EPJ

2021 Impact factor 4.991
EPJ C - Particles and Fields
Particles and Fields
Eur. Phys. J. C 8, 363-366
DOI 10.1007/s100529900983

A dilogarithmic 3-dimensional ising tetrahedron

D.J. Broadhurst

Physics Department, Open University, Milton Keynes MK7 6AA, UK (e-mail: D.Broadhurst@open.ac.uk)

Received: 6 May 1998 / Published online: 22 March 1999

Abstract
In 3 dimensions, the Ising model is in the same universality class as $\phi^4$-theory, whose massive 3-loop tetrahedral diagram, $C^{\rm Tet}$, 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 $C^{\rm Tet}/2^{5/2} = {\rm Cl}_2 (4\alpha ) - {\rm Cl}_2 (2\alpha )$,with ${\rm Cl}_2 (\theta ) := \sum_{n \gt 0} \sin (n \theta )/n^2$and $\alpha := \arcsin \frac{1}{3}$.This empirical relation has been checked at 1,000-digit precision and readily yields 50,000 digits of $C^{\rm Tet}$,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 $\pi/\sqrt 3$.

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