Understanding $W^{(n)}$, $\Gamma^{(n)}$, and $\Sigma$ in Feynman diagrams

In quantum field theory (specifically $$\phi^4$$ theory), $$W$$ is the sum of all connected Feynman diagrams and the effective action $$\Gamma$$ is the sum of all 1PI Feynman diagrams. They are related by a Legendre transform. Then $$W^{(n)}$$ and $$\Gamma^{(n)}$$ represent $$n$$-point connected correlation functions and $$n$$-point 1PI functions obtained from $$W$$ and $$\Gamma$$ by taking functional derivatives.

How can one interpret these correlation functions and represent them using diagrams? Is $$W^{(n)}$$ the sum of all connected diagrams with $$n$$ external legs? Likewise, is $$\Gamma^{(n)}$$ the sum of all 1PI diagrams with $$n$$ external legs?

There also exists an object $$\Sigma$$, which is the sum of all 1PI diagrams with 2 external legs (Peskin & Schroeder page 219). Does this mean that $$\Sigma = \Gamma^{(2)}$$?

I often see $$\Sigma$$ written with an argument, for example $$\Sigma(p)$$ or $$\Sigma(p^2)$$, is $$p$$ the momentum associated with the two external legs?

There already exists a similar question (What is, diagrammatically, the 2-vertex $\Gamma^{(2)}$?) but I do not fully understand the accepted answer, specifically why the diagram they show represents $$\Gamma^{(n)}$$.

• The connected $$n$$-point function $$\langle \phi^{k_1}\ldots \phi^{k_n}\rangle^c_{J=0}~=~\left(\frac{\hbar}{i}\right)^{n-1} W_{c,n}^{k_1\ldots k_n}$$ is the sum of connected Feynman diagrams with $$n$$ external legs.

• For $$n\geq 3$$ the $$n$$-point function$$^1$$ $$\Gamma_{n,k_1,\ldots k_n}$$ of the effective/proper action $$\Gamma[\phi_{\rm cl}]$$ is the sum of 1PI Feynman diagrams with $$n$$ external amputated legs, cf. e.g. this Phys.SE post.

• The $$2$$-point functions $$i^{-1}(W_{c,2})^{k\ell}$$ and $$i^{-1}(\Gamma_2)_{k\ell}$$ are inverses of each other, cf. e.g. eq. (8) in my Phys.SE answer here.

• For $$n\geq 3$$ an $$n$$-point function$$^1$$ $$W_{c,n}^{k_1\ldots k_n}$$ of connected Feynman diagrams is a sum of possible trees consisting of 1PI $$m$$-vertices $$\Gamma_{m,\ell_1,\ldots\ell_m}$$ with $$m\geq 3$$ and lines made of connected propagators $$(\Gamma_2^{-1})^{k\ell}=-(W_{c,2})^{k\ell}$$, cf. e.g. this Phys.SE post. Note that the 2-point function $$(\Gamma_2)_{k\ell}$$ here plays a very different role than the higher point functions.

• The difference between on one hand the $$2$$-point function/inverse connected propagator $$(\Gamma_2)_{k\ell}=-(W_{c,2}^{-1})_{k\ell}$$, and on the other hand the self-energy $$\Sigma$$, is that the former contains an inverse free propagator, cf. e.g. this Phys.SE post. (Note that a free propagator is not 1PI.)

• For further description of the Legendre transformation between $$W_c[J]$$ and $$\Gamma[\phi_{\rm cl}]$$, see e.g. this Phys.SE post.

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$$^1$$ We assume for simplicity in this answer that there are no tadpoles.

• Thank you for your answer. Sorry if it was unclear in my post but my main questions are: (1) Can $W^{(n)}$ be interpreted as the sum of all connected diagrams with $n$ external legs? (2) Can $\Gamma^{(n)}$ be interpreted as the sum of all 1PI diagrams with $n$ external legs? (3) Given that $\Sigma$ is defined to be the sum of all 1PI diagrams with two external legs I would think that $\Sigma = \Gamma^{(2)}$, but this is not true as we have that to first order $\Gamma^{(2)} = p^2 + m^2 + \Sigma(p^2)$, so what is difference between $\Sigma$ and $\Gamma^{(2)}$ in terms of diagrams? Commented Mar 29 at 4:25
• I updated the answer. Commented Mar 29 at 4:48
• Thank you very much! If I may ask a follow up question, why is the case $n = 2$ exceptional for $\Gamma^{(n)}$? How does its interpretation as the sum of all $2$-point amputated 1PI diagrams introduce a propagator? I understand the proof of this fact using geometric series, but how does this follow diagrammatically? Commented Mar 29 at 5:51
• Another way to word my question would be why does the interpretation of $\Gamma^{(n)}$ being the sum of all 1PI diagrams with $n$ external amputated legs only hold for $n \geq 3$ and not $n = 2$? Why does the case $n = 2$ contain a free propagator but $n \geq 3$ does not? Commented Mar 29 at 6:01
• Well, apart from the perhaps most honest answer That's what the calculation tell us, note that we never think of the entire 2-point function $(\Gamma_2)_{k\ell}$ as a 2-vertex-interaction. Commented Mar 29 at 7:04