Theory

The dominant production process of heavy quarks at HERA has been seen to be the boson-gluon fusion process (BGF) which is depicted in fig.1.

Figure 1: Diagram of the process of Boson-Gluon Fusion in $ep$-scattering.

Theory calculations use a factorization ansatz in order to arrive at cross section predictions which can be compared with experimental data. In these factorization approaches, the cross section is composed of a non-perturbative part, which is given by the parton distributions of the initial state particles $f_{j}^{p}$, the perturbative evolution according to the evolution equation and a perturbative hard scattering process, i.e.the photon-parton cross section $\hat{\sigma}_{\gamma j}$, which is calculable in perturbative QCD:

$$\sigma^{\gamma p}_{dir}(P_{\gamma},P_{p})= \sum_{i}\int dx \; f_{... ...\gamma j} (P_{\gamma},xP_{p},\alpha_{s}(\mu_{R}),\mu_{R},\mu_{F},\mu_{\gamma}).$$ (1)

In addition, to predict production cross sections for heavy hadrons and/or other exclusive final states, the fragmentation of the heavy quark into a hadron and additional final state particles are considered. Based on the assumption of universality, the non-calculable non-perturbative parts, i.e.usually the parton distributions and fragmentation functions are taken from measurements of other processes at HERA or other experiments. A number of conditions apply to HERA:

• Heavy quark processes are particularly well measurable at HERA in the region of Bjorken-$x$ between $10^{-4}$ and $10^{-2}$. In this region, gluons are the dominant partons in the proton. The heavy quark cross section is thus directly sensitive to the gluon density distribution in the proton. The kinematic range probed by HERA is relevant for the calculations of cross sections at future experiments, e.g.at the LHC.

• In photoproduction the incoming almost real photon may fluctuate into a hadronic state before undergoing a hard collision. The corresponding contribution to the cross section is referred to as resolved, in contrast to the case where the photon directly interacts with the hadron (direct or point-like). Separation of a cross section into a direct and resolved component is ambiguous beyond leading order: different choices of the factorization schemes lead to different definitions of the two components.

• For heavy quark processes the calculation of the 'hard' matrix element using perturbative QCD is facilitated by the heaviness of the quarks. Already the charm quark mass provides an energy scale, which allows perturbative expansion. Other possible scales are the photon virtuality $Q^2$ (for DIS) or the transverse momentum $p_t$ of the final state particles (jets). Calculations for multi-scale processes, such as the production of charmed jets in DIS, have become available recently. Different approaches for the choice of the scale exist and are discussed below.

• The fragmentation of the heavy quark into a hadron and additional final state particles is usually described by phenomenological parameters as determined from experimental data. Fragmentation functions describing the longitudinal momentum transfered from the quark to the heavy hadron, and fragmentation fractions, describing the probability of a quark to hadronize into a particular hadron, have been measured primarily at $e^+e^-$ colliders (such as LEP). Recent results at HERA have shown that the uncertainty by the use of parameters as determined at LEP is reasonably small.