Asymptotic Pade-Approximant Predictions for Renormalization-Group Functions of Massive φ⁴ Scalar Field Theory

Within the context of massive N-component $\phi^4$ 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 $\beta_g$, the anomalous mass dimension $\gamma_m$, the vacuum-energy beta-function $\beta_v$, and the anomalous dimension $\gamma_2$ of the scalar field propagator. These estimates are then compared with explicit calculations of the five-loop contributions to $\beta_g$, $\gamma_m$, $\beta_v$, 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 $\beta_g$, $\gamma_m$, and $\beta_v$. These predictions, as well as the six-loop prediction for $\gamma_2$, provide a test of asymptotic Pade-approximant methods against future calculations.