Questions tagged [renormalization]

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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Show equation equivalent in RG of 1d Ising Model

I know the Ising model is given by $$Z= \sum_{\sigma_i=\pm 1} e^{-F + J \sum_i \sigma_i \sigma_{i+1} - h\sum_i \sigma_i}$$ and that if h=0I can sum over even spins and get the partition function ...
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Polchinski's toy model of renormalization group flow: significance of main steps

In "Renormalization and Effective Lagrangians" (Nucl. Phys. B, 231 p269, 1984; preprint), Polchinski begins section 2 with a toy model to demonstrate the renormalization group with a relevant and ...
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Why is $T\overline{T}$ deformation so exciting?

I keep an ear to high energy physics discussions, and one of the things I've heard a lot about recently in these channels is the TTbar deformation (stylized $T\overline{T}$)$^1$. Wikipedia is lacking ...
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Why can't we interpret the W, Z bosons as massive vector bosons not arising from a gauge theory? [duplicate]

The standard story goes as follows: gauge bosons cannot have a mass term because it would break gauge invariance in the lagrangian. This is clear, but why can't we just have massive vector bosons ...
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Asymptotic freedom both in IR and UV

I am wondering if there are any (insightful) examples for models which exhibit asymptotic freedom both in the UV and the IR. I know it sounds odd, but if anyone has come across something like that, ...
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Virtual electron contribution to electron charge

Renormalization, or "a dippy way to to sweep all this stuff under the rug", makes QED the most accurate science ever. I came across a value that explained the difference between the predicted electron ...
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QED infrared divergences

How do infrared divergences arise in QED? What is an example case of such a divergence and how do we usually deal with such divergences? Are they absorbed like ultraviolet divergences?
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Wilsonion Renormalization Group in Asymptotically Free Theories

Consider some correlation function computed at some renormalization scale $\mu_0$ in an asymptotically free theory $$ \langle M(z; \mu_o) \rangle. $$ From what I understand of renormalization-group ...
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If the running coupling constant $\alpha(\mu)$ of QED becomes of order one at high $\mu$, why not changing $\mu$?

In the (modified) MS renormalization scheme, after dimensional regularization, we introduce some parameter $\mu$ with power of mass to keep the dimensionality of integrals under control. The ...
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QED integral is zero in dimensional regularization [closed]

Why is this integral zero in dimensional regularization? $$ \int\frac{d^Dk}{(2\pi)^D}\frac{1}{(k^2)^n} $$
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Divergence of Feynman diagram

Can we say whether the given Feynman diagram is divergent or not by just looking into the Feynman diagram? How to remove these divergences?
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Modified Minimal Subtraction $\overline{MS}$ Scheme advantage

What is the benefit of the $\overline{MS}$ method? Don't we just add some contributions from heavy particles that shouldn't be included in the Vacuum Polarization amplitude since they're way far ...
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Correlation functions under rescaling

I was reading this lecture note on Wilson's renormalization group and have hit a snag. I can't obtain equation 5.22. I tried to do the following: \begin{equation} \Gamma^{(n)}_{s\Lambda}(sx_1,…,sx_n;...
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Doubt about the derivation of the Callan-Symanzik equation

I was reading about the Callan Symanzik equation from Peskin and Schroeder. On page 411, they assume that since $G^{(n)}$, the connected Green's function is renormalized, the $\beta$ and $\gamma$ ...
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Renormalization group and counterterms [duplicate]

While regularizing Feynman diagrams, we first isolate its divergent parts and then add counter terms to the Lagrangian in order to subtract out the divergent parts and render the amplitudes finite. So,...
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Power counting and divergences

Often, in many books such as Peskin and Schroeder, a Feynman diagram or the effective potential is expanded as a function of the external momenta or the classical fields respectively. Consider the ...
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Higgs mechanism and phase transition

Generally speaking, phase transitions divide into two types: First order and second order. To me, Higgs field's SSB sounds like a second-order one though I don't know the dependency of Higgs field's ...
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How do you reconcile quark masses with notion of confinement?

In trying to understand exactly what confinement means, I have been reading 't Hooft s original paper on 2D QCD at large $N$. In the paper he shows that the quark propagator pole is moved to infinity, ...
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Inverted propagator in Peskin [duplicate]

Given the Lagrangian (10.18) shouldn't the third diagram in figure 10.3 be the inverse of what has been written?
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Counterterms cancelling divergences

Consider the $\phi^4$ theory. The two divergent Feynman diagrams, namely the two point function and the 4 point function have been isolated and by putting a cut off on their momentum integrations, ...
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Why is gauge invariance so important?

Quantizing the electromagnetic field (without ghosts or gauge fixing terms) using path integrals is not possible because the propagator is not well defined. Textbooks such as P&S or Ashok Das say ...
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What is meant exactly by “renormalization” in condensed matter physics, specifically in density matrix renormalization group (DMRG)?

I first encountered the concept of renormalization in the context of statistical physics. Here, the renormalization "group" is a set of transformations of the system such that the Hamiltonian $H(J,\...
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Is this the picture of a renormalization group flow?

Please look at the cover of this book written by Le Bellac. the book I guess that the picture is about a renormalization group flow (the arrows on the lines). i found a similar picture here here . ...
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Why do the Euler-Mascheroni constant $\gamma$ and $\ln 4\pi$ not show up in observables (renormalisation of electric charge)?

The one-loop contribution of the vacuum polarisation of the photon after using dimreg is given by $$\Pi_2^{\mu\nu}= e^2 J(q) \left(\eta^{\mu\nu} - \frac{q^\mu q^\nu}{q^2}\right),$$ with the metric ...
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A superficially divergent diagram in $\phi^4$ interaction, rarely appeared in the literature

I'm studying superficial degree of divergence of Feynman diagram and I am confused about some concept. In particular, its scope is on scalar $\phi^4$ interaction in $3+1$ dimension. Some literature ...
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QFT in in the asymptotic region

Let $\phi(x)$ be a scalar field operator. It often postulate in text books that in the asymptotic region we have $$\lim_{x_0\to-\infty} \phi(x)=\sqrt Z \phi_{in}(x)$$ where $Z$ is a constant. The ...
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How can the ground state charge be renormalized?

When we calculate the total charge and energy of a quantum field by using Noether's theorem, we find that they are infinitely large, even if we consider a finite spacetime volume_ $$H = \int_V \...
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Do vacuum bubbles exist in theories with normal ordered Hamiltonian? [duplicate]

When we calculate the Hamiltonian in the free theory, we notice that it contains an infinitely large term \begin{align} H &= \int_V \mathrm{d }k^3 \frac{\omega_k}{(2\pi)^3 } a^\dagger(\vec ...
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Is there some truth to the often told story that the running of couplings is the result of screening through virtual particles?

It's a well established fact that coupling parameters changes with the energy scale at which we probe a given process: A popular way to explain this phenomenon goes as follows. Particles are ...
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How can we tell a theory is confining?

Physically, I understand what it means for a theory to be confining. The elementary particles are not observable, but only composite particles are. The classic example is QCD, where quarks are ...
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What is the idea behind coarse-graining?

I don't think I fully understand the idea behind coarse-graining. I will elaborate. I was reading some lecture notes on statistical field theory and the text begins with some previous analyses on the $...
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Using the Born rule to measure UV divergence?

A typical use of cutoffs is to prevent singularities from appearing during calculation. If some quantities are computed as integrals over energy or another physical quantity, these cutoffs ...
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Derivation of the Renormalization Group from Renormalized Coupling?

At page 303 in the book Quantum Field Theory for the Gifted Amateur by Blundell and Lancaster, they argue that the renormalization group equation for the coupling $\lambda$ in $\phi^4$ theory can be ...
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What is computable and independent of subtraction scheme?

I am trying to compute, using mathematica, the renormalization $Z$s (of the field, mass and coupling) in $\phi^4$ theory (using dimentional regularization). I have done so in two different ...
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What does vanishing critical mass exponent mean?

I would like to understand the reason behind the vanishing critical mass exponents. I've written a program that calculates the fixed points and then the eigenvalues corresponding to the fixed point ...
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Confusion about choice of renormalization scale in $\overline{\rm MS}$ mass

The $\overline{\rm MS}$ mass is function of the renormalization scale $\mu$. What does it mean to choose this scale $\mu$ as the $\overline{\rm MS}$ mass itself? I am giving some details below to make ...
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Is it possible to construct a real-space renormalization group for a system with limited self-similarity?

I'm trying to understanding multiscale dynamics in a system which can be loosely mapped to a lattice model embedded in real space, in which the types of entities at the lattice points, and their ...
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Effective $T$ matrix in Kondo Hamiltonian

Consider the Kondo Hamiltonian $$H=\sum \epsilon_k c^\dagger_{k\sigma} c_{k\sigma} + J^z S^z \sum c^\dagger_{k'\alpha} \sigma_{\alpha\beta}^z c_{k\beta} + J^{\pm} \sum \left( S^+ c^\dagger_{k',-} c_{k,...
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Mass Dimension 6 QED-Lagrangian

Consider the QED Lagrangian $$\mathcal{L}_{\text{QED}}=-\frac{1}{4} F^{\mu \nu} F_{\mu \nu} + \bar{\psi}(i D_{\mu} \gamma^\mu -m) \psi.$$ I need to extend the Lagrangian up to mass dimension 6, of ...
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Renormalization of irrelevant operator in the effective field theory

I have some troubles in understanding the renormalization of the irrelevant operators in the effective field theory approach. In the standard approach, one usually writes down the most general ...
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How do self-loop diagrams end up contributing nothing to observables?

The probability amplitude for single particle to enter our system with momentum $k$ and leaving with momentum $q$ can be calculated as \begin{align} A(k\to q ) = \langle q| T e^{i \int_{-\infty}^\...
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What is a dressing function?

Consider, for example, the gluon propagator $$D^{\mu\nu}(q)=-\frac{i}{q^2+i\epsilon}[D(q^2)T^{\mu\nu}_q+\xi L^{\mu\nu}_q]$$ What exactly is the renormalized gluon dressing function $D(q^2)$ and what ...
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Renormalization group fixed points

Suppose we are given two parameters $K,L\geq 0$ such that $$K'= b^{-1} K + (b-1)K^2$$ $$L'=L/b.$$ Where $b$ is the scaling factor. I found all the fixed points of this recursion relation: in ...
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Renormalisibility and renormalisation flow of the $\phi^4$-theory in Peskin & Schroeder p. 402

P&S make a curious remark when they compare relevant & marginal operators to superrenormalizable and renormalizable interaction terms. I will put it in the context. Under the renormalization ...
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How do super-renormalizable theories renormalize?

This question is about a conflict between two facts about scalar field theories in 2D. The same sort of question will apply to any scalar field theory with a polynomial potential, but let's specialize ...
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How do physicists deal with fields at the location of charges?

In the Feynman Lectures Vol 1, Chapter 28 (at the end of section "28–1 Electromagnetism"), it is mentioned: For those purists who know more (the professors who happen to be reading this), we should ...
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Definition of dangerously irrelevant operator

(Disclaimer: There is already a question about dangerous irrelevant operators which has not been very successful. However, the question there is quite broad, and here I aim to ask a more precise ...
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Some details on Wilson's method of integrating out high momenta in Peskin & Schroeder p.397

When P&S explain Wilson's method of integrating out high momenta they start from the Euclidean path integral of the $\phi^4$-theory (eq. (12.3)) and then define in the following: $$\hat{\phi}(k):=...
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Fixed points in triangular lattice 2d Ising model

Given the following recursion relations: $$K^\prime = 2K \Big(\frac{e^{3K}+e^{-K}}{e^{3K}+3e^{-K}}\Big)^2$$ and $$h^\prime = 3h\Big(\frac{e^{3K}+e^{-K}}{e^{3K}+3e^{-K}}\Big),$$ where $K = J/k_B ...
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Critical Mass Exponents in $d=3$

I'm just a Bachelor student, so forgive me if my questions seem too silly. I want to show the convergence of the critical exponents in the Renormalization Group equations when $d=3$. When I construct ...