Non(anti)commutative SYM theory: renormalization in superspace

We present a systematic investigation of one-loop renormalizability fornonanticommutative N=1/2, U(N) SYM theory in superspace. We first discussclassical gauge invariance of the pure gauge theory and show that incontradistinction to the ordinary anticommutative case, differentrepresentations of supercovariant derivatives and field strengths do not leadto equivalent descriptions of the theory. Subsequently we develop backgroundfield methods which allow us to compute a manifestly covariant gauge effectiveaction. One-loop evaluation of divergent contributions reveals that the theorysimply obtained from the ordinary one by trading products for star products isnot renormalizable. In the case of SYM with no matter we present a N=1/2improved action which we show to be one-loop renormalizable and which isperfectly compatible with the algebraic structure of the star product. For thisaction we compute the beta functions. A brief discussion on the inclusion ofchiral matter is also presented.Comment: Latex file, 59 pages, 10 figures, One reference adde