Mass dependences in renormalization-group solutions

A method is proposed for the approximation solution of the renormalization-group equations with allowance for the dependences on the particle masses. The method is used to obtain expressions of high accuracy for the invariant charge and the singleargument renormalization-group solutions corresponding to the allowance for twoand three-loop contributions in the momentum subtraction scheme. These expressions are of interest for the case of masses of different scales and, in particular, for the topical problem of taking into account the thresholds for the production of heavy quarks in modern quantum chromodynamics.