$k_t$ Factorization

In the $k_t$ factorization approach [118,119,120,121], to be used with the BFKL [61,62,63,64] or CCFM [65] evolution equations, parameters additional to $x$ and $Q^2$ are used to describe the distribution of the partons in the proton. The unintegrated gluon density as a function of $x$, $Q^2$ and $k_t$, folded with off-shell matrix elements, is determined through fits to proton structure function data as measured at HERA [66,67], where $k_t$ denotes the transverse parton momentum emitted along the cascade. In unintegrated parton distributions, the dependence on the transverse parton momentum $k_t$ emitted along the cascade is not integrated out. This is in contrast to the DGLAP approach in which, usually, the gluon density is integrated in that it only depends on the energy fraction $x$ and on the squared transverse momentum transfer $Q^2$.

The partons entering the hard scattering matrix element are free to be off-shell, in contrast to the collinear approach (DGLAP) which treats all partons entering the hard subprocess as massless. Off-shell matrix elements of heavy flavor lepto- and hadroproduction processes have been calculated in [122,123]. The CCFM evolution equations can be used with $k_t$ factorization and they apply angular ordering which is a consequence of color coherence, i.e. due to the interference properties of the radiated gluons. As a result in the appropriate limit they reproduce the DGLAP [59] and the BFKL [61,62,63,64] approximation. At small values of the parton momentum fractions $z$, a random walk of the transverse parton momenta $k_t$ is obtained.