TTP05-21 Four-Loop Decoupling Relations for the Strong Coupling

We compute the matching relation for the strong coupling constant

 within the framework of QCD up to four-loop order.
 This allows a consistent five-loop running (once the $\beta$
 function is available to this order) taking into account threshold effects.
 As a side product we obtain the effective coupling of a Higgs boson
 to gluons with five-loop accuracy.