Uncertainties of predictions from parton distribution functions. II. The Hessian method

Abstract

We develop a general method to quantify the uncertainties of parton distribution functions and their physical predictions, with emphasis on incorporating all relevant experimental constraints. The method uses the Hessian formalism to study an effective chi-squared function that quantifies the fit between theory and experiment. Key ingredients are a recently developed iterative procedure to calculate the Hessian matrix in the difficult global analysis environment, and the use of parameters defined as components along appropriately normalized eigenvectors. The result is a set of 2D eigenvector basis parton distributions ͑where dϷ16 is the number of parton parameters͒ from which the uncertainty on any physical quantity due to the uncertainty in parton distributions can be calculated. We illustrate the method by applying it to calculate uncertainties of gluon and quark distribution functions, W boson rapidity distributions, and the correlation between W and Z production cross sections.

A meta-analysis of parton distribution functions

“…Here O ij is an orthogonal matrix, and λ i are the eigenvalues of the covariance matrix. The eigenvalues λ i in the PDF analysis span many orders of magnitude [12]. Those directions in {a ′ } space that are associated with very small λ i have negligible contributions to the PDF uncertainties, so that the corresponding parameters a ′ i can be fixed at their central values, thus reducing the number of independent eigenvectors.…”

“…When combining the PDF ensembles, one follows two common methods used for estimating the PDF uncertainty, the Hessian method [12,13] and the Monte Carlo (MC) sampling method [14,15]. We summarize the core relations of the two methods for completeness.…”

“…In the Hessian approach [12,13] adopted by CT10 and MSTW, a small number of independent PDF eigenvector sets spans only the boundary of the 68% or 90% c.l. hypervolume in their respective PDF parameter space.…”

“…The tolerance (error) ellipse is introduced to study correlations between two observables, e.g., X and Y . In the Hessian approach, we first define the correlation angle ∆ϕ as [12,32,33] cos…”

“…The PDF uncertainty in the Hessian method estimates the boundaries of the confidence interval on X, usually at the 68% or 90% c.l. Both the symmetric [12,13] uncertainty, δ H , and asymmetric uncertainties δ H ± [32] can be estimated:…”