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

TL;DR

This paper derives a simplified integral form for the O(1/N^3) correction to the critical exponent η in large-N models, revealing connections to knot theory and enabling new epsilon-expansion terms involving complex Euler sums.

Contribution

It provides a compact integral representation for non-zeta terms in critical exponents at order 1/N^3, facilitating advanced epsilon-expansion calculations and insights into transcendental structures.

Findings

Confirmed relations between knots and counterterms in sigma-models and phi^4-theory.

Developed 8 new epsilon-expansion terms involving alternating Euler sums.

Highlighted the transcendental complexity of massless diagrams in odd dimensions.

Abstract

We simplify, to a single integral of dilogarithms, the least tractable O(1/N^3) contribution to the large-N critical exponent $\eta$ of the non-linear sigma-model, and hence $\phi^4$-theory, for any spacetime dimensionality, D. It is the sole generator of irreducible multiple zeta values in epsilon-expansions 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 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 epsilon-expansion with $D=3-2\epsilon$. 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 electron's magnetic moment and the electroweak $\rho$-parameter. Consequences for the perturbative sector of Chern-Simons theory are discussed.