Implications of a Froissart bound saturation of $\gamma^*$-$p$ deep inelastic scattering. Part II. Ultra-high energy neutrino interactions

TL;DR

This paper uses a Froissart-bounded model fitted to deep inelastic scattering data to accurately estimate ultra-high energy neutrino-nucleon cross sections up to 10^17 GeV, providing a more reliable prediction than previous power-law extrapolations.

Contribution

It introduces a Froissart-bounded form for structure functions to derive ultra-high energy neutrino cross sections, establishing a theoretical bound on their energy dependence.

Findings

Neutrino cross sections are accurate within 2% at energies up to 10^17 GeV.

The Froissart bound implies a maximum $ ext{ln}^3 E_ u$ growth for neutrino cross sections.

Compared to NLO PDF-based estimates, our results are more conservative and data-driven.

Abstract

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 $E_\nu \sim 10^{16}$-$10^{17}$ 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 $\nu N$ 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 $\ln^3E_\nu$ 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 cms (center-of-mass) energies not otherwise accessible.