Understanding the Non-Perturbative Deep-Inelastic Scattering: Instanton-induced Inelastic Dipole-Dipole Cross Section

We derive the semiclassical (instanton-induced) contribution to the inelastic cross section of two color dipoles at large $\sqrt{s}$. We study its dependence on the dipole sizes, orientations and, most importantly, the impact parameter. The inelastic cross section is approximately quadratic in the dipole sizes, and Gaussian-like in the impact parameter with a width of the order of the instanton size. These results are directly relevant to double DIS ${\gamma }^{\ast }{\gamma }^{\ast }$, as well as ${\gamma }^{\ast }\gamma$ and standard DIS ${\gamma }^{\ast }h$ at small x when a real photon and a hadron can be approximated by a dipole. For such cases, with one small dipole scattering on a large dipole, the impact parameter profile exhibits a width of about 1/2 fm, which is in good agreement with the impact parameter profile recently extracted from DIS HERA data, including diffractive ${\gamma }^{\ast }\to J\mathrm{/}\psi$.