Matching-invariant running of quark masses in quantum chromodynamics

The conventional quark mass is not continuous at thresholds. In this paper, we derive matching-invariant quark masses which are continuous everywhere. They are generally expanded as an obvious function of the logarithmic A scaled energy. The expansion coefficients are related to the gamma and beta functions, with concretization to four-loop level. Due to the maximal elimination of their direct dependence on the coupling with large uncertainties, the new expressions for quark masses converge much fast and accurate.