Star product and the general Leigh–Strassler deformation

We extend the definition of the star product introduced by Lunin andMaldacena to study marginal deformations of N=4 SYM. The essential differencefrom the latter is that instead of considering U(1)xU(1) non-R-symmetry, withcharges in a corresponding diagonal matrix, we consider two Z_3-symmetriesfollowed by an SU(3) transformation, with resulting off-diagonal elements. Fromthis procedure we obtain a more general Leigh-Strassler deformation, includingcubic terms with the same index, for specific values of the coupling constants.We argue that the conformal property of N=4 SYM is preserved, in both beta-(one-parameter) and gamma_{i}-deformed (three-parameters) theories, since thedeformation for each amplitude can be extracted in a prefactor. We alsoconclude that the obtained amplitudes should follow the iterative structure ofMHV amplitudes found by Bern, Dixon and Smirnov.Comment: 21 pages, no figures, JHEP3, v2: references added, v3: appendix A added, v4: clarification in section 3.