# What are “combinatorial factors” in the Lagragian?

I'm reading "Renormalization" by John C. Collins and in the fifth chapter "Renormalization" under the subchapter 5.6 "Relation to $$\mathcal L$$", he defines the Lagrangian and its components. Then he says:

Now that we have developed a convenient notation, the most difficult part of the proof is to ensure that the combinatorial factors come out right.

What are "combinatorial factors" here?

• He is referring to the combinatorial factors in the Feynman graphs whose divergences you're trying to cancel. If the factors you get from building counter terms with the additions to the Lagrangian don't match up in the end, you won't get cancelation. – Richard Myers Jun 8 at 22:38
• Thank you. Do you know why they are called combinatorial? – Michael T Chase Jun 9 at 0:04
• Because they are essentially the order of the automorphism group for the graph...this is pointed out in simple examples in any QFT text. For example, Peskin & Schroeder do a couple examples in the first couple chapters. There are also many questions on this site about these. They are called the combinatorial (or sometimes symmetry) factors of graphs. They have nothing to do with the Lagrangian beyond what vertices are allowed in the first place. – Richard Myers Jun 9 at 0:22
• Yes. As I said, they are the order of the automorphism group of the graph. This calculation is combinatorial. I really suggest looking in Peskin & Schroeder for this. Collins is great, but not necessarily as a first source on perturbation theory. – Richard Myers Jun 9 at 0:53
• They are related to counting the number of ways to pair fields via Wick's theorem so it is very directly combinatorics – octonion Jun 9 at 2:19