Implications of a Froissart bound saturation of $\gamma^*$-$p$ deep inelastic scattering. Part I. Quark distributions at ultra small $x$

TL;DR

This paper explores the hadronic nature of the structure function in deep inelastic scattering, applying the Froissart bound to small-x behavior, and derives quark distributions for high-energy neutrino interactions, challenging previous assumptions.

Contribution

It introduces a Froissart-bounded fit to structure functions, extrapolates to ultra-small x, and refines quark distribution calculations for neutrino cross sections at extremely high energies.

Findings

Froissart bound limits the growth of structure functions at small x.

Quark distributions at ultra-small x are derived without assuming the wee parton limit.

Results show slight differences in neutrino structure functions compared to previous models.

Abstract

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 energies, giving a $\ln^2 (1/x)$ bound on $F_2^{\gamma p}$ as Bjorken $x\rightarrow 0$. 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 $E_\nu=10^{17}$ 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 $F_2^{\nu(\bar{\nu})}$ directly in terms of $F_2^{\gamma p}$. We show that the correct distributions nevertheless give results for $F_2^{\nu(\bar{\nu})}$ 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.