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A systematic study is performed on the finite harmonic sums up to level four. These sums form the general basis for the Mellin transforms of all individual functions f i (x) of the momentum fraction x emerging in the quantities of massless QED and QCD up to two-loop order, as the unpolarized and polarized splitting functions, coefficient functions, and hard scattering cross sections for space and timelike momentum transfer. The finite harmonic sums are calculated explicitly in the linear representation. Algebraic relations connecting these sums are derived to obtain representations based on a reduced set of basic functions. The Mellin transforms of all the corresponding Nielsen functions are calculated. ͓S0556-2821͑99͒04411-2͔

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In this talk results from a new QCD analysis in leading (LO) and next-toleading (NLO) order are presented. New parametrizations of the polarized quark and gluon densities are derived together with parametrizations of their fully correlated 1σ error bands. Furthermore the value of α s (M 2 Z ) is determined. Finally a number of low moments of the polarized parton densities are compared with results from lattice simulations. All details of the analysis are given in Ref. [1].

The computation of Feynman integrals in massive higher order perturbative calculations in renormalizable Quantum Field Theories requires extensions of multiply nested harmonic sums, which can be generated as real representations by Mellin transforms of Poincaré-iterated integrals including denominators of higher cyclotomic polynomials. We derive the cyclotomic harmonic polylogarithms and harmonic sums and study their algebraic and structural relations. The analytic continuation of cyclotomic harmonic sums to complex values of N is performed using analytic representations. We also consider special values of the cyclotomic harmonic polylogarithms at argument x = 1, resp., for the cyclotomic harmonic sums at N → ∞, which are related to colored multiple zeta values, deriving various of their relations, based on the stuffle and shuffle algebras and three multiple argument relations. We also consider infinite generalized nested harmonic sums at roots of unity which are related to the infinite cyclotomic harmonic sums. Basis representations are derived for weight w = 1,2 sums up to cyclotomy l = 20.Dedicated to Martinus Veltman on the occasion of his 80th birthday.

We present a determination of parton distribution functions (ABM11) and the strong coupling constant s at next-to-leading order and next-to-next-to-leading order (NNLO) in QCD based on world data for deep-inelastic scattering and fixed-target data for the Drell-Yan process. The analysis is performed in the fixed-flavor number scheme for n f ¼ 3, 4, 5 and uses the MS scheme for s and the heavy-quark masses. At NNLO we obtain the value s ðM Z Þ ¼ 0:1134 AE 0:0011. The fit results are used to compute benchmark cross sections at hadron colliders to NNLO accuracy and to compare to data from the LHC.

We calculate the O(α 3 s ) heavy flavor contributions to the Wilson coefficients of the structure function F 2 (x, Q 2 ) and the massive operator matrix elements (OMEs) for the twist-2 operators of unpolarized deeply inelastic scattering in the region Q 2 ≫ m 2 . The massive Wilson coefficients are obtained as convolutions of massive OMEs and the known light flavor Wilson coefficients. We also compute the massive OMEs which are needed to evaluate heavy flavor parton distributions in the variable flavor number scheme (VFNS) to 3-loop order. All contributions to the Wilson coefficients and operator matrix elements but the genuine constant terms at O(α 3 s ) of the OMEs are derived in terms of quantities, which are known for general values in the Mellin variable N . For the operator matrix elements A /µ 2 , m 2 i /µ 2 ) this approximation is only valid for Q 2 /m 2 > ∼ 800, [29]. The 3-loop corrections were calculated in Ref. [30].3 massive operator matrix elements A jk contributing to the heavy flavor Wilson coefficients for the structure function F 2 (x, Q 2 ) in the region Q 2 /m 2 > ∼ 10 to 3-loop order for fixed moments of the Mellin variable N. In case of the flavor non-singlet (NS) contributions, we also present the odd moments of the −-projection. We further calculate the operator matrix elements, which are required to define heavy quark densities in the VFNS [31]. Due to renormalization, higher order contributions in ε to corrections of lower order in a s , cf. [29,[31][32][33][34][35], and other renormalization terms, such as the anomalous dimensions and the expansion coefficients of the QCD β-function and mass anomalous dimensions, contribute. For these reasons, the present calculation yields also the moments of the complete 2-loop anomalous dimensions and the terms ∝ T F of the 3-loop anomalous dimensions γ ij (N). In the pure singlet (PS) case, γ +,PS qq (N), and for γ qg (N), these are the complete anomalous dimensions given in [18,19], to which we agree. Since the present calculation is completely independent by method, formalism, and codes, it provides a check on the previous results. Except for the constant part of the unrenormalized heavy flavor operator matrix elements, we obtain the heavy quark Wilson coefficients in the asymptotic region for all values of the Mellin variable N. The analytic continuation of these expressions to complex values of N can be performed with the help of the representations in [36] and those given for the anomalous dimensions and massless Wilson coefficients in [18,19,26].The paper is organized as follows. In Section 2, a brief outline of the basic formalism is given. The renormalization of the different massive operator matrix elements is described in Section 3. In Section 4, we present details on the unrenormalized and renormalized operator matrix elements. Technical details of the calculation and the main results are discussed in Section 5. Depending on the CPU time and storage size required, the moments up to N = 10, 12, and 14 of the different operator matrix eleme...

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