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 -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
.
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