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

TL;DR

The paper develops and applies a comprehensive computer-algebra framework to compute massive three-loop ladder and V-topologies contributing to the OME $A_{Qg}$, expressing results as Laurent series in $\varepsilon$ with coefficients given by indefinite nested sums in Mellin space. It integrates generalized hypergeometric and Mellin-Barnes representations, difference-ring summation, and a refined Almkvist-Zeilberger approach to derive recurrences, uncouple coupled systems, and obtain analytic continuations in $N$. Master integrals are computed through Beta-function, hypergeometric, MB, and differential-equation methods, then assembled into final results in terms of harmonic, generalized harmonic, cyclotomic, and binomial sums, including new constants beyond classical multiple zeta values. The framework is automated and scalable, enabling the explicit $N$-dependence and asymptotic behavior of the diagrams, with potential impact on NNLO analyses of PDFs, $\alpha_s$, and heavy-quark masses. Overall, the work demonstrates a powerful, largely automatic pipeline for tackling complex single-scale multi-loop integrals via a synergy of modern symbolic-sum/integration techniques and differential-equation methods.

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.

Figures (4)

• Figure 1: Sample of ladder diagrams.
• Figure 2: Diagrams with a central triangle.
• Figure 3: Representation showing the origin of the two terms appearing in the $q\bar{q}gg$-operator insertion. The gluon momenta is taken incoming.
• Figure 4: A depiction of the two terms appearing in diagram 12 of Figure \ref{['samplediagrams']}. The internal momenta have been chosen so that the operator insertion term is the same in both diagrams on the right hand side. The bottom middle propagator then differs in each case.

Theorems & Definitions (7)

• Remark 4.1
• Remark 4.2
• Remark 4.3
• Remark 4.4
• Remark 4.5
• Remark 4.6
• Remark 5.1