  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 \beta 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 \beta. 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.
