The Single-Mass Variable Flavor Number Scheme at Three-Loop Order

Abstract: The matching relations in the unpolarized and polarized variable flavor number scheme at three-loop order are presented in the single-mass case. They describe the process of massive quarks becoming light at large virtualities $Q^2$. In this framework, heavy-quark parton distributions can be defined. Numerical results are presented on the matching relations in the case of the single-mass variable flavor number scheme for the light parton, charm and bottom quark distributions. These relations are process independent. In the polarized case we generally work in the Larin scheme. To two-loop order we present the polarized massive OMEs also in the $\overline{\rm MS}$ scheme. Fast numerical codes for the single-mass massive operator matrix elements are provided.