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Questions tagged [1pi-effective-action]

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6
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0answers
135 views

Must the mean field, in the context of the background field method, satisfy the classical equations of motion?

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 ...
0
votes
1answer
40 views

1-loop Correction

Given a Lagrangian: $$L = \frac{1}{2}(\partial_u\phi)^2 + g (\partial_u\phi)^4.$$ Does anyone have idea how to write down the 1-loop correction? The derivative coupling is the part that confuses me.
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0answers
22 views

What happens to the (Wilsonian) effective action if a symmetry is spontaneously broken?

A spontaneously broken symmetry is a symmetry of the action which does not manifest itself in physical states. Since the action is still invariant under this symmetry, can we say the same about the ...
1
vote
1answer
63 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{\...
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0answers
20 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 $$S_{\epsilon}[\phi]=\int{dxdy} \phi(x)\phi(y)[K(x,y) + \epsilon ] + V(\phi).$$ ...
0
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0answers
37 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 ...
1
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0answers
135 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?...
0
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0answers
17 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 ...
1
vote
0answers
47 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 ...
3
votes
1answer
65 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 ...
7
votes
1answer
159 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[\...
2
votes
1answer
143 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)}...
1
vote
1answer
190 views

Definition of one particle irreducible diagrams

Text books often defines 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 ...
1
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1answer
108 views

Proof of 1-particle irreducible (1PI) diagrams

If we split the effective action into $$Γ[Φ] =\frac{1}2ΦiG_0^{-1}Φ + Γ^{int} [Φ]$$ we can show that the full propagator is given by $$G= i[iG − Σ]^{-1}$$ With $$Σ=-Γ_{ΦΦ}^{int} [Φ]$$ Here $Γ_{...
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0answers
106 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 ...
3
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2answers
321 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 ...
1
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1answer
128 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\...
1
vote
1answer
262 views

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) \...
1
vote
1answer
115 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 ...
2
votes
1answer
216 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 ...
5
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3answers
354 views

Srednicki QFT Chapter 29: Feynman diagrams for calculating the effective action

I am trying to work my way through Srednicki Chapter 29 on Wilson's approach to renormalisation. However I am unsure why the Feynman diagrams Srednicki considers and calculates in this chapter are the ...
1
vote
1answer
151 views

Peskin and Schroeder Eq. 11.103: how does it relate to one-loop diagrams?

I am a bit confused by how Peskin & Schroeder describe the corrections to the two point function of the linear sigma model from the second functional derivative of the effective action. Without ...
4
votes
1answer
469 views

Difference Between Vertex Function and Self Energy

I am trying to understand the difference between the 2-point vertex function and the self energy. In many presentations, they are described in ways that seem nearly equivalent, yet as I work through ...
2
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1answer
191 views

About the calculation of one-particle-irreducible two-point diagram

This is derived from the answer and comments of this Phys.SE question concerning the calculation of two-point one-particle-irreducible diagram. On the one hand, according to the discussion on P.236 ...
2
votes
1answer
128 views

Is the sum of one-particle-irreducivle two-point diagrams always a real number?

On page 388 in section 11.6 of Peskin and Shroeder. There appears an equation of the inverse propagator(the second functional derivative of the effective action) for a theory that contains several ...
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2answers
162 views

Can one forget about the contribution of 1PR diagrams in computing a scattering amplitude?

From the LSZ reduction formula, it is clear that only the connected Feynman diagrams that contribute to a scattering amplitude. However, connected diagrams are of two types: 1PR and 1PI. 1PR diagrams ...
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0answers
93 views

How is $\Gamma^{(n)}$ equal to the sum of all 1PI Feynman diagrams with $n$ external legs? [duplicate]

The quantum effective action functional can be expanded in a functional Taylor series as $$ \Gamma[\phi_c]=\sum\limits_{n=0}^{\infty}\frac{1}{n!}\int d^4x_1 \int d^4x_2...\int d^4x_n \Gamma^{(n)}(x_1,...
1
vote
0answers
203 views

Is Schrieffer-Wolff transformation equivalent to Feynman diagram and Path integral?

In high energy community, people usually use path integral (or Feynman diagram) to derive effective action (or effective Hamiltonian). However, in condensed matter or AMO community, people usually use ...
3
votes
2answers
2k views

How to correctly understand these “1-particle-irreducible insertions”?

In QED, when dealing with the vacuum polarization and the photon propagator, some authors like Peskin & Schroeder introduce the so-called "1-particle irreducible" diagrams. These are defined as: ...
1
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1answer
75 views

Concerning a functional of a functional of the former - classical fields in Quantum Action

Let $\varphi(x)$ and $j(x)$ be two field configurations. Let $\Gamma[\varphi]$ be a functional of the field $\varphi$ defined by: $$ \Gamma[\varphi] := \inf_j \ F[\varphi, j] = F[\varphi, j_\varphi] \...
12
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2answers
249 views

Probabilistic Intuition behind connected correlations and 1PI vertex function

In the context of statistical field or quantum field theory, one encounters so called generating function(al) for connected correlations, aka the following function(al): $$ W(J) = \ln (Z(J))$$ $$ Z(...
0
votes
1answer
125 views

A question concerning the effective quantum action for a scalar field

Define the quantum action $\Gamma[\varphi]$ by: $$ \Gamma[\varphi] := -\frac{1}2\int \frac{d^Dk}{(2\pi)^D} \varphi(-k)\Big(k^2 + m^2 - \Pi(k^2)\Big)\phi(k) \\+ \sum_{n=3}^\infty \frac{1}{n!}\int \...
10
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2answers
861 views

In what sense is the proper/effective action $\Gamma[\phi_c]$ a quantum-corrected classical action $S[\phi]$?

There is a difference between the classical field $\phi(x)$ (which appears in the classical action $S[\phi]$) and the quantity $\phi_c$ defined as $$\phi_c(x)\equiv\langle 0|\hat{\phi}(x)|0\rangle_J$$ ...
4
votes
1answer
329 views

Defining a classical field corresponding to a quantum field

Why is the expectation value of the quantum field in the vacuum state $$\phi_c(x)=\langle0|\hat{\phi}(x)|0\rangle_J=\frac{\delta W}{\delta J}$$ referred to as the classical field? Why not the ...
6
votes
2answers
419 views

Why do we need the supremum when performing Legendre transformations?

Legendre transforms appear all over physics. For instance, in statistical mechanics, they allow us to move between descriptions in terms of different thermodynamic potentials. Similarly, in quantum ...
1
vote
1answer
58 views

Incorporation of adiabatic phase into quantum effective action

Suppose we have a system (or a subsystem) in the quantum state $|\text{in}\rangle$ and the same system in the state $|\text{out}\rangle$, which differs from $|\text{in}\rangle$ only by a phase: $$ \...
2
votes
0answers
844 views

Symmetry factor for 1PI Feynman diagrams in $\phi^4$ theory

I am trying to understand the various factors that the Feynman amplitude will carry corresponding to the Feynman diagrams of Fig. 1 of this reference. I understand that the $n^{th}$ diagram containing ...
4
votes
1answer
197 views

Deriving the equality $\frac{\delta \Gamma[\phi_c]}{\delta\phi_c(x)}=0=\langle 0|\frac{\delta S[\phi,J]}{\delta\phi(x)}|0\rangle$?

I'm trying to convince myself that $$\Gamma[\phi_c]=W[J]-\int d^4x\hspace{0.2cm} j(x)\phi_c(x)$$ is the effective action i.e., it contains all quantum corrections to the classical action $S[\phi]$. ...
1
vote
1answer
733 views

A question about two-point 1-particle-irreducible diagram

I have a simple question about 1-particle-irreducible (1PI) diagram, I know I misunderstood something trivial but I just can not figure it out. Following Introduction to quantum field theory by ...
2
votes
0answers
139 views

Interpretation of the chiral anomaly a-la Alvarez-Gaume

In the paper "The topological meaning of non-abelian anomalies" written by Alvares-Gaume and Ginsparg they argue the appearing of the (gauge) anomaly in a theory with chiral fermions in the following ...
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0answers
165 views

Effective potential in Lagrangian

I have two question related to the steps in equations 3-7 in this paper: Question 1 They find the effective potential in eq. (5) as the negative of the effective Lagrangian (eq. (3)). I don't see how ...
6
votes
1answer
336 views

Confusion about the calculation of 1PI effective action using path-integrals

The Lagrangian of the $\phi^4$-theory can be written in terms of bare parameters as $$\mathcal{L}=\frac{1}{2}(\partial_\mu\phi_0)^2-\frac{1}{2}m_0^2\phi_0^2+\frac{\lambda_0}{4!}\phi_0^4\tag{1}.$$ The ...
2
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1answer
138 views

How does one ensure that effective action includes all possible quantum corrections to the clasical action?

Consider a classical scalar field theory for a real scalar field $\phi$ given by $$\mathcal{L}=\frac{1}{2}(\partial_\mu\phi)^2-V(\phi)$$ where $V(\phi)$ is the classical potential. In quantum field ...
2
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0answers
112 views

Locality of Wess-Zumino terms and Goldstone bosons

Suppose a theory with a fermion sector $\psi$ having some global chiral symmetry group $G$ without internal anomalies (i.e., a group whose algebra generators $t_{a}$ give zero coefficients $D_{abc}\...
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0answers
214 views

The locality of Wess-Zumino terms

Suppose the simple theory with chiral fermions possessing non-trivial gauge anomalies cancellation: $$ S = \int d^4 x \big(\bar{\psi}i\gamma_{\mu}D^{\mu}_{\psi}\psi + \bar{\kappa}i\gamma_{\mu}D^{\mu}_{...
0
votes
1answer
111 views

Determinant of a propagator in Effective potential

Last days I have a hard time calculating the Effective potential in Scalar Quantum Electrodynamics. Right now I stuck in the following determinant $$ Det\left[(-k^2+e^2\phi_c^2)g_{\mu\nu}+k_\mu k_\nu ...
3
votes
1answer
467 views

The non-abelian chiral anomaly and one-loop diagrams higher than the triangle one

Suppose chiral fermions $\psi$ interacting with gauge fields $A_{\mu,L/R}$. With $P_{L/R} \equiv \frac{1\mp\gamma_{5}}{2}$ and $t_{a,L/R}$ denoting the generators, the corresponding action reads $$ S =...
1
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1answer
247 views

Effective potential in scalar-vector interaction

I am reading the thesis of Erick James Weinberg from arXiv: https://arxiv.org/abs/hep-th/0507214. On page 22 he finds the effective potential in scalar-vector interaction in one-loop. I am having a ...
2
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0answers
139 views

How to find the CP violating effective operator?

I've been meeting with one not-very-small problem in arriving at (3.3) in Shaposhnikov's "Structure of the high temperature gauge ground state and electroweak production of the barygon asymmetry" (...
2
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0answers
68 views

chiral partition function - group integral

I am reading this Thermodynamics of chiral symmetry and i really want to know how to perform a group integral, and why it is proportional to the effective action in this case? I have not been able to ...