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Text books often defines one-Particle Irreducible diagram (1PI diagram) as a connected diagram which does not fall into two pieces if you cut one internal line. Is this internal line the full propagator or the free propagator?

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It formally amounts to the same notion of one-particle irreducible$^\dagger$ since we can rewrite a full propagator $$G~=~G_0\sum_{n=0}^{\infty}(\Sigma G_0)^n$$ as a geometric series of bare propagators $G_0$, and vice-versa $$G_0~=~G\sum_{n=0}^{\infty}(-\Sigma G)^n,$$ where $$\Sigma~=~G_0^{-1}-G^{-1} $$ is the self-energy.

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$^\dagger$ Note that 1PI is called 2-connected by mathematicians.

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