Why the Feynman diagrams contributing to the effective action $\Gamma[\phi_{\rm cl}]$ are stripped/amputated/have no external lines?

Question

I am reading P&S Chapter 11 and specifically I am trying to understand the derivation of $\Gamma[\phi_{\rm cl}]$. All the algebra is okay, but I am failing to understand the connection to Feynman diagrams. I have also read Chapter 9 from Srednicki and the reply on this stack-exchange question, which I find very illuminating: Perturbation expansion of effective action

My question, however, is:

Why does the author say that the "the Feynman diagrams contributing to $\Gamma[\phi_{\rm cl}]$ have no external lines"? How can I understand that (pictorially or algebraically)? My guess is that it has something to do with the source term missing from the expression of the effective action, but I do not understand it that much.

Answer 1

Here is one argument:

1. Recall that the 1PI effective/proper action$^1$ $$\Gamma[\phi_{\rm cl}]~=~W_c[J]-J_k \phi^k_{\rm cl} \tag{1} $$ is the Legendre transformation of the generator $W_c[J]$ of connected diagrams.

2. We can recursively construct higher and higher $n$-point 1PI correlator functions $\Gamma_{n,k_1\ldots k_n}$ from pertinent combinations of connected $m$-point correlation functions $W_{c,m}^{k_1,\ldots k_m}$, where $m\leq n$, cf. e.g. my Phys.SE answer here.

3. Notice that in this context the connected 2-point function $W_{c,2}^{k\ell}$ plays the role of an (inverse) metric that raises and lowers the DeWitt indices.

4. The connected $m$-point correlation function $W_{c,m}^{k_1,\ldots k_m}$ has upper indices because it includes its external legs (which are attached to the sources $J_{k_1}\ldots J_{k_m}$ with lower indices).

5. The $n$-point 1PI correlator function $\Gamma_{n,k_1\ldots k_n}$ has lower indices because its external legs are stripped/amputated. Instead it is attached to the classical fields $\phi_{\rm cl}^{k_1}\ldots \phi_{\rm cl}^{k_n}$ with with upper indices in the effective action $\Gamma[\phi_{\rm cl}]$.

6. Conversely, and perhaps more illuminating diagramatically, the connected $m$-point correlation function $W_{c,m}^{k_1,\ldots k_m}$ is a sum of all possible trees made from connected propagators $W_{c,2}^{k\ell}$ and (amputated) 1PI vertices $\Gamma_{n,k_1\ldots k_n}$, where $n\leq m$, cf. e.g. this Phys.SE post.

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$^1$ We use DeWitt condensed notation to not clutter the notation.