Real versus complex beta-deformation of the N=4 planar super Yang-Mills theory

This is a sequel of our paper hep-th/0606125 in which we have studied the {cal N}=1 SU(N) SYM theory obtained as a marginal deformation of the {cal N}=4 theory, with a complex deformation parameter eta and in the planar limit. There we have addressed the issue of conformal invariance imposing the theory to be finite and we have found that finiteness requires reality of the deformation parameter eta. In this paper we relax the finiteness request and look for a theory that in the planar limit has vanishing beta functions. We perform explicit calculations up to five loop order: we find that the conditions of beta function vanishing can be achieved with a complex deformation parameter, but the theory is not finite and the result depends on the arbitrary choice of the subtraction procedure. Therefore, while the finiteness condition leads to a scheme independent result, so that the conformal invariant theory with a real deformation is physically well defined, the condition of vanishing beta function leads to a result which is scheme dependent and therefore of unclear significance. In order to show that these findings are not an artefact of dimensional regularization, we confirm our results within the differential renormalization approach.