In Part I (in this journal) 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 ultra-high energy (UHE) cross sections for neutrino scattering on an isoscalar nucleon, $N=(n+p)/2$, up to laboratory neutrino energies $E_ u sim 10^{16}$-$10^{17}$ 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 NLO 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 $ u 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 $ln^3E_ u$ for leading-order-weak $ u N$ scattering. The energy dependence of $ u N$ total cross section measurements consequently has important implications for hadronic interactions at enormous cms (center-of-mass) energies not otherwise accessible.
We argue that the deep inelastic structure function $F_2^{gamma p}(x, Q^2)$, regarded as a cross section for virtual $gamma^*p$ scattering, is hadronic in nature. This implies that its growth is limited by the Froissart bound at high hadronic energie
s, giving a $ln^2 (1/x)$ bound on $F_2^{gamma p}$ as Bjorken $xrightarrow 0$. The same bound holds for the individual quark distributions. In earlier work, we obtained a very accurate global fit to the combined HERA data on $F_2^{gamma p}$ using a fit function which respects the Froissart bound at small $x$, and is equivalent in its $x$ dependence to the function used successfully to describe all high energy hadronic cross sections, including $gamma p$ scattering. We extrapolate that fit by a factor of $lesssim$3 beyond the HERA region in the natural variable $ln(1/x)$ to the values of $x$ down to $x=10^{-14}$ and use the results to derive the quark distributions needed for the reliable calculation of neutrino cross sections at energies up to $E_ u=10^{17}$ GeV. These distributions do not satisfy the Feynman wee parton assumption, that they all converge toward a common distribution $xq(x,Q^2)$ at small $x$ and large $Q^2$. This was used in some past calculations to express the dominant neutrino structure function $F_2^{ u(bar{ u})}$ directly in terms of $F_2^{gamma p}$. We show that the correct distributions nevertheless give results for $F_2^{ u(bar{ u})}$ which differ only slightly from those obtained assuming that the wee parton limit holds. In two Appendices, we develop simple analytic results for the effects of QCD evolution and operator-product corrections on the distribution functions at small $x$, and show that these effects amount mainly to shifting the values of $ln(1/x)$ in the initial distributions.
We have analytically solved the LO pQCD singlet DGLAP equations using Laplace transform techniques. Newly-developed highly accurate numerical inverse Laplace transform algorithms allow us to write fully decoupled solutions for the singlet structure f
unction F_s(x,Q^2)and G(x,Q^2) as F_s(x,Q^2)={cal F}_s(F_{s0}(x), G_0(x)) and G(x,Q^2)={cal G}(F_{s0}(x), G_0(x)). Here {cal F}_s and cal G are known functions of the initial boundary conditions F_{s0}(x) = F_s(x,Q_0^2) and G_{0}(x) = G(x,Q_0^2), i.e., the chosen starting functions at the virtuality Q_0^2. For both G and F_s, we are able to either devolve or evolve each separately and rapidly, with very high numerical accuracy, a computational fractional precision of O(10^{-9}). Armed with this powerful new tool in the pQCD arsenal, we compare our numerical results from the above equations with the published MSTW2008 and CTEQ6L LO gluon and singlet F_s distributions, starting from their initial values at Q_0^2=1 GeV^2 and 1.69 GeV^2, respectively, using their choices of alpha_s(Q^2). This allows an important independent check on the accuracies of their evolution codes and therefore the computational accuracies of their published parton distributions. Our method completely decouples the two LO distributions, at the same time guaranteeing that both G and F_s satisfy the singlet coupled DGLAP equations. It also allows one to easily obtain the effects of the starting functions on the evolved gluon and singlet structure functions, as functions of both Q^2 and Q_0^2, being equally accurate in devolution as in evolution. Further, it can also be used for non-singlet distributions, thus giving LO analytic solutions for individual quark and gluon distributions at a given x and Q^2, rather than the numerical solutions of the coupled integral-differential equations on a large, but fixed, two-dimensional grid that are currently available.
