Alternative Evaluation of a ln tan Integral Arising in Quantum Field Theory

Abstract

A certain dilogarithmic integral I 7 turns up in a number of contexts including Feynman diagram calculations, volumes of tetrahedra in hyperbolic geometry, knot theory, and conjectured relations in analytic number theory. We provide an alternative explicit evaluation of a parameterized family of integrals containing this particular case. By invoking the Bloch-Wigner form of the dilogarithm function, we produce an equivalent result, giving a third evaluation of I 7 . We also alternatively formulate some conjectures which we pose in terms of values of the specific Clausen function Cl 2 .