All Questions
Tagged with renormalization dimensional-analysis
74
questions
4
votes
2
answers
190
views
Role of the natural temperature scale in the anomalous dimension of the renormalization group
In David Tong's lecture notes on statistical field theory, the concept of anomalous dimensions is introduced by considering the scaling of the correlation function $$\langle \phi(\mathbf{x}) \phi(\...
- 307
16
votes
3
answers
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Why is finding a mathematical basis for the fine-structure constant meaningful?
I was reading QED by Richard Feynman and at the end he mentions that:
There is a most profound and beautiful question associated with the observed coupling constant, $e$ – the amplitude for a real ...
- 169
1
vote
2
answers
105
views
Why is Perturbative expansion of gravity in terms of $GE^2$?
From General Relativity by Weinberg p.797 edited by Hawking & Israel:
This is to be used to generate a perturbation series in powers of $GE^2$ or $G/r^2$ (where $E$ and $r$ are an energy and a ...
- 23
5
votes
1
answer
363
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Where does Planck's constant come from in non-renormalizability of quantum gravity?
I am trying to understand the idea that gravity breaks down at the Planck scale, but I am confused by the use of natural units ($c = \hbar = 1$). The Einstein-Hilbert action in natural units is:
\...
- 61
3
votes
2
answers
169
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Difference between renormalizable and super-renormalizable theories
In $\phi^n$ theory in Peskin & Schroeder the superficial degree of divergence is:
$$D = d - V[\lambda] - \big(\frac{d-2}{2}\big)N \tag{10.13}$$
where $d$ is the dimension, $V$ is the number of ...
- 3,350
1
vote
1
answer
64
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Charge renormalization choice in QED
In the lectures on QFT I'm following we define the renormalized QED Lagrangian as
$$\mathcal{L} = \dfrac{1}{4} (F_0)_{\mu\nu} (F_0)^{\mu\nu} + \bar{\psi}_0 (i \bar{\partial} - (m_0)_e) \psi_0 - e_0 \...
- 1,611
2
votes
1
answer
88
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How is dimensionality of $S$ preserved term by term in a perturbative expansion?
In a schematic notation, the scattering matrix element $$\langle p_{out}|S|p_{in}\rangle := 1 + i (2 \pi)^4 \delta^4(p_{in} -p_{out}) M$$ between an incoming state with momentum $|p_{in}\rangle$ and ...
- 307
2
votes
0
answers
52
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Can a relevant operator's OPE with itself only include the identity and irrelevant operators?
I am interested in correlation functions in critical spin chains, and I'm trying to understand the consequences of conformal field theory for these correlation functions. I should warn that I do not ...
- 2,292
0
votes
1
answer
186
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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 ...
- 357
4
votes
2
answers
131
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Miraculous cancellations in a-priori non-renormalizable theories
Einstein's gravity is non-renormalizable since its coupling constant in 4D (I would like to limit the discussion to 4D) has negative mass dimension of -2.
Nevertheles it has been hoped that -- may be ...
- 9,778
3
votes
0
answers
104
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Critical exponent from powercounting of the action through the renormalization group
This will be a very basic question. For example, when we write down a $\phi^4$ action in condensed matter, let's say for an Ising magnet:
$$F[\phi] = \int d^Dx \dfrac{1}{2}(\nabla \phi)^2 + \dfrac{1}{...
- 1,168
0
votes
1
answer
109
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Dimensional Analysis and Power Counting in $R$ and $R^2$ Gravity Perturbation Expansions
In the context of $R$ gravity, the perturbation expansion appears as:
$$
S=\int \left( \partial \tilde{h} \partial \tilde{h} + X \tilde{h} \partial \tilde{h} \partial \tilde{h} + ... \right) d^4 x
$$
...
- 1,548
6
votes
5
answers
368
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Does pure Yang-Mills have a scale?
Consider pure Yang-Mills (YM) in 4 dimensions. The YM mass gap problem (as described in https://www.claymath.org/wp-content/uploads/2022/06/yangmills.pdf) tells us that this is supposed to have a mass-...
- 742
4
votes
1
answer
125
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Degree of divergence of subdiagram
The proof of the Appelquist-Carazzone theorem involves analyzing the behaviour of an internal subdiagram which is a fermion loop with $F$ incoming vector lines. The mass of the fermion is assumed much ...
- 1,742
3
votes
0
answers
50
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Divergent verticies in mesonic scalar theory [closed]
Considering the following Lagrangian density:
$$ \mathcal{L} = - \frac{1}{2} ( \partial_{\mu} \phi \partial^{\mu} \phi + m^2 \phi^2) + \bar{\psi} (i \gamma^{\mu} \partial_{\mu} - m) \psi + g \bar{\psi}...
- 1,124