Comparison between variable flavor number schemes for charm quark electroproduction

A comparison is made between two variable flavor number schemes which describe charm quark production in deep inelastic electron-proton scattering. In these schemes the coefficient functions are derived from mass factorization of the heavy quark coefficient functions presented in a fixed flavor number scheme. Since the coefficient functions in the variable flavor number schemes have to be finite in the limit $\overrightarrow{m}0$ we have defined a prescription for those processes where the virtual photon is attached to a light quark. Furthermore one has to construct a parton density set with four active flavors $\mathrm{}(u,d,s,c)\mathrm{}$ out of a set which only contains three light flavors $\mathrm{}(u,d,s).$ In order ${\alpha }_s^2$ the two sets are discontinuous at $\mu {=m}_c$ which follows from mass factorization of the heavy quark coefficient functions. The charm component of the structure function $F_{2,c}$ is insensitive to the different variable flavor number schemes. In particular in the threshold region they both agree with the description in fixed order perturbation theory presented in a three flavor scheme. However one version does not lead to a correct description of the threshold behavior of the longitudinal structure function $F_{L,c}.$ This happens when one requires a non-vanishing zeroth order longitudinal coefficient function.