Large radiative corrections to the effective potential and the gauge hierarchy problem

We study the higher order corrections to the effective potential in a simple toy model and in the SU(5) grand unified theory, with a view to seeing what their effects are on the stability equations, and hence on the gauge hierarchy problem for these theories. These corrections contain powers of log (v2/h2), where v and h are the large and small vacuum expectation values respectively, and hence cannot a priori be neglected. Nevertheless, after summing these large logarithms we find that the stability equations always contain two equations for v (i.e. these equations are independent of h) and hence can only be satisfied by a special (and hence unnatural) choice of parameters. This we claim is the precise statement of the gauge hierarchy problem. (orig.)