Solution for the BFKL Pomeron Calculus in zero transverse dimensions

In this paper the exact analytical solution is found for the BFKL Pomeron calculus in zero transverse dimensions, in which all Pomeron loops have been included. The comparison with the approximate methods of the solution is given, and the kinematic regions are discussed where they describe the behaviour of the scattering amplitude quite well. In particular, the semi-classical approach is considered, which reproduces the main properties of the exact solution at large values of rapidity ($Y \geq 10$). It is shown that the mean field approximation leads to a good description of the scattering amplitude only if the amplitude at low energy is rather large. However, even in this case, it does not lead to the correct asymptotic behaviour of the scattering amplitude at high energies.