Gauge hierarchies

It is shown that a hierarchial spontaneous breakdown of a gauge symmetry can occur with enormous ratios of ``superheavy'' and ``ordinary'' gauge boson masses. A sufficient condition is that the initial spontaneous breakdown of the over-all gauge group should leave some scalars with small or zero masses. If these scalars have zero masses, the ratios of superheavy and ordinary gauge boson masses will automatically be of the form exp (C/g²), where C is typically of order unity, and g is a representative gauge coupling. Numerical calculations show that these ratios can easily be as large as needed in a grand unification of QCD with SU(2) x U(1). For a range of parameters, there is an absolute upper bound on these ratios, which is larger by a factor exp (1/4) than the values for zero scalar mass.