Uncertainties of Predictions from Parton Distribution Functions I: the Lagrange Multiplier Method

TL;DR

This paper introduces the Lagrange Multiplier method to quantify how uncertainties in parton distribution functions propagate to hadron collider predictions, using W and Z production as archetypal observables. By linking an effective global chi-squared to constrained variations of a target observable, the authors generate optimal PDF sets that reveal the full range of allowed cross-section values beyond traditional Hessian-based error estimates. They demonstrate this for Tevatron and LHC kinematics, finding approximately 4% PDF-induced uncertainty for W production at the Tevatron and 8–10% at the LHC, with Z production showing similar trends. The work provides concrete PDF sets and a principled framework for benchmarking uncertainty estimates and for testing approximate error-propagation methods in precision QCD. It also highlights that true global uncertainties depend on the collider and process due to differing sensitivities to small- and large-$x$ PDFs, and it presents a suite of extreme PDFs to guide future phenomenology.

Abstract

We apply the Lagrange Multiplier method to study the uncertainties of physical predictions due to the uncertainties of parton distribution functions (PDFs), using the cross section for W production at a hadron collider as an archetypal example. An effective chi-squared function based on the CTEQ global QCD analysis is used to generate a series of PDFs, each of which represents the best fit to the global data for some specified value of the cross section. By analyzing the likelihood of these "alterative hypotheses", using available information on errors from the individual experiments, we estimate that the fractional uncertainty of the cross section due to current experimental input to the PDF analysis is approximately 4% at the Tevatron, and 8-10% at the LHC. We give sets of PDFs corresponding to these up and down variations of the cross section. We also present similar results on Z production at the colliders. Our method can be applied to any combination of physical variables in precision QCD phenomenology, and it can be used to generate benchmarks for testing the accuracy of approximate methods based on the error matrix.

Figures (12)

• Figure 1: Left: The LM method provides sample points along a single curve $L_{X}$ in the multi-dimensional PDF parameter space, relevant ro the observable $X$. Right: For a given tolerance $\Delta\chi_\mathrm{global}^{2}$, the uncertainty in the calculated value of $X$ is $\pm\Delta{X}$. The solid points correspond to the sample points on the curve $L_{X}$ in the left plot.
• Figure 2: Calculated cross section for $W^{\pm}$ boson production (multiplied by the branching ratio for $W^{-}\rightarrow e\overline{\nu}$) at the Tevatron, for various current and historical PDFs. The two plots are from Refs. MRST2 and RunII respectively.
• Figure 3: Minimum $\chi_{\rm global}^2$ versus $\sigma_{W}$, the inclusive $W^{\pm}$ production cross section at the Tevatron ($\bar{p}p$ collisions at $\sqrt{s}=1.8$ TeV) in nb. The points were obtained by the Lagrange Multiplier method. The curve is a polynomial fit to the points.
• Figure 4: $\chi^{2}/N$ of the H1 data, including error correlations, for sample PDFs obtained by the Lagrange Multiplier method for constrained values of $\sigma_{W}$ at the Tevatron. The arrow indicates the global minimum.
• Figure 5: The abcissa is $\sigma_{W}$ in nb, at the Tevatron. The ordinate is $\chi_{n}^{2}-\chi_{n,0}^{2}$. The number in parentheses is the number of data points. The horizontal lines are explained in the text.