Conclusions

The production of heavy quarks in electron proton collisions at HERA has proven to be a complex and fascinating field of experimental, phenomenological and theoretical challenges. The measurements performed by the H1 and ZEUS collaborations at the HERA collider at DESY have made strong contributions to the quantitative understanding of quantum chromodynamics, the theory of the strong interactions. Since the first measurements of charm at HERA in 1995, new theoretical approaches have been developed and new experimental opportunities have been opened up which allow for detailed tests of the theoretical predictions.

Experimentally, the measurement of heavy quarks poses specific difficulties as the heavy quark cross sections and heavy hadron branching ratios are small compared to the inclusive $ep$ event rates. To cope with the challenge of measuring charm and also beauty, the experiments H1 and ZEUS have continuously refined both the online and offline selection criteria and analysis techniques for heavy quark events. Special purpose detectors, such as the silicon vertex detectors, and dedicated trigger electronics have been implemented in the H1 and ZEUS detectors in order to enhance the efficiencies for identifying and recording events with heavy quarks. Novel analysis methods have been introduced to determine the rates and distributions of events with heavy quarks down to very low transverse momenta and photon virtualities.

Several theoretical calculations are available and are commonly used to describe heavy quark production. In Monte Carlo generators, the matrix element is usually calculated at leading order and parton showers are used to simulate the presence of parton radiations, thus approximating higher order contributions. Calculations at next-to-leading order perturbation theory are available in the collinear factorization approach. These calculations are complemented by predictions using the $k_t$ factorization ansatz. The complexity of the perturbative calculations increases when the one or both of the scales $Q^2$ and/or $p_t$ are of similar size as the quark mass. The most recent theoretical calculations have been seen to be successful in dealing with this multi-scale problem of heavy quark production.

In particular for charm production, the theories provide predictions that are in reasonable agreement with the H1 and ZEUS measurements. In these calculations the charm mass is typically set to 1.5 GeV or lower, thus allowing the predictions to match with the normalization of the data. In general, the achieved overall precision of the data and the general agreement of the theory suggest the use of the charm data for constraints in global fits of the parton distributions of the proton.

Measurements of charm with jets in photoproduction, when compared to predictions from event generators with leading order matrix elements in the collinear factorization approach, show large contributions from processes with resolved photons, described by photon structure functions. Next-to-leading order calculations in the collinear factorization approach, as well as calculations using $k_t$ factorization are able to give a reasonable description of the contributions from resolved photons already at leading order. The contribution from processes with resolved virtual photons, i.e.in electroproduction, has not yet been precisely determined.

HERA has also provided results on charm fragmentation fractions and momentum distributions which show general agreement with the results from $e^+e^-$ experiments. The measurements of fragmentation, together with the cross section measurements should be used for a (model-dependent) determination of the parameter choices for the charm mass, the renormalization and factorization scales and the fragmentation functions and parameters.

In more exclusive measurements, some deviations between data and predictions, and among the predictions, have been observed which are situated in distinct areas of phase space. Most prominently, in the region of forward rapidity, the data tend to be higher than the predictions in the collinear factorization approach. New instrumentation of the experiments in the forward region will provide access to this problem in the HERA-II data.

Over the last few years, differential measurements of beauty production have become available. The measurements using events with jets and muons are reasonably reproduced by QCD calculations at next-to-leading order perturbation theory, with a slight excess ( $\sim 2\sigma$) of the data in normalization. The analyses make use of new techniques in order to enhance the fraction of $b$-events in the data samples. In particular the long lifetime of the $B$-hadrons is exploited using the silicon vertex detectors to determine the beauty production cross sections. H1 measurements using inclusive lifetime tagging have led to the first measurement of the structure function $F_2^{b\bar{b}}$ which describes the contribution of events with $b$-quarks to the inclusive $ep$ cross section. The results are in good agreement with the predictions based on global fits to inclusive data.

First measurements in which both heavy quarks are identified have been performed. These events allow the simultaneous extraction of the charm and beauty cross sections from angular and charge correlations, and provide access to new tests of higher order QCD effects. Although the available statistics are still poor, there is a general trend for the measured beauty cross sections to be higher than the expectations while the charm results agree with the predictions.

The HERA collider experiments have also contributed to the study of the process of inelastic charmonium production. The description of the data by leading order calculations based on non-relativistic QCD (NRQCD) is yet to be clarified.

While the picture of heavy quark production in $ep$ collisions is taking on shape, it is clear that many more measurements are needed to reach a precision which allows to discriminate between the existing theories and to establish precision predictions. At HERA-II, increased luminosity and new detectors will allow for the data to match or exceed the precision of the theory calculations. The goals for HERA-II are to complete the picture of charm production towards the highest possible photon virtualities $Q^2$, transverse momenta $p_t$, and pseudo-rapidities $\eta$ and to verify the models using beauty production. With increased luminosity, ample statistics of charmed mesons will be produced allowing to answer the open questions in the field of spectroscopy, the most prominent of which is the apparent contradiction between H1 who have reported evidence for an exotic resonance with charm contents and ZEUS who have not confirmed this observation.

In summary, the understanding of heavy-quark production is currently one of the many intriguing issues in $ep$ collision physics. During HERA-I the field has been outlined and many questions have been raised and provisionally answered. The HERA-II data will be instrumental in bringing many of the still open questions to a conclusion.