Monte Carlo Simulations Using $k_t$ Factorization

The Monte Carlo generator CASCADE [104] contains an implementation of the $k_t$ factorization approach (see section 2.5) using the CCFM evolution equation [65]. In CASCADE the direct heavy quark production processes $\gamma g \rightarrow c\bar{c}$ and $b\bar{b}$ are implemented using off-shell matrix elements convoluted with $k_t$-unintegrated parton distributions in the proton. In CASCADE, higher order parton emissions based on the CCFM [65] evolution equations are matched to ${\cal O}(\alpha_s)$ matrix elements in which the incoming parton can be off-shell. This implies that the parton kinematics are treated correctly. For the final state parton showers PYTHIA [108] is used. The unintegrated gluon density in the proton has been determined in fits to the inclusive structure function data, in the range $x<10^{-2}$ and $Q^2 > 5$ GeV$^2$. With this input CASCADE has been able to correctly reproduce the $b$ quark production cross-sections in $p\bar{p}$-collisions at the Tevatron [128].