Questions tagged [1pi-effective-action]

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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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25 views

Scattering matrix element working with an effective action

I've obtained the Euler-Heisenberg effective action and I'm trying to obtain it's photon-photon scattering cross section in order to compare it with the complete QED cross section. I was able to ...
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1answer
82 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: $$...
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80 views

QED electron self energy in 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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54 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): $$\int \frac{d^4x}{2g}\phi^2(x)+\int d^4x \ \log(-\nabla^2+\mu^2+i\phi(x)). $$ ...
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155 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 ...
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1answer
126 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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1answer
74 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
29 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 $$S_{\epsilon}[\phi]=\int{dxdy} \phi(x)\phi(y)[K(x,y) + \epsilon ] + V(\phi).$$ ...
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46 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 ...
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266 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?...
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40 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 ...
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0answers
59 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
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1answer
179 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
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1answer
248 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
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1answer
253 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)}...
2
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1answer
559 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 ...
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1answer
160 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_{...
4
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1answer
155 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}...
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142 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 ...
4
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2answers
711 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
178 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\...
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1answer
440 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) \...
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1answer
143 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
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1answer
315 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 ...
6
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3answers
498 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
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1answer
204 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 ...
5
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1answer
633 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
218 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
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1answer
140 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
223 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
242 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 ...
4
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2answers
3k 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: ...
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1answer
78 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] \...
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2answers
278 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(...
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1answer
145 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 \...
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2answers
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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 $$\phi_c(x)\equiv\langle 0|\hat{\phi}(x)|0\rangle_J$$ ...
5
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1answer
367 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 ...
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2answers
539 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 ...
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1answer
62 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: $$ \...
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0answers
1k 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
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1answer
220 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 $S[\phi]$. ...
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1answer
948 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 ...
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0answers
159 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
176 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 ...
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1answer
438 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
167 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
128 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
247 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}_{...
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1answer
136 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 ...