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

Theoretical physics

## 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 $$\zeta(\Omega^2\omega_3\omega_0)=\sum_{m> n>0}(-1)^{m+n}/m^3n$$; in three cases it combines with the square of Clausen's $${\rm Cl}_2(\pi/3)=\Im\,\zeta(\Omega\omega_1)=\sum_{n>0}\sin(\pi n/3)/n^2$$. The case with 6 masses involves no further constant; with 5 masses a Deligne-Euler-Zagier sum appears: $${\frak R} \zeta(\Omega^2\omega_3\omega_1)= \sum_{m>n>0}(-1)^m\cos(2\pi n/3)/m^3n$$. 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 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.

## Keywords

Feynman Diagram Finite Part Iterate Integral Zero Mass Remarkable Simplicity
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