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This question will fully refer to the presentation ref. 1, from which I'll take the numbering. Since it involves also diagrams and it appears as a fairly basic question about Wilsonian renormalization, I won't be exhaustive.

After stating their programme, in ref. 1 they proceed to integrate out the "fast" modes in the partition function of $\phi^4$ theory, which will lead to an effective theory $(b<1)$ $$Z[J]=\int [D\phi]_{b\Lambda}\exp\left\{-\int d^dx\mathcal{L}_\text{eff}\right\}\tag{12.6}$$ where $\exp\left\{-\int d^dx\mathcal{L}_\text{eff}\right\}$ arises after an integration of $\exp\left\{-\int d^dx\mathcal{L}\right\}$ in the momentum shell $[b\Lambda,\Lambda]$. At this moment P&S proceed to treat the problem perturbatively, i.e. they consider the non-kinetic part of the lagrangian as a perturbation. Casting things in terms of diagrams, at first order in $\lambda$ we are left with a tadpole correction to the mass $m^2$ term of the theory, while at second order two diagrams arise (c.f. eq. $(12.13)$ in ref.1). One is the squared tadpole diagram, which is disconnected and the other is a fish diagram, giving a correction to the coupling $\lambda$. Then, they claim that with an argument analogous to $(4.52)$ (which was the exponentiation of disconnected diagrams) here we get an exponential of the sum of connected (???) diagrams. $$\mathcal{L}_\text{eff}=\mathcal{L}+(\text{connected diagrams})\tag{12.18}.$$

I can't see why disconnected diagrams do not contribute in the present case and where the analogy they mention really is, so I'd be grateful if you helped me flesh this out. It would be ok to restrict to second order, as I did.

Reference

  1. An Introduction to Quantum Field Theory, M. Peskin, D. Schroeder. CRC Press, 1995. Chapter 12, pp. 394-399.
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    $\begingroup$ I think that, basically, integrating out the fast modes produces a kind of generating functional of connected diagrams $W^\Lambda[J]$. This is the same reasoning as $Z[J] = e^{-W[J]}$. So you have the integrand of (12.6) that is simply $e^{-S-W[J]}$, where now we can identify $S+W[J]=S_\text{eff}$. In a sense, you have $\mathcal{L}_\text{eff} = \mathcal{L}+(\text{connected diagrams})$, but $e^{-S_\text{eff}} = e^{-S}\times (\text{all diagrams})$. $\endgroup$ Commented Mar 7 at 11:33

1 Answer 1

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  1. The Wilsonian effective action is defined as $$\begin{align} \exp&\left\{-\frac{1}{\hbar}W_c[J^H,\phi_L] \right\} \cr ~:=~& \int_{\Lambda_L\leq |k|\leq \Lambda_H} \! {\cal D}\frac{\phi_H}{\sqrt{\hbar}}~\exp\left\{ \frac{1}{\hbar} \left(-S[\phi_L+\phi_H]+J^H\phi_H\right)\right\},\end{align}\tag{W}$$ cf. eqs. (12.5) & (12.6) in Ref. 1.

  2. The right-hand side of eq. (W) has an interpretation as the sum of all Feynman diagrams of heavy/high modes $\phi_H$ using arguments similar to my Phys.SE answer here.

  3. The Wilsonian effective action $W_c[J^H,\phi_L]$ consists of connected Feynman diagrams of $\phi_H$, due to the linked cluster theorem, cf. e.g. this Phys.SE post.

References:

  1. M.E. Peskin & D.V. Schroeder, An Intro to QFT, 1995; section 12.1.
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  • $\begingroup$ Thanks, Qmechanic. 1. If I underdstand correctly, this is the same theorem as in chapter 2 of Weinberg vol. 2, concerning the effective action. 2. Could you please comment on how it's related to the exponentiation of vacuum bubbles ($(4.52)$) as P&S say? In that case it appears to me that the exponentiated ones are not the connected diagrams. $\endgroup$ Commented Mar 7 at 12:35
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    $\begingroup$ 1. Yes, eq. (16.1.3). 2. One should be aware that eq. (4.52) refers to a numerator of a 2-point function rather than the partition function. $\endgroup$
    – Qmechanic
    Commented Mar 7 at 13:08

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