On non-commutativeU_⋆(1) gauge models and renormalizability

Abstract

Based on our recent findings regarding (non-)renormalizability of non-commutative U⋆(1) gauge theories [1, 2] we present the construction of a new type of model. By introducing a soft-breaking term in such a way that only the bilinear part of the action is modified, no interaction between the gauge sector and auxiliary fields occurs. Demanding in addition that the latter form BRST doublet structures, this leads to a minimally altered non-commutative U⋆(1) gauge model featuring an IR damping behavior. Moreover, the new breaking term is shown to provide the necessary structure in order to absorb the inevitable quadratic IR divergences appearing at one-loop level in theories of this kind. In this paper we compute Feynman rules, symmetries and results for the vacuum polarization together with the one-loop renormalization of the gauge boson propagator and three-point functions.

One-loop structure of the U(1) gauge model on the truncated Heisenberg space

“…The latter was implemented for the scalar field theory in [13]. Several generalizations were studied for the gauge fields, for example models defined in [33,34] and [36,37]; however, the complexity of the actions prevented the complete analysis so it remains unclear which nonlocal operators could render the gauge theory renormalizable. Our present result shows that −1 terms appear in quantization even in a local version of gauge theory.…”

Involutive representations of coordinate algebras and quantum spaces

“…Informally, this framework amounts (among other tasks) to represent the abstract involutive algebras of operators stemming from the above mentioned coordinate algebras on well chosen involutive algebras of functions equipped with a deformed product, i.e star-product, which can be achieved through the introduction of some suitable (invertible) quantization map. One well-known example heavily used in earlier studies of quantum properties of NCFT on Moyal spaces [31][32][33] (for a review on gauge theories on Moyal spaces see [34]; families of star products on the Moyal space R 4 θ have been constructed in [35]) is the Weyl quantization map, linked to the Wigner-Weyl transform, giving rise to the Moyal product. Other star-products related to κ-Minkowski spaces as well as deformations of R 3 with su(2) noncommutativity have also appeared and used to construct and study NCFT on these spaces [36][37][38][39][40][41][42][43][44][46][47][48] (for a general construction see [45]).…”

Noncommutative gauge theories on ℝ λ 3 $$ {\mathrm{\mathbb{R}}}_{\uplambda}^3 $$ : perturbatively finite models

“…Unfortunately, its complicated vacuum structure explored in [36,37] forbids the use of any standard perturbative treatment. 1 Alternative based on the implementation of a IR damping mechanism have been proposed and studied [39][40][41][42]. Although this damping mechanism is appealing, it is not known if it can produce a renormalisable gauge theory on R 4 θ .…”