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

  • D.J. Broadhurst
Theoretical physics


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.


Feynman Diagram Finite Part Iterate Integral Zero Mass Remarkable Simplicity 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • D.J. Broadhurst
    • 1
  1. 1. Physics Department, Open University, Milton Keynes MK7 6AA, UK GB

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