Transition from soft- to hard-Pomeron in the structure functions of hadrons at small-$x$ from holography

We study the nucleon and pion structure functions at small Bjorken-$x$ region in the framework of holographic QCD with a special emphasis on the roles of AdS space wave functions. Using the BPST kernel for the Pomeron exchange and calculating its coupling to target hadrons in the AdS space, we obtain $F_2$ structure functions at the small-$x$. Results for the proton $F^p_2$ as well as the pion $F^\pi_2$ are consistent with experimental data of the deep inelastic scattering and the forward electroproduction of a neutron. Observed $Q^2$ dependence of the Pomeron intercept is well reproduced from soft non-perturbatibve $(Q^2 \sim 0)$ to hard perturbative $(Q^2 \gg 1 GeV^2)$ region. We find the interplay between soft and hard Pomerons is closely related with behavior of AdS wave functions of hadrons and the virtual photon.