Calculating three loop ladder and Vtopologies for massive operator matrix elements by computer algebra
Abstract
Three loop ladder and Vtopology diagrams contributing to the massive operator matrix element A_{Qg} 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 ε. Given these representations, the desired Laurent series expansions in ε can be obtained with the help of our computer algebra toolbox. Here we rely on generalized hypergeometric functions and MellinBarnes representations, on difference ring algorithms for symbolic summation, on an optimized version of the multivariate AlmkvistZeilberger 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 productsum expressions. This structural result is based on new results of our difference ring theory. In the cases discussed we deal with iterative sum and integralsolutions over general alphabets. The final results are expressed in terms of special sums, forming quasishuffle 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 Vtopologies.
 Publication:

Computer Physics Communications
 Pub Date:
 May 2016
 DOI:
 10.1016/j.cpc.2016.01.002
 arXiv:
 arXiv:1509.08324
 Bibcode:
 2016CoPhC.202...33A
 Keywords:

 Massive 3loop integrals in QCD;
 Master integrals;
 Automation and method of differential equations;
 AlmkvistZeilberger algorithm;
 MellinBarnes method;
 High Energy Physics  Phenomenology;
 Computer Science  Symbolic Computation;
 High Energy Physics  Theory;
 Mathematical Physics
 EPrint:
 110 pages Latex, 4 Figures