{"paper":{"title":"Massive 3-loop Feynman diagrams reducible to SC* primitives of algebras of the sixth root of unity","license":"","headline":"","cross_cats":["hep-ph","math.CA"],"primary_cat":"hep-th","authors_text":"D. J. Broadhurst","submitted_at":"1998-03-11T09:33:09Z","abstract_excerpt":"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 $\\zeta(\\Omega^3\\omega_0)=1/90\\pi^4$. 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 $Cl_2(\\pi/3)=\\Im \\zeta(\\Omega\\omega_1)=\\sum_{n>0}\\sin(\\pi n/3)/n^2$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"hep-th/9803091","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}