Massive 3-loop Feynman diagrams reducible to SC* primitives of algebras of the sixth root of unity
Physics Department, Open University, Milton Keynes MK7 6AA, UK
Received: 13 March 1998 / Published online: 22 March 1999
In each of the 10 cases with propagators of unit or zero mass, the finite part of the scalar 3-loop tetrahedral vacuum diagram is reduced to 4-letter words in the 7-letter alphabet of the 1-forms and , where is the sixth root of unity. Three diagrams yield only . In two cases combines with the Euler-Zagier sum ; in three cases it combines with the square of Clausen's . The case with 6 masses involves no further constant; with 5 masses a Deligne-Euler-Zagier sum appears: . The previously unidentified term in the 3-loop rho-parameter of the standard model is merely . The remarkable simplicity of these results stems from two shuffle algebras: one for nested sums; the other for iterated integrals. Each diagram evaluates to 10000 digits in seconds, because the primitive words are transformable to exponentially convergent single sums, as recently shown for and , familiar in QCD. Those are SC*(2) constants, whose base of super-fast computation is 2. Mass involves the novel base-3 set SC*(3). All 10 diagrams reduce to SCSC* (2) constants and their products. Only the 6-mass case entails both bases.