On the perturbative chiral ring for marginally deformed Script N=4 SYM theories

For \cal{N}=1 SU(N) SYM theories obtained as marginal deformations of the\cal{N}=4 parent theory we study perturbatively some sectors of the chiral ringin the weak coupling regime and for finite N. By exploiting the relationbetween the definition of chiral ring and the effective superpotential wedevelop a procedure which allows us to easily determine protected chiraloperators up to n loops once the superpotential has been computed up to (n-1)order. In particular, for the Lunin-Maldacena beta-deformed theory we determinethe quantum structure of a large class of operators up to three loops. Weextend our procedure to more general Leigh-Strassler deformations whose chiralring is not fully understood yet and determine the weight-two and weight-threesectors up to two loops. We use our results to infer general properties of thechiral ring.Comment: LaTex, 40 pages, 4 figures, uses JHEP3; v2: minor correction