Star product and the general Leigh-Strassler deformation
We extend the definition of the star product introduced by Lunin and Maldacena to study marginal deformations of N=4 SYM. The essential difference from the latter is that instead of considering U(1)xU(1) non-R-symmetry, with charges in a corresponding diagonal matrix, we consider two Z_3-symmetries followed by an SU(3) transformation, with resulting off-diagonal elements. From this procedure we obtain a more general Leigh-Strassler deformation, including cubic 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 the deformation for each amplitude can be extracted in a prefactor. We also conclude that the obtained amplitudes should follow the iterative structure of MHV amplitudes found by Bern, Dixon and Smirnov.