The Green Function for the BFKL Pomeron and the Transition to DGLAP Evolution

We consider the (process-independent) Green function for the BFKL equation with running coupling, and explain how, within the semi-classical approximation, it is related to Green function of the Airy equation. The unique Green function is obtained from a combination of its required ultraviolet behaviour compatible with asymptotic freedom and an infrared limit phase imposed by the non-perturbative sector of QCD. We show that at sufficiently large gluon transverse momenta the corresponding gluon density matches that of the DGLAP analysis, whereas for relatively small values of the gluon transverse momentum the gluon distribution is sensitive to the Regge poles, whose positions are determined both by the non-perturbative QCD dynamics and physics at large transverse momenta