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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Checking gauge invariance with a hard cutoff

When calculating loop diagrams the result is usually divergent, so some regularization scheme should be used. If I use a hard cutoff, which excludes the momentum modes with momentum larger than ...
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Questions regarding $D=4 $ ${\cal N}=4$ supersymmetric Yang-Mills

I have some questions regarding the $D=4 $ ${\cal N}=4$ super-Yang-Mills theory (the one with a really long action which can be acquired by compactifying the 10-dimensional ${\cal N}=1$ theory). I ...
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Field renormalization of scalar Yang-Mills

In most books, one can find the field renormalization $Z_3$ in Yang-Mills with fermionic matter in the fundamental. In the $\overline{MS}$ scheme, tt is given by $$ Z_3 = 1 + \frac{g^2}{16\pi^2 ...
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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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Momentum Space Renormalization of $\phi ^6 $ Model

I'm trying to find the RG flow to lowest order in $\epsilon = 3 -d $ for the energy functional: $$ f=\frac{1}{2} \phi ^2 +u \phi ^6 +\frac{c}{2} (\nabla \phi ) ^2 $$ where $\ d$ is the dimension ...
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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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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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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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Charge dependence of operators in QED renormalization

Consider a UV cutoff regulator $\Lambda$ with an effective QED lagrangian: $\mathcal{L}_{\Lambda} = \bar{\psi}_{\Lambda}(i\not \partial - m_{\Lambda})\psi_{\Lambda} - ...
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$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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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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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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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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Aren't $\phi^4$ composite operators?

I have this trouble with terminology. I wonder why authors introduce the concept of composite operators after they've already talked about eg phi four theory, it phi cubed. Aren't these operators ...
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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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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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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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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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Renormalization condition

Can any on explain to me, why renormalization condition $$\Sigma(\gamma_\mu p^\mu=m)=0,$$ for one loop implies $$\Sigma_2(m)=m\delta_2-\delta_m~?$$ In the original $\Sigma_2$ we had $ m_0$ which is ...
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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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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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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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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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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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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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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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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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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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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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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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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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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 ...
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Constants of infinity [duplicate]

Many times in physics when we analyze a physical system mathematicly we get divergences, but when those divergences has no dependence on any actual physical quantity of interest we tend to disregard ...
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A Conceptual Problem With the Field Equations of General Relativity

I have two questions: Suppose that we have an amount of energy in the form of a perfect fluid in the right hand side of Einstein field equations (energy momentum tensor), this will lead to a ...
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Lack of scale in Schrödinger equation for square-inverse potential

I see that if we set our potential in schrodinger equation to be a inverse-square dependence we don't have a typical unit of length as we have for hydrogen atom. $$-{\hbar^2\over 2m}\nabla^2\psi + ...
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Is there a way to obtain an RG flow equation for Quantum spin systems using MERA

We restrict ourselves to ground states of translationally invariant 1d quantum systems. I understand that there is the scale invariant MERA(multiscale entanglement renormalization ansatz) which ...