Renormalized coupling constants for the three-dimensional scalar $\lambda\phi^4$ field theory and pseudo-$\epsilon$-expansion

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-$\epsilon$-expansion approach to calculate them together with the ratios $R_{2k}^{}=g_{2k}^{}/ g_4^{k-1}$ for the three-dimensional scalar $\lambda\phi^4$ field theory. We derive pseudo-$\epsilon$-expansions for $g_6^*$, $g_8^*$, $R_6^*$, and $R_8^*$ in the five-loop approximation and present numerical estimates for $R_6^*$ and $R_8^*$. The higher-order coefficients of the pseudo-$\epsilon$-expansions for $g_6^*$ and $R_6^*$ are so small that simple Padé approximants turn out to suffice for very good numerical results. Using them gives $R_6^*= 1.650$, while the recent lattice calculation gave $R_6^*=1.649(2)$. The pseudo-$\epsilon$-expansions of $g_8^*$ and $R_8^*$ are less favorable from the numerical standpoint. Nevertheless, Padé–Borel summation of the series for $R_8^*$ gives the estimate $R_8^*=0.890$, differing only slightly from the values $R_8^*=0.871$ and $R_8^*=0.857$ extracted from the results of lattice and field theory calculations.