Compact analytical form for non-zeta terms in critical exponents at order 1/N3

We simplify, to a single integral of dilogarithms, the least tractable O(1/N3) contribution to the large-N critical exponent η of the non-linear σ-model, and hence φ4-theory, for any spacetime dimensionality, D. It is the sole generator of irreducible multiple zeta values in ε-expansions with D=2−2ε, for the σ-model, and D=4−2ε, for φ4-theory. In both cases we confirm results of Broadhurst, Gracey and 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 ε-expansion with D=3−2ε. These involve alternating Euler sums, for which the basis of irreducibles is larger. We conclude that massless Feynman diagrams in odd spacetime dimensions share the greater transcendental complexity of massive diagrams in even dimensions, such as those contributing to the electronʹs magnetic moment and the electroweak ρ-parameter. Consequences for the perturbative sector of Chern-Simons theory are discussed.