  Taking the dominant couplings in the standard model to be the quartic scalar
coupling, the Yukawa coupling of the top quark, and the SU(3) gauge coupling,
we consider their associated running couplings to one loop order. Despite the
non-linear nature of the differential equations governing these functions, we
show that they can be solved exactly. The nature of these solutions is
discussed and their singularity structure is examined. It is shown that for a
sufficiently small Higgs mass, the quartic scalar coupling decreases with
increasing energy scale and becomes negative, indicative of vacuum instability.
This behavior changes for a Higgs mass greater than 168 GeV, beyond which this
couplant increases with increasing energy scales and becomes singular prior to
the ultraviolet (UV) pole of the Yukawa coupling. Upper and lower bounds on the
Higgs mass corresponding to new physics at the TeV scale are obtained and
compare favourably with the numerical results of the one-loop and two-loop
analyses with inclusion of electroweak couplings.
