A dilogarithmic 3-dimensional Ising tetrahedron

In 3 dimensions, the Ising model is in the same universality class as φ-theory, whose massive 3-loop tetrahedral diagram, C, 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/2 = Cl2(4α)− Cl2(2α), with Cl2(θ) := ∑ n>0 sin(nθ)/n 2 and α := arcsin 1 3 . This empirical relation has been checked at 1,000-digit precision and readily yields 50,000 digits of C, 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 π/ √ 2, in the manner that 4-dimensional results build on that for π/ √ 3. ) [email protected]; http://physics.open.ac.uk/ d̃broadhu