Elastic scattering of virtual photons via quark loop in Double-Logarithmic Approximation

We calculate the amplitude of elastic photon-photon scattering via a single quark loop in the Double-Logarithmic Approximation, presuming all external photons to be off-shell and unpolarized.. At the same time we account for the running coupling effects. We consider this process in the forward kinematics at arbitrary relations between t and the external photon virtualities. We obtain explicit expressions for the photon-photon scattering amplitudes in all double-logarithmic kinematic regions. Then we calculate the small-x asymptotics of the obtained amplitudes and compare them with the parent amplitudes, thereby fixing the applicability regions of the asymptorics, i.e. fixing the applicability region for the non-vacuum Reggeons. We find that these Reggeons should be used at x < 10^{-8} only.