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

Abstract

The renormalized coupling constants g_2k that enter the equation of state and determine nonlinear susceptibilities of the system have universal values g_2k^* at the Curie point. We use the pseudo-ε-expansion approach to calculate them together with the ratios R_2k = g_2k/g₄^k-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.