# Does every fermion loop have two contributions?

Suppose I have a Feynman Diagram with a closed fermion loop. This introduces a negative sign, and a trace over the product of the propagators.

Is the same Feynman diagram, but with the arrows in the loop going in the opposite direction, a distinct diagram? In other words, are there really two diagrams, which we can take into account by introducing a factor of 2?

If the answer to the above is “yes,” then does this also apply to ghost diagrams?

• Well, Dirac fermion lines, ghost lines & complex scalar lines are oriented. Apr 23, 2022 at 18:44
• It's the other way around. A loop diagram with Majorana fermions has no arrows on the fermion lines and gets a symmetry factor of $1/2$ Aug 9, 2022 at 16:29

I have been wondering a similar thing about symmetry factors for fermion loops. Based on this solution for the beta function of the Gross-Neveu model (problem 12.2), for diagrams where the two fermion propagators are oriented in the same direction you get a factor of 2 since you can exchange the two propagators and get an identical diagram. If the arrows are oppositely oriented, you don't get a symmetry factor since you can't exchange them while preserving the diagram. So for example, diagram (b) below gets an overall $$1/2$$ while the others don't. This is consistent with the rules in for example equation 30 in this paper.