Asymptotic Pade-Approximant Predictions for Renormalization-Group Functions of Massive $ϕ^4$ Scalar Field Theory

Abstract

Within the context of massive N-component $ϕ^4$ scalar field theory, we use asymptotic Pade-approximant methods to estimate from prior orders of perturbation theory the five-loop contributions to the coupling-constant beta-function $β_g$, the anomalous mass dimension $γ_m$, the vacuum-energy beta-function $β_v$, and the anomalous dimension $γ_2$ of the scalar field propagator. These estimates are then compared with explicit calculations of the five-loop contributions to $β_g$, $γ_m$, $β_v$, and are seen to be respectively within 5%, 18%, and 27% of their true values for $N$ between 1 and 5. We then extend asymptotic Pade-approximant methods to predict the presently unknown six-loop contributions to $β_g$, $γ_m$, and $β_v$. These predictions, as well as the six-loop prediction for $γ_2$, provide a test of asymptotic Pade-approximant methods against future calculations.