Variable-flavor number scheme for next-to-next-to-leading order

At NNLO it is particularly important to have a Variable-flavor Number Scheme (VFNS) to deal with heavy quarks because there are major problems with both the zero-mass variable-flavor number scheme and the fixed-flavor number scheme. I illustrate these problems and present a general formulation of a Variable-flavor Number Scheme (VFNS) for heavy quarks that is explicitly implemented up to NNLO in the strong coupling constant αS, and may be used in NNLO global fits for parton distributions. The procedure combines elements of the ACOT(χ) scheme and the Thorne-Roberts scheme. Despite the fact that at NNLO the parton distributions are discontinuous as one changes the number of active quark flavors, all physical quantities are continuous at flavor transitions and the comparison with data is successful.