Renormalizability of Non(anti)commutative Gauge Theories with N=1/2 Supersymmetry

TL;DR

This paper proves that four-dimensional non(anti)commutative gauge theories with N=1/2 supersymmetry are renormalizable to all orders, using operator analysis and symmetry arguments, despite nontrivial deformation effects.

Contribution

It establishes the all-order renormalizability of N=1/2 supersymmetric gauge theories with non(anti)commutative deformation, a result not previously demonstrated.

Findings

Theories are renormalizable to all orders in perturbation theory.

Deformation operators do not lead to power divergences due to their Hermitian properties.

Ultraviolet-infrared mixing affects renormalization in noncommutative cases.

Abstract

Non(anti)commutative gauge theories are supersymmetric Yang-Mills and matter system defined on a deformed superspace whose coordinates obey non(anti)commutative algebra. We prove that these theories in four dimensions with N=1/2 supersymmetry are renormalizable to all orders in perturbation theory. Our proof is based on operator analysis and symmetry arguments. In a case when the Grassman-even coordinates are commutative, deformation induced by non(anti)commutativity of the Grassman-odd coordinates contains operators of dimension-four or higher. Nevertheless, they do not lead to power divergences in a loop diagram because of absence of operators Hermitian-conjugate to them. In a case when the Grassman-even coordinates are noncommutative, the ultraviolet-infrared mixing makes the theory renormalizable by the planar diagrams, and the deformed operators are not renormalized at all. We also elucidate relation at quantum level between non(anti)commutative deformation and N=1/2 supersymmetry. We point out that the star product structure dictates a specific relation for renormalization among the deformed operators.