Uncertainties of predictions from parton distribution functions II: the Hessian method

TL;DR

This work develops a Hessian-based framework to quantify uncertainties in parton distribution functions (PDFs) and their physical predictions within a global QCD analysis. It introduces an iterative Hessian calculation and normalized eigenvector coordinates to produce a set of $2d$ Eigenvector Basis PDFs ($d\approx16$) that enable straightforward propagation of PDF uncertainties to any observable. The authors formulate master equations that relate observable changes to eigen-directions in PDF parameter space and demonstrate the method by quantifying uncertainties in PDFs themselves and in predictions such as gluon/quark distributions, $W$ rapidity distributions, and $W$–$Z$ cross-section correlations. They discuss the tolerance $T$ (estimated to be about 10–15) governing the allowed neighborhood around the global minimum and provide guidance on applying the method to current and future collider phenomenology.

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 d=16 is the number of parton parameters) 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.

Figures (11)

• Figure 1: Illustration of the basic ideas of our implementation of the Hessian method. An iterative procedure Paper0 diagonalizes the Hessian matrix and rescales the eigenvectors to adapt the step sizes to their natural scale. The solid points represent the resulting eigenvector basis PDFs described in Sec. \ref{['sec:Eigenvectors']}. Point $\mathbf{p(i)}$ is explained in Sec. \ref{['sec:UncParamAi']}.
• Figure 2: Distribution of eigenvalues of the Hessian matrix for fits using $d=13$, $16$ (standard), and $18$ free PDF parameters.
• Figure 3: Two extreme gluon distributions (left) and $u$-quark distributions (right) for $Q \! = \! 2 \, {\rm GeV}$ (long dash) and $Q \! = \! 100 \, {\rm GeV}$ (short dash) with $T\!=\!10$. Each curve is calculated to give the minimum or maximum value for some particular $x$. The entire allowed region, which is the envelope of all such curves, is shaded.
• Figure 4: Ratio of gluon (left) and $u$-quark (right) distributions to Best Fit $S_0$ at $Q = 10 \, {\rm GeV}$.
• Figure 5: Left: Predicted rapidity distribution for $p \bar{p} \to W^{+} \, + \, X$ at $\sqrt{s} = 1.8 \, {\rm TeV}$. The curves are extreme predictions for the integrated cross section $\sigma$ (solid), or the rapidity moments $\langle y \rangle$ (long dash), or $\langle y^2 \rangle$ (short dash). Right: same except the Best Fit prediction is subtracted to show the details better.