Massless scalar Feynman diagrams: five loops and beyond

Abstract

Several powerful techniques for evaluating massless scalar Feynman diagrams are developed, viz: the solution of recurrence relations to evaluate diagrams with arbitrary numbers of loops in $n=4-2\omega$ dimensions; the discovery and use of symmetry properties to restrict and compute Taylor series in $\omega$; the reduction of triple sums over Chebyshev polynomials to products of Riemann zeta functions; the exploitation of conformal invariance to avoid four-dimensional Racah coefficients. As an example of the power of these techniques we evaluate all of the 216 diagrams, with 5 loops or less, which give finite contributions of order $1/k^2$ or $1/k^4$ to a propagator of momentum $k$ in massless four-dimensional scalar field theories. Remarkably, only 5 basic numbers are encountered: $\zeta (3)$, $\zeta (5)$, $\zeta (7)$, $\zeta (9)$ and the value of the most symmetrical diagram, which is calculated to 14 significant figures. It is conceivable that these are the only irrationals appearing in 6-loop beta functions. En route to these results we uncover and only partially explain many remarkable relations between diagrams.

report of The Open University, Milton Keynes, England (UK), 1985, copy provided by John Gracey, http://cds.cern.ch/record/164890, LaTeX reproduction by Erik Panzer

Cited by in corpus (4)

• Minimally subtracted six loop renormalization of $O(n)$-symmetric ${\varphi }^4$ theory and critical exponents
• Embedding Feynman Integral (Calabi-Yau) Geometries in Weighted Projective Space
• Hepp's bound for Feynman graphs and matroids
• Glue-and-Cut at Five Loops