Issues in the Determination of Parton Distribution Functions

TL;DR

This note develops a rigorous framework for error analysis in global fits of parton distribution functions, incorporating correlated experimental and theoretical uncertainties through a full covariance structure. It derives how to compute and interpret the parameter error matrix $E_{\alpha\beta}$ and how to propagate these uncertainties to predictions, proposing practical ways to publish error sets and to include prior information or sum-rule constraints. The authors also address computational feasibility and advocate using χ^2 on data subsets to diagnose inconsistencies, ultimately providing a comprehensive methodology to quantify and utilize uncertainties in high-dimensional fits. The work lays groundwork for more reliable uncertainty estimates in PDFs and their predictions, with clear guidance for both theorists and experimental analysts.

Abstract

The CTEQ and MRS parton distributions involve a substantial number (~30) of parameters that are fit to a large number (~900) of data. Typically, these groups produce fits that represent a good fit to the data, but there is no substantial attempt to determine the errors associated with the fits. Determination of errors would involve consideration of the experimental statistical and systematic errors and also the errors in the theoretical formulas that relate the measured cross sections to parton distributions. We discuss the principles that would be needed in such an error analysis. These principles are standard. However, certain aspects of the principles appear counter-intuitive in the case of a large number of data. Accordingly, we strive to devote careful attention to the logic behind the methods.