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

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.

• To reopen this post (v1), consider to only ask 1 question. Commented Feb 11, 2022 at 14:23
• I have thought about the answer to one of the questions. If you let me answer it below without deleting any of the two questions asked, maybe it would be beneficial for someone reading the questions in the future Commented Feb 12, 2022 at 12:42
• Please remove 1 question so that it can be reopened. (Both are still in the revision history.) Commented Feb 12, 2022 at 12:49
• Okay, I will do that. However, is there a particular reason as to why only one question should be asked at a time? Commented Feb 12, 2022 at 12:51
• See e.g. this & this meta discussions. Commented Feb 12, 2022 at 13:52

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.

--

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