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

Understand

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.