Connection of the virtual $\gamma^*p$ cross section of $ep$ deep inelastic scattering to real $\gamma p$ scattering, and the implications for $\nu N$ and $ep$ total cross sections

We show that it is possible to fit all of the HERA DIS (deep inelastic scattering) data on $F_2^{\gamma p}$ at small values of Bjorken $x$, including the data at {\em 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 parameterization, 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 $\alpha +\beta\ln s+\gamma\ln^2s$. 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 $W\ge 25$ 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 {\em strong interactions at incredibly high energies}.