Massive 3-loop Feynman diagrams reducible to SC* primitives of algebras of the sixth root of unity

TL;DR

This work reveals that all ten single-scale massive 3-loop tetrahedral diagrams reduce to a remarkably small set of constants derived from polylogarithms at sixth-root-of-unity arguments. By combining a dispersive calculation with two shuffle algebras (nested sums and iterated integrals), the finite parts decompose into a basis of SC*(2) and SC*(3) constants and their products, with the mass configuration determining which bases appear. Key findings include exact expressions for D3 in terms of ζ(3) and Cl2(π/3), star-triangle-line relations among three-mass diagrams, and the absence of new primitives in many topologies; the totally massive case links back to Deligne-type primitives. The study also develops comprehensive algebraic machinery for reducing two-, three-, and four-letter words and shows that certain constants admit exponentially convergent single-sum representations, enabling ultra-fast high-precision computation. Overall, the paper uncovers a deep structure linking Feynman diagram topology to a small, highly computable set of constants derived from roots of unity, with implications for QCD rho-parameter calculations and beyond.

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 $Ω:=dz/z$ and $ω_p:=dz/ (λ^{-p}-z)$, where $λ$ is the sixth root of unity. Three diagrams yield only $ζ(Ω^3ω_0)=1/90π^4$. In two cases $π^4$ combines with the Euler-Zagier sum $ζ(Ω^2ω_3ω_0)=\sum_{m> n>0}(-1)^{m+n}/m^3n$; in three cases it combines with the square of Clausen's $Cl_2(π/3)=\Im ζ(Ωω_1)=\sum_{n>0}\sin(πn/3)/n^2$. The case with 6 masses involves no further constant; with 5 masses a Deligne-Euler-Zagier sum appears: $\Re ζ(Ω^2ω_3ω_1)= \sum_{m>n>0}(-1)^m\cos(2πn/3)/m^3n$. The previously unidentified term in the 3-loop rho-parameter of the standard model is merely $D_3=6ζ(3)-6 Cl_2^2(π/3)-{1/24}π^4$. 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 $ζ(3)$ 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$^*(3)\cup$SC$^* (2)$ constants and their products. Only the 6-mass case entails both bases.