  We analyze soft and collinear gluon resummation effects at the N$^3$LL level
for Standard Model Higgs boson production via gluon fusion $gg\to H$ and the
neutral scalar and pseudoscalar Higgs bosons of the minimal supersymmetric
extension at the N$^3$LL and NNLL level, respectively. We introduce refinements
in the treatment of quark mass effects and subleading collinear gluon effects
within the resummation. Soft and collinear gluon resummation effects amount to
up to about 5% beyond the fixed-order results for scalar and pseudoscalar Higgs
boson production.

High Energy Physics - Phenomenology;High Energy Physics - Experiment