  We analyze the one-loop renormalisation group equations for the parameters of
the Higgs potential of a supersymmetric SU(5) model with first step of symmetry
breaking involving an adjoint Higgs. In particular, we investigate the running
of the parameters that decide the first step of symmetry breaking in an attempt
to establish which symmetry-breaking scenarios would be most likely if the
model is the effective low-energy description of some more fundamental theory.
An infra-red fixed point is identified analytically. We show that it is located
at the boundary between the region of Higgs parameter space corresponding to
unbroken SU(5) and the region corresponding to the breaking of SU(5) to the
Standard Model, and we elaborate on its implications. We also observe that
certain forms of the Higgs potential discussed at tree level in the literature
are not renormalisation group invariant.
