The BFKL Pomeron with running coupling constant: how much of its hard nature survives?

Abstract

We discuss the BFKL equation with a running gauge coupling and identify in its solutions the contributions originating from different transverse momentum scales. We show that for a running coupling constant the distribution of the gluons making up the BFKL Pomeron shifts to smaller transverse momenta so that the dominant part of Pomeron can have a nonperturbative origin. It is demonstrated how this soft physics enters into the BFKL solution through the boundary condition. We consider two kinematical regimes leading to different behaviour of the rapidity and transverse momentum dependence of the gluon distribution. In the diffusion approximation to the BFKL kernel with running $\alpha_s$, we find a sequence of poles which replaces the cut for fixed $\alpha_s$. The second regime corresponds to the singular part of the kernel, which gives the dominant contribution in the limit of very large transverse momenta. Finally, a simple more general picture is obtained for the QCD Pomeron in hard processes: it is of soft, nonperturbative nature, but has hard ends of DGLAP-type.Comment: 23 pages REVTEX and 4 figures include