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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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Meaning of Regularization

In renormalization theory, we choose a certain type of regularization method in order to study the analytic behavior of UV divergence (whether it diverges as $\lim_{\epsilon\rightarrow 0} 1/\epsilon$, ...
Ting-Kai Hsu's user avatar
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Photon Mass Regulator in IR divergences

On Schwartz's QFT page 333, he metions that there is infrared divergence when we try to renormalized the two-point function of electron field in on-shell substraction scheme, $$\frac{d}{d\,p_{\mu}\...
Ting-Kai Hsu's user avatar
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$\phi^4$ quantum fields theory with vanishing physical mass

Let us consider the $\phi^4$ theory, where $\phi$ is a real scalar field, such that the physical mass vanishes. Is it true that the bare mass also vanishes?
MKO's user avatar
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Konishi operator anomalous dimension [closed]

The Konishi operators are operators in the ${\cal N}=4$ SYM theory and are given by: $$ K = \sum _{i=1}^6Tr\ (\phi^i\phi^i) $$ The 2 point function of this operator is: $$ \big\langle K(x)K(y)\big\...
BVquantization's user avatar
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Infrared regularizing the harmonic oscillator path integral

This is from Laine and Vuorinen’s Basics of Thermal Field Theory. I do not understand why the fact that the integral over $x(\tau)$ implies the following regularization scheme. That is, I don’t ...
Vimal Rajan's user avatar
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Regularization of black hole singularities

Hi I have a question: when dealing with the gravitational Lorentz factor from schwarzchild solution to EFE, used in defining gravitaional time dilation and one encounters singularities at $r=0$ or $r=...
Precious Adegbite's user avatar
3 votes
2 answers
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Some integrals in QED Renormalisation

I am currently leaning the renormalisation of QED and I have met some tricky integral that seems unsolvable. The integrals are shown in Quantum Field Theory and the Standard Model by Schwartz, page ...
quantumology's user avatar
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3 answers
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Closed form expression of 2D CFT integral

I am currently working on a 2d CFT and am wanting to compute a complex plane integral, making sure I take into consideration potential contact terms as well. The integral in question is $$ \int_{\...
NoName's user avatar
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Residue theorem application

First I want to provide a little bit of context: I finished my undergrad degree in physics in 2008 and after that I moved into strategic consulting and into the financial world. Right now, at 41 years ...
ateixeira82's user avatar
3 votes
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Question about RG from QFT perspective, in particular Weinbergs book

This must be fairly basic I fail to understand. According to Weinberg, QFT Vol2, Ch.18 (The preamble) When we replace bare couplings and fields with renormalized couplings and fields defined in terms ...
Confuse-ray30's user avatar
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Renormalization in quantum field theory by discretizing space (but not time)

I'm a mathematician slowly trying to learn quantum field theory and I have a small question about renormalization, which I still have a shaky understanding of. One common way to explain what's ...
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Wilsonian RG in QFT: what is the difference between renormalized and bare couplings?

I want to understand the relation between the Wilsonian RG and the usual QFT RG approach. Several questions have been asked, such as this and many others, yet I don't find a conceptual answer to what ...
Mr. Feynman's user avatar
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Partition function of Hydrogen atoms problem

I know there are several questions asking this problem, but I found this problem has not been solved yet to me. I will repeat the problem and state my view. Consider the statistical mechanics of a ...
TOAA's user avatar
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Analytical continuation as regularization in Quantum Field Theory, the remaining questions

There is an old question posted (Regularization) which did not get an answer, about the validation of analytic continuation as regularization. It did get some discussion in the comments, referring to ...
Jos Bergervoet's user avatar
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Expression of $\langle 0 | 0 \rangle _{f,h}$ in the Srednicki's quantum field theory book (eq. (6.21), p.47) [duplicate]

I am reading the Srednicki's quantum field theory book and stuck at some statement. In the book p.46, the author worte that : "Now consider modifying the lagrangian of our theory by including ...
Plantation's user avatar
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Why do different contours give different answers in the limit $\epsilon \rightarrow 0$ when calculating propagators?

Let $\phi$ denote the Klein-Gordon field. Then its propagator $\langle 0 \mid [\phi(x), \phi(y)] \mid 0 \rangle$ can be calculated as $$\int \frac{d^4}{(2\pi)^3} \frac{-e^{-ip(x-y)}}{p^2 -m ^2}. \tag{...
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Can $\gamma^5$ matrices be ignored in $q\bar{q}\to ZZ$ processes?

In the $q\bar{q}\to ZZ$ process, the following Feynman diagram in LO appears: This means for each vertex, the Feynman amplitude contains a term proportional to $(g_V-g_A\gamma^5)$, which makes $D$-...
Ozzy's user avatar
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How small is $\eta$ when we say $\eta\to 0^+$ in Green's functions

When we convert Matsubara's imaginary time Green's function to the retarded Green's function, we perform an analytical continuation by substituting $i\omega_n$ with $\omega + i\eta$, with $\eta\to0^+$....
Luqman Saleem's user avatar
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Point-splitting regularization for anomaly in curved spacetime

In flat spacetime, the point-splitting regularization for (chiral) anomaly is discussed in great details in Peskin and Schroeder's QFT. Does anyone know any good references for calculating anomaly ...
4 votes
2 answers
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Derivation of Peskin & Schroeder eq. (4.29)

Background material: These are the parts that I can follow. Previously Peskin & Schroeder have derived already the expression of the interaction ground state $|\Omega\rangle$ in terms of the free ...
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How can the linear combination of infinite normalized Klein-Gordon fields be a normalizable field?

In the context of a Klein-Gordon field with charge $e$, mass $m$, immersed in an external classical electric field $A_\mu = (A_0(z), 0)$, I am asked to calculate the charge density of the field ...
dolefeast's user avatar
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1 answer
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Massless tadpole integrals in dimensional regularization

I'm trying to prove the following: \begin{equation} \int_0^\infty x^a dx = 0, \hspace{2pt} \forall a\in \mathbb{R}. \end{equation} This should work in dimensional regularization. I found a lot of ...
Alex's user avatar
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Vertex correction cut-off

I'm trying to retrieve the Vertex cut-off solved by Bjorken and Drell in their book (J.D.Bjorken S.D.Drell-Relativistic Quantum Mechanics Bjorken Drell (1964)-McGraw-Hill (1964). The main issue is ...
Leonardo's user avatar
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Wilsonian effective action and dimensional regularization

In the Wilsonian approach to QFTs, QFTs are treated as effective field theories which are reliable at some UV cut-off $\Lambda_{eff}$, We then integrated out high energy modes and see how couplings ...
Arian's user avatar
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1 answer
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Expectation value of the exponential of a quadratic term in fields

I have the following relation in this paper (J.B. Kogut: Introduction to Lattice Gauge Theory and Spin Systems, equation 8.39, page 709) (RG), where the author while doing an RG calculation writes $$\...
QFTheorist's user avatar
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2-loop correction to the Beta function in Fradkin's OPE-based RG

In chapter 4.5 of Fradkin's book "Field Theories for Condensed Matter Physics" (page 81 in my version) he claims that "it is straightforward to see that this [the perturbative ...
Noctis's user avatar
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Can $\int e^{ix^2 t} dx$ be defined without going to imaginary time? [duplicate]

The motivation is trying to define the path integral, where at some point we get the integral $$ Z=\int e^{iS}Dx $$ which is then taken to imaginary time $$ Z_E=\int e^{-S_E}Dx $$ such that $Z_E$ can ...
Toby Peterken's user avatar
10 votes
1 answer
225 views

Wick Rotation vs Sokhotski-Plemeli Method to compute internal loop of Feynman correlators

When computing loop integrals in QFT, one often encounters integrals of the form $$\int_{-\infty}^\infty\frac{dp^4}{(2\pi)^4}\frac{-i}{p^2+m^2-i\epsilon},$$ where we are in Minkowski space with metric ...
Sean's user avatar
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If quantum fields are operator valued distributions, why aren't they always smeared?

I don't completely understand the distributional character of a quantum field because I never see them "smeared" in basic textbooks. As I understand it, if $\chi : \mathcal{F} \rightarrow \...
R. M.'s user avatar
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1 answer
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How do Dedekind's eta function arise while computing the partition function of a compact scalar field over circle?

I am following the book String Theory in a nutshell (From Elias Kiritsis). In chapter 4.18, it takes a theory following the action: $$S=\frac{1}{4\pi l_s^2}\int X\square X\ d\sigma,\tag{4.18.1}$$ $$ \...
R. Á. Candás's user avatar
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Stuck on proof of $\epsilon$ prescription of integral [duplicate]

\begin{equation} f(\infty)+f(-\infty)=\lim_{\epsilon\rightarrow 0^{+}}\epsilon \int_{-\infty}^{\infty}dt f(t) e^{-\epsilon|t|}\tag{9.2.15} \end{equation} I assume this is a smooth function $f(\tau)$ ...
wong tom's user avatar
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Understanding the renormalization process in QFT

I am having problems understanding the process of regularization and renormalization in quantum field theory. As far as I understand it, this process of taming the infinites could be summarized in the ...
SrJaimito's user avatar
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2 answers
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Does that the regularized sums of series and integrals divergent to infinity appear in measurements prove that they represent actual infinite values?

There is a philosophic debate about whether there could be infinite quantities in nature. Definitely we cannot measure infinite quantities with measurement instruments. But we know the regularized ...
Anixx's user avatar
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Proof that $-\partial^2 G(x, y) = \delta(x-y)$ for free field propagator

I recently realized that there is a slightly pedantic issue when one normally proves that the equations of motion acting on the free field propagator gives a delta function which I have become ...
pseudo-goldstone's user avatar
2 votes
1 answer
95 views

Loop diagrams with derivative couplings

Consider the following Lagrangian of two massless scalars in 3+1D interacting through derivative interaction: $$ {\scr L} = \frac12(\partial a)^2(1 + g b) + \frac12(\partial b)^2, $$ where $g$ is a ...
Guy's user avatar
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1 answer
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Kinetic mixing, and a bare mass?

I've been reading the following classic paper by Bob Holdom "Two $U(1)$s and $\epsilon$ charge shifts", and I'm attempting to derive the expression for $\chi$. In particular, I am computing ...
Guy's user avatar
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2 votes
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How to integrate a Gaussian path integral of free particle using zeta function regularization?

I am attempting to integrate this path integral in Euclidean variable $\tau $ (but this need not be the same as the $X^0$ field): $$Z=\int _{X(0)=x}^{X(i)=x'}DX\exp \left(-\int _0^i d\tau \left[\frac{...
Andrew Dynneson's user avatar
1 vote
1 answer
84 views

A simple question on creation and annihilation operators

We know that the KG solution for a Spin-0 particle has the following Hamiltonian $$\hat{H}=∫ d^{3}p\frac{ω_{p}}{2}(\hat{a}_{p}\hat{a}^{\dagger}_{p}+\hat{a}^{\dagger}_{p}\hat{a}_{p})\hspace{2cm}[\hat{a}...
Filippo's user avatar
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0 answers
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Renormalization group equation and method of characteristics

All of this question refers to ref. 1. The equation are numbered alike. The author claims to solve a renormalization group (RG) equation using the Method of characteristics, but there is a passage ...
Mr. Feynman's user avatar
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2 votes
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Pauli-Villars regularization and self-energy

In the calculation of the electron self-energy in QED (one-loop level), there is a UV and IR divergent integral that needs to be regularized. A common choice for the regularization is the Pauli-...
Mr. Feynman's user avatar
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4 votes
1 answer
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LSZ Reduction formula: Peskin and Schroeder

On page 224 of P&S, they have the following expression (7.36), The integral over $d^3q$ gives us all the $q \to p$, then the integral over $dx^0$ is computed. The RHS given matches when only the $...
QFT_groupie's user avatar
6 votes
2 answers
486 views

One-loop Integral for a Tensor Quantity

I am considering one-loop integrals in quantum field theory. I am happy with how these calculations work with scalar integrals, but I am a bit lost in the details of what happens when the quantity ...
Tom's user avatar
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2 votes
1 answer
258 views

Why does non-perturbative QCD need to be regularized and renormalized?

The $n$-point correlation functions of QCD, which define the theory, are computed by performing functional derivatives on $Z_{QCD}[J]$, the generating functional of QCD, $$\frac{\delta^nZ_{QCD}[J]}{\...
orochi's user avatar
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0 answers
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Kustaanheimo-Stiefel (KS) transformation and initial conditions

I understand how Kustaanheimo-Stiefel (KS) transformations work and I have derived my equations of motion as well. However, I am struggling to find initial conditions in the KS space. How do I convert ...
Roshan Thomas Eapen's user avatar
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42 views

On the computation of natural logarithm in dimensional regularization

When calculating the integral \begin{equation} \int\frac{d^4q}{(2\pi)^4\left(q^2+\Delta+i\epsilon\right)^2} \end{equation} We encounter a term of $\ln\Delta$ and I am not sure how does one treat it ...
JavaGamesJAR's user avatar
3 votes
0 answers
167 views

Beta function of $\phi^4$ from RG equation

I am currently following David Tongs lecture notes on statistical field theory (Link to the script) and I have an issue with the calculation of the beta-function from the RG equations (equation (3.45))...
VerwirrterStudent's user avatar
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0 answers
70 views

Can gravity be the high-energy regularizer for QFTs?

In order for QFTs to be mathematically well-defined some sort of regularization scheme needs to be introduced, which typically looks like a discrete cutoff of all modes outside of some shell with ...
Panopticon's user avatar
5 votes
0 answers
110 views

Questioning a calculation in Kapusta / Infinite entropy of a Fermi gas?

I am going through Kapusta's calculation of the free energy of a Fermi gas, and I find one of his steps dubious (and if I'm right, it would mean the free energy of a Fermi gas is either infinite or ...
WillG's user avatar
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3 votes
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Vacuum bubble with the $Z$-boson

I'm interested in computing the vacuum bubble diagram of a single $Z$-boson. The propagator for a massive vector is $$ D_{\mu\nu}(k_\mu) = \frac{i}{k^2 - m_Z^2}\left(g_{\mu\nu} - \frac{k_\mu k_\nu}{...
Guy's user avatar
  • 1,291
2 votes
1 answer
102 views

Explicit form of cutoff dependent bare coupling at first order in $\phi^4$ with cutoff regularization

I would like to renormalize $\phi^4$ theory with Lagrangian \begin{equation} \mathcal{L} = \frac{1}{2} \partial_\mu \phi \partial^\mu \phi - \frac{1}{2} m_0^2 \phi^2 - \frac{\lambda_0}{4!} \phi^4 \end{...
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