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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$\operatorname{O}(N)$ sigma model at large $N$

I would like to better understand the main principles of large-$N$ expansion in quantum field theory. To this end I decided to consider simple toy-model with lagrangian (from Wikipedia) $ \mathcal{L} ...
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494 views

Understanding the $\phi^4$ phase diagram

I'm having trouble making sense of this phase diagram. The model is a $V(\phi)=g_2 \phi^2+g_4\phi^4$ scalar field theory. Here's what I think I understand: the capital letters represent different ...
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2answers
401 views

Link between anomalous dimensions and fractal dimensions

I just realized that anomalous dimensions in quantum/statistical field theory is not that different from fractal dimensions of objects. They both describe how quantitaive objects transform under a ...
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269 views

Renormalisation and the Fisher-Rao metric

The renormalisation group (I'm talking about classical, statistical physics here, I'm not familiar with field theory too much) can be though of as a flux in a space of possible Hamiltonians for a ...
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How do we know for sure a theory is non-renormalizable?

In quantum field theory, we are looking for a Lagrangian that is, amongst other, renormalizable. But how do we determine whether or not a theory is renormalizable? Is this purely done by power ...
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606 views

Instantons and Borel Resummation

As explained in Weinberg's The Quantum Theory of Fields, Volume 2, Chapter 20.7 Renormalons, instantons are a known source of poles in the Borel transform of the perturbative series. These poles are ...
8
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1answer
355 views

Anomalous Ward Identities and anomalous dimensions

Let us consider an action $S[\phi,\partial\phi]$ which is classically invariant under a transformation group $G$. The associated Noether current $\mathcal{J}^\mu$ is classically conserved, namely $\...
8
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1answer
418 views

Cutoff-Scheme Renormalization and Order of Integration in QFT

The following is the result of Fubini's Theorem, describing when you can replace a double integral with an iterated integral safely: For a set $X \times Y \subset \mathbb{R}^2$, if $\iint |f(x,y)| d(...
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1answer
180 views

How does gauge invariance protect the SM gauge boson masses in SUSY from divergent radiative corrections?

The W and Z gauge bosons receive radiative corrections in loop from the heavy SUSY scalars. There is an argument using gauge invariance which explains how the masses remains protected. I am not able ...
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410 views

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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336 views

$d=2$ 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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506 views

Effective Field Theory (EFT) decoupling top

The decoupling theorem of Appelquist-Carazzone says that if you want to decouple a particle, the low energy resulting theory need to be renormalizable. You can't do that for the top, because you break ...
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446 views

Dimensional regularization and IR divergences and scale invariance

I want to know if dimensional regularization has any issues if the theory has IR divergences or is scale invariant. Does dimensional regularization see "all" kinds of divergences? I mean - what ...
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514 views

Drawing the RG flow diagram

In real-space renormalization group how does one find the complete RG flow exactly, (not schematically)? I understand it needs to be done on a computer. For example, I have the ising model on a ...
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279 views

Understanding the Renormalization Group for Entanglement Entropy

I'm trying to understand the renormalization group (RG), and in particular, how it is used to study entanglement entropy (EE) and c-theorems in quantum field theory (QFT). But I'm having trouble with ...
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320 views

In QED/Yang Mills, why do fermions contribute 4 times as much as scalars to vacuum polarization?

Consider a Yang-Mills theory in $4D$ over a gauge group $G$ $$ \mathcal{L} = - \frac{1}{4} F^{a\mu\nu}F_{\mu\nu}^a + \bar \psi i D_\mu \gamma^\mu \psi + (D_\mu \phi)^\dagger D^\mu \phi $$ where $\...
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Renormalization group and minimum substraction

I have several questions about renormalization group and minimum substraction scheme in particular. My first question is: 1) Why is the beta function typically just a function of coupling? In other ...
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211 views

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 ...
7
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146 views

Finite corrections to the Higgs mass in SUSY

I am worried here about specific supersymmetric scenarios in which an energy scale above the TeV is introduces. An example would be the Seesaw extension of the Minimal Supersymmetric Standard Model. ...
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Are irrelevant terms in the Kahler potential always irrelevant, even at strong coupling?

I've been reading about the duality cascade in Strassler's TASI '03 lectures (hep-th/0505153). He reminds us of the non-renormalization theorem theorem for the superpotential so that the beta ...
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303 views

Geometric entropy vs entanglement entropy (dependent on curvature coupling parameter)

I have a quick question. In hep-th/9506066, Larsen and Wilczek calculated the geometric entropy (which I believe is just another name for entanglement entropy) for a non-minimally coupled scalar field ...
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213 views

Renormalization group approach to renormalization

Given a $n$-point bare Green function in a massless asymptotically free theory, we have that the following limit exists and is finite \begin{equation} \lim_{\Lambda\rightarrow\infty} Z^{-n/2}(g_0,\...
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323 views

The “dangerous” fixed points for Renormalization Group

What is the definition of dangerously irrelevant renormalization-group (RG) fixed point? What are some examples of dangerously irrelevant RG fixed points? Do we also have the use of dangerously ...
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Higher category theory, renormalization, and non-perturbative QFTs

I'm (vaguely) aware of certain uses of higher category theory in attempts to mathematically understand quantum field theories -- for example, Lurie's work on eTQFTs, the recent-ish book by Paugam, and ...
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LSZ reduction theorem derivation in Weinberg QFT

When deriving LSZ reduction theorem Weinberg in his QFT book have assumed n-point generalized Green functions, $$ G(q_{1},...,q_{n}) = \int d^{4}x_{1}...d^{4}x_{n}e^{-i\prod_{i =1}^{n}q_{j}x_{j}} \...
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How does the renormalization scale $\mu$ cancel in all finite observables?

In dimensional regularization, we must shift the dimensionless coupling $g$ by the renormalization scale $\mu$ (which has unit mass dimension): \begin{equation} g \rightarrow \mu^{4-d} g \tag{1} \end{...
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conformal anomaly of free scalar in 2D

I'm trying to calculate the conformal anomaly $c$ of a free scalar on a 2-sphere. I've seen other, indirect ways to do this, but since this is a free theory I feel like it should be possible to see ...
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270 views

Do we care about CFTs in particle physics?

This question is related to these others: mostly this one, but also this one and this one. Do we care about CFTs in particle physics? Let me explain. Suppose we don’t know anything about string ...
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Renormalization Group - Scaling fields and physical critical exponents (1D Ising model)

This is related to this question: Critical exponents and scaling dimensions from RG theory. TLDR: How to compute physical critical exponents $\alpha, \beta, \gamma, etc$ from the RG exponents when ...
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271 views

Renormalization, Phase transitions and order parameters

Renormalization is the phenomenon for which, once a finite number of parameters, which are the couplings with positive-mass dimension, are fixed, then it is possible to express any $n$-point ...
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1answer
185 views

Lagrangians in field theory and ignorance

The thing that has always bothered me while taking my QFT course was the seemingly arbitrary nature of Lagrangians. For the Klein Gordon equation we just wrote down the simplest Lorentz invariant ...
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455 views

Divergent self-energy of point charges in Classical Electrodynamics

Assuming the electron to be a classical point particle, if one calculates the self-energy one finds $$U=-\frac{e^2}{8\pi\epsilon_0r}$$ which diverges as $r\rightarrow 0$. Therefore, the measured mass ...
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Renormalization of $\phi^3$-theory in 3+1 dimensions

Consider the Lagrangian $$\mathcal{L} = \frac{1}{2}(\partial_{\mu}\phi)(\partial^{\mu}\phi) - \frac{1}{2} m^{2} \phi^2 -\frac{1}{3!}\lambda \phi^{3}.$$ Following Michio Kaku's discussion in his QFT ...
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685 views

How to handle the infrared divergence of massless $\phi^4$ in scattering

For massless $\phi^4$ theory, if exterior momentums are going to zero, then this diagram will be $$\int \frac{dk^4}{k^4}$$ will suffer from infrared divergence. Because the infrared divergence, ...
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109 views

Is there a maximum number of fixed points that a QFT can have?

I was wondering: is there a maximum number of (trivial and non-trivial) fixed points that a QFT can have (as a function of the space-time dimension and field content in the QFT)?
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Is the Higgs bare mass larger than the physical mass?

The Higgs boson propagator can be written $$\frac{1}{p^2-m^2+\Sigma(p^2)}$$. If we take $p^2=m_P^2$ the physical mass, we get $m_P^2=m^2-\Sigma(m_P^2)$. Now, if $\Sigma\sim \Lambda^2$, we get $m^2\sim\...
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Equality of electric charges of all leptons

What does it precisely mean the often repeated statement that the electric charges of all leptons are the same. Let's consider QED with two leptons: electron and muon. The interaction part of the ...
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Some questions about the large-N Gross-Neveu-Yukawa model

Consider the following action with a fermionic field $\psi$ and a scalar field $\sigma$, $S = \int d^dx \{ -\bar{\psi}(\gamma^\mu \partial_\mu +\sigma )\psi + \Lambda^{d-4}[ \frac{(\partial_\mu \...
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Why do people rule out zeta regularization for renormalization?

Using zeta regularization one can get a formula for regularizing the integral $ \int_{a}^{\infty}x^{m-s}\text dx $ for any $m$. However, I have not seen anywhere. For example, I do not know why in ...
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Exact Beta Functions in Statistical Mechanics

I'm looking for analytically solvable models in statistical mechanics (classical or quantum) or related areas such as solid state physics in which the beta function for a certain renormalization ...
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297 views

From vertex function to anomalous dimension

In a $d$ dimensional space-time, how does one argue that the mass dimension of the $n-$point vertex function is $D = d + n(1-\frac{d}{2})$? Why is the following equality assumed or does one prove ...
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Field strength renormalization and the energy-momentum tensor

This question is about the connection between the energy-momentum tensor, dilation transformations, and field renormalization. From a Wilsonian perspective on renormalization we start out with a ...
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35 views

Why/When would one study Renormalization Group flow of a system?

It is not that I am looking for a cheap way out of reading a book about RG flow, but I would like to know few key insights that RG flow study provides, backed with some specific examples. I know a ...
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How to calculate $n$-point functions of interacting fields in curved spacetime (Schwarzschild metric)?

How to renormalize quantum field theory in curved spacetime? (or in Schwarzschild spacetime?)  I want to calculate n-point functions $$<0|Tφ(x_1)...φ(x_n)|0>$$ in massless $φ^4$ theory in ...
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Does the background shift affects the renormalization group equations?

In Section 21 of "Quantum Field theory" by Mark Srednicki, it is shown that there are two equivalent ways to get the quantum action of the shifted field $\phi'= \phi-\tilde{\phi}$, where $\phi$ is the ...
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Closed set of operators under renormalization

While reading the article http://inspirehep.net/record/61135, I came across the concept of "closed set under renormalization". The definition they give is the following. In any renormalizable field ...
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How does renormalization relate to emergence?

In statistical mechanics renormalization is often related to coarse-graining which in turn allows to calculate some macroscopic states. The resulting macroscopic description is sometimes called ...
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When was the phrase “beta function” of renormalization first used?

My question is a historical one: when was the phrase "beta function", as it pertains to the renormalization-group equations, used in physics? I am talking about this beta function: $$\beta_g\equiv \...
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How do the renormalization factors disappear from the computation recipe of the S-matrix in Peskin & Schroeder (p. 229 (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) $\phi_r= (\sqrt{Z})^{-1} \phi$ in eq. (...
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123 views

Understanding field strength renormalization

When it comes to renormalization in QFT I always considered the renormalization of mass and coupling constant as rather intuitive, whereas I found the field strength renormalization rather counter-...