A dilogarithmic integral arising in quantum field theory

TL;DR

This paper demonstrates that a specific dilogarithmic integral from quantum field theory can be expressed using Clausen function values, linking it to hyperbolic volume and related conjectures.

Contribution

It provides a simple, direct proof connecting the integral to Clausen function values, clarifying its relation to hyperbolic geometry and conjectures.

Findings

Integral expressed in terms of Clausen function values

Connection to hyperbolic volume of ideal tetrahedron

Relation to Borwein and Broadhurst conjectures

Abstract

Recently, an interesting dilogarithmic integral arising in quantum field theory has been closed-form evaluated in terms of the Clausen function $\text{Cl}_2(\theta)$ by Coffey [J. Math. Phys.} 49 (2008), 093508]. It represents the volume of an ideal tetrahedron in hyperbolic space and is involved in two intriguing equivalent conjectures of Borwein and Broadhurst. It is shown here, by simple and direct arguments, that this integral can be expressed by the triplet of the Clausen function values which are involved in one of the two above-mentioned conjectures.