Massless scalar Feynman diagrams: five loops and beyond

Understand

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.