Gamma^* - Gamma^* Scattering: Saturation and Unitarization in the BFKL approach

In this paper ${\gamma }^{\ast }-{\gamma }^{\ast }$ scattering with large, but more or less equal virtualities of two photons is discussed using BFKL dynamics, emphasizing the large impact parameter behavior ($b_t$) of the dipole-dipole amplitude. It is shown that the non-perturbative contribution is essential to fulfill the unitarity constraints in the region of $b_t>1\mathrm{/}2m_{\pi }$, where $m_{\pi }$ is pion mass. The saturation and the unitarization of the dipole-dipole amplitude is considered in the framework of the Glauber-Mueller approach. The main result is that we can satisfy the unitarity constraints introducing the non-perturbative corrections only in initial conditions (Born amplitude).