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Questions tagged [dimensional-regularization]

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

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

148 questions
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3 answers

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 continuing" this result in the number of dimensions $d$. …
tparker
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24
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2 answers

Intuition for parameter $\mu$ in dimensional regularization

In dimensional regularization, a dimensionless coupling $g$ is replaced by $\mu^{4-d}g$ where $\mu$ has dimension of mass, so that $g$ can remain dimensionless. $\mu$ is unphysical, though its choice affects the values of counterterms. By setting…
knzhou
  • 107,105
24
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2 answers

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 ill-defined. It can be shown, however, that no…
innisfree
  • 15,425
19
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2 answers

Can dimensional regularization solve the fine-tuning problem?

I have recently read that the dimensional regularization scheme is "special" because power law divergences are absent. It was argued that power law divergences were unphysical and that there was no fine-tuning problem. I was immediately…
innisfree
  • 15,425
17
votes
1 answer

Dimensional regularization: removing more than just logarithmic divergencies?

I have followed two courses on QFT, which both involved renormalization by dimensional regularization. My confusion is that one of the professors claimed that dimensional regularization can only be used to regularize logarithmically divergent…
Funzies
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16
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1 answer

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 external momentum limit and denote $p$ as the…
JeffDror
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12
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1 answer

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. I'm interested to know whether there's a…
vik
  • 621
11
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0 answers

$d=2$ 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 discussed about in $\lambda\phi^4$ theory[2]? For…
LYg
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11
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2 answers

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)=\int \frac{\mathrm{d}^d…
Physics_maths
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10
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0 answers

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 $d$ dimensions, we could compute the simple trace…
Bob Knighton
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1 answer

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 a complete set of anomalous dimensions…
moha
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8
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2 answers

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 $$ I(p)=\int\frac{d^D p_1}{(2\pi)^D}\frac{d^Dp_2}{(2\pi)^D}\frac{1}{(p_1^2+m^2)(p_2+m^2)[(p-p_1-p_2)^2+m^2]}. $$ I try to use trick like…
maplemaple
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8
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2 answers

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^{2} \int \frac{d^{4}k}{(2 \pi…
user8817
8
votes
1 answer

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 we can…
PPR
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0 answers

Renormalization Group and Dimensional Regularization

Currently I am learning about regularization, renormalization and the renormalization group. In particular, a lot of detail is devoted to dimensional regularization. There are a couple of things I struggle with when discussing the renormalization…
maarten442
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