Uncertainties of predictions from parton distribution functions. I. The Lagrange multiplier method

We apply the Lagrange multiplier method to study the uncertainties of physical predictions due to the uncertainties of parton distribution functions (PDF’s), using the cross section ${\sigma }_W$ for W production at a hadron collider as an archetypal example. An effective ${\chi }^2$ function based on the CTEQ global QCD analysis is used to generate a series of PDF’s, each of which represents the best fit to the global data for some specified value of ${\sigma }_W.$ By analyzing the likelihood of these “alterative hypotheses,” using available information on errors from the individual experiments, we estimate that the fractional uncertainty of ${\sigma }_W$ due to current experimental input to the PDF analysis is approximately $\pm 4\%$ at the Fermilab Tevatron, and $\pm 8–10\%$ at the CERN Large Hadron Collider. We give sets of PDF’s corresponding to these up and down variations of ${\sigma }_W.$ 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.