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The effective 1PI action $\Gamma$ generates the sum of all proper diagrams, which are:

1. Connected
2. Truncated
3. Irreducible on the internal lines. This means that if we cut any one of the internal lines of the diagram, it remains connected.

The truncated Green's function is obtained by multiplying the Green's function with the exact inverse propagators corresponding to the external lines: $$G^{(n)}_{\text {trunc}}(p_1,\dots ,p_n)=\Delta'(p_1)^{-1}\Delta'(p_2)^{-1}\cdots \Delta'(p_n)^{-1} G^{(n)}(p_1,\dots ,p_n)\qquad (\sum _i p_i =0).$$ Ref. [1] gives the above definition for $n>2$ (cfr. eq. (6-70)). In fact, for $n=2$, sticking to the above recipe gives: $$G^{(2)}_{\text {trunc}} (p)=\Delta'(p)^{-1} \Delta'(p)\Delta'(p)^{-1}=\Delta'(p)^{-1}=\Gamma_2(p),$$ so that $G^{(2)}_{\text {trunc}}$ is in fact just the proper (1PI) function.

The connection of truncated functions with $S$-matrix elements is found by nothing that, near the pole of the propagator $p^2\approx m^2$: $$\Delta'(p^2)^{-1}\approx iZ^{-1}(p^2-m^2).$$ A glance to the LSZ formulas shows that the connection of truncated Green functions with connected $S$-matrix elements is: $$\langle p_1 ,p_2,\dots ,p_n\,\text {out}\vert q_1,q_2,\dots,q_m \text{in}\rangle = (2\pi)^4\delta^4(\sum _i p_i -\sum _j q_j)\times \\ \times Z^{\frac{n+m}{2}}G^{(n+m)}_{\text {trunc}}(-p_1,-p_2,\dots,-p_n,q_1,q_2,\dots ,q_m)$$ where all the momenta are on shell (cfr. Ref. [1], after eq. (6-70)).

[1] Itzykson&Zuber, "Quantum Field Theory", 6-2-2.

The effective 1PI action $\Gamma$ generates the sum of all proper diagrams, which are:

1. Connected
2. Truncated
3. Irreducible on the internal lines. This means that if we cut any one of the internal lines of the diagram, it remains connected.

The truncated Green's function is obtained by multiplying the Green's function with the exact inverse propagators corresponding to the external lines: $$G^{(n)}_{\text {trunc}}(p_1,\dots ,p_n)=\Delta'(p_1)^{-1}\Delta'(p_2)^{-1}\cdots \Delta'(p_n)^{-1} G^{(n)}(p_1,\dots ,p_n)\qquad (\sum _i p_i =0).$$ Ref. [1] gives the above definition for $n>2$ (cfr. eq. (6-70)). In fact, for $n=2$, sticking to the above recipe gives: $$G^{(2)}_{\text {trunc}} (p)=\Delta'(p)^{-1} \Delta'(p)\Delta'(p)^{-1}=\Delta'(p)^{-1}=\Gamma_2(p),$$ so that $G^{(2)}_{\text {trunc}}$ would be just the proper function. I believe that the theorem is that "the $\Gamma$ functional generates proper function for $n>2$".

The connection of truncated functions with $S$-matrix elements is found by nothing that, near the pole of the propagator $p^2\approx m^2$: $$\Delta'(p^2)^{-1}\approx iZ^{-1}(p^2-m^2).$$ A glance to the LSZ formulas shows that the connection of truncated Green functions with connected $S$-matrix elements is: $$\langle p_1 ,p_2,\dots ,p_n\,\text {out}\vert q_1,q_2,\dots,q_m \text{in}\rangle = (2\pi)^4\delta^4(\sum _i p_i -\sum _j q_j)\times \\ \times Z^{\frac{n+m}{2}}G^{(n+m)}_{\text {trunc}}(-p_1,-p_2,\dots,-p_n,q_1,q_2,\dots ,q_m)$$ where all the momenta are on shell (cfr. Ref. [1], after eq. (6-70)).

[1] Itzykson&Zuber, "Quantum Field Theory", 6-2-2.

The effective 1PI action $\Gamma$ generates the sum of all proper diagrams, which are:

1. Connected
2. Truncated
3. Irreducible on the internal lines. This means that if we cut any one of the internal lines of the diagram, it remains connected.

The truncated Green's function is obtained by multiplying the Green's function with the exact inverse propagators corresponding to the external lines: $$G^{(n)}_{\text {trunc}}(p_1,\dots ,p_n)=\Delta'(p_1)^{-1}\Delta'(p_2)^{-1}\cdots \Delta'(p_n)^{-1} G^{(n)}(p_1,\dots ,p_n)\qquad (\sum _i p_i =0).$$ Ref. [1] gives the above definition for $n>2$ (cfr. eq. (6-70)). In fact, for $n=2$, sticking to the above recipe gives: $$G^{(2)}_{\text {trunc}} (p)=\Delta'(p)^{-1} \Delta'(p)\Delta'(p)^{-1}=\Delta'(p)^{-1}=\Gamma_2(p),$$ so that $G^{(2)}_{\text {trunc}}$ is in fact just the proper (1PI) function. A way to see that $\Delta'(p)^{-1}$ is the proper function is as follows: write the sum of all 1PI two point diagrams as: $$\frac{1}{p^2-m^2}- \frac{1}{p^2-m_0^2}M^2(p^2)\frac{1}{p^2-m_0^2}.$$ Truncating with the bare inverse propagator $p^2-m_0^2$ we obtain $\Gamma_2(p)=\Delta'(p)^{-1}$.

The connection of truncated functions with $S$-matrix elements is found by nothing that, near the pole of the propagator $p^2\approx m^2$: $$\Delta'(p^2)^{-1}\approx iZ^{-1}(p^2-m^2).$$ A glance to the LSZ formulas shows that the connection of truncated Green functions with connected $S$-matrix elements is: $$\langle p_1 ,p_2,\dots ,p_n\,\text {out}\vert q_1,q_2,\dots,q_m \text{in}\rangle = (2\pi)^4\delta^4(\sum _i p_i -\sum _j q_j)\times \\ \times Z^{\frac{n+m}{2}}G^{(n+m)}_{\text {trunc}}(-p_1,-p_2,\dots,-p_n,q_1,q_2,\dots ,q_m)$$ where all the momenta are on shell (cfr. Ref. [1], after eq. (6-70)).

[1] Itzykson&Zuber, "Quantum Field Theory", 6-2-2.

The effective 1PI action $\Gamma$ generates the sum of all proper diagrams, which are:

1. Connected
2. Truncated
3. Irreducible on the internal lines. This means that if we cut any one of the internal lines of the diagram, it remains connected.

The truncated Green's function is obtained by multiplying the Green's function with the exact inverse propagators corresponding to the external lines: $$G^{(n)}_{\text {trunc}}(p_1,\dots ,p_n)=\Delta'(p_1)^{-1}\Delta'(p_2)^{-1}\cdots \Delta'(p_n)^{-1} G^{(n)}(p_1,\dots ,p_n)\qquad (\sum _i p_i =0).$$ Ref. [1] gives the above definition for $n>2$ (cfr. eq. (6-70)). In fact, for $n=2$, sticking to the above recipe gives: $$G^{(2)}_{\text {trunc}} (p)=\Delta'(p)^{-1} \Delta'(p)\Delta'(p)^{-1}=\Delta'(p)^{-1}=\Gamma_2(p),$$ so that $G^{(2)}_{\text {trunc}}$ is in fact just the proper (1PI) function.

The connection of truncated functions with $S$-matrix elements is found by nothing that, near the pole of the propagator $p^2\approx m^2$: $$\Delta'(p^2)^{-1}\approx iZ^{-1}(p^2-m^2).$$ A glance to the LSZ formulas shows that the connection of truncated Green functions with connected $S$-matrix elements is: $$\langle p_1 ,p_2,\dots ,p_n\,\text {out}\vert q_1,q_2,\dots,q_m \text{in}\rangle = (2\pi)^4\delta^4(\sum _i p_i -\sum _j q_j)\times \\ \times Z^{\frac{n+m}{2}}G^{(n+m)}_{\text {trunc}}(-p_1,-p_2,\dots,-p_n,q_1,q_2,\dots ,q_m)$$ where all the momenta are on shell (cfr. Ref. [1], after eq. (6-70)).

[1] Itzykson&Zuber, "Quantum Field Theory", 6-2-2.

The effective 1PI action $\Gamma$ generates the sum of all proper diagrams, which are:

1. Connected
2. Truncated
3. Irreducible on the internal lines. This means that if we cut any one of the internal lines of the diagram, it remains connected.

The truncated Green's function is obtained by multiplying the Green's function with the exact inverse propagators corresponding to the external lines: $$G^{(n)}_{\text {trunc}}(p_1,\dots ,p_n)=\Delta'(p_1)^{-1}\Delta'(p_2)^{-1}\cdots \Delta'(p_n)^{-1} G^{(n)}(p_1,\dots ,p_n)\qquad (\sum _i p_i =0).$$ Ref. [1] gives the above definition for $n>2$ (cfr. eq. (6-70)), but it seems to me that they then implicitly use the same definition for $n=2$ (cfr. the discussion after eq. (6-75)), and I think that it's correct. In fact, sticking to the above recipe, we obtain: $$\Gamma _2 (p)=\Delta'(p)^{-1} \Delta'(p)\Delta'(p)^{-1}=\Delta'(p)^{-1},$$ since $G^{(2)}(p)=\Delta'(p)$, which is exactly the answer found by differentiating the $\Gamma$ functional.

The connection with $S$-matrix elements is found by nothing that, near the pole of the propagator $p^2\approx m^2$: $$\Delta'(p^2)^{-1}\approx iZ^{-1}(p^2-m^2).$$ A glance to the LSZ formulas shows that the connection of truncated Green functions with connected $S$-matrix elements is: $$\langle p_1 ,p_2,\dots ,p_n\,\text {out}\vert q_1,q_2,\dots,q_m \text{in}\rangle = (2\pi)^4\delta^4(\sum _i p_i -\sum _j q_j)\times \\ \times Z^{\frac{n+m}{2}}G^{(n+m)}_{\text {trunc}}(-p_1,-p_2,\dots,-p_n,q_1,q_2,\dots ,q_m)$$ where all the momenta are on shell (cfr. Ref. [1], after eq. (6-70)).

EDIT: in my previous post, I was truncating with the bare propagator because the terms $$\frac{1}{p^2-m^2},\quad \frac{1}{p^2-m_0^2}M^2(p^2)\frac{1}{p^2-m_0^2}$$ are already 1PI on the external lines. However, the above definition is the correct one and it shows more clearly the connection with the LSZ formulas.

[1] Itzykson&Zuber, "Quantum Field Theory", 6-2-2.

The effective 1PI action $\Gamma$ generates the sum of all proper diagrams, which are:

1. Connected
2. Truncated
3. Irreducible on the internal lines. This means that if we cut any one of the internal lines of the diagram, it remains connected.

The truncated Green's function is obtained by multiplying the Green's function with the exact inverse propagators corresponding to the external lines: $$G^{(n)}_{\text {trunc}}(p_1,\dots ,p_n)=\Delta'(p_1)^{-1}\Delta'(p_2)^{-1}\cdots \Delta'(p_n)^{-1} G^{(n)}(p_1,\dots ,p_n)\qquad (\sum _i p_i =0).$$ Ref. [1] gives the above definition for $n>2$ (cfr. eq. (6-70)). In fact, for $n=2$, sticking to the above recipe gives: $$G^{(2)}_{\text {trunc}} (p)=\Delta'(p)^{-1} \Delta'(p)\Delta'(p)^{-1}=\Delta'(p)^{-1}=\Gamma_2(p),$$ so that $G^{(2)}_{\text {trunc}}$ is in fact just the proper (1PI) function. A way to see that $\Delta'(p)^{-1}$ is the proper function is as follows: write the sum of all 1PI two point diagrams as: $$\frac{1}{p^2-m^2}- \frac{1}{p^2-m_0^2}M^2(p^2)\frac{1}{p^2-m_0^2}.$$ Truncating with the bare inverse propagator $p^2-m_0^2$ we obtain $\Gamma_2(p)=\Delta'(p)^{-1}$.

The connection of truncated functions with $S$-matrix elements is found by nothing that, near the pole of the propagator $p^2\approx m^2$: $$\Delta'(p^2)^{-1}\approx iZ^{-1}(p^2-m^2).$$ A glance to the LSZ formulas shows that the connection of truncated Green functions with connected $S$-matrix elements is: $$\langle p_1 ,p_2,\dots ,p_n\,\text {out}\vert q_1,q_2,\dots,q_m \text{in}\rangle = (2\pi)^4\delta^4(\sum _i p_i -\sum _j q_j)\times \\ \times Z^{\frac{n+m}{2}}G^{(n+m)}_{\text {trunc}}(-p_1,-p_2,\dots,-p_n,q_1,q_2,\dots ,q_m)$$ where all the momenta are on shell (cfr. Ref. [1], after eq. (6-70)).

[1] Itzykson&Zuber, "Quantum Field Theory", 6-2-2.