Renormalized coupling constants for the three-dimensional scalar λϕ4 field theory and pseudo-ε-expansion

The renormalized coupling constants g2k that enter the equation of state and determine nonlinear susceptibilities of the system have universal values g2k* at the Curie point. We use the pseudo-ε-expansion approach to calculate them together with the ratios R2k = g2k/g4k-1 for the three-dimensional scalar λϕ4 field theory. We derive pseudo-ε-expansions for g6*, g8*, R6*, and R8* in the five-loop approximation and present numerical estimates for R6* and R8*. The higher-order coefficients of the pseudo-ε-expansions for g6* and R6* are so small that simple Padé approximants turn out to suffice for very good numerical results. Using them gives R6* = 1.650, while the recent lattice calculation gave R6* = 1.649(2). The pseudo-ε-expansions of g8* and R8* are less favorable from the numerical standpoint. Nevertheless, Padé–Borel summation of the series for R8* gives the estimate R8* = 0.890, differing only slightly from the values R8* = 0.871 and R8* = 0.857 extracted from the results of lattice and field theory calculations.