Dipole-dipole scattering at high energy in the Pomeron field theory with Braun Hamiltonian and beyond

In this paper we find that the scattering matrix for dipole-dipole interaction in the saturation region has the form: ${\cal S}^d_d \xrightarrow{z \,\gg\,1} \exp\Lb - C^d_d z^2\Rb \propto \Lb {\cal S}_{BK}\Rb^4 = \exp\Lb - 4\,C_{BK} z^2\Rb$, where $z = \bas \kappa Y\,+\,\ln\Lb \frac{r^2}{r'^2}\Rb$ for interaction of a dipole $r$ at rapidity $Y$ with dipole $r'$ at rest. $S_{BK}$ is the S-matrix for the Balitsky-Kovchegov amplitude. All constants are determined in the text. The proof is given in the same theoretical framework for both cases (${\cal S}^d_d$ and ${\cal S}_{BK}$): the Pomeron interaction which takes into account the Pomeron vertices in the leading $1/N_c$ approximation ($N_c$ is the number of colours). This result is in striking contradiction with 'rare' fluctuation approach as well as with summing the large Pomeron loops. These lead to ${\cal S}^d_d \propto \sqrt{{\cal S}_{BK}}$ at large $z$. In the paper we collect arguments supporting the idea that the sum of large Pomeron loops can be trusted in a wide region of energy: $Y \leq 1/\as^4$. In addition we discuss the influence of the structure of the Hamiltonian on the asymptotic behaviour of the scattering amplitudes using the exactly solvable one dimensional model as our theoretical ground.