Questions tagged [regularization]

In QFT, regularization is a method of addressing divergent expressions by introducing an arbitrary regulator, such as a minimal distance *ϵ* in space, or maximal energy *Λ*. While the physical divergent result is obtained in the limit in which the regulator goes away, *ϵ* → 0 or *Λ* → ∞, the regularized result is finite, allowing comparison and combination of results as functions of *ϵ, Λ*. Use for dimensional regularization as well.

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4
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1answer
101 views

$\varphi^4$ via renormalization group with hard cut-off

I am studying the application of the renormalization group to the $\varphi^4$ theory: $$\mathcal{L} = -\frac{1}{2} \varphi (\Box + m^2)\varphi -\frac{\lambda}{4!}\varphi^4.$$ In particular I wanted to ...
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3answers
76 views

Perturbative expansion and self-contractions in functional integral

Consider a one-dimensional integral $$I(g)=\int dx\, e^{-x^2-gx^4}$$ One can formally expand it perturbatively order by order in $g$ so that $$I(g)=\left<1\right>-g\left<x^4\right>+\frac{g^...
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1answer
180 views

One-loop effective action for scalar field on the curved background in large potential

I hope to compute a functional integral $Z=\int \mathcal{D}\phi\,\, e^{-S[\phi]}$ with an action $$S[\phi]=\int d^2x \sqrt{g}\Big((\nabla \phi)^2+\frac{1}{\lambda}M^2(x) \phi^2\Big)$$ The scalar field ...
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2answers
87 views

Harmonic oscillator partition function via Matsubara formalism

I am trying to understand the solution to a problem in Altland & Simons, chapter 4, p. 183. As a demonstration of the finite temperature path integral, the problem asks to calculate the partition ...
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1answer
69 views

Taylor Expansion of Feynman Propagator with regulator masses and gamma matrices

I am currently trying to understand a really old paper of Jackiw and Coleman: "Why dilatation generators do not generate dilatations". There, at some point they arrive at the following ...
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1answer
57 views

About sending time to infinity in a slightly imaginary direction in QFT

I am going through the Peskin and Schroeder QFT book. While proving the Gell-Mann and Low theorem in chapter 4 of their book, the authors started with the equation \begin{equation} e^{-iHt}|0\rangle = ...
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1answer
58 views

Cut-off Regularization - Renomalization - Definition of counter-terms - in Curved Spacetimes

I'm trying to study renormalization in QFT in curved spacetime. So let's say we have a fixed de Sitter background and we have an interacting theory (e.g. massive $\lambda \phi^4$) and I'm going to ...
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26 views

Regulators and Transversality (Wards identity)

In QED, why some regulators obey transversality (Ward identity) and some not?
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1answer
124 views

Renormalization group in statistical mechanics: (1) rescaling of parameters and (2) calculating the free energy

I have some questions about the momentum space renormalization group procedure as described in the textbook "Statistical Mechanics of Fields" by Kardar (Ch5). The first is about the rescaling of ...
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49 views

Gauge invariance of the regulator when calculating the chiral (ABJ) anomaly by the Fujikawa method

I am currently studying the calculation of chiral anomaly using fermionic path integral. In all texts I looked at, the authors simply use a regulator of the following form $e^{(\gamma_{\mu}D^{\mu})^...
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Wilson's approach to renormalisation according to Peskin & Schroeder

Although Peskin & Schroeder treats Wilson's approach to renormalisation theory in some depth, I don't get one of its main points. According to P&S (p.401): Imagine that we wish to compute ...
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1answer
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Dimensional regularization and (dimension) compactification [closed]

I believe I read that "additional dimension" in dimensional regularization can be understood as spatial dimensions compactified, but I could not find resources related to this. Is this view correct? ...
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2answers
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Notation for current operator in a paper

Equation (3) of https://arxiv.org/abs/cond-mat/0606800 is the current operator $$j_i(x)=e\tilde{\psi}^\dagger(x')v_{x'x}^i\tilde\psi(x)-\frac{e^2}{c}\tilde\psi^\dagger(x')(m^{-1})^{ij}_{x'x}\tilde\...
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1answer
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Confusion about dimensional regularization

I am recently trying to understand dimensional regularization in the context of quantum field theory. So to solve an integral $$ \int_{\mathbb R^d} \frac{\text d ^d p}{(2 \pi)^d} \frac{1}{(p^2 + m^2)^...
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Does lattice field theory also have infinity like perturbation before regularizations?

In loop corrections, the integrals are often infinity before regularization. The reason is attributed to the applicability of field theory. Does lattice approach also have infinity? If yes, where can ...
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Photon mass Infrared divergence regulization in the one-loop electron self-energy in QED

So basically I'm trying to calculate the one-loop mass and field strength counterterms from the electron's self-energy in QED using Pauli-Villars regularization (i.e. some heavy particle of mass $\...
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47 views

Slow and fast variables in Non-linear Sigma model

I am following Peskin Section 13.3, where they solve the nonlinear sigma model using Polyakov method. This system has Lagrangian \begin{equation} \mathcal{L}=\frac{1}{2g^2}|\partial_\mu\vec{n}|^2,\tag{...
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What is the Fourier transform of this expression in $2\omega$ dimensions?

I would like to perform the following Fourier transform in $2\omega$ (Euclidean) dimensions: $$A(x_1,q) = \int d^{2\omega} p_1\ e^{i p_1 \cdot x_1} \frac{\delta^{(1)}(v \cdot (p_1 + q))}{p_1^2 (p_1^2 ...
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1answer
97 views

Why is renormalization (instead of regularization) is needed in QFT?

This question looks like a duplicate question but I will try to make it different. In QFT, since divergence arise in the calculation of some quantities, we need regularization to remove the ...
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1answer
89 views

Integral divergent in Peskin and Schroeder eq. (7.90)

I'm working on the Eq. (7.90) in Peskin (page 252). However, I don't understand why it diverges logarithmically. Does $\Gamma(0)$ mean logarithmically divergence?
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1answer
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Polchinski massless vertex operator in the Polyakov approach p105 Eq. (3.6.16)

I am trying to check the Weyl transformation of the massless vertex operator in Polchinski closed bosonic string in the Polyakov approach (p105, Eq 3.6.16).To do that one needs to calculate something ...
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1answer
82 views

Evaluating Wilson loop in Abelian theory (Srednicki)

In chapter 82 https://web.physics.ucsb.edu/~mark/ms-qft-DRAFT.pdf Srednicki comes to the following form for the Wilson loop for free electromagnetic theory: $$\langle 0|W_C|0\rangle=\exp\left[-\frac{...
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Regularization of an integral

I really need to properly evaluate the following integral. \begin{equation}\label{1} \frac{1}{2}\int dx \, dy \,\frac{\rho(x)\rho(y)}{(x-y)^2} \end{equation} Where $\rho$ is some density function ...
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1answer
76 views

Is this box integral divergent or finite when “pinched” at one point?

Let us define the following conformal integral: $$X_{1234} = \int \frac{d^4 x_5}{(2\pi)^8} \frac{1}{x_{15}^2 x_{25}^2 x_{35}^2 x_{45}^2}\tag{1}$$ This is the box integral in position space, and it ...
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1answer
47 views

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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107 views

$\phi^4$-theory: Feynman diagrams loop integral calculation [closed]

I am studying quantum field theory by myself, could anyone help me with this integral? How can I get this result? Be more specific?
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2answers
162 views

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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1answer
116 views

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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0answers
41 views

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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1answer
94 views

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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Are Feynman diagrams analytic functions of the external momenta?

I was reading Ashok Das's book on QFT and came across this statement: Thus we see explicitly that any Feynman amplitude can be Taylor expanded so that the divergent parts can be separated out as ...
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1answer
105 views

If a regularization procedure respects a symmetry, is this symmetry unbroken in perturbation theory?

I read in this paper the statement that a proof that SUSY is preserved in perturbation theory would be the existence of a regularization procedure which respects SUSY (for a particular theory). Is ...
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27 views

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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1answer
217 views

“Bad” behavior of propagator in $O(N)$ model

In Polyakov's book about gauge fields & strings, in chapter devoted to non-linear sigma model he emphasizes problem with large $N$ expansion of this model. Lagrangian of 2D model is $$\frac{1}{2g^...
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44 views

Determining Feynman one-loop integral $I_{2,1}$

I need to determine Feynman one-loop integrals to work out some Feynman diagrams, in particular $I_{2,1}$. Starting from the general formula: $$I_{n,m}=\frac{1}{(4\pi)^2}\frac{\Gamma(m+2-\frac{\...
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1answer
63 views

Convergence of the path integral

In P&S 9.3 the path integral $$ Z[J]=\int {\cal D}\phi \exp[i\int d^4x ({\cal L} + J\phi)]$$ of the (Minkowski) $\phi^4$-theory when subjected to a Wick-rotation (change of the integration path ...
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1answer
68 views

How can I relate this integral to dimensional regularization?

In the paper "Scattering into the Fifth Dimension of $\mathcal{N}=4$ Super Yang-Mills", the authors give the following result for an integral: $$\begin{align} I^{(1)}(x_{13}^2,x_{24}^2,m) =& \...
3
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1answer
77 views

Regulating a divergent integral in QED

When we try to regulate a divergent integral, we introduce another parameter, say $\lambda$ and then compute the integral. We finally take a limit (either $\lambda \rightarrow 0, \infty $) to restore ...
2
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1answer
48 views

Simple question on computing commutation relation

In bosonization, one faces with the following commutator: $$[\phi(x_1), \theta(x_2)]=\sum_{q\neq 0} \frac{\pi}{Lq} e^{iq(x_2-x_1)-\alpha |q|}\tag{1}$$ where $q$ is an non-zero integer multiple of $2\...
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0answers
57 views

Chiral anomaly: gauge covariance and regularization

I am looking at the treatment of the chiral anomaly in Fujikawa and Suzuki's "Path Integrals and Quantum Anomalies." To illustrate the quantum breaking of chiral symmetry (section 4.3), they start ...
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Renormalization of the $O(N)$ vector model - integral

I am following the steps made in the review https://arxiv.org/abs/1512.06784. there, at page 14 the author proceeds to find a specific logarithmic term for $\frac{\partial U_{eff}}{\partial \sigma}$ ...
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77 views

Gaussian oscillatory integral evaluation using regularization

To evaluate the Gaussian integral $$ \int_{-\infty}^\infty dx e^{iax^2} = \sqrt{\frac{\pi i}{a}}, $$ one can use an appropriate contour as here, or use the method of "regularization", contained for ...
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0answers
72 views

Anomalies depend on how they are calculated. How is this satisfactory?

If we have a set of linear symmetry currents $J^{\mu}_{\alpha}$ and attempt to find if they are anomalous, we find that if we change the regularization procedure, the anomaly will get mixed around the ...
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Regularization methods' equivalence and the hierarchy problem

There are several questions related to this one but whose answers just raise the doubt I'm going to describe here. Some facts that are everywhere are (i) The hierarchy problem results from the fact ...
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1answer
123 views

The central charge and normal ordering

This question is about how the normal ordering in the energy momentum tensor for a free field is consistent with a non-vanishing vacuum expectation value implied by the transformation rules for a CFT. ...
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1answer
248 views

On scheme dependence in QFT renormalization

I searched for the answer to my question quite a while and it seems nobody ever asked similar questions or it is written explicitly in any textbooks. The question is, If physical parameters of any ...
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1answer
101 views

Regularization is mandatory. What about renormalization?

We need to regularize in order to declare with confidence that infinities drop out from measurable quantities, e.g. in the form of a cutoff scale. In general, the amplitudes in QFT depend on the ...
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0answers
55 views

Wick rotation graphically

We have to evaluate $$i\int_{-\infty}^{\infty} f(t) dt.$$ We can make a change of variable $t\mapsto -i\tau$, which results in $$\int_{-i\infty}^{i\infty} f(-i\tau) d\tau.$$ If we now multiply the ...
4
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1answer
112 views

Wick rotation convergence. Functions in the integrand

Performing a Wick rotation over an integral is not equivalent to just a change of variable $t \to \mathrm{i}t = \tau$, after that we rotate the complex plane so that $$\mathrm{i} \int_{-\infty}^{\...
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1answer
93 views

Questions about perturbation theory

In time-dependent perturbation theory, when assuming $\hat{V}$ is time-independent, the time development operator is as: $$\hat{U}(t,0)\theta(t)=e^{-i(\hat{H_{0}}+\hat{V})t}=\int \frac{dw}{2\pi}\...

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