Massive 3-loop Ladder Diagrams for Quarkonic Local Operator Matrix Elements

TL;DR

This paper computes complex 3-loop ladder diagrams for quarkonic operator matrix elements using advanced mathematical tools, revealing generalized harmonic sums and their asymptotic behavior relevant for deep-inelastic scattering.

Contribution

It introduces a direct computation method for 3-loop ladder diagrams with Mellin variable N, utilizing Appell functions and modern summation techniques, including generalized harmonic sums.

Findings

Derived explicit expressions for 3-loop ladder diagrams.

Identified generalized harmonic sums beyond nested sums.

Confirmed well-behaved asymptotic behavior for large N.

Abstract

3-loop diagrams of the ladder-type, which emerge for local quarkonic twist-2 operator matrix elements, are computed directly for general values of the Mellin variable $N$ using Appell-function representations and applying modern summation technologies provided by the package {\sf Sigma} and the method of hyperlogarithms. In some of the diagrams generalized harmonic sums with $\xi \in \{1,1/2,2\}$ emerge beyond the usual nested harmonic sums. As the asymptotic representation of the corresponding integrals shows, the generalized sums conspire giving well behaved expressions for large values of $N$. These diagrams contribute to the 3-loop heavy flavor Wilson coefficients of the structure functions in deep-inelastic scattering in the region $Q^2 \gg m^2$.