Noncommutative field theories on $R_{\lambda }^3$: Towards UV/IR mixing freedom

Abstract

We consider the noncommutative space ${\mathbb{R}}_{\lambda }^3$, a deformation of the algebra of functions on ${\mathbb{R}}^3$ which yields a "foliation" of ${\mathbb{R}}^3$ into fuzzy spheres. We first construct a natural matrix base adapted to ${\mathbb{R}}_{\lambda }^3$. We then apply this general framework to the one-loop study of a two-parameter family of real-valued scalar noncommutative field theories with quartic polynomial interaction, which becomes a non-local matrix model when expressed in the above matrix base. The kinetic operator involves a part related to dynamics on the fuzzy sphere supplemented by a term reproducing radial dynamics. We then compute the planar and non-planar 1-loop contributions to the 2-point correlation function. We find that these diagrams are both finite in the matrix base. We find no singularity of IR type, which signals very likely the absence of UV/IR mixing. We also consider the case of a kinetic operator with only the radial part. We find that the resulting theory is finite to all orders in perturbation expansion.

31 pages, 4 figures. Improved version. Sections 5.1 and 5.2 have been clarified. A minor error corrected. References added

References in corpus (16)

• Algebras of distributions suitable for phase-space quantum mechanics. I
• Vanishing of Beta Function of Non Commutative ${\Phi }_4^4$ Theory to all orders
• Noncommutative Induced Gauge Theory
• Algebras of distributions suitable for phase-space quantum mechanics. II. Topologies on the Moyal algebra
• Induced Gauge Theory on a Noncommutative Space
• Vacuum configurations for renormalizable non-commutative scalar models
• Star products, duality and double Lie algebras
• Renormalization of the Orientable Non-commutative Gross-Neveu Model
• Non-Commutative U(1) Gauge Theory on R**4 with Oscillator Term and BRST Symmetry
• Duality symmetry for star products
• Translation Invariance, Commutation Relations and Ultraviolet/Infrared Mixing
• On the $\star$-product quantization and the Duflo map in three dimensions
• Duality covariant quantum field theory on noncommutative Minkowski space
• One-loop Beta Functions for the Orientable Non-commutative Gross-Neveu Model
• Noncommutative Induced Gauge Theories on Moyal Spaces
• Twisted Quantum Fields on Moyal and Wick-Voros Planes are Inequivalent

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• Four-Dimensional Gravity on a Covariant Noncommutative Space (II)
• Double Quantization
• Lie-Poisson gauge theories and $\kappa$-Minkowski electrodynamics
• Intertwining noncommutativity with gravity and particle physics
• Quantum instability of gauge theories on $\kappa$-Minkowski space
• Conserved symmetries in noncommutative quantum mechanics
• Three-dimensional noncommutative Yukawa theory: Induced effective action and propagating modes
• On the UV/IR mixing of Lie algebra-type noncommutatitive ${\varphi }^4$-theories