Solution of the Kwiecinski evolution equations for unintegrated parton distributions using the Mellin transform

Abstract

The Kwiecinski equations for the QCD evolution of the unintegrated parton distributions in the transverse-coordinate space (b) are analyzed with the help of the Mellin-transform method. The equations are solved numerically in the general case, as well as in a small-b expansion which converges fast for b Lambda_QCD sufficiently small. We also discuss the asymptotic limit of large bQ and show that the distributions generated by the evolution decrease with b according to a power law. Numerical results are presented for the pion distributions with a simple valence-like initial condition at the low scale, following from chiral large-N_c quark models. We use two models: the Spectral Quark Model and the Nambu--Jona-Lasinio model. Formal aspects of the equations, such as the analytic form of the b-dependent anomalous dimensions, their analytic structure, as well as the limits of unintegrated parton densities at x -> 0, x -> 1, and at large b, are discussed in detail. The effect of spreading of the transverse momentum with the increasing scale is confirmed, with <k_\perp^2> growing asymptotically as Q^2 alpha(Q^2). Approximate formulas for <k_\perp^2> for each parton species is given, which may be used in practical applications.