The effective 1PI action $\Gamma$ generates the sum of all proper diagrams, which are:
- Connected
- Truncated
- 1 particle irreducible in the internal lines.
For the two point correlation functions, the connected and 1PI diagrams give, apart from $i$'s and $i\varepsilon$'s: $$G_{\text {c,1PI}}(p^2)=\frac{1}{p^2-m^2}- \frac{1}{p^2-m^2}M^2(p^2)\frac{1}{p^2-m^2}.$$ Truncating we obtain: $$G_{\text {proper}}(p^2)=(p^2-m^2)^2G_{\text {c,1PI}}(p^2)=p^2-m^2 -M^2(p^2).$$