A dilogarithmic 3-dimensional Ising tetrahedron

Abstract

In 3 dimensions, the Ising model is in the same universality class as ${\varphi }^4$-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 $C^{Tet}/2^{5/2}=C{l}_2(4\alpha )-C{l}_2(2\alpha )$, with $C{l}_2(\theta ):={\sum }_{n>0}\sin (n\theta )/n^2$ and $\alpha :=\arcsin \frac{1}{3}$. 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 $\pi /\sqrt{2}$, in the manner that 4-dimensional results build on that for $\pi /\sqrt{3}$.

Comments: 8 pages, LaTeX; Eq(25) simplified; Eqs(27,33) and refs[3,18] added