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What is the precise definition of a $n$-loop one-particle irreducible ($1$PI) diagram?

For example, consider the following diagrams.

  • Is the first diagram a $0$-loop $1$PI diagram?
  • Is the second diagram $1$-loop $1$PI ?
  • Is the third diagram $2$-loop $1$P (i.e. reducible)?
  • Are the fourth and fifth diagrams $2$-loop $1$PI?

enter image description here

Qmechanic
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TaeNyFan
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  • Can any close voter please explain their reason in a comment, thank you – doublefelix Dec 06 '22 at 23:21
  • Does this help explain the definition: https://physics.stackexchange.com/q/342994/ – hft Dec 07 '22 at 01:36
  • Shouldn't this definition just be plainly stated in your textbook? What textbook are you using? – hft Dec 07 '22 at 01:36
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    @doublefelix The reason is: A community-specific reason. The sub-reason is: This question doesn’t meet a Physics Stack Exchange guideline. The sub-sub reason is: Homework-like or check-my-work type question. – hft Dec 07 '22 at 01:42
  • @doublefelix Can you explain whether or not that makes sense to you? – hft Dec 07 '22 at 01:44
  • It does make sense to me. It looks like the question could have been copied from homework problems. On first look I couldn't be sure whether it was written by OP or copied, and so I thought OP would at least like to know the reason for close votes. But on second look I think there is a high chance. – doublefelix Dec 07 '22 at 10:39

1 Answers1

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A 1PI diagram is a connected graph that cannot be disconnected into two pieces by cutting a single line, cf. Wikipedia. In particular its external legs must be amputated, i.e. not part of the diagram.

For the last reason, none of OP's examples are actually 1PI diagrams. [However, if we strip/amputate their external legs, then there is nothing left from the 1st diagram (the free propagator), and the 2nd diagram, the 4th diagram (the sunset diagram) and 5th diagram become 1PI.] Nevertheless they are all connected diagrams of loop-order 0,1,2,2,2, respectively.

Qmechanic
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