β functions of orbifold theories and the hierarchy problem

We examine a class of gauge theories obtained by projecting out certain fields from an N=4 supersymmetric SU(N) gauge theory. These theories are non-supersymmetric and in the large N limit are known to be conformal. Recently it was proposed that the hierarchy problem could be solved by embedding the standard model in a theory of this kind with finite N. In order to check this claim one must find the conformal points of the theory. To do this we calculate the one-loop β functions for the Yukawa and quartic scalar couplings. We find that with the β functions set to zero the one-loop quadratic divergences are not canceled at sub-leading order in N; thus the hierarchy between the weak scale and the Planck scale is not stabilized unless N is of the order 10²⁸ or larger. We also find that at sub-leading orders in N renormalization induces new interactions, which were not present in the original Lagrangian.