A dilogarithmic 3-dimensional Ising tetrahedron

TL;DR

This paper uncovers an explicit dilogarithmic formula for the 3D Ising model's tetrahedral diagram, linking it to polylogarithms and providing highly precise numerical validation.

Contribution

It presents the first analytical expression for the 3D Ising tetrahedral diagram using polylogarithms, connecting it to known mathematical constants and demonstrating high-precision numerical agreement.

Findings

Derived an explicit dilogarithmic formula for the diagram.

Achieved 1,000-digit numerical validation of the relation.

Extended the understanding of 3D Ising model integrals via polylogarithms.

Abstract

In 3 dimensions, the Ising model is in the same universality class as $\phi^4$-theory, whose massive 3-loop tetrahedral diagram, $C^{Tet}$, was of an unknown analytical nature. In contrast, all single-scale 4-dimensional tetrahedra were reduced, in hep-th/9803091, to special values of exponentially convergent polylogarithms. Combining dispersion relations with the integer-relation finder PSLQ, we find that $C^{Tet}/2^{5/2} = Cl_2(4\alpha) - Cl_2(2\alpha)$, with $Cl_2(\theta):=\sum_{n>0}\sin(n\theta)/n^2$ and $\alpha:=\arcsin\frac13$. This empirical relation has been checked at 1,000-digit precision and readily yields 50,000 digits of $C^{Tet}$, after transformation to an exponentially convergent sum, akin to those studied in math.CA/9803067. It appears that this 3-dimensional result entails a polylogarithmic ladder beginning with the classical formula for $\pi/\sqrt2$, in the manner that 4-dimensional results build on that for $\pi/\sqrt3$.