A. Behring

Calculating three loop ladder and V-topologies for massive operator matrix elements by computer algebra

Abstract Three loop ladder and V -topology diagrams contributing to the massive operator matrix element A Q g are calculated. The corresponding objects can all be expressed in terms of nested sums and recurrences depending on the Mellin variable N and the dimensional parameter e . Given these representations, the desired Laurent series expansions in e can be obtained with the help of our computer algebra toolbox. Here we rely on generalized hypergeometric functions and Mellin–Barnes representations, on difference ring algorithms for symbolic summation, on an optimized version of the multivariate Almkvist–Zeilberger algorithm for symbolic integration, and on new methods to calculate Laurent series solutions of coupled systems of differential equations. The solutions can be computed for general coefficient matrices directly for any basis also performing the expansion in the dimensional parameter in case it is expressible in terms of indefinite nested product–sum expressions. This structural result is based on new results of our difference ring theory. In the cases discussed we deal with iterative sum- and integral-solutions over general alphabets. The final results are expressed in terms of special sums, forming quasi-shuffle algebras, such as nested harmonic sums, generalized harmonic sums, and nested binomially weighted (cyclotomic) sums. Analytic continuations to complex values of N are possible through the recursion relations obeyed by these quantities and their analytic asymptotic expansions. The latter lead to a host of new constants beyond the multiple zeta values, the infinite generalized harmonic and cyclotomic sums in the case of V -topologies.

The 3-loop non-singlet heavy flavor contributions and anomalous dimensions for the structure function F2(x,Q2) and transversity

We calculate the massive flavor non-singlet Wilson coefficient for the heavy flavor contributions to the structure function F2(x,Q 2 ) in the asymptotic region Q 2 ≫ m 2 and the associated operator matrix element A (3),NS qq,Q (N) to 3-loop order in Quantum Chromodynamics at general values of the Mellin variable N. This matrix element is associated to the vector current and axial vector current for the even and the odd moments N, respectively. We also calculate the corresponding operator matrix elements for transversity, compute the contributions to the 3-loop anomalous dimensions to O(NF) and compare to results in the literature. The 3-loop matching of the flavor non-singlet distribution in the variable flavor number scheme is derived. All results can be expressed in terms of nested harmonic sums in N space and harmonic polylogarithms inx-space. Numerical results are presented for the non-singlet charm quark contribution to F2(x,Q 2 ).

The 3-loop pure singlet heavy flavor contributions to the structure function F2(x,Q2) and the anomalous dimension

The pure singlet asymptotic heavy flavor corrections to 3–loop order for the deep–inelastic scattering structure function F2(x,Q 2 ) and the corresponding transition matrix element A (3),PS Qq in the variable flavor number scheme are computed. In Mellin-N space these inclusive quantities depend on generalized harmonic sums. We also recalculate the complete 3-loop pure singlet anomalous dimension for the first time. Numerical results for the Wilson coefficients, the operator matrix element and the contributionto the structure function F2(x,Q 2 ) are presented.

The logarithmic contributions to the O(\alpha _s^3) asymptotic massive Wilson coefficients and operator matrix elements in deeply inelastic scattering

We calculate the logarithmic contributions to the massive Wilson coefficients for deep-inelastic scattering in the asymptotic region \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}

_s^3

Q^2 \gg m^2

xF_3(x,Q^2)

\end{document}Q2≫m2 to 3-loop order in the fixed flavor number scheme and present the corresponding expressions for the massive operator matrix elements needed in the variable flavor number scheme. Explicit expressions are given in Mellin \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}

The 3-loop non-singlet heavy flavor contributions to the structure function g1(x,Q2) at large momentum transfer

N