A $N_F$-Dependent VFNS for Heavy Flavors: Merging the FFNS and VFNS

We introduce a Hybrid Variable Flavor Number Scheme (H-VFNS) for heavy flavors, which incorporates the advantages of both the traditional Variable Flavor Number Scheme (VFNS) as well as the Fixed Flavor Number Scheme (FFNS). We include an explicit dependence on number of active flavors $N_F$ in both the Parton Distribution Functions (PDFs) and the strong coupling constant $\alpha_{S}$. This results in sets of coexisting PDFs and $\alpha_{S}$ for $N_F=\{3,4,5,6\}$, that are related analytically by the $\overline{\rm MS}$ matching conditions. The H-VFNS resums the heavy quark contributions and provides the freedom to choose the optimal $N_F$ for each particular data set. Thus, we can fit selected HERA data in a FFNS framework, while retaining the benefits of the VFNS to analyze LHC data at high scales. We illustrate how such a fit can be implemented for the case of both HERA and LHC data.