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.
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