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Renormalizable nonanticommutative SYM theories with chiral matter in theadjoint representation of the gauge group have been recently constructed in[arXiv:0901.3094]. In the present paper we focus on the U*(1) case with matterinteracting through a cubic superpotential. For a single flavor, in asuperspace setup and manifest background covariant approach we perform thecomplete one-loop renormalization and compute the beta-functions for allcouplings appearing in the action. We then generalize the calculation to thecase of SU(3) flavor matter with a cubic superpotential viewed as a nontrivialNAC generalization of the ordinary abelian N=4 SYM and its marginaldeformations. We find that, as in the ordinary commutative case, the NAC N=4theory is one-loop finite. We provide general arguments in support of all-loopfiniteness. Instead, deforming the superpotential by marginal operators givesrise to beta-functions which are in general non-vanishing. We study thespectrum of fixed points and the RG flows. We find that nonanticommutativityalways makes the fixed points unstable.

Nonanticommutative U(1) SYM theories: Renormalization, fixed points and infrared stability