Massless scalar Feynman diagrams: five loops and beyond

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 $1\mathrm{/}{k}^2$ or $1\mathrm{/}{k}^4$ 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.