In part I [M. M. Block et al., preceding article, Phys. Rev. D 88, 014006 (2013)] we argued that the structure function $F_2^{\gamma p}(x,Q^2)$ in deep inelastic $ep$ scattering, regarded as a cross section for virtual ${\gamma }^{*}p$ scattering, has a saturated Froissart-bounded form behaving as ${ln}^2(1/x)$ at small $x$. This form provides an excellent fit to the low $x$ HERA data, including the very low $Q^2$ regions, and can be extrapolated reliably to small $x$ using the natural variable $\ln (1/x)$. We used our fit to derive quark distributions for values of $x$ down to $x={10}^{-14}$. We use those distributions here to evaluate ultrahigh energy (UHE) cross sections for neutrino scattering on an isoscalar nucleon, $N=(n+p)/2$, up to laboratory neutrino energies $E_{\nu }\sim {10}^{16}–{10}^{17}\mathrm{GeV}$ where there are now limits on neutrino fluxes. We estimate that these cross sections are accurate to $\sim 2\%$ at the highest energies considered, with the major uncertainty coming from the errors in the parameters that were needed to fit $F_2^{\gamma p}(x,Q^2)$. We compare our results to recently published neutrino cross sections derived from next-to-leading order parton distribution functions, which become much larger at high energies because of the use of power-law extrapolations of quark distributions to small $x$. We argue that our calculation of the UHE $\nu N$ cross sections is the best one can make based the existing experimental deep inelastic scattering data. Further, we show that the strong interaction Froissart bound of ${ln}^2(1/x)$ on $F_2^{\gamma p}$ translates to an exact bound of ${ln}^3{E}_{\nu }$ for leading-order-weak $\nu N$ scattering. The energy dependence of $\nu N$ total cross section measurements consequently has important implications for hadronic interactions at enormous center-of-mass energies not otherwise accessible.

• Received 24 February 2013

DOI:https://doi.org/10.1103/PhysRevD.88.013003