# Renormalized coupling constants for the three-dimensional scalar *λ* *ϕ* ^{4} field theory and pseudo-*ε*-expansion

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## 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 }/*g* _{4} ^{ k-1} for the three-dimensional scalar *λ* *ϕ* ^{4} field theory. We derive pseudo-*ε*-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-*ε*-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-*ε*-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.

### Keywords

nonlinear susceptibility effective coupling constant Ising model renormalization group pseudo-*ε*-expansion

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