Calculating Three Loop Ladder and V-Topologies for Massive Operator Matrix Elements by Computer Algebra

TL;DR

This paper develops a computer algebra framework to compute three-loop ladder and V-topology diagrams for massive operator matrix elements, expressing results in nested sums and recurrences for arbitrary Mellin variable N.

Contribution

It introduces new algorithms and methods for symbolic summation, integration, and Laurent series expansion of complex Feynman diagrams in quantum field theory.

Findings

Explicit calculations of three-loop diagrams for massive operator matrix elements.

Representation of results in terms of nested harmonic sums and generalized sums.

New constants beyond multiple zeta values identified in the process.

Abstract

Three loop ladder and $V$-topology 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 $\varepsilon$. Given these representations, the desired Laurent series expansions in $\varepsilon$ 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.