Renormalization of the commutative scalar theory with harmonic term to all orders

Axel de Goursac

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Renormalization of the commutative scalar theory with harmonic term to all orders

Axel de Goursac

High Energy Physics - Theory · Submitted 26 Jul 2012

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim In this paper, we prove the renormalizability of the commutative scalar field theory with harmonic term to all orders by using multiscale analysis in the momentum space.

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What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.09 · human participants 0.01 · controlled comparison 0.35 · randomized assignment 0.01 · states quantitative result 0.37 · compares with prior results 0.94 · mentions code or data availability 0.01.

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The noncommutative scalar theory with harmonic term (on the Moyal space) has a vanishing beta function. In this paper, we prove the renormalizability of the commutative scalar field theory with harmonic term to all orders by using multiscale analysis in the momentum space. Then, we consider and compute its one-loop beta function, as well as the one on the degenerate Moyal space. We can finally compare both to the vanishing beta function of the theory with harmonic term on the Moyal space.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The noncommutative scalar theory with harmonic term (on the Moyal space) has a vanishing beta function.	background 0.98	0.18	0.08	asserted 1.00
s1	In this paper, we prove the renormalizability of the commutative scalar field theory with harmonic term to all orders by using multiscale analysis in the momentum space.	contribution 0.92	0.95	0.87	asserted 1.00
s2	Then, we consider and compute its one-loop beta function, as well as the one on the degenerate Moyal space.	contribution 0.89	0.94	0.06	asserted 0.98
s3	We can finally compare both to the vanishing beta function of the theory with harmonic term on the Moyal space.	interpretation 0.49	0.61	0.12	asserted 0.66

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Original abstract

The noncommutative scalar theory with harmonic term (on the Moyal space) has a vanishing beta function. In this paper, we prove the renormalizability of the commutative scalar field theory with harmonic term to all orders by using multiscale analysis in the momentum space. Then, we consider and compute its one-loop beta function, as well as the one on the degenerate Moyal space. We can finally compare both to the vanishing beta function of the theory with harmonic term on the Moyal space.

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1 arXiv paper cite this in 1 sentence

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

What it cites · 29 sentences without a confident label

Uses the method or tool of 1 paper

We use below the multiscale analysis [ref] .

Renormalisation des theories de champs non commutatives · Power-counting · 1.00

Compares with 4 papers

These results coincide with [ref] up to the change of regularization scheme.

The \beta-function in duality-covariant noncommutative \phi^4-theory · Theory on the Moyal space · 1.00
Since the effective action takes the form [equation] in function of the physical (renormalized) constants: $λ_R$, $m^2_R$, $Ω_0,R$, $Ω_*,R$, the renormalization of the wave function is given by: [equation] We expand as before the bare constants $λ$, $m^2$, $Ω_0$, $Ω_*$ and $Z$ in terms of $λ_R$, and we obtain the beta functions: [equation] These results have also been obtained in [ref] for the case $Ω_0=0$.

Degenerate noncommutativity · Theory on the degenerate Moyal space · 0.94
The renormalization of the wave function does not take place at one-loop like without the harmonic term. * Contrary to this, in the Moyal case [ref], the parameter $Ω$ has a strong effect on the flow of the coupling constant $λ$ due to the non-vanishing renormalization of the wave function. $Ω=1$ is a fixed point of the renormalization group and for this value, $β_λ=0$ [ref] , so that the theory is asymptotically safe.

Vanishing of Beta Function of Non Commutative $\Phi^4_4$ Theory to all orders · Discussion · 0.95

Cites as background 16 papers

Otherwise, a new divergence, called UV-IR mixing [ref] , spoils the renormalizability.

Noncommutative Perturbative Dynamics · Introduction · 0.90
See also chapter 1 of [ref] for a good overview on this procedure.

Renormalisation des theories de champs non commutatives · Renormalization · 0.98
This theory has other remarkable properties, like a new symmetry called Langmann-Szabo duality [ref] and interpreted as a grading symmetry [ref] .

Duality in Scalar Field Theory on Noncommutative Phase Spaces · Introduction · 0.98

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