The Running coupling BFKL anomalous dimensions and splitting functions.

I explicitly calculate the anomalous dimensions and splitting functions governing the Q2 evolution of the parton densities and structure functions which result from the running coupling BFKL equation at LO, i.e. I perform a resummation in powers of ln(1/x) and in powers of β0 simultaneously. This is extended as far as possible to NLO. These are expressed in an exact, perturbatively calculable analytic form, up to small power-suppressed contributions which may also be modelled to very good accuracy by analytic expressions. Infrared renormalons, while in principle present in a solution in terms of powers in αs(Q2), are ultimately avoided. The few higher twist contributions which are directly calculable are extremely small. The splitting functions are very different from those obtained from the fixed coupling equation, with weaker power-like growth ∼ x−0.25, which does not set in until extremely small x indeed. The NLO BFKL corrections to the splitting functions are moderate, both for the form of the asymptotic power-like behaviour and more importantly for the range of x relevant for collider physics. Hence, a stable perturbative expansion and predictive power at small x are obtained. March 2001 1 Roya

Uncertainties in parton related quantities.

I discuss the issue of uncertainties in parton distributions and in the physical quantities which are determined in terms of them. While there has been significant progress on the uncertainties associated with errors on experimental data, there are still outstanding questions. Also, I demonstrate that in many circumstances this source of errors may be less important than errors due to underlying assumptions in the fitting procedure and due to the incomplete nature of the theoretical calculations

A Leading-Order, But More Than One-Loop, Calculation of structure functions

I present a full leading-order calculation of F_2(x,Q^2) and F_L(x,Q^2), including contributions not only from leading order in \alpha_s, but also from the leading power of \alpha_s for each order in ln(1/x). The calculation is ordered according to the inputs and evolution of the structure functions, and the perturbative form of the inputs is determined. I compare the results of fits to data to those using conventional LO and NLO order calculations, and the correct inclusion of leading ln(1/x) terms is clearly preferred. A prediction for F_L(x,Q^2) is produced which is smaller at small x than that obtained from the conventional approach.Comment: 5 pages, Latex, 1 figure, uses epsfig.sty and aipproc.sty. Talk presented at the 5th International Workshop on ``Deep Inelastic Scattering and QCD'' (DIS 97), Chicago, USA, April, 1997. A couple of minor tyops correcte

Global fits of parton distributions.

I discuss recent developments in the determination of parton distributions from global fits. I concentrate on the errors associated with these parton distributions and with the physical quantities which are determined in terms of them. I outline the various techniques used to quantify the uncertainties due to errors on experimental data used in the fits, and provide a number of examples of predictions with their uncertainties. However, I demonstrate that this source of errors may, in some circumstances, be less important than errors due to underlying assumptions in the fitting procedure and due to the incomplete nature of the theoretical framework currently employed.Comment: Invited talk at the 14th Topical Conference on Hadron Collider Physics (HCP2002), Sept 30th - Oct 4th 2002, Karlsruhe, Germany. 13 pages, 11 figures, uses svmult.cls and physmubb.st

Comparison of NNLO DIS scheme splitting functions with results from exact gluon kinematics at small x.

We consider the effect of exact gluon kinematics in the virtual photon-gluon impact factor at small x. By comparing with fixed order DIS scheme splitting and coefficient functions, we show that the exact kinematics results match the fixed order results well at each order, which suggests that they allow for an accurate NLL analysis of proton structure functions. We also present, available for the first time, x-space parameterisations of the NNLO DGLAP splitting functions in the DIS scheme, and also the longitudinal coefficients for neutral current scattering

Importance of a measurement of F(L)(X,Q**2) at HERA.

I investigate what a direct measurement of the longitudinal structure function FL(x,Q2) could teach us about the structure of the proton and the best way in which to use perturbative QCD for structure functions. I assume HERA running at a lowered beam energy for approximately 4-5 months and examine how well the measurement could distinguish between different theoretical approaches. I conclude that such a measurement would provide useful information on how to calculate structure functions and parton distributions at small x

A Variable-flavor 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 S, and may be used in NNLO global fits for parton distributions. The procedure combines elements of the ACOT( ) 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

A Variable flavor 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

Gluon distributions and fits using dipole cross-sections.

I investigate the relationship between the gluon distribution obtained using a dipole model fit to low-x data on F2(x,Q2) and standard gluons obtained from global fits with the collinear factorization theorem at fixed order. I stress the necessity to do fits of this type carefully, and in particular to include the contribution from heavy flavours to the inclusive structure function. I find that the dipole cross-section must be rather steeper than the gluon distribution, which at least partially explains why dipole model fits produce dipole cross-sections growing quite strongly at small x, while DGLAP based fits have valence-like, or even negative, small-x gluons as inputs. However, I also find that the gluon distributions obtained from the dipole fits are much too small to match onto the conventional DGLAP gluons at high Q2 ∼ 50GeV2, where the two approaches should coincide. The main reason for this discrepancy is found to be the large approximations made in converting the dipole cross-sections into structure functions using formulae which are designed only for asymptotically small x. The shortcomings in this step affect the accuracy of the extracted dipole cross-sections in terms of size and shape, and hence also in terms of interpretation, at all scales