Two-loop matching coefficients for the strong coupling in the minimal supersymmetric standard model

When relating the strong coupling α s , measured at the scale of the Z boson mass, to its numerical value at some higher energy, for example the scale of Grand Unification, it is important to include higher order corrections both in the running of α s and the decoupling of the heavy particles. We compute the two-loop matching coefficients for α s within the Minimal Supersymmetric Standard Model (MSSM) which are necessary for a consistent three-loop evolution of the strong coupling constant. Different scenarios for the hierarchy of the supersymmetric scales are considered and the numerical effects are discussed. We find that the three-loop effects can be as large as and sometimes even larger than the uncertainty induced by the current experimental accuracy of α s ( M Z ).