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

D.J. Broadhurst

doi:doi:10.1007/s100529900935

2021 Impact factor 4.991

Particles and Fields

Eur. Phys. J. C 8, 311-333
DOI 10.1007/s100529900935

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

D.J. Broadhurst

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 $\Omega:=dz/z$ and $\omega_p:=dz/ (\lambda^{-p}-z)$ , where $\lambda$ is the sixth root of unity. Three diagrams yield only $\zeta(\Omega^3\omega_0)=\frac1{90}\pi^4$ . In two cases $\pi^4$ combines with the Euler-Zagier sum $\zeta(\Omega^2\omega_3\omega_0)=\sum_{m\gt n\gt}(-1)^{m+n}/m^3n$ ; in three cases it combines with the square of Clausen's ${\rm Cl}_2(\pi/3)=\Im\,\zeta(\Omega\omega_1)=\sum_{n\gt}\sin(\pi n/3)/n^2$ . The case with 6 masses involves no further constant; with 5 masses a Deligne-Euler-Zagier sum appears: $\Re\,\zeta(\Omega^2\omega_3\omega_1)= \sum_{m\gt n\gt}(-1)^m\cos(2\pi n/3)/m^3n$ . The previously unidentified term in the 3-loop rho-parameter of the standard model is merely $D_3=6\zeta(3)-6{\rm Cl}_2^2(\pi/3)-\frac{1}{24}\pi^4$ . The remarkable simplicity of these results stems from two shuffle algebras: one for nested sums; the other for iterated integrals. Each diagram evaluates to 10000 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.