Three-loop massive tadpoles and polylogarithms through weight six

TL;DR

This work advances the analytic evaluation of three-loop massive vacuum bubbles by expressing results up to weight six in terms of polylogarithms, facilitated by constructing a minimal basis of harmonic polylogarithms at sixth roots of unity. Using dimensional regularization and a differential-equation approach, the authors reduce all relevant diagrams to a fixed set of master integrals and reconstruct their epsilon expansions via PSLQ in the ${\cal A}_{H(\omega)}$ basis, with several integrals given in closed Gamma-function form. The results include explicit pole structures and finite parts for diagrams BN, BN$_1$, E$_3$, D-type, and the two-loop T$_{111}$, alongside high-precision numerical values, providing a robust, universal framework for single-scale three-loop tadpoles and related form-factor calculations. The methods and basis are potentially applicable to higher-loop calculations and other single-scale diagrams, offering a reproducible route to weight-six polylogarithmic representations in quantum field theory.

Abstract

We evaluate the three-loop massive vacuum bubble diagrams in terms of polylogarithms up to weight six. We also construct the basis of irrational constants being harmonic polylgarithms of arguments $e^{k i π/3}$.

Figures (2)

• Figure 1: Full set of non-factorizing master integrals. Solid and dashed lines correspond to massive and massless scalar propagators, respectively.
• Figure 2: Two-loop and one-loop master integrals. The line coding is the same as in figure \ref{['fig:tad1234']}.