Star product and the general Leigh-Strassler deformation

TL;DR

This paper generalizes the star product for N=4 SYM to include Leigh-Strassler deformations with off-diagonal elements, preserving conformality and suggesting the amplitudes follow known iterative structures.

Contribution

It introduces a broader class of Leigh-Strassler deformations using a generalized star product involving Z_3-symmetries and SU(3) transformations, extending previous U(1) based approaches.

Findings

Deformation preserves conformal invariance in beta- and gamma-deformed theories.

Amplitudes can be factorized to reveal underlying structure.

Deformed amplitudes are expected to follow Bern-Dixon-Smirnov iterative patterns.

Abstract

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.