Questions tagged [1pi-effective-action]
In quantum field theory, an effective action is a modified version of the action which takes into account quantum mechanical corrections. The effective action action is a generating functional of the one-particle-irreducible (1PI) diagrams, which are diagrams that cannot be broken into two disconnected diagrams by cutting an internal propagator.
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Schwinger–Dyson equation represented in a different form?
This is a follow-up question to my earlier post. The Schwinger–Dyson equation on Peskin and Schroeder reads (p.308):
$$
\left\langle\left(\frac{\delta}{\delta\phi(x)}\int d^4x'\mathcal L\right)\phi(...
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Why gives trace of Hessian only 1-loop contributions?
Why does the term $\frac{i\hbar}{2}\log\det\left[\frac{\delta^2S}{\delta\phi^2}\right]$ in the expansion
$$\Gamma[\phi]=S[\phi]+ \frac{i\hbar}{2}\log\det\left[\frac{\delta^2S}{\delta\phi^2}\right] +…\...
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One-loop Effective potential, Higgs VEV and renormalization condition
On Peskin & Schroeder's QFT, chapter 11.4, the book discusses the computation of the effective action of linear sigma model.
I am troubled for the relation between renormalization condition and ...
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3-point 1PI vertex function in pseudoscalar Yukawa theory
Consider pseudoscalar Yukawa theory in 4D:
$$ S =\int d^4x\ \frac{1}{2}(\partial\phi)^2 - \frac{1}{2}m_\phi^2\phi^2 +\bar\psi(i\gamma^\mu\partial_\mu-m_e)\psi - ig\bar\psi\gamma^5\psi\phi -\frac{\...
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Coleman-Weinberg mechanism at two-loop
I'm trying to understand how to perform the CW mechanism (http://www.scholarpedia.org/article/Coleman-Weinberg_mechanism) to scalar theories at two-loop order. More specifically how to find the ...
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One-Loop Effective Potential for Scalar Theory in $d=3$: why does dimensional regularization leads to a finite expression?
I've been playing around with scalar theories as a personal exercise in getting beta functions by calculating the effective potential. Hollowood's Renormalization Group and Fixed Points in Quantum ...
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How to understand this diagram, and is it relevant to the renormalizability of a theory?
I'm having trouble understanding this diagram from my lecture note:
Each grey-shaded circle represents the diagrams for the two-point function $D(k)$. In equation, the diagram reads$$ \bar G_n(k_1,......
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How is the quantum effective action defined in a theory with more than one field?
How is the one-loop quantum effective action derived in a theory with more than one interacting field? When looking at some books and my course notes I find that the expression for the one-loop ...
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What is the definition of $n$-loop 1PI diagrams in QFT? [closed]
What is the precise definition of a $n$-loop one-particle irreducible ($1$PI) diagram?
For example, consider the following diagrams.
Is the first diagram a $0$-loop $1$PI diagram?
Is the second ...
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Getting units right when computing the effective action
In QFT in Euclidean signature, the one-loop effective action is given by
$$\Gamma[\Phi] = S[\Phi] + \frac{1}{2} \mathrm{STr}\log S^{(2)}, \tag{1}$$
where $S[\Phi]$ is the theory's classical action, $\...
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Functional determinant: linking Series, Heat-Kernel and Zeta function
I would like to express a functional determinant as a series of diagrams, using the zeta function renormalization applied to the heat-kernel method, but I don't know if it's possible. Let me explain:
...
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Equivalence of effective potential from 1PI effective action and Wilsonian effective action for QFT with a mass gap
I was reading https://arxiv.org/abs/0909.0859 and came across this statement at the end of Chapter 2:
"It can be shown that the effective potential defined in terms of the Wilsonian effective ...
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Relation between Feynman diagrams and the effective action [duplicate]
I have learned that the effective action can be obtained from legendre transforming the generating function $$W(J) = \ln(Z(J)),$$ which corresponds to connected graphs. However I don't quite get an ...
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$2PI$ contribution to the $2PI$ effective action
I have an issue understanding a point that my professor made in the lecture. We started with the following derivation:
Consider an even Theory, i.e. the action takes the form:
$$S[\phi] = \phi \cdot C^...
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Renormalization as adding new operators to the effective action
I found the following paper online: http://www2.mathematik.hu-berlin.de/publ/pre/2013/P-2014-01.pdf. I also attach a screenshot of the paragraph I am confused by (page 63-64).
What is meant by the ...
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Explicit calculation of the effective action for $\phi^4$ by a Legendre-transformation
Let's say my generating functional for the connected moments is given by
\begin{align}
W[J]&=\underbrace { -\frac { 1 }{ 2N } \ln { ( } Na) }_{ { ring } } +\underbrace { \frac { 1 }{ N^{ 2 } } \...
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$\Gamma^{1PI}$ contribution to the effective action $\Gamma$
So I am currently taking a class on advanced quantum field theory, and I came across something that I don't understand. I have been thinking about it for quite some time and I cannot get my head ...
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Effective Action behaviour in $SU(N)$ Gauge Theory in Solodukhin
In Solodukhins paper https://arxiv.org/abs/0802.3117
he says that the effective Action of a 4D CFT has the general structure:
$$W = \frac{a_0}{\epsilon^4} +\frac{a_1}{\epsilon^2}+a_2 \ln{\epsilon} +\...
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Can the Functional Renormalization Group not generate a flow that is generated perturbatively?
I think I might have stumbled on a calculation that appears to undergo renormalization when you compute it perturbatively, but not when you compute it using the FRG. Consider, for the sake of argument,...
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The effective action in the linear sigma model
I am reading the section 11.4 of Peskin and Schroeder's book (page 373), and there is a step I could not follow.
To calculate the effective action of linear sigma model, the determinant of $\frac{\...
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Effective action as a generating functional and its derivative expansion
On page 381 of Peskin and Schroeder, equation (11.90) reads
$$ \frac{\delta^2 \Gamma}{\delta \phi_{cl}(x)\delta \phi_{cl}(y)} = iD^{-1}(x,y).\tag{11.90}$$
I am having a bit of trouble interpreting ...
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Why is the functional derivative of the Lagrangian wrt. field evaluated at the classical field equal to negative of source current at lowest order?
I am having difficulty showing this equation in Peskin & Schroeder's Introduction to Quantum Field Theory (Section 11.4 p.340):
We wish to compute $\Gamma$ as a function of $\phi_{\text{cl}}$. ...
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Resummation of single class of diagrams vs all 1PI diagrams
In the book A Modern Introduction to Quantum Field Theory, Maggiore considers the resummation of tadpole diagrams as its own individual geometric series to give
$$\frac{i}{p^2-m^2-B}\tag{1}\label{1}$$
...
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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 ...
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Computation of the effective action in Peskin & Schroeder
For the computation of the effective action Peskin & Schroeder choose the perturbative expansion of the generating functional. For this purpose they split the full Lagrangian up in two parts:
$$ {\...
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Equation 13.20 of Peskin & Schroeder
I don't quite understand some skipped steps in the book An Introduction to Quantum Field Theory, by Peskin & Schroeder. Here is an example, I don't know why Taylor series would lead to this.
In ...
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What is, diagrammatically, the 2-vertex $\Gamma^{(2)}$?
I know that the 2-vertex $\Gamma^{(2)}$ is the second derivative of the effective action, but I fail to see what it is diagrammatically: is it the truncated 1PI diagram? The non-truncated one?
If this ...
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Can 1-loop corrections generate a neutrino mass?
I'm looking at the Electroweak theory with massless neutrinos. The neutrino has an interaction with the $W$-boson and electron which gives 1-loop corrections to the propagator via the self-energy. I'm ...
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What exactly is effective interaction with regards to Feynman diagrams?
What does it actually mean to calculate to calculate the effective interaction using Feynman diagrams? To be concrete, let us consider the example of random phase approximation (RPA) calculation of ...
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One-loop 1PI effective action and dressed propagators
I'm working through the notes of https://www.hef.ru.nl/~kleiss/qft.pdf
and I'm currently on page 46: Computing the 1PI effective action and I'm having a lot of problems with the language.
We start by ...
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Similarity between 1PI effective action and Wilsonian effective action
A similar question about the difference between 1PI and Wilsonian effective actions was asked and answered here. Now I ask, when are they the same? Particularly, Seiberg says here (Pg 6, Sec 2.3) that ...
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Why one-particle irreducible functional is closely related to pressure (electroweak phase transition)?
Consider the following system: the SM lagrangian somewhat below the EW transition, where we keep only bilinear terms, only the heaviest fermion -- $t$-quark, and plus the potential terms for a VEV $\...
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How to perform a derivative of a functional determinant?
Let us consider a functional determinant
$$\det G^{-1}(x,y;g_{\mu\nu})$$
where the operator $G^{-1}(x,y;g_{\mu\nu})$ reads
$$G^{-1}(x,y;g_{\mu\nu})=\delta^{(4)}(x-y)\sqrt{-g(y)}\left(g^{\mu\nu}(y)\...
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Diagrammatic representations of generating functionals $Z[J]$, $W[J]$, and $\Gamma[\varphi]$
The book Boulevard of Broken Symmetries by Adriaan Schakel gives an excellent, if not exceedingly brief, overview of the path integral approach to perturbation theory. In particular, pages 47-58 give ...
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Methods for deriving goldstone theorem
I am reading chapter 19 of Weinberg Vol. 2, and in section 19.2 he gives two proof of goldstone theorems - one uses functional method involving quantum effective action and other is operator method ...
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Why do we need effective action $\Gamma$ given the connected generating functional $W$?
I have just learnt the path integral formalism in QFT, up to the point where we computed the generating functionals $\mathcal{Z}[J] := Z[J]/Z[0]$, $W[J]$, and $\Gamma[\varphi]$. Here $J(x)$ is the ...
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One-loop approximation in chiral sigma models
Consider a principal chiral model $$S[g] = \frac{1}{4\pi\lambda^2}\int(|g^{-1}dg|^2) = \frac{1}{4\pi\lambda^2}\int(g^{-1}dg\wedge\star g^{-1}dg) ,\tag{2.1}$$ where $g:\Sigma \rightarrow G $ is the so ...
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From 1PI function to cross section
The full propagator in QFT is the inverse of this two-point 1PI function defined as $$
\tilde{\Gamma}^{(2)}(p)=p^{2}-m^{2}-\Sigma(p)
$$
where $\Sigma(p)$ is the self-energy. I am have calculated the ...
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Fermionic Version of the effective Action
For a scalar field theory one introduces the partition function with external sources
$$
Z[j] = \int \mathscr{D} \varphi \, \exp \left( -S[\varphi] + \int j \, \varphi \right) \text{,}
$$
the analogon ...
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Meson operator treated as an elementary field in effective action?
I have seen in many papers, with this as the earliest example I can find, that the meson chiral order operator $M_j^{\,\,\,i}(x)=\overline{\psi}^i(1+\gamma_5)\psi_j(x)$ is treated as an elementary ...
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Conditions for renormalization of effective potential
Currently I am reading Das' Finite Temperature Field Theory. After computing the first order correction to the potential with a cut-off (eq. 6.77) Das states that, "we add counterterms and ...
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Effective action quantum field
I am reading Peskin and Schroeder Section 11.4. They derive a formula for the effective action p.372 Equation 11.63 using a scalar field interaction,
$$
\Gamma \left ( \phi _{cl} \right )=\int d^{4}...
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Why does the background field effective action generate only vacuum graphs?
I refer to LF Abbott's "Introduction to the background field method". The background field generating functional is
$$ \tilde{Z}[J,\phi] = \int \mathcal{D}Q \exp i[S[Q+\phi] + J.Q], \text{ ...
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How to compute the quantum effective action from 1PI Feynman diagrams?
On page 33 of these notes by David Skinner, it is claimed that
[starting from a connected graph and removing the bridges] tells us how to compute $\Gamma(\Phi)$ perturbatively from the original ...
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Confusions on effective action and 1PI 3-point Green's function [duplicate]
In Peskin's QFT book chapter 11.5, the author gives a graph(see below) and claims that the third functional derivative for effective action $\Gamma$ gives the 1PI (one-particle-irreducible) 3-point ...
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Why don't counterterms appear in one-loop correction for 1PI effective action?
In Zee's "QFT in a Nutshell: Second Edition", section IV.3, the author calculates the 1PI effective potential for a single real scalar field. The full Lagrangian is given by equation (1):
$$\mathcal{...
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Srednicki's explanation of the 1PI quantum action
In chapter 21 (p.127-129) of Srednicki's book the quantum action $\Gamma(\phi)$ is defined in formula (21.1) I won't repeat here (it's quite long). Then he considers the following path integral:
$$...
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QED electron self-energy in 1PI effective action
The electron self-energy at one-loop is given by the one-particle irreducible graph
I know how to calculate it using the Feynman rules but I was wondering how this diagram appears in the QED ...
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Feynman rules for space-dependent coupling
Let's say I have an effective action which looks like (I got this action from large $N$ method for $\varphi^4$ theory):
$$\int \frac{d^4x}{2g}\phi^2(x)+\int d^4x \ \log(-\nabla^2+\mu^2+i\phi(x)). $$
...
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Must the mean field, in the context of the background field method, satisfy the classical equations of motion? [duplicate]
When deriving the effective action $\Gamma$ in the background field method, one splits the field $\phi = \phi_b + \phi_f$ into a background (or mean field) $\phi_b$ and fluctuations $\phi_f$, then ...
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