Compact analytical form for non-zeta terms in critical exponents at order 1/N-3

We simplify, to a single integral of dilogarithms, the least tractable O(1/ N-3) contribution to the large-hi critical exponent eta Of the non-linear s igma-model, and hence phi(4)-theory, for any spacetime dimensionality, D. I t is the sole generator of irreducible multiple zeta values in epsilon-expa nsions with D = 2 - 2 epsilon, for the sigma-model, and D = 4 - 2 epsilon, for phi(4)-theory. In both cases we confirm results of Broadhurst, Gracey a nd Kreimer (BGK) that relate knots to counterterms. The new compact form is much simpler than that of BGK. It enables us to develop 8 new terms in the epsilon-expansion with D = 3 - 2 epsilon. These involve alternating Euler sums, for which the basis of irreducibles is larger. We conclude that massl ess Feynman diagrams in odd spacetime dimensions share the greater transcen dental complexity of massive diagrams in even dimensions, such as those con tributing to the electron's magnetic moment and the electroweak rho-paramet er. Consequences for the perturbative sector of Chern-Simons theory are dis cussed. (C) 1998 Published by Elsevier Science B.V. All rights reserved.