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Questions tagged [dimensional-regularization]

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

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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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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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183 views

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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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
132 views

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
253 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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264 views

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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969 views

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
313 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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173 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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581 views

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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185 views

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 ...
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111 views

Simple feynman parameters question

I have the following integral $$\int d^D l \frac{1}{p^2 (p-l)^2 l^2}$$ which I want to rexpress using feynman parameters. I can write as a first step, $$2 \int_0^1 dx \int_o^{1-x} dy \int d^D l \frac{...
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1answer
352 views

On shell and off shell simultaneously?

I am considering the following one loop virtual correction in the DIS process: where I have a quark of momentum $p$ coming in, emitting a gluon before interacting with a photon of momentum $q$ to ...
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268 views

What justifies the dependence of the coupling renormalization constant in the dimensional regularization regulator?

I wanna clarify some issues about renormalization in the $\bar{MS}$ scheme that I glossed over when I first learnt about this stuff. I am following http://arxiv.org/abs/1411.7853 section 3.1. The ...
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1answer
298 views

Can I use Pauli-Villars and dimensional regularization together?

There are at least two ways to compute the electron-self energy. You can use Pauli-Villars or dimensional regularization, for example. On Weinberg's book, it's chosen the first method, while on my ...
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1answer
160 views

Computation of the QCD vector two point function

I am following some notes on the computation of the vector two point function in QCD and I would like somebody to make some intermediate steps more explicit. Let's consider $$\Pi_{\mu\nu}=i\mu^{2\...
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126 views

Dimensional Regularization of the Higgs Mass Correction

I've found plenty of blog posts and papers where the authors claim that the Higgs mass divergence (usually presented with a momentum cutoff) doesn't show up under dimensional regularization. ...
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83 views

Dependence of finite part of loop integral on regularization

Recently I've calculated some process in which arise triangle loop with running two $W$ bosons and one massless fermion. The expression for integral is following: $$ I_{\alpha \beta}(r, q) = \int \...
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1answer
414 views

Conversion of results between cutoff regularization and dimensional regularization

Generally it would be expected that a renormalizable/physical quantum field theory (QFT) would be regularization independent. For this I would first fix my regularization scheme and then compute stuff....
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$d=2$ 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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633 views

The integral is zero! $\int \frac{\mathrm{d}^d k}{(2\pi)^d} = 0$

In using dimensional regularization in QFT calculations, one comes across integrals over propagators, they might look like $(d = \text{dimension of spacetime}, n = \text{a number})$ $$\tag{1}I(d,n)=\...
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77 views

Regularization ambiguity for leading singularity in dimensional regularization

I have a question with a perhaps well-known answer. Consider a two-loop sunset (log divergent) integral in two dimensions: $$ I_S = \int \frac{d^2k d^2l}{(2\pi)^4} \frac{ k^2}{(k^2-m^2)(l^2-m^2)((p-k-...
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Electron's self-energy in QED in arbitrary gauge

Recently I've tried to evaluate electron's self-energy in QED in the second order of perturbation theory by using dimensional regularization. Corresponding 1PI-diagram leads to $$ \Sigma_{1loop} = -ie^...
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Massless integrals in dim-reg

Consider the massless divergent integral $$ \int dk^4 \frac{1}{k^2}, $$ which occurs in QFT. We can't regularize this integral with dim-reg; the continuation from the massive to the massless case is ...
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1answer
127 views

Dimensionial Regularization (integrating over extra dimensions)

I tried to replicate a result about dimensional regularizations in the review article of Howard Georgi, Effective Field Theory at: people.fas.harvard.edu/~hgeorgi/review.pdf. On page 17, he writes ...
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1answer
2k views

Renormalizing IR and UV divergences

In lectures on effective field theory the professor wanted to find the correction to the four point vertex in massless $\phi^4$ theory by calculating the diagram, $\hspace{6cm}$ We consider the zero ...
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1answer
1k views

Gamma Matrices in Dimensional Regularization

Prove that $tr\left(\gamma_\mu\gamma_\nu\gamma_\rho\gamma_\sigma\gamma_5\right)=0$ when the spacetime dimension is not 4. What I have tried: We know that $\gamma_\alpha\gamma^\alpha=d\mathbb{1}$, so ...
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369 views

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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279 views

Dimensional regularization and the finite part

Let be a dimensional regularized integral $$ \int d^{4-\epsilon}kF(k,m,s)= \frac{2}{\epsilon}+\frac{m^{2}}{3}(\gamma +log(4\pi)-\frac{1}{\epsilon}))$$ then formally if we elmiinate the divergent ...