The photon-photon interaction at low $x$ in the theory of reggeized gluons with a running coupling and $N_c\to \mathrm{\infty }$

The forward elastic amplitude for scattering of real and weakly virtual photons is studied in the framework of the theory of reggeized gluons, with large $N_c$ and a running coupling constant introduced in the manner which preserves the bootstrap condition. Transition from a single to multiple pomeron exchanges is observed as $x$ gets smaller. For very low $x$ the amplitude aquires an eikonal form. The photon structure function reveals a strong violation of scaling: it grows as $Q^2$. As a function of $x$ it behaves as $(\ln (1\mathrm{/}x)\ln \ln (1\mathrm{/}x){)}^2$. Correspondingly the cross-section for physical photons grows with energy as $(\ln s\ln \ln s{)}^2$. Hadronic structure functions and cross-sections are also briefly discussed.