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:
$$ \Pi_2^{\mu \nu} = \frac{-2 e^2}{(4 \pi )^{d/2}}(p^2g^{\mu\nu}-p^{\mu}p^{\nu})\Gamma(2- \frac{d}{2}) \mu^{4-d} \...
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Polarization tensor of graviton in $d$ dimensions
Take the following tensor, that is the sum over the two polarizations of a gravitational wave in 3 spatial dimensions:
$$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 $$-\lambda^2 \frac{1}{6} (\mu)^{2(4-d)}\int \frac{d^dk_1}{(2\pi)^d} \int \frac{d^dk_2}{(2\...
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1
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Massless tadpole integrals in dimensional regularization
I'm trying to prove the following:
\begin{equation}
\int_0^\infty x^a dx = 0, \hspace{2pt} \forall a\in \mathbb{R}.
\end{equation}
This should work in dimensional regularization. I found a lot of ...
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Wilsonian effective action and dimensional regularization
In the Wilsonian approach to QFTs, QFTs are treated as effective field theories which are reliable at some UV cut-off $\Lambda_{eff}$, We then integrated out high energy modes and see how couplings ...
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Are the one-loop beta functions in bosonic string theory written in terms of bare or renormalized background fields?
Given a bosonic string theory defined by the action
$$\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}{...
- 704
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votes
2
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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 ...
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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 ...
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