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

Physics – High Energy Physics – High Energy Physics - Theory

Scientific paper

Rate now

  [ 0.00 ] – not rated yet Voters 0   Comments 0

Details

41 pages, LaTeX

Scientific paper

10.1007/s100529900935

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 $Cl_2(\pi/3)=\Im \zeta(\Omega\omega_1)=\sum_{n>0}\sin(\pi n/3)/n^2$. The case with 6 masses involves no further constant; with 5 masses a Deligne-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 the 3-loop rho-parameter of the standard model is merely $D_3=6\zeta(3)-6 Cl_2^2(\pi/3)-{1/24}\pi^4$. 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.

No associations

LandOfFree

Say what you really think

Search LandOfFree.com for scientists and scientific papers. Rate them and share your experience with other people.

Rating

Massive 3-loop Feynman diagrams reducible to SC* primitives of algebras of the sixth root of unity does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Massive 3-loop Feynman diagrams reducible to SC* primitives of algebras of the sixth root of unity, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Massive 3-loop Feynman diagrams reducible to SC* primitives of algebras of the sixth root of unity will most certainly appreciate the feedback.

Rate now

     

Profile ID: LFWR-SCP-O-516579

  Search
All data on this website is collected from public sources. Our data reflects the most accurate information available at the time of publication.