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Two-loop matching coefficients for the strong coupling in the MSSM

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 $α_s(M_Z)$.