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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Quantum effective action for Yang-Mills theories

During a course I came across a formula for the quantum effective action of a Yang-Mills theory in euclidean space and it appears like this (some indices may be dropped but I hope that won't be a ...
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In QFT, is the effective potential at tree-level always the same as the potential? Why?

If I have a simple scalar theory $$ \mathcal{L}(\phi) = \frac{1}{2} (\partial_\mu \phi)^2 - V(\phi), $$ the effective potential $V_{eff}$, derived from $Z\rightarrow W \rightarrow \Gamma \rightarrow ...
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Renormalization conditions and proper vertices at tree level

I'm trying to understand a statement of my teacher of TQFT: he said that when expanding the effective action in terms of proper vertices, we can choose a new theory with only tree diagrams in order to ...
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Generating functional for fields with non-zero expectation value

When doing QFT or statistical field theory of, say $N$ scalar fields $\varphi_i$, we consider the the generating functional $$ W[J] = - \ln Z[J], \quad Z[J] = \int \mathcal D \varphi \, e^{-S[\varphi] ...
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Evaluation of functional derivative of effective action

I'm trying to understand a calculation in appendix A of this paper https://arxiv.org/abs/2204.04197, however I don't understand how they end up with equation (125) and I think I am going wrong in the ...
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1PI effective action and Action generated through Hubbard-Stratonovich transformation

In standard lectures on advanced QFT one learnt that performing Legendre transformation leads to effective actions generating one-particle-irreducible (1PI) diagrams, which is encoded by Schwinger-...
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Clarification about Weinberg effective action and effective potential

I'm going over Weinberg Quantum effective action section in his second book, I understood his derivation and reasoning up to the equation: $$iW[J]=\int_{\substack{Connected\\trees}}\left[\mathcal{D}\...
Raeed Mundow's user avatar
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A question on Schwartz's derivation of the Euler-Heisenberg Lagrangian

In Subsection 33.2.2. of Schwartz's Quantum Field Theory and the Standard Model, he starts to derive the Euler-Heisenberg effective Lagrangian by "replacing" the field which is being ...
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1PI effective potential vs self-energy

Consider the following Lagrangian describing the interaction between a massless field $\phi$ and a massive field $\psi$: $$ {\scr L} = \frac12(\partial\phi)^2 + \frac12 (\partial\psi)^2(1 + f(\phi/M)) ...
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False vacua and the effective action

The effective action $\Gamma[\phi]$ of a (scalar) field theory described by the action $S[\phi]$, can be computed in the "background field method" by shifting $S[\phi + \phi_b]$, where $\...
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Functional derivative of the generating functional with respect to the source term

To be specific, let us use the notations in W. Metzner et al., Rev. Mod. Phys. 84,299 (2012). The generating functional $G[\eta, \bar\eta]$ is given by [Eq. 4] \begin{align} G[\eta, \bar\eta] = - \ln \...
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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 ...
Níckolas Alves's user avatar
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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, $\...
Níckolas Alves's user avatar
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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,...
Níckolas Alves's user avatar
2 votes
2 answers
248 views

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{\...
David Shaw's user avatar
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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 $\phi_{\text{cl}}$. ...
Andrew Dynneson's user avatar
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Resummation of single class of diagrams vs all 1PI diagrams

Maggiore considers on page 136 in Section 5.6 Renormalization in the book A Modern Introduction to Quantum Field Theory, the resummation of tadpole diagrams as its own individual geometric series to ...
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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 in section 11.4 Peskin & Schroeder choose the perturbative expansion of the generating functional. For this purpose they split the full Lagrangian up in ...
Frederic Thomas's user avatar
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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 ...
LWC's user avatar
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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 ...
Mauro Giliberti's user avatar
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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 ...
Некто's user avatar
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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 ...
BVquantization's user avatar
3 votes
1 answer
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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 ...
theoreticalphysics's user avatar
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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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