English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In each of the 10 cases with propagators of unit or zero mass, the finitepart of the scalar 3-loop tetrahedral vacuum diagram is reduced to 4-letterwords 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 diagramsyield only $\zeta(\Omega^3\omega_0)=1/90\pi^4$. In two cases $\pi^4$ combineswith 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$Cl_2(\pi/3)=\Im \zeta(\Omega\omega_1)=\sum_{n>0}\sin(\pi n/3)/n^2$. The casewith 6 masses involves no further constant; with 5 masses aDeligne-Euler-Zagier sum appears: $\Re \zeta(\Omega^2\omega_3\omega_1)=\sum_{m>n>0}(-1)^m\cos(2\pi n/3)/m^3n$. The previously unidentified term in the3-loop rho-parameter of the standard model is merely $D_3=6\zeta(3)-6Cl_2^2(\pi/3)-{1/24}\pi^4$. The remarkable simplicity of these results stemsfrom two shuffle algebras: one for nested sums; the other for iteratedintegrals. Each diagram evaluates to 10 000 digits in seconds, because theprimitive words are transformable to exponentially convergent single sums, asrecently shown for $\zeta(3)$ and $\zeta(5)$, familiar in QCD. Those areSC$^*(2)$ constants, whose base of super-fast computation is 2. Mass involvesthe 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.Comment: 41 pages, LaTe

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