[Paper Review] Issues in the Determination of Parton Distribution Functions

John C. Collins, Davison E. Soper|ArXiv.org|Nov 2, 1994

Particle physics theoretical and experimental studies2 references3 citations

TL;DR

This paper addresses the critical challenge of quantifying uncertainties in global fits of parton distribution functions (PDFs), which typically involve ~30 parameters fitted to ~900 data points without rigorous error estimation. The authors outline a systematic framework for error analysis that incorporates experimental statistical and systematic errors, as well as theoretical uncertainties in the perturbative QCD formulas, emphasizing logical consistency despite counterintuitive aspects in high-dimensional fits.

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.

Motivation & Objective

• To address the lack of systematic error estimation in global fits of parton distribution functions (PDFs), which typically report good fits without quantifying uncertainties.
• To develop a principled approach for propagating experimental statistical and systematic errors into PDF uncertainties.
• To incorporate theoretical uncertainties from perturbative QCD calculations into the error analysis of PDF fits.
• To clarify the logical foundations of error estimation in high-dimensional parameter spaces, where intuition may fail.
• To provide a foundation for reliable uncertainty quantification in global PDF fits used in high-energy physics phenomenology.

Proposed method

• Adopt a frequentist statistical framework to propagate experimental errors through the fitting procedure.
• Integrate both statistical and systematic errors from experimental data into the χ² minimization process.
• Account for theoretical uncertainties by considering the sensitivity of cross-section predictions to higher-order corrections and factorization scale variations.
• Use a consistent treatment of parameter space dimensionality to avoid misleading error estimates in high-dimensional fits.
• Apply standard error propagation principles while carefully analyzing their counterintuitive behavior when fitting many parameters to many data points.
• Formalize the error analysis using a likelihood-based approach that treats the full covariance structure of data and theory.

Experimental results

Research questions

• RQ1 How can experimental statistical and systematic errors be consistently propagated into uncertainties of parton distribution functions in global fits?
• RQ2 What is the proper treatment of theoretical uncertainties in the perturbative QCD formulas that relate cross sections to PDFs?
• RQ3 Why do standard error estimation techniques become counterintuitive when fitting a large number of parameters to a large number of data points?
• RQ4 What logical principles must underlie a reliable error analysis in high-dimensional PDF fits?
• RQ5 How can one distinguish between fitting quality and actual uncertainty estimation in global PDF fits?

Key findings

• The paper establishes that rigorous error estimation in PDF fits requires a consistent treatment of both experimental and theoretical uncertainties.
• It identifies that standard error propagation methods can yield misleading results in high-dimensional parameter spaces, necessitating careful logical scrutiny.
• The authors demonstrate that theoretical uncertainties—such as those from scale dependence and higher-order corrections—must be explicitly included in the error budget.
• The framework proposed ensures that error estimates are not merely statistical but also reflect the reliability of the underlying theoretical model.
• The work provides a foundational methodological guide for future PDF global fits aiming to report not just best-fit parameters but also their uncertainties.
• The study highlights that a good fit does not imply small uncertainties, and that uncertainty quantification must be explicitly modeled rather than assumed.