Renormalization is an ensemble of techniques which serves to treat the infinities which appear in quantum field theory or statistical mechanics.

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Correlation length in d>1 Ising model, at zero temperature

I am studying the renormalization group approach to the Ising model using as a reference Cardy's book "Scaling and renormalization in statistical mechanics". I cannot understand what happens in the ...
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Renormalization in Classical Field Theory

1) The statement that general relativity (GR) is not renormalizable - is it a statement only about the quantization of GR or is it non-renormalizable also as a classical field theory? 2) More ...
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Why does renormalization need an unbroken symmetry?

Common wisdom is that for a QFT to be renormalizable it must be invariant under a symmetry transformation. Why does renormalization need an unbroken symmetry? Which is the first publication that ...
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Conversion of results between cutoff regularization and dimensional regularization

Generally it would be expected that a renormalizable/physical quantum field theory (QFT) would be regularization independent. For this I would first fix my regularization scheme and then compute ...
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Physical meaning of RG transformation

When we do RG transformation in Statistical mechanics we eliminate unnecessary degrees of freedom and it leads us to the fixed point. How can I visualize it physically?
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A pedestrian explanation of Renormalization Groups - from QED to classical field theories

shortly after the invention of quantum electrodynamics, one discovered that the theory had some very bad properties. It took twenty years to discover that certain infinities could be overcome by a ...
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Scaling with the Ising Model

I am stuck with one formula in the CFT book by Di Francesco and al. Chapter 3. Equation 3.46 third step, for those who don't have the book, he integrates out degrees of freedom from the Ising Model by ...
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Fast and slow modes in renormalization group of nonlinear sigma model

A general nonlinear sigma model can be expressed as \begin{equation} S[g] = \frac{1}{\lambda} \int d^dr\ \text{tr}[\triangledown g\triangledown g^{-1}] \end{equation} where $g$ takes value in a matrix ...
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Cut-off Regularisation and Renormalisation in Scalar Field Theory, Deriving the Cutoff Independent Physical Mass

I'm having trouble reproducing Equation 42: \begin{equation}\tag{1} m^{2}_{\text{phys}}= m^{2}_{r} + m^{2}_{r} \tilde{\lambda} \text{log} \left( \dfrac{m^{2}_{r}}{\mu^{2}} \right) \end{equation} ...
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Anomalous dimension for bare actions with a standard kinetic term

In this paper on p42, it is explained that when starting with a bare action that contains a standard kinetic term, this kinetic term attains a correction in the course of the RG flow which can be ...
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Charge dependence of operators in QED renormalization

Consider a UV cutoff regulator $\Lambda$ with an effective QED lagrangian: $\mathcal{L}_{\Lambda} = \bar{\psi}_{\Lambda}(i\not \partial - m_{\Lambda})\psi_{\Lambda} - ...
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Reconciling two interpretations of renormalization

I know of two fascinating and perfectly reasonable explanations of renormalization. However, I'm having difficulty reconciling the two. The first is to say that when we initially write down a ...
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$d=2$ pole argument of quadratic divergences in Peskin & Schroeder's book

Q1: My question is, in the context of dimensional regularisation(DREG, in dimension $d$), why do they mention the absence of $d=2$ pole in the gauge theory cases[1], whereas the $d=2$ pole is not ...
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Wavefunction renormalisation in first order perturbation theory

I just read the following in the context of scattering amplitudes in QFT: Note that the wavefunction renormalisation factor $Z$ itself is of the form $1 + \mathcal{O}(\lambda)$ in perturbation ...
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Introducing cut-off in a renormalisation procedure for quantum mechanics

I've been reading a paper on renormalisation theory as applied to a simple one-particle Coulombic system with a short-range potential. In the process of renormalisation, the authors introduce an ...
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A question about Ising model

If $H$ is the Hamiltonian of an Ising model of $n$ spins on a lattice then is the following quantity look like something one has seen? $([uI-H]^{-1})_{ii} - \frac{1}{n}Tr[[uI - H]^{-1}]$ where $u$ ...
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What is the relation between renormalization and self-adjoint extension?

What is the relation between renormalization and self-adjoint extension? It seems that a renormalization scheme can be rigorously treated mathematically using the self-adjoint extension theory. Is ...
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Fast and slow modes, and the vanishing of certain diagrams during re-normalization

In the middle of pg. 452 of Atland and Simonss Condensed Matter Field Theory, they state the following: Terms of $\mathcal{O}(\phi _{\text{s}}^3\phi _{\text{f}})$ do not arise because the addition ...
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Evaluation of the anomalous dimensions of fields in SUSY $SU(5)$

The general formula for the anomalous dimension can be found in Martin΄s review article (hep-ph/9709356), on page 62 relation (6.5.4). In the case of $SU(5)$ and especially in the paper of Kobayashi, ...
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Relationship between the on-shell and BPHZ renormalization schemes

In his book Quantum Field Theory - A Tourist Guide for Mathematicians, Gerald Folland introduces the on-shell renormalization scheme for the $ \phi^{4} $-scalar field theory. According to my ...
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Isolating the divergences in the stress energy tensor

In DeWitt's report "Quantum Field Theory in Curved Spacetime" (B. S. DeWitt, Phys. Rep. 19C, 292 (1975)), he states that in Eq.(175) $$\langle in, vac| T^{\mu\nu}|in,vac\rangle = 2 \frac{\delta ...
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Is it possible to derive the effective potential of a given theory by only using the RGE equations?

I know that it is possible to derive the RGE equations from the effective potential by requiring that the first derivative with respect to the renormalization scale $\mu$ vanishes: $$ ...
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what happen if we have a set of divergent constant $ a_{n} $ but in our theory we have only the $ m $ mass and the coupling constant?

Let us suppose we have 4 divergent integrals: $$ \int_{0}^{\infty}x^{m}dx = u_{m},\ \ \ \ \ \ \ \ \ m=0,1,2,3 $$ but our only free parameters are the $ m$ mass and $ q $ charge. Can we 'invent' two ...
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Which renormalisation techniques are available for 3+1 QED?

I hope my question is not too naive, but I would like to know what are the available renormalisation techniques for 3+1 QED. I have read a bit about Pauli-Villars, but I am wondering if there are ...
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Regarding Randall Sundrum model

In Randall Sundrum model 2, that is the one with non compact fifth dimension, there is only one brane, which is the Planck brane. The TeV brane is removed by taking the radius of the fifth dimension ...
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Non-abelian bosonization

Reading this review about non-abelian bosonization, Non-abelian bosonization by I.Karmazin, I stumbled about two questions Below equation 6, I don't get the final point in the statement about the ...
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Renormalization in non-relativistic quantum mechanics

I read many articles about renormalization in the Internet, but as I currently don't know much of QFT (currently just studying classical field theory and QM), and as all this looks quite interesting, ...
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Renormalizability of standard model

I'm wonder what precisely is meant by the renormalizability of the standard model. I can imagine two possibilities: The renormalizability of all of the interaction described by the Lagrangian before ...
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Renormalization, integrating out high momenta Wilson way

In equation $(12.5)$ in Peskin and Schroeder, they write out the generating function but leave out all quadratic terms of the form $\phi\hat{\phi}$ arguing that they vanish since Fourier ...
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What is the exact renormalization regularization for divergent harmonic serise?

given the harmonic series $$ \sum_{n=0}^{\infty}\frac{1}{n+a} $$ what is the correct option for the regularization ? a) $ \sum_{n=0}^{\infty}\frac{1}{n+a}= -\Psi (a) $ Digamma function b) $ ...
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Convergence of light by light scattering amplitude

Perhaps I'm too exhausted to see the answer of why the photon-photon scattering should contain no divergences. In Peskin and Schroeder page 320 we find that because of the Ward identity the ...
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Zamolodchikov's c-theorem paper

I am reading the 1986 paper [1] where Zamolodchikov proves the c-theorem and I would like to understand how equations (7a), (7b) and (8) are derived from the Callan-Symanzik equation. For ...
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Is QED valid for arbitrarily short length scale?

Solving the Renormalization Group equation the running coupling constant in quantum electrodynamics is given by $$\bar{\alpha}(q)=\frac{\alpha}{1-\frac{\alpha}{3\pi}\ln{\frac{q^2}{M^2}}}$$ (i) It is ...
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Gauge choice after Spontaneous Symmetry Breaking

After the spontaneous breakdown of local symmetry in presence of gauge fields (Higgs Mechanism), we can always choose a gauge where the Goldstone bosons are eaten up by the gauge field (also called ...
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Holographic Renormalization in non-AdS/non-CFT

In AdS/CFT, the story of renormalization has an elegant gravity dual. Regularizing the theory is done by putting a cutoff near the conformal boundary of AdS space, and renormalization is done by ...
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Are critical exponents below and above the critical point always same?

The scaling relations don't distinguish the the critical exponents below and above the critical value. In the mean field level, I understand these critical exponents are same whatever one approaches ...
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Is the stability matrix of a linearised RG flow always diagonalisable?

This is a follow up on "Why are the eigenvalues of a linearized RG transformation real?". My question is simple: Is there some physical (or mathematical) reason for the stability matrix of ...
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List of known universality classes

I am working with RG and have a pretty good idea of how it works. However I have noticed that even though the idea of universality class is very general and makes it possible to classify critical ...
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Correlation functions scale as powers of external momenta

In the 1st paragraph of section 12.2 on page 406 of Peskin and Schroeder, it says "the correlation functions turn out to scale as powers of their external momenta, with power laws that do not appear ...
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Why are the eigenvalues of a linearized RG transformation real?

The RG transformation $R_\ell$ maps a set of coupling constants $[K]$ of a model Hamiltonian to a new set of coupling constants $[K']=R_\ell[K]$ of a coarse-grained model where the length scale is ...
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What is IR CFT and UV CFT?

What is IR CFT and UV CFT? In many physics related materials, they often mention IR, and UV. I think it is related with regularization (I remember in QFT, there is UV cutoff in some regularization ...
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Regarding Non-renormalizatibility of GR

I've been doing some reading trying to get to a better understanding of some renormalization issues with the Einstein-Hilbert action. But, something odd came into mind that I'm hoping some users may ...
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Beta function of the non-linear sigma model

In chapter 7.1.1. inTong's notes about String Theory could someone sketch how can I show the statements that he nmakes around eq. 7.5 That the addition of the counterterm can be absorbed by ...
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What are renormalons from a physics point of view?

This is again a question in the context of this paper about the Exact Renormalization Group. On p 23 and the following few pages, it is explained that for a $\lambda \phi^4$ bare action at the bare ...
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Beta function of pure $SU(N_\text{c})$ Yang-Mills theory

What is the dependence of the beta function of pure $SU(N_\text{c})$ Yang-Mills theory on the number of colors? I guess ...
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Unitarity and renormalizability

What is the difference between the unitarity of the theory and its renormalizability? Can we say that renormalizable theory is unitary after renormalization? The questions have arisen after I have ...
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Anomalously broken conformal symmetry

I'm trying to understand an argument made by Bardeen in On Naturalness in the Standard Model. The argument is about quadratic divergences in Standard Model. My notation is that the SM Higgs potential ...
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Typical form of the beta function of the renormalization group

Why in "typical" cases (according to some non-English text I read), does the $\beta$-function have the form $$ \beta (g) = ag^{2} + bg^{3} + O(g^{4})\ ? $$ I.e., why are there no linear or logarithmic ...
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How to get the relation for dependence of anomalous dimension on regularization?

Here is the anomalous dimension: $$ \gamma_{\Gamma}(t, g) = \left[\frac{\partial }{\partial t}\ln \left(Z_{\Gamma}(t , g) \right)\right]_{t = 1}, $$ where $Z_{\Gamma}$ is renormalization factor which ...
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Asymptotic freedom in QCD

From renormalization group equation $$ t \frac{d \bar{g}(t , g)}{dt} = \beta (\bar{g}(t , g)), \quad \bar{g}(1 , g) = g $$ (here $t$ is momentum scale factor, $g$ is initial coupling constant and ...