# Three-loop massive tadpoles and polylogarithms through weight six

## TL;DR

This paper evaluates complex three-loop vacuum diagrams with massive particles using polylogarithms up to weight six, and constructs a basis of irrational constants involving harmonic polylogarithms with specific arguments.

## Contribution

It introduces a method to express three-loop massive tadpole diagrams in terms of polylogarithms and constructs a basis of irrational constants involving harmonic polylogarithms.

## Key findings

- Explicit expressions for three-loop massive tadpoles in polylogarithms up to weight six.
- A constructed basis of irrational constants involving harmonic polylogarithms at specific roots of unity.
- Enhanced understanding of the mathematical structure of multi-loop Feynman diagrams.

## 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 $e^{k i \pi/3}$.