  The Higgs couplings to matter fields are proportional to their masses. Thus
Higgs amplitudes can be obtained by differentiating amplitudes without Higgs
with respect to masses. We show how this well-known statement can be extended
to higher order when renormalization effects are taken into account. We
establish the connection with the Callan-Symanzik and renormalization group
equations and consider also pseudoscalar Higgs couplings to fermions.
Furthermore, we address the case where the Higgs couples to a heavy particle
that is integrated out from the low-energy effective Lagrangian. We derive
effective interactions where mass logarithms are resummed by
renormalization-group methods, and give expansions of the results up to
next-to-leading order.
