We show that it is possible to fit all of the HERA deep inelastic scattering data on $F_2^{\gamma p}$ at small values of Bjorken $x$, including the data at very low $Q^2$, using a new model for $F_2^{\gamma p}$ which both includes an asymptotic (high-energy) part that satisfies a saturated Froissart bound behavior, with a vector-dominance-like mass factor in the parametrization, and extends smoothly to $Q^2=0$. We require that the corresponding part of the virtual ${\gamma }^{*}p$ cross section match the known asymptotic part of the real $\gamma p$ cross section at $Q^2=0$, a cross section which is determined by strong interactions and asymptotically satisfies a saturated Froissart bound of the form $\alpha +\beta \ln s+\gamma {\ln }^{2}s$. Using this model for the asymptotic part of $F_2^{\gamma p}$ plus a known valence contribution, we fit the asymptotic high–energy part of the HERA data with $x\le 0.1$ and $W\ge 25\mathrm{GeV}$; the fit is excellent. We find that the mass parameter in the fit lies in the region of the light vector mesons, somewhat above the $\rho$-meson mass, and is compatible with vector dominance. We use this fit to obtain accurate results for the high-energy $ep$ and isoscalar $\nu N$ total cross sections. Both cross sections obey an analytic expression of the type $a+b\ln E+c{\ln }^{2}E+d{\ln }^{3}E$ at large energies $E$ of the incident particle, reflecting the fact that the underlying strong interaction parts of the ${\gamma }^{*}p$, $Z^{*}N$ and $W^{*}N$ cross sections satisfy the saturated Froissart bound. Since approximately 50% of the $\nu N$ center-of-mass (cms) energy is found in $W$—the cms energy of the strongly interacting intermediate vector boson–nucleon system—a study of ultra-high-energy neutrino-nucleon cross sections would allow us, for the first time, to explore strong interactions at incredibly high energies.

• Received 17 April 2014

DOI:https://doi.org/10.1103/PhysRevD.89.094027