Calculating Massive 3-loop Graphs for Operator Matrix Elements by the Method of Hyperlogarithms

TL;DR

The paper extends the method of hyperlogarithms to massive 3-loop Feynman diagrams with local operator insertions, enabling the calculation of massive operator matrix elements relevant for deep-inelastic scattering. By encoding operator insertions with generating functions and employing a generating-function–driven, linear-reducibility approach, the authors obtain both fixed-N moments and general-N representations for complex topologies, including Benz and V-type diagrams, as well as crossed boxes. The work reveals new nested sums with binomial weights and root-valued iterated integrals, provides asymptotic representations for complex-N, and demonstrates the feasibility and limitations of the method through extensive moment calculations. These results broaden the computational toolkit for heavy-quark Wilson coefficients at three loops and inform Mellin-inversion strategies, albeit with significant computational resource demands and the appearance of constants beyond standard multiple zeta values.

Abstract

We calculate convergent 3-loop Feynman diagrams containing a single massive loop equipped with twist $τ=2$ local operator insertions corresponding to spin $N$. They contribute to the massive operator matrix elements in QCD describing the massive Wilson coefficients for deep-inelastic scattering at large virtualities. Diagrams of this kind can be computed using an extended version to the method of hyperlogarithms, originally being designed for massless Feynman diagrams without operators. The method is applied to Benz- and $V$-type graphs, belonging to the genuine 3-loop topologies. In case of the $V$-type graphs with five massive propagators new types of nested sums and iterated integrals emerge. The sums are given in terms of finite binomially and inverse binomially weighted generalized cyclotomic sums, while the 1-dimensionally iterated integrals are based on a set of $\sim 30$ square-root valued letters. We also derive the asymptotic representations of the nested sums and present the solution for $N \in \mathbb{C}$. Integrals with a power-like divergence in $N$--space $\propto a^N, a \in \mathbb{R}, a > 1,$ for large values of $N$ emerge. They still possess a representation in $x$--space, which is given in terms of root-valued iterated integrals in the present case. The method of hyperlogarithms is also used to calculate higher moments for crossed box graphs with different operator insertions.

Figures (9)

• Figure 1: The operators expressed in terms of the graph-polynomial $\Psi_G$ and different Dodgson polynomials DODGSONBrown:2009taBrown:2010bw of the graph $\tilde{G}$.
• Figure 2: A $V$-topology diagram.
• Figure 3: The 3-loop Benz diagram for $I_1(N)$, Eq. (\ref{['EX1a']}).
• Figure 4: The 3-loop Benz diagram for $I_2(N)$, Eq. (\ref{['EX1b']}).
• Figure 5: The 3-loop Benz diagram for $I_3(N)$, Eq. (\ref{['EX1c']}).