A Variable-Flavour Number Scheme for NNLO

At NNLO it is particularly important to have a Variable-Flavour Number Scheme (VFNS) to deal with heavy quarks because there are major problems with both the zero mass variable-flavour number scheme and the fixed-flavour number scheme. I illustrate these problems and present a general formulation of a Variable-Flavour Number Scheme (VFNS)for heavy quarks that is explicitly implemented up to NNLO in the strong coupling constant alpha_S, and may be used in NNLO global fits for parton distributions. The procedure combines elements of the ACOT(chi) scheme and the Thorne-Roberts scheme. Despite the fact that at NNLO the parton distributions are discontinuous as one changes the number of active quark flavours, all physical quantities are continuous at flavour transitions and the comparison with data is successful.Comment: 17 pages, 5 figures included as .ps files, uses axodraw. One additional explanatory sentence after eq. (25). Correction of typos and updated references. To be published in Phys. Rev.

A Variable Flavour Number Scheme at NNLO

I present a formulation of a Variable Flavour Number Scheme for heavy quarks that is implemented up to NNLO in the strong coupling constant and may be used in NNLO global fits for parton distributions.Comment: 4 pages, 6 figures included as .ps files. To appear in proceedings of DIS05, XIII International Workshop on Deep Inelastic Scatterin

A Complete Leading-Order, Renormalization-Scheme-Consistent Calculation of Small--x Structure functions, Including Leading-ln(1/x) Terms

We present calculations of the structure functions F_2(x,Q^2) and F_L(x,Q^2), concentrating on small x. After discussing the standard expansion of the structure functions in powers of \alpha_s(Q^2) we consider a leading-order expansion in ln(1/x) and finally an expansion which is leading order in both ln(1/x) and \alpha_s(Q^2), and which we argue is the only really correct expansion scheme. Ordering the calculation in a renormalization-scheme- consistent manner, there is no factorization scheme dependence, as there should not be in calculations of physical quantities. The calculational method naturally leads to the ``physical anomalous dimensions'' of Catani, but imposes stronger constraints than just the use of these effective anomalous dimensions. In particular, a relationship between the small-x forms of the inputs F_2(x,Q_0^2) and F_L(x,Q_0^2) is predicted. Analysis of a wide range of data for F_2(x,Q^2) is performed, and a very good global fit obtained, particularly for data at small x. The fit allows a prediction for F_L(x,Q^2) to be produced, which is smaller than those produced by the usual NLO-in-\alpha_s(Q^2) fits to F_2(x,Q^2) and different in shape.Comment: 106 pages, 4 figures as ps files, includes a variation of harmac. Corrections to some typos in references, and form of some references changed, in particular hep-ph(ex) numbers included for papers not yet published. No changes to body of tex

The role of uncertainties in parton distribution functions

I consider the uncertainties in parton distributions and the consequences for hadronic cross-sections. There is ever-increasing sophistication in the relationship between the uncertainties of the distributions and the errors on the experimental data used to extract them. However, I demonstrate that this uncertainty is frequently subsumed by that due to the choice of data used in fits, and more surprisingly by the precise details of the theoretical framework used. Variations in heavy flavour prescriptions provide striking examples.Comment: 10 pages, 15 figures as .ps or .eps files, invited talk at PHYSTAT-LHC Workshop on Statistical Issues for LHC Physics, June 200

Study of Monte Carlo approach to experimental uncertainty propagation with MSTW 2008 PDFs

We investigate the Monte Carlo approach to propagation of experimental uncertainties within the context of the established "MSTW 2008" global analysis of parton distribution functions (PDFs) of the proton at next-to-leading order in the strong coupling. We show that the Monte Carlo approach using replicas of the original data gives PDF uncertainties in good agreement with the usual Hessian approach using the standard Delta(chi^2) = 1 criterion, then we explore potential parameterisation bias by increasing the number of free parameters, concluding that any parameterisation bias is likely to be small, with the exception of the valence-quark distributions at low momentum fractions x. We motivate the need for a larger tolerance, Delta(chi^2) > 1, by making fits to restricted data sets and idealised consistent or inconsistent pseudodata. Instead of using data replicas, we alternatively produce PDF sets randomly distributed according to the covariance matrix of fit parameters including appropriate tolerance values, then we demonstrate a simpler method to produce an arbitrary number of random predictions on-the-fly from the existing eigenvector PDF sets. Finally, as a simple example application, we use Bayesian reweighting to study the effect of recent LHC data on the lepton charge asymmetry from W boson decays.Comment: 37 pages, 17 figures. v2: version published in JHEP. Supplementary material at http://mstwpdf.hepforge.org/random

Parton Distributions

I discuss our current understanding of parton distributions. I begin with the underlying theoretical framework, and the way in which different data sets constrain different partons, highlighting recent developments. The methods of examining the uncertainties on the distributions and those physical quantities dependent on them is analysed. Finally I look at the evidence that additional theoretical corrections beyond NLO perturbative QCD may be necessary, what type of corrections are indicated and the impact these may have on the uncertainties.Comment: Invited talk at "XXI International Symposium on Lepton and Photon Interactions at High Energies," (Fermilab, Chicago, August 2003). 12 pages, 21 figure