Universal effective coupling constant ratios of 3D scalar ϕ⁴ field theory and pseudo-ϵ expansion

¹ Department of Quantum Mechanics, Saint Petersburg State University, Ulyanovskaya 1, Petergof, Saint Petersburg 198504, Russia
² ITMO University, Kronverkskii ave 49, Saint Petersburg 197101, Russia

^* e-mail: ais2002@mail.ru
^** e-mail: andrewkudlis@gmail.com

Published online: 28 October 2016

Abstract

The ratios R_2k = g_2k/g^{k − 1}₄ of renormalized coupling constants g_2k entering the small-field equation of state approach universal values R*_2k at criticality. They are calculated for the three-dimensional λϕ⁴ field theory within the pseudo-ϵ expansion approach. Pseudo-ϵ expansions for R*₆, R*₈, R*₁₀ are derived in the five-loop approximation, numerical estimates are obtained with a help of the Padé–Borel–Leroy resummation technique. Its use gives R*₆ = 1.6488, the number which perfectly agrees with the most recent lattice result R*₆ = 1.649. For the octic coupling the pseudo-ϵ expansion is less favorable numerically. Nevertheless the Padé–Borel–Leroy resummation leads to the estimate R*₈ = 0.890 close to the values R*₈ = 0.87, R*₈ = 0.857 extracted from the lattice and field-theoretical calculations. The pseudo-ϵ expansion for R*₁₀ turns out to have big and rapidly increasing coefficients. This makes correspondent estimates strongly dependent on the Borel–Leroy shift parameter b and prevents proper evaluation of R*₁₀