Four loop renormalization of phi^3 theory in six dimensions

TL;DR

The paper delivers a four-loop renormalization of six-dimensional $\phi^3$ theory in the $\overline{MS}$ scheme and derives the corresponding $\beta$-function and anomalous dimensions to extract critical exponents for Lee–Yang and percolation problems. It extends to an $O(N)$-symmetric sector with full mass-operator mixing and validates results against large-$N$ expansions and exact $d=2$ CFT data to improve cross-dimensional extrapolations. Padé and constrained Padé resummations yield consistent estimates in $d=3,4,5$, supporting the utility of high-loop perturbation theory in mapping conformal fixed points. The work also illuminates conformal-window structure in five dimensions and suggests connections to Banks–Zaks-type fixed points in related theories, pointing to future extensions to gauge or supersymmetric contexts.

Abstract

We renormalize six dimensional phi^3 theory in the modified minimal subtraction (MSbar) scheme at four loops. From the resulting beta-function, anomalous dimension and mass anomalous dimension we compute four loop critical exponents relevant to the Lee-Yang edge singularity and percolation problems. Using resummation methods and information on the exponents of the relevant two dimensional conformal field theory we obtain estimates for exponents in dimensions 3, 4 and 5 which are in reasonable agreement with other techniques for these two problems. The renormalization group functions for the more general theory with an O(N) symmetry are also computed in order to obtain estimates of exponents at various fixed points in five dimensions. Included in this O(N) analysis is the full evaluation of the mass operator mixing matrix of anomalous dimensions at four loops. We show that its eigen-exponents are in agreement with the mass exponents computed at O(1/N^2) in the non-perturbative large N expansion.