Symmetry factor of a Wick diagram

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Discussion Overview

The discussion revolves around the computation of the symmetry factor of a Wick diagram, specifically in the context of a hermitian field interacting with a complex field. Participants explore the reasoning behind the symmetry factors assigned to different diagrams, including a specific example from a book and a double bubble diagram.

Discussion Character

  • Technical explanation, Debate/contested

Main Points Raised

  • One participant expresses confusion regarding the symmetry factor of 2 for a Wick diagram, questioning the automorphisms that lead to this value.
  • Another participant clarifies that a specific propagator loop does not have an arrow, which is relevant to the symmetry considerations.
  • A subsequent reply refines the understanding by stating that the hermitian propagator loop can only map to itself, contributing to the symmetry factor.
  • Another participant proposes that the direction of the hermitian loop can be changed, which supports the symmetry factor of 2.
  • One participant initially questions the symmetry factor of a double bubble diagram, suggesting it should be 4 based on previous reasoning, but later corrects themselves, stating it actually has a symmetry factor of 8.

Areas of Agreement / Disagreement

Participants exhibit some agreement on the nature of the symmetry factors but also demonstrate disagreement and confusion regarding the specific values assigned to different diagrams, indicating that the discussion remains unresolved in terms of clarity on these factors.

Contextual Notes

There are limitations in the assumptions made about the automorphisms of the diagrams and the specific characteristics of the propagators involved, which may affect the understanding of the symmetry factors.

pscplaton
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Hello !

In the book Quantum Field for mathematician, there is this Wick diagram as an example to understand how to compute the symmetry factor (I am sorry, I draw it with paint...)

wick diagram.png


This is about a hermitian field interacting with a complex field. The book says it has a symmetry factor of 2 but I don't understand why, even after reading about symmetry factor on wikipedia (http://en.wikipedia.org/wiki/Feynman_diagram#Symmetry_factors). I don't see what are the two automorphisms of this Feynman diagram... As far as I understand, for a given automorphism, "a" can only be mapped to "a" because of the complex field loop, and the same applies for "d". "c" cannot be mapped to "b" because it has incoming complex propagator from "a", which is not the case of "b"...

Where am I wrong ?

Thanks!
 
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d is not a complex propagator loop, it has no arrow.
 
Yes you are right. I mean that like "a", "d" can only be mapped to itself, because it has a hermitian propagator loop attached, and this is the only hermitian propagator loop of the diagram
 
This is the point, you can make the symmetry transformation of changing the direction of this loop without changing the diagram. Hence a symmetry factor of 2.
 
Thanks for your reply.

But then I don't understand why the double bubble diagram has a symmetry factor of 2.
bubble.png

According to what you are saying, it should have a symmetry factor of 4, shouldn't it ?

edit: Sorry I was confused. It works. There is actually a symmetry factor of 8
 
Last edited:

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