertex ABC.

He counts explicitly the number of higher-order diagrams for the process $A+A\to B+B$ at order $g^4$, i.e. with 4 vertices.

What confuses me is that his result changes in the 2nd edition:

In the first edition of his book (p. 207) he finds that there are 15 because each additional line can start at one of the 5 lines of the original $A+A\to B+B$ diagram and end on the same or another line, so enumerating

• $1\to1$, $1\to2$, ..., $1\to5$,
• $2\to2$, ..., $2\to5$,
• ...
• $5\to5$,

we end up with 15. $\Rightarrow$ This makes sense naively.

However, in the second, revised edition, he talks about 8 diagrams only (p. 217): 5 self-energy diagrams ($i\to i$, a line "sprouts a loop"), two vertex corrections ("a vertex becomes a triangle") and one box diagram.

My question: How many diagrams are there? 15 or 8? Where have the other 7 gone, e.g. a line connecting an initial and a final-state particle? Was it wrong to include them? If so why? Are they equivalent to another diagram or ruled out for some reason?

• And please add a comment if you downvote so I can improve the question! :/ – fuenfundachtzig May 4 '15 at 9:35