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I'm studying QFT, and we've recently introduced 1PI (one-particle-irreducible) diagrams, which are diagrams that remain connected even after cutting a single internal line.

We also showed that only fully connected diagrams contribute to $S$-matrix elements via the LSZ reduction formula, so we restrict our attention to connected diagrams.

Recently, we studied the exact two-point function and computed its 1-loop correction. We then saw that the full propagator can be organized as a sum of 1PI diagrams inserted in series, order by order in perturbation theory.

My question is: by building the full propagator this way (using only 1PI diagrams), are we losing any generality?

In other words: Are there any fully connected diagrams contributing to the 2-point function that cannot be built from 1PI pieces?

Qmechanic
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Lip
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1 Answers1

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Well, a formal way to show this is by exploring the relation between the generating function of connected diagrams W and the generating function of 1PI diagrams (the effective action). However, there is an easy way to show it diagrammatically. Take an arbitrary (connected) diagram $\Gamma$. Then this diagram is either 1PI (in which case you are done), or there is one line that, when cut, splits the diagram into two disconnected diagrams, $\Gamma_1, \Gamma_2$. Thus the diagram $\Gamma$ can be constructed by connecting $\Gamma_1$ and $\Gamma_2$ with a single propagator. Keep repeating the argument for $\Gamma_1$ and $\Gamma_2$. Since the number of lines is finite, this process must eventually finish, so you can write your initial diagram $\Gamma$ as a set of 1PI diagrams $\Gamma_i$ joined by propagators.

There is, however, one subtle detail that is usually not explained when deriving the full propagator from 1PI diagrams, which is that you assume they form a "chain" (-o-o-o-) where '-' are propagators and 'o' are 1PI diagrams. This is not necessarily true.

However, if you take any diagram for the 2-point function that is not of the form of the "chain", you can easily prove that you need at least one tadpole (a 1PI diagram with a single external leg). Tadpoles should vanish in any model describing physical particles, so you are good ignoring those diagrams. If your tadpoles don't vanish, then you should first do a field redefinition to remove them.

Gaussian97
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