  Higher loop calculations in the Higgs sector of the standard model at the
Higgs mass scale have shown that perturbation theory diverges very badly at
about 1 TeV in the on-shell renormalization scheme. The prediction of the
position of the Higgs pole in the complex s-plane becomes unreliable. We show
that in the pole renormalization scheme this appears to have much better
convergence properties, while showing good agreement with the on-shell scheme
over the validity range of the latter. This suggests that the pole scheme
should be preferable for phenomenological studies of heavy Higgs bosons. We
discuss whether this behaviour can be the result of a certain relation between
the on-shell mass and the pole mass at the nonperturbative level.
