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Questions tagged [dimensional-regularization]

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

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Some calculation in Schwartz's Quantum field theory eq. (16.39)

In Schwartz's Quantum field theory and the standard model, p.307 he derives a formula: $$ \Pi_2^{\mu \nu} = \frac{-2 e^2}{(4 \pi )^{d/2}}(p^2g^{\mu\nu}-p^{\mu}p^{\nu})\Gamma(2- \frac{d}{2}) \mu^{4-d} \...
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How does Superconductiviy change in 3D to 2D or vice versa? [closed]

I can't really understand how these transition works, maybe someone could recommend me reading something about these tranisiontions in superconductors. How does a 3d supercon transition to 2d or vice ...
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Polarization tensor of graviton in $d$ dimensions

Take the following tensor, that is the sum over the two polarizations of a gravitational wave in 3 spatial dimensions: $$E_{ijkl}(\vec{k})\equiv\sum_{\lambda = +,\times} \epsilon^\lambda_{ij}(\vec{k})\...
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Massless Sunset Diagram $\phi^4$ [closed]

I should compute an explicit calculation for the sunset diagram in massless $\phi^4$ theory. The integral is $$-\lambda^2 \frac{1}{6} (\mu)^{2(4-d)}\int \frac{d^dk_1}{(2\pi)^d} \int \frac{d^dk_2}{(2\...
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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 ...
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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 ...
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Are the one-loop beta functions in bosonic string theory written in terms of bare or renormalized background fields?

Given a bosonic string theory defined by the action $$\tag1 S = \frac{1}{4\pi \alpha'}\int_\Sigma \! \mathrm{d}^2 \sigma \, \sqrt{|g|} \, \left[ G_{\mu\nu} \partial_\alpha X^\mu \partial_\beta X^\nu ...
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RG equations for renormalized metric in string theory

I'm studying these PDF notes on strings on curved backgrounds and the author introduces the dimensional regularization of the theory by first defining the bare and renormalized target space metric, $...
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How is the dimensionful renornalization scale $\mu$ related to break of scale invariance in String Theory?

In the $7.1.1$ of David Tong's String Theory notes it is said the following about regularization of Polyakov action in a curved target manifold: $$\tag{7.3} S= \frac{1}{4\pi \alpha'} \int d^2\sigma \ ...
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Is it possible to expand the measure in dimensional regularization?

In the dimensional regularization scheme, four-dimensional integrals are analytically continued from their $d$-dimensional counterparts, i.e., $$\int d^4 x\, f(x) \longrightarrow d^d x\, f(x)\,, \tag{...
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Dimensional regularization order of integration

I simply have a question of which integration I should perform first. Consider the typical integral from some loop calculation that has had the Feynman-trick and the typical dim-reg procedure perform, ...
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Peskin and Schrorder's QFT eq.(12.131), $\beta$ function of $\phi^4$ theory

On Peskin and Schroeder's QFT, page 435, they derived the $\beta$ function for $\phi^4$ theory in a general $d-$dimensional spacetime. $$\beta=(d-4)\lambda +\beta^{(4)}(\lambda)+\cdots \tag{12.131}$$ ...
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Odd number of momentum should vanish but doesn't? [closed]

I have the following integral found within a loop-calculation (the actual content of the Feyman diagram, this is purely a math question) \begin{equation} J_\mu = \int\frac{d^4l}{(2\pi)^4}\frac{l_\mu}{...
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Expanding functions with poles in QFT Calculation

I am using Series function in Mathematica on $(1/z)(-k^2)^z$. Up to $z^0$, the function gives me $1/z + \log[-k^2]$. But in the standard textbook on QFT, it turns out the expansion should give $1/z + \...
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Analyzing the one-loop self-energy graph in $\phi^3$ model

Consider the $\phi^3$ model with a real scalar field $\phi(x)$ in $3+1$ dimensional Minkowski spacetime with metric $(-,+,+,+)$. Its Lagrangian density is $$ \mathcal{L}=-\frac{1}{2} \partial_\mu \phi ...
Ho-Oh's user avatar
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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
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Regularization of loop integrals with Feynman slash

As the title suggests, I am trying to compute a loop integral with a Feynman slash in the numerator, like $$\int\frac{d^Dq}{(2\pi)^D}\cdot\frac{q_\mu\gamma^\mu}{\left(q^2+\Delta+i\epsilon\right)^3}$$ ...
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Dimensional regularization yields finite result for loop integral in 3d $\phi^4$ theory [duplicate]

As an exercise I wanted to compute the mass renormalization in 3d $\phi^4$ theory (1-loop). At 1-loop the field requires no renormalization (as in the 4d case) and the mass renormalization is ...
VerwirrterStudent's user avatar
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Renormalisation of the fermionic triangle loop

I am trying to renormalise the following loop diagram in the Standard Model: $\qquad\qquad\qquad\qquad\qquad\qquad$ Using the Feynman rules, we can write the amplitude as follows: $$ \Gamma_f \sim - ...
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Renormalization of Two Scalar Field Theory at Tree-Level and One-Loop Levels

I have been given a problem on renormalization, but due to my inexperience, I don't understand what to do with it. Here is the statement: Consider a theory of two real scalar fields $\phi$ and $\Phi$ ...
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Are large logarithms tied to dimensional regularization?

When introducing the renormalization group many textbooks start by stating that scattering amplitudes often contain logarithms of the form $\log\frac{\mu^2}{p^2}$, where $p^2$ is the characteristic ...
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$\gamma_5$ in non-integer dimension $D$

I need to calculate a trace containing a single $\gamma_5$ in $D$ dimensions. The trace is given by $$\displaylines{ \text{Tr}\bigg[ \gamma^{\alpha\tau\eta} (p_1-p_2)_\tau (p_1)_\eta \left(\gamma\cdot ...
Nik's user avatar
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Connection between the Beta Function and Residue Theorem?

When we define the bare coupling in Minimal Subtraction we write it as a Laurent series where the analytic part is identified with the finite, renormalized coupling and the nonanalytic part is ...
user119706's user avatar
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How to understand this field redefinition example from path integral formalism?

I'm studying the Lagrangian $$ \mathcal{L} = \frac{1}{2}\partial_\mu\phi \partial^\mu\phi+\lambda\phi\partial_\mu\phi\partial^\mu\phi~=~\frac{1}{2}(1+2\lambda \phi)\partial_\mu\phi \partial^\mu\phi.\...
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Expansion at first order ${\cal O}(\alpha_s)$ in counterterms for the QCD vertex renormalization at 1-loop

What is the meaning of the expansion at first order ${\cal O}(\alpha_s)$ in $\delta_2$ and $\delta_3$ at the second step in the last line? These quantities are not "small" - on the contrary,...
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Epsilon dependence of the beta function

For some time now, I’ve been trying to prove that the beta function for a quantum field theory with coupling $g$ (in the typical case of a coupling with mass dimension $\varepsilon$ in dimensional ...
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When should UV regulators be removed?

I have been working in QFT for a few years now, and I cannot believe I've never come across this problem. When considering an effective field theory, the allowed operators mix under renormalization: $$...
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Peskin and Schroeder's QFT eq. (7.88)

On Peskin and Schroeder's QFT book, page 251, the book discussed how things will be changed in $d$ dimensions. For example $g_{\mu \nu}g^{\mu \nu}=d$. In eq. (7.88), the book gave how Dirac matrices ...
Daren's user avatar
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Reproduction of the explicit calculation of the Sunset diagram from the book Critical Properties of $\phi^4$ Theories (Kleinert)

The book starts with the equation \begin{equation} I(D) = \lambda^2 \int \frac{d^Dp_1}{(2\pi)^D}\frac{d^Dp_2}{(2\pi)^D} \frac{1}{\mathbf{p}_1^2 + m^2} \frac{1}{\mathbf{p}_2^2+m^2} \frac{1}{(\mathbf{p}...
Feynman Diagrms's user avatar
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Peskin and Schroeder p. 409 “quadratically divergent mass renormalization”

I have three questions about page 409 of Peskin and Schroeder. First, they state that the diagram where $\Delta = m_f^2 - x(1-x)p^2|_{m_f=0}$, has a pole at $d=2$ “corresponding to the quadratically ...
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Table of integrals for dimensional regularization

Is there any reference (book or paper) that contains a list of integrals useful for dimensional regularization? I would need it to approach integrals like these $$ \int d^dx \frac{x^\mu}{|x|^{2d-4}}, \...
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What is the origin of these log terms in dimensional regularization?

The following limit is implied on page 250 of Peskin and Schroeder: $$\Gamma\left(2-\frac d2\right)\left(\frac 1 \Delta\right)^{2-\frac d2} \xrightarrow{d\rightarrow 4} \frac 2\epsilon - \log \Delta -\...
Rodrigo's user avatar
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Dimensional regularization vs. hard cutoff and their relation to the renormalization scale in 2d vs 4d to find $\beta$ functions

I would like to understand some shortcuts people are using to calculate $\beta$ functions using dim. reg. with mass scale $\mu$ and/or the hard cutoff $\Lambda$. My end goal is to use equation 12.53 ...
mkn's user avatar
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2 answers
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The conditions for a shift in a loop momentum to be allowed

When we evaluate the Feynman diagram containing a loop, we commonly use the identity: \begin{align} \frac{1}{A_{1}^{m_{1}} A_{2}^{m_{2}} \cdots A_{n}^{m_{n}}}= \int_{0}^{1} d x_{1} \cdots d x_{n} \...
tak's user avatar
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3 answers
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Limit of $d\rightarrow 4$ of a function in Peskin & Schroeder

In Peskin & Schroeder section 12.1 equation 12.15 we compute the function $$ \frac{-3\lambda^2}{(4\pi)^{d/2} \Gamma(\frac{d}{2})}\frac{(1-b^{d-4})}{d-4}\Lambda^{d-4} $$ Now when we take the limit $...
twisted manifold's user avatar
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1 answer
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Do "tadpole diagram with two external legs" always vanish in massless theories?

What I think the physics community means when talking about tadpole diagrams is something like this: and I don't understand why these diagrams always cancel (or why they don't for the Higgs boson) ...
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Schwartz loop result in $\bar{c}b\rightarrow \bar{u}d$ EFT

I am trying to reproduce the result of equation $(31.97)$ of Schwart's book Quantum field theory and the Standard Model. The results involves the calculation of the following diagram where the ...
Davide Morgante's user avatar
3 votes
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134 views

Equivalence between wilsonian and non-wilsonian RGE on QFT

My current way of viewing Wilsonian RGE applied to QFT: (1) We start with a lagrangian that accurately models dynamics up to a scale $\Lambda_0$. (2) We fix $\Lambda_0$ as a cutoff to regulate the ...
GaloisFan's user avatar
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1 vote
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Traces in 't Hooft-Veltman scheme

I'm currently looking at the 't Hooft-Veltman regularization scheme and I'm a bit confused on how exactly one calculates traces in this scheme. As far as I understand one has to divide the $D$-...
Sito's user avatar
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7 votes
0 answers
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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 ...
maarten442's user avatar
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Explicit example of dimensional regularization involving $\gamma^5$

I'm currently reading Collins' book Renormalization, Chapter 4. In section 4.5 he introduces $\gamma$-matrices and the trace operation in an arbitrary dimension $d$. In section 4.6 he then talks about ...
Sito's user avatar
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Peskin & Schroeder, section 12.2, page 409, analytic continuing from $d=2$ to $d=4$?

In Peskin & Schroeder section 12.2,page 409, consider the one-loop correction to the propagator in Yukawa theory, it has the form The authors first spot the pole at $d=2$, and find out that it ...
张嘉宝's user avatar
4 votes
1 answer
346 views

Dimensional regularization: order of integration

This is a two-loop calculation in dim reg where I seem to be getting different results by integrating it in different orders. I am expanding it about $D=1$. What rule am I breaking? $$\int \frac{d^{D} ...
octonion's user avatar
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6 votes
1 answer
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Operator mixing in dimensional regularization of EFTs

When renormalizing "non-renormalizable" operators within an effective field theory (EFT) one usually has to introduce additional (higher-dimensional) operators to the Lagrangian which act as ...
Katermickie's user avatar
2 votes
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328 views

Loop integrals in particle physics: Odd number of momenta and tensor decomposition

Given the following two-loop tensor integral, $$ \int \frac{\text{d}^D k_1}{(2\pi)^D} \int \frac{\text{d}^D k_2}{(2\pi)^D} \frac{k_1^\mu k_1^\nu k_2^\rho}{(k_1^2-a)^\alpha (k_2^2-b)^\beta ((k_1 + ...
Albercoc's user avatar
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1 answer
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Why can we set the external momenta to zero while calculating the 1-loop correction to non-abelian theory coupling?

Refer to the second diagram in figure 73.2 on page 440 of Srednicki's Quantum Field Theory book. On page 441 he proceeds to calculate the amplitude and argues that since the divergent part is ...
GaloisFan's user avatar
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1 vote
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What is mass factorisation?

I know that for some observables in QCD, e.g. inclusive DIS cross sections, one can use factorisation by writing said observable as a convolution of hard scattering coefficients with PDFs. But what is ...
Thomas Wening's user avatar
1 vote
2 answers
320 views

Fourier transform of bubble momentum space integral

I encountered the following integral in dimensional regularization $$ I=\int d^d k \,e^{i\vec{k}\cdot \vec{x}}\frac{1}{\vec{k}^2}\frac{1}{(\vec{q}-\vec{k})^2}, $$ say that we already Wick rotated the ...
rootofunity's user avatar
6 votes
1 answer
750 views

How to deal with two-loop integrals in dimensional regularization?

I've done several calculations on one-loop diagrams in dimensional regularization, involving things like Feynman parameters, or using hyperspherical coordinates after a Wick rotation, s.t. you can ...
ersbygre1's user avatar
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7 votes
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$1/\epsilon $ poles order in dimensional regularization of $\dfrac{\lambda}{4!}\phi^4$ theory

I have to find the order of 1/ε poles in dimensional regularization of $\dfrac{\lambda}{4!}\phi^4$ theory. The Feynman integral is the following: \begin{equation} I(p)=-λ^6\int \frac{d^4p_1}{(2\pi)^4}\...
Δανάη Ρουμελιώτη's user avatar