Questions tagged [renormalization]

This tag is for questions which relates with the renormalization, an ensemble of techniques which serves to treat the infinities which appear in quantum field theory or statistical mechanics. Renormalization procedures are based on the requirement that certain physical quantities (such as the mass and charge of an electron) equal observed (experimental) values.

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Questions on RG flows, CFTs, and UV and IR theories

In the space of field theories, Conformal field theories are fixed points in the RG flow. However, a lot of literature on CFT usually talks about a QFT being the RG flow between two CFTs: one UV and ...
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Weinberg, Chapter 12, General Renormalization Theory

Weinberg argues in section 12.2 (Cancellation of divergences) that for the renormalization program to work, it is essential that the Lagrangian includes all interactions that correspond to the ...
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Weinberg QFT Vol 1, charge renormalization

Weinberg says (Chapter 10, section 10.4) that we have to assume that the bare charge, $q_{Bl}$ that appears in the Lagrangian is equal and opposite for the electron and for the particles (two $u$-...
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Normal Ordered Product in Operator Product Expansions

In an example of operator product expansion applied to $\phi^4$ theory of the book QFT an integrated approach, where the Eulerian Lagrangian is $$\mathcal{L}=\frac{1}{2}\left(\partial_{\mu} \phi\right)...
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Why is there still disagreement over the mass of the bottom (or beauty) quark, but none of the others?

Wikipedia (among other places) lists two values for the alleged mass of the B quark, 4.18 and 4.65 GeV. Only one of the two possible masses listed has a link to another Wiki page explaining the ...
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The Beta Function and the Bare Charge

According to "Introductory Lectures on Quantum Field Theory", by L. Álvarez-Gaumé and M. A. Vázquez-Mozo, (P. 84) the dependence of the bare charge $e_{0}(\Lambda)$ on the cutoff $\Lambda$ ...
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Do I have to take the trace over gamma matrices in Yukawa vertex correction?

Given the Yukawa coupling $\mathcal{L}_{\text{int}}=g\phi\bar{\psi}\psi$, if I want to compute the correction to one loop to the vertex, I would write something like this $$\Lambda\sim g\int\frac{d^...
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Can we do regularization by just discarding the divergent part?

Consider the one-loop $\phi^4$ integral to order $\lambda^2$ $$I=\int \frac{d^{4}k}{(2\pi)^4}\frac{1}{k^2-m^2}\frac{1}{(k-p)^2-m^2}.$$ After some transformation (refer to the answer in here) $$I=\int_{...
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On the derivation of Eq. (12.5) in Peskin and Shroeder's QFT

I stuck at the derivation of Eq. (12.5) in Peskin and Schroeder's QFT. The authors tried to from (12.3) $$ Z = \int [\mathcal{D}\phi]_{\Lambda} \exp\left(-\int d^dx \left[\frac{1}{2} (\partial_{\mu} \...
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What causes running coupling constants to converge to one value at high energy?

In this article ("The coupling constants and unification of interactions", sicsmasterclasses.org) we can read: The running coupling constants. Scientists believe that as they push the ...
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MS bar renormalization scheme [duplicate]

Kindly suggest any reference or book in which I can grasp the idea of the MS bar renormalization scheme, and it will be beneficial if it shows some mathematical rigor.
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Is a running coupling constant a natural consequence in QFT, or is it a consequence of the "dressing-up" of particles?

The running coupling constant ("hold that constant!) is a well known phenomenon in quantum field theory. The constant varies with the energy of the interacting particles. I think this is rather ...
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Renormalization of the $\phi^4$ theory

The question is about the renormalization of the $\phi^4$ theory. Starting from the bare Lagrangian density, we can introduce physical parameters and physical variables through suitable ...
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Can we make a massive non-abelian gauge field renormalizable by gauge fixing without Higgs mechanism?

There have been a lot of similar questions about this topic on this website, such as Gauge invariance is just a redundancy. Why is massive abelian gauge field renormalizable but massive non-abelian ...
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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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How to write the amplitude for mixing inducing loop diagrams?

Consider figure T1-i-2-A, on pag. 8, on this paper. It generates a neutrino mass through the mixing of left and right-components of a fermion through its mass term: this is represented by the cross in ...
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Is there a nice derivation for the renormalization group β-functions in String Theory that leads to the Einstein Field Equations?

There are many existing threads here on Physics Stack Exchange which deal with how General Relativity arises from String Theory. However, all of them simply list the renormalization group beta ...
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Renormalization Group Flow

I am reading a book on effective field theory where the following "renormalization group equation" is given: Now a quick search on google shows a bunch of interesting pictures of "...
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Renormalization and virtual soft divergences

I am reading Weinberg's book on QFT. Specifically, chapter 13.2. The author calculates the effect of including infrared quantum corrections (i.e. associated with soft virtual photons) to amplitudes. ...
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What is the physical significance of the running mass $m^2(\mu)$? [duplicate]

In QFT under the $\overline{\text{MS}}$ subtraction scheme the renormalized mass $m_R$ becomes a $\mu$-dependent quantity which "runs" via a differential equation $$\frac{1}{m_R^2}\frac{d ...
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What are critical dimensions in statistical field theories (SFTs) and quantum field theories (QFTs) and how do they relate to divergences?

My question is the following. Statistical field theories (SFTs) and quantum field theories (QFTs) are usually associated with some upper critical dimension (UCD) and lower critical dimension (LCD). ...
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EFT matching: using tree-level to perform 1-loop-level

I'm reading the Saclay Lectures on EFT, and I don't understand how it uses the tree-level matching to compute the 1-loop-level matching. To simplify, in this post I'll put its $C_6,\lambda_1=0$ since ...
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Question about field configurations on the boundary of $\mathcal{I}^+$

I am reading Strominger's lecture notes "The infrared structure of gravity and gauge theory" (https://arxiv.org/abs/1703.05448). In chapter two, while trying to derive an expression about ...
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Implicit renormalization of $\phi^4$ theory

In the lecture notes, I am following on Quantum Field Theory, we want to renormalize $\phi^4$ theory with the Lagrangian: $$\mathcal{L} = -\frac{1}{2}(\partial_\mu \phi^0)^2 - \frac{1}{2} m_0^2 (\phi^...
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Faddeev - Kulish paper questions

I have several questions regarding the paper Asymptotic conditions and infrared divergences in QED, written by Faddeev and Kulish in 1970. This is not a post about one question, but since all the ...
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Virtual soft photon exponentiation

I have a question regarding Weinberg's soft exponentiation result in his 1965 paper called 'Infrared Photons and Gravitons' (https://doi.org/10.1103/PhysRev.140.B516). When he tries to calculate the ...
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Subleading soft theorem and gauge invariance

I am reading a paper by Burnett-Norman and Kroll, called extension of the Low soft photon theorem (https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.20.86). The authors claim that in order to ...
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Dimensional regularization vs. hard cutoff and their relation to the renormalization scale in 2d vs 4d to find $\beta$ functions

I would like to understand some shortcuts people are using to calculate $\beta$ functions using dim. reg. with mass scale $\mu$ and/or the hard cutoff $\Lambda$. My end goal is to use equation 12.53 ...
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Running of the fine structure constant: meaning of $\alpha(Q^2=0)$?

In Measurement of the Running of the Fine-Structure Constant, the L3 collaborations writes At zero momentum transfer, the QED fine structure constant $\alpha(0)$ is very accurately known from the ...
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Did I understand RG correctly?

I am currently self-studying Renormalization Group (RG) in Condensed matter physics (in preparation for graduate school while I'm in Alternative Military Service). While I'm writing bunch of ...
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Does the vacuum really have infinite energy density?

I said: As far as I understand it quantum field theory says that the vacuum has an infinite energy density. r/AskPhysics RedditorAbstractAlgebruh said: But wouldn't that be due to the way we do the ...
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Does Beta function depends on the physical process we consider?

Given a theory, i.e. an interaction, we normally say that we can compute its Callan-Symanzik $\beta $ function. The process can be sketched as(here I follow the notation in Weinberg, Vol 2.): ...
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Does the 1-loop beta function in $\phi^4$ theory depend on the renormalization scheme?

Consider Euclidean $\phi^4$ theory in $D=4$ $$ \mathcal{L}_0 = \frac{1}{2}\partial_{a}\phi_0\partial_{a}\phi_0 + \frac{1}{2}m_0^2\phi_0^2 + \frac{1}{4!}\lambda_0\phi_0^4 $$ The loop integrals are ...
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$\log{(4 \pi)}$ in dimensional regularization integral for vacuum polarization

I have been reading through Peskin's chapter 7 and I have arrived at this expression for the first order correction to the photon propagator. Peskin uses dimensional regularization and even though I ...
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Choosing a renormalization point where the amplitude is not defined?

This lecture note goes through the renormalization of the s-channel one-loop correction to the four-point function $\Gamma(s,t,u)$ in $\lambda \phi^4$-theory. It uses Pauli-Villars regularization, ...
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Is Planck length constant in String Theory? Does it have a renormalization flow?

Is Planck length constant? Planck length $l_p$ is dependent on Newton constant $G_N$ which is related to coupling constant of interaction of gravitons, but from field theory point of view, we know ...
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What boundary/initial values should one take for running the couplings in the Standard Model?

(This is a very naive question!) I am trying to run the Standard Model(SM) couplings at 2-Loop, which includes three gauge couplings, Yukawa couplings, and Higgs quartic coupling. This requires me to ...
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Deriving the momentum Feynman rule for a vertex with a derivative of the field

Consider the following modification of the klein gordon lagrangian: $$S = \int\left(\frac{1}{2}(\partial \phi)^2 - \frac{m^2}{2}\phi^2 + \frac{\delta Z}{2}(\partial \phi)^2 - \frac{\delta m^2}{2}\phi^...
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Are there still objections against the renormalizability of QED?

I read that even Dirac in 1975 and Feynman in 1986 had concerns with the renormalizability of QED: Renormalization. Which objections did they make? What is the present state concerning their ...
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Struggling with Peskin and Schroeder equation (12.49) and the constraint of renormalizability

In peskin and schroeder it's written that any renormalizable massless scalar field theory has a 2-point greens function of the form: I don't get how we can know that the 1-loop diagrams have exactly ...
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Symmetry Factors in $n$-point one-loop function for QCD

I am calculating (the divergent part) of the gluon 3-point function and gluon 4-point function in the QCD Lagrangian. So I have found here what I believe to be all the 1PI Feynman diagrams at one-loop....
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What are all the 1PI gluon 3-point function Feynman diagrams at 1-loop?

As an exercise in renormalization, I want to calculate the divergent part of the gluon 3-point function and gluon 4-point function matrix element. What are all the 1PI one-loop Feynman diagrams?
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Brownian motion and multi-scale stochastic processes

The Stokes-Einstein equation for the diffusion coefficient of small colloidal particles in suspension is canonically derived under the assumption that the primary motion of the particle is ...
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Loop Integrals and Dimensional Regularization

I want to calculate the divergent part of a Feynman diagram using the Feynman parameters: $$\frac{1}{A_1 A_2 \ldots A_n} = \int_0^1 dx_1 ... dx_n \delta (\Sigma x_i -1) \frac{(n-1)!}{[x_1 A_1 + x_2 ...
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Processes at $10^{-13}$ cm and smaller

I am reading the paper by David Bohm, "A Suggested Interpretation of the Quantum Theory in Terms of "Hidden" Variables. I". My question is not about Bohmian mechanics. However, he ...
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How to evaluate effective Feynman diagrams in the standard model?

I was reading the "Weak Hamiltonian, CP Violation and Rare Decays" by Andrzej J. Buras and, in the page 57 I don't understood how he calculate the effective Feynman diagrams for to get the ...
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Computing the anomalous dimension of $\phi^2$ via Peskin and Schroeder

I am trying to understand the example that Peskin and Schroeder present at section 12.4 where they calculate the AD of $\phi^2$. Specifically they give a renormalization condition in 12.113 which does ...
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How to show that in 2D CFT the marginal operator must have $(h,\bar h)=(1,1)$?

A related post might be What are marginal fields in CFT? where Qmechanic♦ pointed to Ginsparg secion 8.6. However, I heard about two argument. Claim 1:In a $D$ dimension CFT, the marginal operator ...
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Solving renormalization group equations when calculating $\bar{MS}$ mass

My textbook gives the RG equation: $$ \frac{d \bar{m}(\mu)}{d \ln \mu}=\gamma_{m} \bar{m}(\mu) ; \quad \gamma_{m}=-\frac{3 \alpha}{2 \pi} $$ And then says this is easy to solve and the solution is: $$ ...
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