2
$\begingroup$

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?

$\endgroup$
2
  • $\begingroup$ Well, Dirac fermion lines, ghost lines & complex scalar lines are oriented. $\endgroup$
    – Qmechanic
    Apr 23, 2022 at 18:44
  • $\begingroup$ 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 $1/2$ $\endgroup$
    – mike stone
    Aug 9, 2022 at 16:29

1 Answer 1

1
$\begingroup$

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.

I explained in more detail how to deduce the symmetry factors here.

Diagrams from the Gross-Neveu model

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.