A dilogarithmic 3-dimensional ising tetrahedron
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
In 3 dimensions, the Ising model is in the same universality class as -theory, whose massive 3-loop tetrahedral diagram, , 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 ,with and .This empirical relation has been checked at 1,000-digit precision and readily yields 50,000 digits of ,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 , in the manner that 4-dimensional results build on that for .