The asymptotic behaviour of parton distributions at small and large $x$

TL;DR

This paper investigates the asymptotic behavior of proton parton distribution functions at small and large x, comparing theoretical expectations with empirical fits, and analyzing their scale dependence and uncertainties.

Contribution

It critically examines the asymptotic exponents of PDFs from global fits, confirming theoretical predictions for valence quarks and highlighting uncertainties for sea quarks and gluons.

Findings

Valence distributions agree with Regge and counting rules within uncertainties.

Sea quarks and gluons show less conclusive results.

Caution advised against overconstrained parametrizations in PDF fits.

Abstract

It has been argued from the earliest days of quantum chromodynamics (QCD) that at asymptotically small values of $x$ the parton distribution functions (PDFs) of the proton behave as $x^\alpha$, where the values of $\alpha$ can be deduced from Regge theory, while at asymptotically large values of $x$ the PDFs behave as $(1-x)^\beta$, where the values of $\beta$ can be deduced from the Brodsky-Farrar quark counting rules. We critically examine these claims by extracting the exponents $\alpha$ and $\beta$ from various global fits of parton distributions, analysing their scale dependence, and comparing their values to the naive expectations. We find that for valence distributions both Regge theory and counting rules are confirmed, at least within uncertainties, while for sea quarks and gluons the results are less conclusive. We also compare results from various PDF fits for the structure function ratio $F_2^n/F_2^p$ at large $x$, and caution against unrealistic uncertainty estimates due to overconstrained parametrisations.