Fixed Order Parton Level Calculations

$ep$ and $\gamma p$ event generators to next-to-leading order are available at the parton level. The program FMNR [80] implements cross section calculations for the photoproduction regime and the program HVQDIS [81] for the DIS case. Both programs are based on calculations in the massive scheme and provide weighted parton level events with two or three outgoing partons, i.e.a $b$ quark, a $\bar{b}$-quark and possibly an additional light parton.

For calculations of 'visible' cross sections, such as $D^*$ cross sections the programs can be extended to include the fragmentation of the heavy quarks into hadrons and optionally decays of the hadrons into final states e.g. with leptons. In a simple approach, the heavy quarks are `hadronized' into a heavy hadrons by rescaling the three momentum of the quark according to the distribution as given by a fragmentation function. Usually, the Peterson fragmentation function [109] is used with parameter choices, e.g.as those determined in [129].

For the calculation of cross sections involving jets, a jet algorithm, e.g. the inclusive $k_t$ algorithm (see section 5.4), is used on the final state partons, yielding parton level jets. For the comparison with experimental measurements - which are usually given at the hadron level, i.e.including fragmentation and hadronization effects - parton-to-hadron level corrections are applied to the parton level results. These corrections should in principle be performed in the same scheme (NLO) as the parton level calculations. However, for lack of more appropriate choices LO+PS Monte Carlo event generators (such as PYTHIA, HERWIG, RAPGAP or CASCADE described above) are commonly used to calculate the parton-to-hadron level corrections. The corrections range typically from $-30\%$ to $+5\%$ in both photoproduction and DIS, decreasing towards larger values of $Q^2$ and/or jet transverse momentum.

The theoretical uncertainties of the NLO calculations are estimated in the following way: For the heavy quark mass $m_q$, typically, central values of $m_c=1.5$ and $m_b=4.75$ GeV are used. The renormalization scales are set to the transverse masses $m_T=\sqrt{m_q^2 + p_{t,q\bar{q}}^2}$, where $p_{t,q\bar{q}}^2$ is the average of the squared transverse momenta of the quark and anti-quark. For beauty, the factorization scale $\mu_f$ is set to $m_T$ while for charm $\mu_f=\sqrt{4(m_c^2 + p_{t,c\bar{c}}^2)}$. Here, $p_{tb\bar{b}}$ is the average of the transverse momenta of the two $b$ quarks. In DIS, the scale $\sqrt{m_q^2+Q^2}$ is used.

The theoretical uncertainties of the NLO calculation are usually estimated by variations of the renormalization and factorization scale parameters up and down by a factor of two and the $c$ ($b$) mass between 1.3 and 1.7 (4.5 and 5.0) GeV. These variations, when combined, typically lead to a change in the cross section predictions of 30-35% for charm photoproduction and 20-30% for beauty photoproduction (FMNR). In DIS (HVQDIS) the uncertainties are typically between 10 and 20%.

The cross section variations when using different proton structure functions are less than 10% for most measurements. Further uncertainties (of order 10%) have been seen to arise from the implementation of muon decay spectra and fragmentation functions. These are usually taken from the spectra as implemented in the Monte Carlo simulations. The uncertainty due to variations of the fragmentation parameter $\epsilon$ by $25\%$ is usually small ($\sim 3\%$).

It should be noted that the parameter choices and variations as described above are conventions which are mainly justified by the fact that the normalization of the cross sections and size of the total systematic error obtained when following this procedure is plausible. A combined analysis using several data sets and measurements could be useful in order to determine the appropriate parameters more precisely, thus reducing the uncertainties due to the quark masses, the scales and the fragmentation functions and parameters for future predictions.