TOPICS
Search

Tetrahedral Vacuum Feynman Diagram

DOWNLOAD Mathematica NotebookDownload Wolfram Notebook

In each of the ten cases with zero or unit mass, the finite part of the scalar 3-loop tetrahedral vacuum Feynman diagram reduces to four-letter "words" that represent iterated integrals in an alphabet of seven "letters" (Broadhurst 1998, Bailey et al. 2007). There are 7^4 such four-letter words, but only two of these are primitive terms. Defining

U=int_0^1(dx_1)/(x_1)int_0^(x_1)(dx_2)/(x_2)int_0^(x_2)(dx_3)/(-(1+x_3))×int_0^(x_3)(dx_4)/(1-x_4)sum_(0<k<j)((-1)^(j+k))/(j^3k)
(1)
V=sum_(0<k<j)((-1)^jcos(2/3pik))/(j^3k)
(2)
C=sum_(k=1)^(infty)(sin(2/3pik))/(k^2)
(3)

gives

V_1=6zeta(3)+3zeta(4)
(4)
V_(2A)=6zeta(3)-5zeta(4)
(5)
V_(2N)=6zeta(3)-(13)/2zeta(4)-8U
(6)
V_(3T)=6zeta(3)-9zeta(4)
(7)
V_(3S)=6zeta(3)-(11)/2zeta(4)-4C^2
(8)
V_(3L)=6zeta(3)-(15)/4zeta(4)-6C^2
(9)
V_(4A)=6zeta(3)-(77)/(12)zeta(4)-6C^2
(10)
V_(4N)=6zeta(3)-14zeta(4)-16U
(11)
V_5=6zeta(3)-(469)/(27)zeta(4)+3/2C^2-16V
(12)
V_6=6zeta(3)-13zeta(4)-8U-4C^2
(13)

(Bailey et al. 2007, p. 41).

See also

Multivariate Zeta Function

Explore with Wolfram|Alpha

References

Bailey, D. H.; Borwein, J. M.; Calkin, N. J.; Girgensohn, R.; Luke, D. R.; and Moll, V. H. "Quantum Field Theory." §2.3.5 in Experimental Mathematics in Action. Wellesley, MA: A K Peters, pp. 40-41, 2007.Broadhurst, D. J. "Massive 3-Loop Feynman Diagrams Reducible to SC^* Primitives of Algebras of the Sixth Root of Unity." March 11, 1998. http://arxiv.org/abs/hep-th/9803091.

Referenced on Wolfram|Alpha

Tetrahedral Vacuum Feynman Diagram

Cite this as:

Weisstein, Eric W. "Tetrahedral Vacuum Feynman Diagram." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/TetrahedralVacuumFeynmanDiagram.html

Subject classifications