  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.
