DOI 10.1007/s100529900935
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
Abstract
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 SC
SC*
(2) constants and their products. Only the 6-mass case entails both bases.
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