1998/03/11 by D. J. Broadhurst, D.J. Broadhurst · 4 citations
Mathematics · Physics and Astronomy · #Algebraic and Geometric Analysis #Base (topology) #Diagram #Feynman diagram #Iterated function #Noncommutative and Quantum Gravity Theories #Propagator #Quantum and Classical Electrodynamics #Scalar (mathematics) #Set (abstract data type) #Square root #hep-ph #hep-th #math.CA
paper · pdf · doi:10.1007/s100529900935
published as Eur.Phys.J.C8:311-333,1999 · 41 pages, LaTeX
arxiv created 1998/03/11 · openalex publication_date 1999/04/01 · arxiv updated 2011/09/13 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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 Ω:=dz/z and ωp:=dz/ (λ-p-z), where λ is the sixth root of unity. Three diagrams yield only ζ(Ω3ω0)=1/90π4. In two cases π4 combines with the Euler-Zagier sum ζ(Ω2ω3ω0)=∑m> n>0(-1)m+n/m3n; in three cases it combines with the square of Clausen's Cl2(π/3)=\Im ζ(Ωω1)=∑n>0sin(πn/3)/n2. The case with 6 masses involves no further constant; with 5 masses a Deligne-Euler-Zagier sum appears: \Re ζ(Ω2ω3ω1)= ∑m>n>0(-1)mcos(2πn/3)/m3n. The previously unidentified term in the 3-loop rho-parameter of the standard model is merely D3=6ζ(3)-6 Cl22(π/3)-1/24π4. The remarkable simplicity of these results stems from two shuffle algebras: one for nested sums; the other for iterated integrals. Each diagram evaluates to 10 000 digits in seconds, because the primitive words are transformable to exponentially convergent single sums, as recently shown for ζ(3) and ζ(5), 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^*(3)∪SC^* (2) constants and their products. Only the 6-mass case entails both bases.