Parton densities and structure functions at next-to-next-to-leading order and beyond

We summarize recent results on the evolution of unpolarized parton densities and deep-inelastic structure functions in massless perturbative QCD. Due to last year's extension of the integer-moment calculations of the three-loop splitting functions, the NNLO evolution of the parton distributions can now be performed reliably at momentum fractions x >= 10^-4, facilitating a considerably improved theoretical accuracy of their extraction from data on deep-inelastic scattering. The NNLO corrections are not dominated, at relevant values of x, by their leading small-x terms. At large x the splitting-function series converges very rapidly, hence, employing results on the three-loop coefficient functions, the structure functions can be analysed at N^3LO for x > 10^-2. The resulting values for alpha_s do not significantly change beyond NNLO, their renormalization scale dependence reaches about +-1% at N^3LO.Comment: 10 pages, LaTeX, 6 figures. Talk presented at the workshops `New Trends in HERA Physics 2001', Ringberg Castle (Germany), June 2001 and `DIS 2001', Bologna (Italy), April 2001. To appear, slightly shortened in the latter case, in the proceeding

Less Singular Terms and Small x Evolution in a Soluble Model

We calculate the effect of the less singular terms at small x on the evolution of the coefficient function in \phi^3 theory in six dimensions, which result from a complete solution of the ladder equation. Scale-invariant next-to-leading order contributions are also studied. We show that the small x approximation does not deliver the dominant contributions.Comment: 7 pages LATEX including 2 eps-file

NNLO evolution of deep-inelastic structure functions: the singlet case

We study the next-to-next-to-leading order (NNLO) evolution of flavour singlet parton densities and structure functions in massless perturbative QCD. Present information on the corresponding three-loop splitting functions is used to derive parametrizations of these quantities, including Bjorken-x dependent estimates of their residual uncertainties. Compact expressions are also provided for the exactly known, but in part rather lengthy two-loop singlet coefficient functions. The size of the NNLO corrections and their effect on the stability under variations of the renormalization and mass-factorizations scales are investigated. Except for rather low values of the scales, the residual uncertainty of the three-loop splitting functions does not lead to relevant effects for x > 10^-3. Inclusion of the NNLO contributions considerably reduces the theoretical uncertainty of determinations of the quark and gluon densities from deep-inelastic structure functions.Comment: 31 pages, LaTeX, 14 eps-figure