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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?

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Qmechanic
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TaeNyFan
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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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