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BPHZ renormalization and its application to non-commutative field theory
High Energy Physics - Theory · Submitted 17 Jul 2013
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Furthermore, we show that the application of the modified BPHZ scheme to the IR-singularities appearing in non-planar graphs (UV/IR mixing problem) leads to the introduction of a 1/p^2 term and thereby to a renormalizable model.
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The abstract, sentence by sentence
In a recent work a modified BPHZ scheme has been introduced and applied to one-loop Feynman graphs in non-commutative phi^4-theory. In the present paper, we first review the BPHZ method and then we apply the modified BPHZ scheme as well as Zimmermann's forest formula to the sunrise graph, i.e. a typical higher-loop graph involving overlapping divergences. Furthermore, we show that the application of the modified BPHZ scheme to the IR-singularities appearing in non-planar graphs (UV/IR mixing problem) leads to the introduction of a 1/p^2 term and thereby to a renormalizable model. Finally, we address the application of this approach to gauge field theories.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | In a recent work a modified BPHZ scheme has been introduced and applied to one-loop Feynman graphs in non-commutative phi^4-theory. | background 0.99 | 0.10 | 0.04 | asserted 1.00 |
| s1 | In the present paper, we first review the BPHZ method and then we apply the modified BPHZ scheme as well as Zimmermann's forest formula to the sunrise graph, i.e. a typical higher-loop graph involving overlapping divergences. | contribution 0.83 | 0.82 | 0.06 | asserted 0.97 |
| s2 | Furthermore, we show that the application of the modified BPHZ scheme to the IR-singularities appearing in non-planar graphs (UV/IR mixing problem) leads to the introduction of a 1/p^2 term and thereby to a renormalizable model. | result 0.94 | 0.93 | 0.63 | asserted 1.00 |
| s3 | Finally, we address the application of this approach to gauge field theories. | contribution 0.77 | 0.74 | 0.05 | unclear 0.60 |
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The paper
Original abstract
In a recent work a modified BPHZ scheme has been introduced and applied to one-loop Feynman graphs in non-commutative phi^4-theory. In the present paper, we first review the BPHZ method and then we apply the modified BPHZ scheme as well as Zimmermann's forest formula to the sunrise graph, i.e. a typical higher-loop graph involving overlapping divergences. Furthermore, we show that the application of the modified BPHZ scheme to the IR-singularities appearing in non-planar graphs (UV/IR mixing problem) leads to the introduction of a 1/p^2 term and thereby to a renormalizable model. Finally, we address the application of this approach to gauge field theories.
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How the literature uses it
5 arXiv papers cite this in 10 sentences
Papers citing it · 7 sentences without a confident label
Uses its method or tool 2 papers
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we find that [ref] [equation] where the operator $t_p^2$ denotes a second order Taylor expansion with respect to $p$ around $p=0$ (but keeping $≠0$ and independent), and we introduced the abbreviation $N:=(k^2+a^2/k^2)$.
Gauge Fields on Non-Commutative Spaces and Renormalization · Non-commutative gauge field theories · 0.85
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After the momentum integration and cancellation of subdiagram divergence à la BPHZ [ref] (see also [ref] ), the self-energy has the form [ref] Σ̃_n(p^2)= ∫dx [ A + c Δ(p^2) + e Δ(p^2) logB/Δ(p^2) ], where $c,e$ are order-one constants and $d x$ is the measure of the Feynman parameters.
Stability of the Scalar Mass against Loop Corrections · Introduction · 0.89
Cites it as background 1 paper
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It was completed by the program by Bogoliubov, Parasiuk, Hepp and Zimmermann (BPHZ) [ref] , extended for the massless fields [ref] (see also [ref] for pedagogical introduction).
Self-Similar Structure of Loop Amplitudes and Renormalization · Introduction · 0.80
What it cites · 27 sentences without a confident label
Uses the method or tool of 1 paper
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By expanding this factor and linearizing the squares of the involved trigonometric functions [ref] and by using the assignments of momenta specified in fig:Sunrise-with-momenta we get the result [equation] Thus, the unrenormalized sunrise graph is described by the integral [equation] where the function $I_Γ$ is the product of propagators ([ref]), and where we suppressed the numerical prefactor involving $λ^2$.
Noncommutative $\Phi^4$ Theory at Two Loops · Renormalization of the sunrise graph · 0.84
Cites as support 3 papers
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Actually [ref] , it is the only non-local counterterm which can appear in a translation invariant non-commutative scalar field model.
On the Renormalization of Non-Commutative Field Theories · Finite renormalizations · 0.98
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However, the damping behaviour ([ref]) allows to overcome this problem [ref] and indeed it has been proven to provide a renormalizable model [ref] .
A translation-invariant renormalizable non-commutative scalar model · Renormalization of the theory · 0.99
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However, the damping behaviour ([ref]) allows to overcome this problem [ref] and indeed it has been proven to provide a renormalizable model [ref] .
Translation-invariant models for non-commutative gauge fields · Renormalization of the theory · 0.99
Cites as background 9 papers
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Before considering the BPHZ approach to the UV/IR mixing problem, it is useful to recall briefly the origin and nature of this problem [ref] .
Noncommutative Perturbative Dynamics · UV/IR mixing problem (Infrared singularities) · 1.00
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Thus, the action has to be modified and, inspired by the results achieved for the scalar models, various approaches have been proposed in recent years [ref] – see also the discussion [ref] and references therein.
Translation-invariant models for non-commutative gauge fields · Non-commutative gauge field theories · 0.80
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Thus, the action has to be modified and, inspired by the results achieved for the scalar models, various approaches have been proposed in recent years [ref] – see also the discussion [ref] and references therein.
On the Renormalizability of Noncommutative U(1) Gauge Theory - an Algebraic Approach · Non-commutative gauge field theories · 0.80
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