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

@article{Broadhurst1999Massive3F,
  title={Massive 3-loop Feynman diagrams reducible to SC\$^*\$ primitives of algebras of the sixth root of unity},
  author={D. Broadhurst},
  journal={The European Physical Journal C - Particles and Fields},
  year={1999},
  volume={8},
  pages={311-333}
}
  • D. Broadhurst
  • Published 1999
  • Physics, Mathematics
  • The European Physical Journal C - Particles and Fields
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… Expand
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