On a three-dimensional symmetric Ising tetrahedron, and contributions to the theory of the dilogarithm and Clausen functions

TL;DR

This paper proves a conjecture related to a symmetric tetrahedral Feynman diagram in quantum field theory, revealing new connections between special functions, hyperbolic geometry, and number theory.

Contribution

It establishes a convergent sum for the symmetric tetrahedral diagram and relates it to Clausen functions, extending the theory of dilogarithm and special functions in mathematical physics.

Findings

Proved a sum for the symmetric tetrahedral diagram C(1,1)

Linked Feynman integrals to hyperbolic volumes and special functions

Extended the mathematical theory of dilogarithm and Clausen functions

Abstract

Perturbative quantum field theory for the Ising model at the three-loop level yields a tetrahedral Feynman diagram C(a,b) with masses a and b and four other lines with unit mass. The completely symmetric tetrahedron C^Tet \equiv C(1,1) has been of interest from many points of view, with several representations and conjectures having been given in the literature. We prove a conjectured exponentially fast convergent sum for C(1,1), as well as a previously empirical relation for C(1,1) as a remarkable difference of Clausen function values. Our presentation includes Propositions extending the theory of the dilogarithm Li_2 and Clausen Cl_2 functions, as well as their relation to other special functions of mathematical physics. The results strengthen connections between Feynman diagram integrals, volumes in hyperbolic space, number theory, and special functions and numbers, specifically including dilogarithms, Clausen function values, and harmonic numbers.