Parton Densities for Heavy Quarks

We compare parton densities for heavy quarks. Reactions with incoming heavy (c,b) quarks are often calculated with heavy quark densities just like those with incoming light mass (u,d,s) quarks are calculated with light quark densities. The heavy quark densities are derived within the framework of the so-called zero-mass variable flavor number scheme (ZM-VFNS). In this scheme these quarks are described by massless densities which are zero below a specific mass scale µ. The latter depends on m c or m b. Let us call this scale the matching point. Below it there are n f massless quarks described by n f massless densities. Above it there are n f + 1 massless quarks described by n f + 1 massless densities. The latter densities are used to calculate processes with a hard scale M ≫ m c , m b. For example in the production of single top quarks via the weak process q i + b → q j + t, where q i , q j are light mass quarks in the proton/antiproton, one can argue that M = m t should be chosen as the large scale and m b can be neglected. Hence the incoming bottom quark can be described by a massless bottom quark density. The generation of these densities starts from the solution of the evolution equations for n f massless quarks below the matching point. At and above this point one solves the evolution equations for n f + 1 mass-less quarks. However in contrast to the parameteri-zation of the x-dependences of the light quarks and gluon at the initial starting scale, the x dependence of the heavy quark density at the matching point is fixed. In perturbative QCD it is defined by convolutions of the densities for the n f quarks and the gluon with specific operator matrix elements (OME's), which are now know up to O(α 2 s) [ 1]. These matching conditions determine both the ZM-VFNS density and the other light-mass quark and gluon densities at the matching points. Then the evolution equations determine the new densities at larger scales. The momentum sum rule is satisfied for the n f + 1 quark densities together with the corresponding gluon density.