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.

Filter by
Sorted by
Tagged with
3 votes
2 answers
90 views

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 ...
user avatar
  • 767
2 votes
1 answer
55 views

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 $\phi_{\text{cl}}$. ...
user avatar
1 vote
1 answer
74 views

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 $$\frac{i}{p^2-m^2-B}\tag{1}\label{1}$$ ...
user avatar
  • 1,287
8 votes
1 answer
110 views

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 ...
user avatar
  • 487
1 vote
0 answers
63 views

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: $$ {\...
user avatar
1 vote
1 answer
102 views

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 ...
user avatar
  • 37
3 votes
2 answers
163 views

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 ...
user avatar
0 votes
0 answers
40 views

One-loop effective potential

Is there a known version of the one-loop effective potential (for e.g. an elementary massive scalar particle with quartic interactions) that doesn't utilize the assumption that the field is in ...
user avatar
  • 1,188
5 votes
2 answers
167 views

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 ...
user avatar
  • 83
1 vote
0 answers
65 views

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 ...
user avatar
  • 1,618
1 vote
1 answer
133 views

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 ...
user avatar
0 votes
0 answers
73 views

Interpreting the derivative of the effective potential with the tadpole

I am currently going through the Finite Temperature Field Theory notes by Mariano Quiros, and I stumbled upon a claim that I have seen often in the literature, namely that the first derivative of the ...
user avatar
1 vote
0 answers
82 views

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 ...
user avatar
  • 167
6 votes
0 answers
79 views

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 $\...
user avatar
  • 8,454
10 votes
0 answers
290 views

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)\...
user avatar
  • 3,441
6 votes
1 answer
195 views

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 ...
0 votes
0 answers
93 views

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 ...
user avatar
1 vote
2 answers
184 views

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 ...
user avatar
  • 1,375
3 votes
0 answers
95 views

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 ...
user avatar
3 votes
1 answer
64 views

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 ...
user avatar
0 votes
2 answers
129 views

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 ...
user avatar
  • 141
1 vote
0 answers
53 views

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 ...
user avatar
  • 3,294
2 votes
0 answers
75 views

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 ...
user avatar
2 votes
0 answers
126 views

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}...
user avatar
  • 65
1 vote
1 answer
127 views

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{ ...
user avatar
  • 900
5 votes
2 answers
382 views

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 ...
user avatar
  • 767
2 votes
0 answers
108 views

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 ...
user avatar
  • 855
3 votes
0 answers
91 views

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{...
user avatar
2 votes
1 answer
231 views

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: $$...
user avatar
5 votes
1 answer
241 views

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 ...
user avatar
  • 103
0 votes
0 answers
67 views

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): $$\int \frac{d^4x}{2g}\phi^2(x)+\int d^4x \ \log(-\nabla^2+\mu^2+i\phi(x)). $$ ...
user avatar
  • 555
7 votes
0 answers
230 views

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 ...
user avatar
  • 585
1 vote
1 answer
86 views

Why do the diagrams in $\Gamma[\Phi]$ differ from those in $\Phi\Gamma^{\rm int}_{\Phi}[\Phi]$ only by numerical prefactors?

Suppose $W$ is the generator of connected Feynman diagrams in $\Phi^4$ theory. We define $$\Gamma[\Phi]=W[j]-W_jJ,\tag{13.37}$$ where $$W_jJ=\int{dxW_j(x)j(x)}\tag{13.38}$$ and $$ \Phi\equiv\frac{\...
user avatar
1 vote
0 answers
36 views

Substitution of propagator to a product in Zinn-Justin

I'm reading Quantum Field Theory and Critical Phenomena, 4th ed., by Zinn-Justin and on page 127 he defines an action $$S_{\epsilon}[\phi]=\int{dxdy} \phi(x)\phi(y)[K(x,y) + \epsilon ] + V(\phi).$$ ...
user avatar
0 votes
0 answers
57 views

Any connected diagram is a tree of full propagators

In P. Etingof, Geometry & QFT, MIT 2002 online lecture notes; Lemma 3.11 (https://physics.stackexchange.com/users/7266/abdelmalek-abdesselam).) He says that any connected diagram is a tree of ...
user avatar
1 vote
0 answers
426 views

Quantum Scalar Field Theory with cubic and quartic interaction

If I have a scalar Lagrangian with and interaction term given by cubic and quartic terms (so a scalar theory + $φ^3+φ^4$ interaction), what are the possible divergent 1PI diagrams at one and two loops?...
user avatar
1 vote
0 answers
65 views

Effective potential and radiation corrections

I'm a bit confused on the idea of adding corrections to the classical potential of $\phi^4$ theory in QFT. From what I understand is that one should add corrections to the potential in order to ...
user avatar
  • 555
2 votes
0 answers
75 views

How does the generalized effective action in Wetterich's exact RG scheme relate to observables at different scales?

I am not familiar with Wetterich's exact RG paradigm, and cannot understand the main idea behind it. I understand that if one could have solved the model and obtained the all the n-point functions ...
user avatar
  • 517
3 votes
1 answer
446 views

2PI-effective action and functional derivatives

I'm trying to work out the 2PI-effective action for complex scalar fields. Introducing a multi field index $(a,b,c...)$ the complex conjugation and all other degrees of freedoms are suppressed, and ...
user avatar
  • 368
8 votes
1 answer
400 views

Anomaly is due to the noninvariance of the path-integral under a symmetry. Is the noninvariance reflected on 1PI effective action?

When a symmetry is anomalous, the path integral $Z=\int\mathcal{D}\phi e^{iS[\phi]}$ is not invariant under that group of symmetry transformations $G$. This is because though the classical action $S[\...
user avatar
  • 24k
3 votes
1 answer
518 views

Proof of geometric series two-point function

In deriving the expression for the exact propagator $$G_c^{(2)}(x_1,x_2)=[p^2-m^2+\Pi(p)]^{-1}$$ for $\phi^4$ theory all books that i know use the following argument: $$G_c^{(2)}(x_1,x_2)=G_0^{(2)}...
user avatar
2 votes
1 answer
1k views

Definition of one-particle irreducible diagrams

Text books often defines a one-particle irreducible diagram (1PI diagram) as a connected diagram which does not fall into two pieces if you cut one internal line. Is this internal line the full ...
user avatar
1 vote
1 answer
317 views

Proof of 1-particle irreducible (1PI) diagrams

If we split the effective action into $$Γ[Φ] =\frac{1}2ΦiG_0^{-1}Φ + Γ^{int} [Φ]\tag{1}$$ we can show that the full propagator is given by $$G= i[iG − Σ]^{-1}\tag{2}$$ With $$Σ=-Γ_{ΦΦ}^{int} [Φ]\tag{3}...
user avatar
4 votes
1 answer
231 views

How do the renormalization factors disappear from the computation recipe of the S-matrix in Peskin & Schroeder (p. 229 eq. (7.45) & p.324)?

In the following I limit my considerations to 4-point diagrams. After the introduction of renormalized field operator (in renormalized perturbation theory) $$\phi_r= (\sqrt{Z})^{-1} \phi\tag{10.15}...
user avatar
2 votes
0 answers
192 views

The background field method of deriving a 2PI effective action. Calzetta and Hu book

I am going through "Nonequilibrium Quantum Field Theory" by Calzetta and Hu right now and it seems that I cannot fully understand the derivations in chapter 6.5. There, they consider the derivation of ...
user avatar
  • 111
4 votes
2 answers
2k views

One-loop Correction to Effective Action

This might be a stupid question. In Bailin and Love's "Cosmology in gauge field theory and string theory", the authors are describing how to calculate the effective potential at a finite temperature ...
user avatar
2 votes
1 answer
216 views

Does $ℏ$ play a role in the 1PI effective action?

In most cases, people discuss the effective action or the effective potential in the convention $\hbar=1$. Occasionally, we see the expression at the 1-loop order as $$\Gamma[\phi]=S[\phi]+\frac{i\...
user avatar
  • 3,441
1 vote
1 answer
826 views

One-loop effective action of QED and the partition function

Given the partition function for QED $$ Z= \int \mathcal{D}A_{\mu}\mathcal{D}\Psi \mathcal{D}\bar{\Psi}\, \text{exp}\left(- \frac{i}{4}\int F_{\mu\nu} F^{\mu\nu} + i \int \bar{\Psi} (i {\not} D-m) \...
user avatar
  • 289
1 vote
1 answer
168 views

What is the intuation of path integral in QFT? [closed]

It is known that the path integral in quantum mechanics means the summation of all probable classical trajectories between first and last measurement of the quantum state. In QFT this formalism leads ...
user avatar
  • 163
2 votes
1 answer
586 views

Field renormalization of $\phi^4$ to second order

In Peskin & Schroeder Problem 10.3 pg. 345 they renormalize the field in $\phi^4$ theory using the following 2-loop sunset diagram. When looking at the correlation function $G^{(2)}_0$ this would ...
user avatar