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

Abstract

We simplify, to a single integral of dilogarithms, the least tractable O(1/N 3 ) contribution to the large-N critical exponent η of the non-linear σ-model, and hence φ 4 -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 φ 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 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 electron's magnetic moment and the electroweak ρ-parameter. Consequences for the perturbative sector of Chern-Simons theory are discussed.

CriticalO(N)models in6−εdimensions

“…The term of order 1/N 3 depends on a non-trivial self-energy integral that was not evaluated for general dimension in [6]. An explicit derivation of this integral in general d was later obtained in [50]. Using that result, we find in d = 5 We note that the coefficients of the 1/N expansion are considerably larger than in the d = 3 case.…”

Weight Systems from Feynman Diagrams

“…Thus, what has to be shown is that neither the horizontal paths diverge at the end, nor the vertical paths apart from the ones in Eq. (7). Not unexpectedly, the horizontal paths will become finite when one considers the paths above and below the real axis together, and the vertical paths cancel in appropriate pairs (or quartets) due to their opposite orientations.…”

Note on an application of the method of uniqueness to reduced quantum electrodynamics

“…Its evaluation for arbitrary indices is however highly nontrivial: the results can be represented [10] as a combination of twofold series. In some particular cases, however, the results can be obtained [2,3,7,[11][12][13][14] in significantly simpler form. In Ref.…”