Massive 3-loop Feynman Diagrams reducible to SC ⋆ primitives of algebras at the sixth root of unity

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 $\\Omega:=dz/z$ and $\\omega_p:=dz/ yield only $\\zeta(Ømega^3ømega_0)=1/90\\pi^4$. In two cases $\\pi^4$ combines with the Euler-Zagier sum $\\zeta(Ømega^2ømega_3ømega_0)=\\sum_{m> n>0}(-1)^{m+n}/m^3n$; in three cases it combines with the square of Clausen's $Cl_2(\\pi/3)=\\Im \\zeta(Ømegaømega_1)=\\sum_{n>0}\\sin(\\pi n/3)/n^2$. The case with 6 masses involves no further constant; with 5 masses a Deligne-Euler-Zagier sum appears: $\\Re \\zeta(Ømega^2ømega_3ømega_1)= 3-loop rho-parameter of the standard model is merely $D_3=6\\zeta(3)-6 Cl_2^2(\\pi/3)-{1/24}\\pi^4$. 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 $\\zeta(3)$ and $\\zeta(5)$, 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$^*(3)\\cup$SC$^* (2)$ constants and their products. Only the 6-mass case entails both bases.