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

Abstract

The ratios R2k = g2k/gk − 14 of renormalized coupling constants g2k 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