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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 $\Gamma^{(n)}$.

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1 Answer 1

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  • 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.

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  • $\begingroup$ 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? $\endgroup$
    – CBBAM
    Commented Mar 29 at 4:25
  • $\begingroup$ I updated the answer. $\endgroup$
    – Qmechanic
    Commented Mar 29 at 4:48
  • $\begingroup$ 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? $\endgroup$
    – CBBAM
    Commented Mar 29 at 5:51
  • $\begingroup$ 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? $\endgroup$
    – CBBAM
    Commented Mar 29 at 6:01
  • $\begingroup$ 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. $\endgroup$
    – Qmechanic
    Commented Mar 29 at 7:04

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