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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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Wavefunction Renormalization in Wess-Zumino Model

In Modern Supersymmetry: Dynamics and Duality, on page 134 and 135 in section 8.2, the authors studied the wavefunction renormalization of the Wess-Zumino model. The kinetic terms are given by $$\...
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Would Color Confinement apply in higher dimensions?

As I understand it color confinement comes from the fact that as the distance between two color charges increases the color potential energy increases, instead of decreasing, and the energy needed to ...
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Why coupling constants with negative mass dimensions lead to non-renormalizable theories?

can somebody explain or point to the relating mathematics showing Why coupling constants with negative mass dimensions lead to non-renormalizable theories?
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Scaling limit of the Ising model with nonzero order parameter

I'm interested in simulating the continuum limit of the 2D Ising model $$H=J\sum_{\langle i j\rangle} s_i s_j+ h \sum_i s_i$$ In one dimension I can fix average magnetization $m=\langle s\rangle$ and ...
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How scale invariance is broken in nature?

By definition a system will exhibit scale invariance at low energies if it has an IR fixed point. I am having some doubts on how to interpret this fact in terms of quantum field theory and to ...
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Dimensional regularisation in $\phi^4$ theory

My question is in regards to the 1-loop corrections of phi 4 theory. The question is in regards to these notes: http://www.damtp.cam.ac.uk/user/dbs26/AQFT/chap5.pdf On page 15 of these notes (PDF ...
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Scaling dimension and system size

I am reading a paper (Sliding Luttinger liquid phases ) which is trying to obtain the scaling dimension of several operators in (condensed matter) field theory. In this paper, the authors mentioned ...
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What Lagrangian counterterms might be needed in a $\phi^ 6$ theory in 3D?

Assuming a massive $\phi^6$ theory in $d=3$ given by the Lagrangian $$\mathcal L=\frac{1}{2}(\partial\phi)^2-\frac{1}{2}m^2\phi^2-\frac{\lambda}{6!}\phi^6 +\mathcal L_\text{ct},$$ what are the counter ...
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A question about renormalization condition and Callan-Symanzik equation

In Ch.12 of the textbook An Introduction to Quantum Field Theory by Peskin and Schroeder, on P.408 the renormalization conditions are given at the energy scale $M$ $$\mathrm{dressed\ 4\ point\ vertex}...
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What happens to renormalisation counter-terms in the classical limit?

My understanding is we add counter-terms to the actions in the process of renormalisation. Presumably these terms don't have a physical effect in a classical interpretation of the action. i.e. they ...
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A Question about In/Out States in Quantum Field Theory

When I was reading the lecture notes Advanced Quantum Field Theory by Jorge Crispim Romao, I accidentally found the following thing that I don't understand. On page 56, section 2.2, the author ...
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Is the term “quantum triviality” defined by the UV or the IR behavior of the RG flow?

The Wikipedia page on quantum triviality seems to give two different definitions for the term that are not obviously equivalent. Some parts of the page seem to define a renormalizable theory as "...
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Partition function in renormalization

When studying statistical mechanics, renormalization is understood from attempts to calculate partition function by simplifying. (For example, David Tong's lecture note) While I understand that ...
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Asymptotically free/flat

What does the expression: "...the theory becomes asymptotically free/conformal" mean? If it means that the spacetime $M$ on which the fields are defined is e invariant under conformal ...
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Renormalisation group flow of the $\phi^4$ theory

I am reading Peskin & Schroeder about the renormalisation group flow of the $\phi^4$ theory: $${\cal L} = \frac{1}{2}(\partial_\mu\phi)^2 +\frac{1}{2}m^2\phi^2 + \frac{\lambda}{4!}\phi^4 $$ P &...
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Why does the universe manifest scale?

I'll try outline my question in clear terms, articulating specific aspects that are its primary motivators. I'm just beginning in my exploration of physics as a student, but a persistent question that ...
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Why does the divergence of a QFT's coupling constant under RG flow trivialize the theory if it occurs in the UV but not in the IR?

When you first learn quantum field theory, at some point you calculate the beta function (to leading order) for a renormalizable coupling constant of some theory like $\varphi^4$ theory, Yukawa theory,...
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Exact FRG flow equation for QED renormalizability proof

I would like to see how can one prove QED renormalizability in terms of Wetterich & Morris exact FRG flow equation for effective action $\Gamma_k$? May be my question is unclear.
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Is string theory self-consistent? (Conformal anomaly)

Recently I attended a very short course on string theory. We went through the standard presentation in light-cone gauge for brevity. We ‘derived’ the Einstein field equation in the following manner. ...
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Hierarchy problems: calculating the mass of the Higgs vs. other SM particles

While reading up on hierarchy problems in particle physics (wikipedia), I stumbled onto the statement that in the Standard Model, there is no hierarchy problem (or at least it cannot be formulated) ...
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Politzer, Gross & Wilczek running formula

I've been told that for any group of SM, the running of the corresponding coupling constant, $g$, is given by: $$ \frac{dg}{d(\ln{Q})} = b·g^3/(16\pi^2) $$ Where $$ b = -\frac{11}{3}C_2(A) + \sum\...
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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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Why do we demand that the counterterms in $\varphi^3$ theory be $O(g^2)$?

In Srednicki's QFT book, section 9, he introduces the $\varphi^3$ lagrangian: $$\mathcal{L}= -\frac{1}{2}Z_\varphi(\partial_\mu\varphi)(\partial^\mu\varphi) -\frac{1}{2}Z_mm^2\varphi^2 +\frac{1}{6}...
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Failing to show $\xi$-gauge-independence in an abelian Spontaneously Broken Gauge Theories (SBGT)

I am studying the following paper: Appelquist, Carazzone, Goldman & Quinn, Renormalization and Gauge Independence in Spontaneously Broken Gauge Theories, https://doi.org/10.1103/PhysRevD.8.1747 ...
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How Does A $\theta$ Angle Shift Affect the Wilsonian Effective Lagrangrian?

Say we have some quantum field theory which includes a gauge field, and some matter, and a topological $\theta$ term so that the Lagrangian reads $$L=(stuff)+\frac{\theta}{64\pi^2}\varepsilon^{\mu\nu\...
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Why doesn't the $\theta$ Angle Renormalize?

The $\theta$ term for Yang-Mills takes the form $$L_{\theta}=\frac{\theta}{64\pi^2}\varepsilon^{\mu\nu\rho\sigma}F^a_{{\mu\nu}}F^a_{\rho\sigma}$$ A fact that I have heard is that $\theta$ does not ...
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Zee's explanation of expressing bare coupling by physical coupling

In terms of bare parameter $\lambda$, the $\phi\phi\to\phi\phi$ scattering amplitude is $\lambda\phi^4$ theory is given by $$\mathcal{M}=-i\lambda+iC\lambda^2\Big[\ln\Big(\frac{\Lambda^2}{s}\Big)+\ln\...
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Terminology: Infrared and Ultraviolet

I am new to high energy physics and string theory. I keep reading the terms infrared and ultraviolet in papers. I assume they aren't talking about electromagnetic radiation. For example, one paper ...
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Question related to electric dipole moment via QFT

My question is related to the following post: Extracting Electric Dipole Moment from Matrix Element via Form Factor There, it is said that the electric dipole moment (EDM) is giving by a term that ...
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rigorous definition of coherence length at mean field theory

so as far as I know, when we are doing mean field theory, in qft, we expand a action of a theory around a classical solution. so we find a classical solution, than we add quantum mechanical ...
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What goes wrong with GR as a lattice gauge theory?

If one tried to formulate General Realativity in a similar manner to say lattice QCD, what goes wrong that makes it not work? My first thoughts are that for any particular grid, we might have a ...
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Why does the renormalizable theory have only those particles with helicity less than or equal to 1?

Let the helicity operator be $\frac{P \cdot J}{P^0}$ with an eigenvalue $\lambda$. Then why do renormalizable theories have $|\lambda| \le 1?$ (in general dimensions or in 4ds?) Also, what is the ...
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IR divergence Feynman diagram topology query

I am trying to calculate the superficial degree of Infrared divergence. To do this I am reading section 12 of this source. It seems you can calculate it by a method involving the 'shrinking' of ...
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Counterterms in quantum brownian motion

In the part "Quantum Brownian motion" of the book, The theroy of open quantum systems written by Breuer, the author investigates on the Caldeira-Leggett model: The Hamiltonian of the particle is $H_{...
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Arbitrary function on the Faddev-Kulish dressing

On this paper the authors review the Faddev-Kulish dressing in QED which is a solution to the IR divergence problem. Given one electron momentum $\mathbf{p}$, They define the soft factor by $$F_\ell(...
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Why does literature list the strong coupling at the scale of the Z-boson's mass?

In the 2004 edition of the book "QCD as a Theory of Hadrons" by S. Narison, the author provides a value for the strong coupling at a scale of the mass of the Z boson, $$ \alpha_s (M_Z) = 0.1181 \pm ...
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Wilsonian RGE: Problem 23.7 in the textbook, M.D. Schwartz's ''QFT and Standard Model'' [closed]

Can anyone give me some hints or directions to work out the solution to the following problem? This problem is from chapter 23 of the textbook written by Professor Schwartz. I can't figure out ...
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Conserved charge during renormalization-group flow

Let us consider a quantum system (at zero temperature) with a continuous (anomaly-free) symmetry $G$ and there exists a corresponding conserved charge $Q$. Then we perturb this (might-be critical) ...
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Quantum expressions for the Virasoro constraints

I am trying to derive the quantum form of the Virasoro constraints. $$ L_{m} = \frac{1}{2} \sum_{n} :\alpha_{m-n}.\alpha_{n}: $$ :...: meaans normal ordering. Using the common commutator between ...
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Divergent integral problem

When expanding the scalar field vacuum energy $$\sum_k \frac{1}{2} \omega_k = \frac{1}{2} (L/2\pi)^{n-1} \int \omega(k) d^{n-1}k = \frac{(L^2/4\pi)^{(n-1)/2}}{\Gamma(\frac{n-1}{2})} \int_0^\infty (k^...
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Euler-Maclaurin formula for path integral

Is there a corresponding Euler-Maclaurin formula for path integral when we divide the path integral into discrete lattice? What is the error correction when we divide the space into lattice of length ...
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Relation between mean field critical point and RG critical point

In the mean field / Landau picture a critical point is where the potential of the order parameter changes curvature. E.g. the mean field potential of a scalar $\phi^4$ theory is $$\mathcal{L} = a t \...
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Nonperturbative results for $\phi^3$ theory in dimensions $d>6$?

The theory is nonrenormalizeable in those dimensions, but can you say anything about the theory anyway? Specifically I am wondering about the status of whether the theory is trivial, i.e. a ...
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Behavior in renormalization group flow that reaches critical point

First question. Does correlation length in renormalization group flow has to be infinite when it eventually reaches critical point? Second question. Why does renormalization group flow keep partition ...
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Can we measure renormalized mass in QFT? [duplicate]

Due to QFT books, we measure pole mass(physical mass) in experiments. From the Lagrangian point of view, renormalized mass is a parameter(in MS bar or some similar renormalization scheme that has an ...
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Renormalization group flow when temperature $T < T_C$, $T_C$ being critical point temperature

Does renormalization group flow have to decrease temperature when $T<T_C$, with $T_C$ being critical point temperature? I think not, but my professor suggests something like that. Maybe I ...
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Taylor expansion in beta function calculation

This post is related to the answer given in Beta function in $\lambda_0\phi^4$ theory The beta function calculus for that theory provides you of $$ \beta(\lambda_p) = - \frac{\epsilon \lambda_p + z\...
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Effective Lagrangians

I get the impression from reading, e.g., this paper, that the term "effective Lagrangian" refers to a Lagrangian derived from a Taylor series expansion of an arbitrary function of known invariants. ...
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Non-renormalizeable Interaction Implies Trivial Interaction?

It has been rigorously proved that the $\phi^4$ theory is trivial, i.e. is a generalized free field, in spacetime dimensions $d>4$. It is also the case that this theory is non-renormalizeable in ...
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Beta function in $\lambda_0\phi^4$ theory

For a real scalar field $\phi$ after performing all the 1-loop renormalization for dimensional regulator $d = 4 - \epsilon,\ \epsilon \rightarrow 0^+$, I have found that the renormalized coupling $\...