Renormalizability of non(anti)commutative gauge theories with Script N = 1/2 supersymmetry

Non(anti)commutative gauge theories are supersymmetric Yang-Mills and mattersystem defined on a deformed superspace whose coordinates obeynon(anti)commutative algebra. We prove that these theories in four dimensionswith N=1/2 supersymmetry are renormalizable to all orders in perturbationtheory. Our proof is based on operator analysis and symmetry arguments. In acase when the Grassman-even coordinates are commutative, deformation induced bynon(anti)commutativity of the Grassman-odd coordinates contains operators ofdimension-four or higher. Nevertheless, they do not lead to power divergencesin a loop diagram because of absence of operators Hermitian-conjugate to them.In a case when the Grassman-even coordinates are noncommutative, theultraviolet-infrared mixing makes the theory renormalizable by the planardiagrams, and the deformed operators are not renormalized at all. We alsoelucidate relation at quantum level between non(anti)commutative deformationand N=1/2 supersymmetry. We point out that the star product structure dictatesa specific relation for renormalization among the deformed operators.Comment: 21 pages, Latex, 2 figs; v2. minor correction