High energy evolution for Gribov-Zwanziger confinement: Solution to the equation

Abstract

In this paper, we solve the new evolution equation for high energy scattering amplitude that stems from the Gribov-Zwanziger approach to the confinement of quarks and gluons. We found that (1) the energy dependence of the scattering amplitude turns out to be the same as for QCD Balitsky, Fadin, Kuraev and Lipatov (BFKL) evolution, (2) the spectrum of the new equation does not depend on the details of the Gribov-Zwanzinger approach, and (3) all eigenfunctions coincide with the eigenfunctions of the QCD BFKL equation at large transverse momenta $\ensuremath{\kappa}\ensuremath{\ge}1$. The numerical calculations show that there exist no new eigenvalues with the eigenfunctions which decrease faster than solutions of the QCD BFKL equation at large transverse momenta. The structure of the gluon propagator in the Gribov-Zwanziger approach, that stems from the lattice QCD and from the theoretical evaluation, results in the exponential suppression of the eigenfunctions at long distances and in the resolution of the difficulties, which the color glass condensate and some other approaches, based on perturbative QCD, face at large impact parameters. We can conclude that the confinement of quark and gluons, at least in the form of the Gribov-Zwanziger approach, does not influence on the scattering amplitude except for solving the long-standing theoretical problem of its behavior at large impact parameters.