Questions tagged [dimensional-regularization]

Dimensional regularization is a method of isolating divergencies in scattering amplitudes.

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Do all unitary-preserving regulators necessarily turn real loop integrals into pure imaginary numbers?

The optical theorem, which results from the unitarity of the $S$-matrix, relates the imaginary part of the forward scattering amplitude to the total cross section. When using this theorem in practice, ...
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Why can I not asymptotically expand a Feynman integral this way?

I would like to asymptotically expand a series of Feynman diagrams in Euclidean space, and as a toy I started with the following integral, for which I know the full solution in $4d$ ($\omega \to 2$): ...
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What evidence exists to show that hyperdimensions are spacially perpindicular to the dimensions before it?

I've heard of a tesseract which is supposedly spacially perpendicular to the other 3 dimensions. Is there evidence this is possible in our universe or another?
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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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Dimensional regularization integral

I'm new here. Are there places to put specific problems or do they just go to a general list? There are some similar problems posted, but they are all a bit different and I can't see how to use the ...
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1answer
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Analytical continuation in QFT

My question is quite basic and generic. It is known that scaleless integrals that appear in QFT such as $\int \frac{d^dk}{(k^2)^2} = \frac{1}{\epsilon_{\mathrm{UV}}} - \frac{1}{\epsilon_{\mathrm{IR}}} ...
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1answer
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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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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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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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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) =& \...
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Is this useful identity valid only under the integral sign?

Studying dimensional reugularization one often encounters the following identity: $$ \int d^d q\, \, q^\mu q^\nu f(q^2) = \frac{1}{d}g^{\mu\nu}\int d^d q\,\, q^2 f(q^2) $$ often justified by some ...
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Dimensional regularization of a divergent integral

Suppose there is an integral in four dimension Euclidean space \begin{equation} I_{d=4}=\int_0^\infty d^4x\frac{1}{|x|^2},~ \end{equation} which is divergent. $|x|$ is the length of the vector. Can ...
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Can I use dimensional regularization with this integral?

I would like to extract the divergence of this integral in 4d Euclidean space: $$\int d^4z \frac{1}{(x-z)^4}\tag{1}$$ This divergence is expected to cancel with other divergences, which I got using ...
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1answer
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When can I set $d=4$ in dimensional regularization?

I am using dimensional regularization to extract the divergence of some complicated integral. I work in $d=2\omega$ dimensions, with $\omega\approx 2$. After I extract the divergence, I have an ...
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1answer
52 views

How to take the Fourier transform of the propagator of a vector field?

In the paper Wilson Loops in N=4 Supersymmetric Yang--Mills Theory, the authors give the following generalized Fourier transform for a propagator in $d=2\omega$ dimensions: $$\int \frac{d^{2\omega}p}{...
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1answer
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What's the point of dimensional regularization?

I'm studying regularization of divergent integrals in QFT from Here: Roberto Soldati - Field Theory 2. Intermediate Quantum Field Theory (A Next-to-Basic Course for Primary Education) I think I'm ...
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Momentum replacement in the axial anomaly calculation in dimensional regularisation (‘t Hooft prescription)

I have been studying the axial anomaly and everywhere I see the calculation of the triangle loop using dimensional regularisation (see for example pages 661-664 of section 19.2 of Peskin). In the ‘t ...
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Vector current conservation in vertex correction

Recently, I was calculating this observable: $\langle p s|\bar\psi(0)\gamma^{\mu}\psi(0)|ps\rangle$ Where we only consider the QED case. $\psi$ corresponds to massless Dirac fermion field, p is the ...
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What Dimensionality Reduction method I can follow on a dataset that has Physics Parameter?

I am trying to model data related to Locomotive Train. We have a various set of parameters and we have the possibility to generate a few more parameters from this. Our model is currently using a lot ...
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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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Dimensional regularization and expansion of gamma function

In my calculations, I used dimensional regularization, i.e. replace $d\rightarrow d-\epsilon$ and calculated the divergent integral. Then, I would like to expand the answer into seriers by $\epsilon$ ...
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71 views

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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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 $\...
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Explicit computation of singular part of two-loop sunrise diagram

For $\phi^4$, there is two-loop self-energy contribution from sunrise (sunset) diagram. The integration is $$ I(p)=\int\frac{d^D p_1}{(2\pi)^D}\frac{d^Dp_2}{(2\pi)^D}\frac{1}{(p_1^2+m^2)(p_2+m^2)[...
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Dimensional regularization - Expansion of powers of $\epsilon$ turns into logarithms

Looking into Schwartz's book on QFT at the appendices, it seems that when doing a dimensional regularization, one expands around $\epsilon=0$ and usually obtains $$ x^\epsilon=\log x+O(\epsilon), $$ ...
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Problem with loop Integral (HQET)

I have come across the Integral: $$ \int_0^{\infty}dx [x^2-ixa+c]^{n-\frac{d}{2}}e^{-bx},$$ where $n = 1,2 ; a,b,c,d \in \mathbb{R}; b,d > 0$. This integral should contain some divergences for $d ...
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Problem with converting Integral to Gamma functions (from HQET heavy quark self-energy diagram)

In the calculation of HQET radiative correction, I came across the Equation: $$\int_0^{\infty}d\lambda ~ \lambda^{-\epsilon}(\lambda+\omega)^{-\epsilon} = \frac{1}{2\sqrt{\pi}}\Gamma(\epsilon-\frac{1}{...
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Dimension of gamma matrices in dimensional regularization

When performing loop integrals in theories containing Dirac fermions, one almost always confronts terms of the form $$\text{Tr}\left[\gamma^{\mu_1}\cdots\gamma^{\mu_n}\right].$$ For instance, in $d$ ...
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1answer
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Dirac matrices in dimensional regularization, get correct order epsilon

Let us work in dimension $D = 4-2\epsilon$. In 4-dimension, we can write $\text{Tr}[A B]$, where $A$ and $B$ are string of gamma matrices, as $\sum_m \text{Tr}[A~\Gamma^m]\text{Tr}[B~\Gamma^m]$, ...
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Integration by parts in dimensional regularisation

I have a question concerning integration by parts identities in dimensional regularisation. Appearently, almost every textbook about dimensional regularisation claims that $$\int d^Dl_1...d^Dl_L \...
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1answer
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How to extract a finite answer after applying dimensional regularization in QED?

When one applies dimensional regularization in QED, in the end, one often gets an expression like $$\Gamma(n/2)\left(\frac{s}{\mu^{2}}\right)^{-n/2}$$, where $n$ is a small number, $\Gamma()$ is the ...
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Massless tadpoles vanish in Baikov representation?

In dim-reg, massless tadpoles vanish. Is it the case also if we use Baikov representation of the Feynman integral?
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Feynman Parameterization for the three-propagator problem

I am considering the three particle scattering problem, where I have the loop integral $$ \int \frac{d^dp}{(2\pi)^d} \frac{1}{((p_1-k)^2-m^2+i\varepsilon)((p_3+k)^2-m^2+i\varepsilon)(k^2-m^2+i\...
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Why are dimReg divergences power-like? Or are they?

An implicit assumption when working with dimensional regularisation is that the divergences are always of the form $\varepsilon^{-n}$ for some integer $n$ (e.g. refs.1&2). Is there any way to ...
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The $\Delta$ in Dimension Regularization

In dimension regularization, we often encounter the integral: $$\int \frac{d^Dl_E}{(2\pi)^D}\frac{l_E^k}{(l_E^2+\Delta)^n}$$ where the subscript $E$ stands for Euclidean signature. The result of this ...
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Does the angular measure matter in dimensional regularization?

In dimensional regularization, we replace a momentum integral $I= \int d^nk f(|k|)$ with the family of regularized integrals $$\mu^{n-d}\int d^dk f(|k|) = \mu^{\epsilon}\Omega_d \int p^{d-1} f(p)dp.\...
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1answer
359 views

Derivative interaction for one-loop diagrams

I'am not sure about some things about derivative interactions. Lets say I have to following 2 interaction terms: $$\mathcal{L} = \left(a\phi + b\phi^{2}\right)\partial_{\mu}\phi\partial^{\mu}\phi.$$ ...
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Observable Masses and Couplings in a Renormalized Theory

I am struggling against the running of coupling constants in QFT. I will consider $\lambda \phi^4_4$-Theory in order to express my point of view and to ask for further explanations. Since most of the ...
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1answer
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Trace technology with polarisation vectors

Consider $d$-dimensional gamma matrix structures. I have an expression like $$ \sum_{h_2=\pm}\text{Tr}(\not{\xi}_2\not{p}_3\bar{\not{\xi}}_2\not{p}_1), $$ where $\not p=p^\mu \eta_{\mu\nu}\gamma^\nu$ ...
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1answer
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This One-Loop diagram for $\phi^{4}$ theory - renormalization and going to position space

This is somewhat related to an earlier question I asked about the following diagram in $\phi^{4}$ theory: I've been following these lecture notes by H. Kleinert and V. Schulte-Frohlinde. Saying we'...
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Calculate in Method of dimensional regularization with normalization on physical masses [closed]

I have this question that says Calculate in Method of dimensional regularization with normalization on physical masses. any idea how should I begin?
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1answer
393 views

Schroeder's Minkowski Space Integral - Concerns about Wick Rotations

In the Appendix of Peskin & Schroeder's "An Introduction to Quantum Field Theory" there is a list of integrals in Minkowski space. Of particular interest to me is the integral (A.44): $$ I(\Delta) ...
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$\phi^{4}$ Propagator - Feynman Diagram: internal vertex that loops back to itself

In all that follows I'll be dealing with everything massless. The free, massless propagator ($\mathcal{L} = \int d^{4}x \left(\partial \phi(x) \right)^{2} $) is supposedly given by $G_{0}(x,y) = c (x-...
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Momentum space integrals in dimensional regularization

I am trying to go through some derivations in this paper (http://arxiv.org/abs/0804.3170). I am unsure about how they got from eq 56 to eq 57. Specifically the step is from $$i\int \frac{d^dq}{(2\...
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1answer
212 views

Renormalization group invariant objects of a quantum field theory

Consider an arbitrary QFT with $g_b$ as the bare coupling constant. After dimensional regularization, is $g_b \mu^\epsilon$ a renormalization group invariant object of the theory? In other words, is ...
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The relation between anomalous dimensions and renormalization constants

I am trying to understand the general strategy and technical details of calculating $\beta$-function at higher orders. $\beta$-function is the anomalous dimension of the coupling constant and there is ...
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How can dimensional regularization “analytically continue” from a discrete set?

The procedure of dimensional regularization for UV-divergent integrals is generally described as first evaluating the integral in dimensions low enough for it to converge, then "analytically ...
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Srednicki - computing divergent piece of loop integral

I was reading through Srednicki and didn't quite understand one of the paragraphs in Section $51$ on loop corrections in the Yukawa theory on P.$322$. It's the fermion loop correction to the local ...
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A box loop-integral [closed]

I am trying to evaluate the integrate $$ \int\frac{d^Dk}{(2 \pi)^D} \frac{1}{(k^2)^2(k^2-m^2)} $$ using dimensional regularisation ($D=4-2\epsilon$). From various references it appears that it should ...