The Leigh-Strassler deformation and the quest for integrability

In this paper we study the one-loop dilatation operator of the full scalarfield sector of Leigh-Strassler deformed N=4 SYM theory. In particular we mapit onto a spin chain and find the parameter values for which the Reshetikhinintegrability criteria are fulfilled. Some years ago Roiban found an integrablesubsector, consisting of two holomorphic scalar fields, corresponding to theXXZ model. He was pondering about the existence of a subsector which would formgeneralisation of that model to an integrable su_q(3) model. Later Berensteinand Cherkis added one more holomorphic field and showed that the subsectorobtained this way cannot be integrable except for the case when q=e^{i beta},beta real. In this work we show if we add an anti-holomorphic field to the twoholomorphic ones, we get indeed an integrable su_q(3) subsector. We find itplausible that a direct generalisation to a su_q(3,2) one-loop sector willexist, and possibly beyond one-loop.Comment: 2 figures, fixed some typos, and improved the notation a bi