Error Estimates on Parton Density Distributions

TL;DR

This work evaluates error estimation for parton density distributions in global QCD fits, emphasizing Gaussian error assumptions and covariance-based propagation. It contrasts the Hessian (covariance) and offset methods for incorporating systematic uncertainties, discusses computational techniques to handle large χ^2 calculations, and promotes exploring the χ^2 landscape via Minos and Lagrange multipliers. It also discusses the limits of the quadratic approximation and the interpretation of Δχ^2 in complex fits. Finally, it surveys public parton distribution sets (Alekhinfit, epdflib) that encode uncertainties for practical predictions at high-energy colliders.

Abstract

Error estimates on parton density distributions are presently based on the traditional method of least squares minimisation and linear error propagation in global QCD fits. We review the underlying assumptions and the various mathematical representations of the method and address some technical issues encountered in such a global analysis. Parton distribution sets which contain error information are described.

Figures (5)

• Figure 1: (a) Ellipsoid contours defined by constant values of $\chi^2$ in parameter space. (b) Hyper-sphere contours of constant $\chi^2$ after an orthogonal transformation of the parameters defined by the eigenvectors and eigenvalues of the Hessian matrix. Figure taken from ref:hmethod.
• Figure 2: Left: The eigenvalue spectrum of the Hessian matrix from a typical global QCD fit with (13,16,18) free parameters. Taken from ref:hmethod. Right: Distribution of $\Delta \chi^2$ calculated by minuit (dashed histogram) and by an improved calculation from ref:cteqpap1 (full histogram) on a 10-dimensional ellipsoid defined by $\Delta \chi^2 = 5$. The spread is caused by numerical errors in the calculation of the Hessian matrix.
• Figure 3: The gluon density versus the Bjorken scaling variable $x$ from a LO QCD fit to the BCDMS structure function data. The dotted curves show the statistical error band. The full curves indicate the systematic error band calculated using the offset method (a) and the Hessian method (b). Figure taken from ref:alekhinstudy.
• Figure 4: Left: The $\chi^2_g$ (see text) versus the cross-section $\sigma_W$ of inclusive $W$ production at Tevatron energies from the QCD analysis of ref:lmethod. Right: Optimal values of $\sigma_W$ (dots) and 90% confidence levels (bars) for each data set included in the fit. The full line shows the best prediction for $\sigma_W$ and the dashed lines represent the confidence bounds described in the text. Figures taken from ref:lmethod.
• Figure 5: Left: The parton momentum densities from the QCD analysis of ref:mbfit versus $x$ at $Q^2 = 10$ GeV$^2$. Right: The errors on the gluon and quark densities from the various sources described in the text. Figures taken from ref:mbfit.