Noncommutative field theories on $ \mathbb{R}_{\lambda}^3 $: towards UV/IR mixing freedom

Abstract

Abstract:We consider the noncommutative space R 3 λ , a deformation of the algebra of functions on R 3 which yields a "foliation" of R 3 into fuzzy spheres. We first construct a natural matrix base adapted to R 3 λ . 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.

Noncommutative gauge theories on $ \mathbb{R}_{\theta}^2 $ as matrix models

“…Therefore, the diagonalization of (3.34) can be achieved by using a suitable family of Jacobi orthogonal polynomials. Note that a similar situation arises within the scalar Grosse-Wulkenhaar model [22] as well as in noncommutative scalar field theory on R 3 λ constructed in [36]. It is useful to recall here some technical points that will clarify the computation.…”

“…The Grosse-Wulkenhaar model has vanishing of the β-function to all orders [32,33] when it is self-dual under the Langmann-Szabo duality [34], and is very likely to be non-perturbatively solvable [35]. Scalar field theories on the noncommutative space R 3 λ , a deformation of R 3 , which are free of UV/IR mixing have been built recently in [36]. Whether this simply comes from the low "dimension" of the space or reflects a specific property of the underlying NCG remains to be seen.…”

Closed star product on noncommutative ℝ 3 and scalar field dynamics

“…For a review on the related literature, see [46,47] (see also [48]- [51] and references therein). Recently, scalar field theories on R 3 λ , a deformation of R 3 introduced in [52] (see also [53]), have been studied in [54]. Some of these NCFT have been shown to be free of perturbative UV/IR mixing [54] and characterized by the occurrence of a natural UV cut-off, stemming from the group algebra structure underlying the R 3 λ algebra [55].…”

κ -Poincaré invariant quantum field theories with Kubo-Martin-Schwinger weight

“…This latter corresponds, in the Moyal case, to the algebra for the Heisenberg group, which actually underlies the Weyl quantization, as it will be recalled below. For the space R 3 λ , it is the convolution algebra of SUð2Þ, which has been shown to play an essential role in originating the special properties of R 3 λ [14,16,17]. In the case of κ-Minkowski space-time, the relevant group algebra is the convolution algebra of the affine group as it will be shown below.…”

“…Doing this, one can take advantage of the machinery of the harmonic analysis on Lie groups and, in particular, measures involved in action functionals are provided by Haar measures. Note that such a viewpoint has also been intensively used in [16,17] for R 3 λ , the relevant group being SUð2Þ reflecting the suð2Þ noncommutativity of the quantum space and has provided the relationship between R 3 λ and the convolution algebra of SUð2Þ, the determination of the natural measure in the action functionals and by the way clarified the origin of the matrix basis used in [14]. In the case of κ-Minkowski space, we note that such a natural construction has already been used in [35,36] to derive a star product for a 2-dimensional κ-Minkowski space and to characterize a related multiplier algebra [35].…”