Renormalized coupling constants for 3D scalar \lambda\phi^4 field theory and pseudo-\epsilon-expansion

TL;DR

This paper calculates universal renormalized coupling constants for 3D scalar  field theory using pseudo--expansion, providing numerical estimates that closely match lattice and field-theoretical results.

Contribution

It derives pseudo--expansions for higher-order couplings in 3D  theory and demonstrates their effectiveness in numerical estimation.

Findings

Pseudo--expansions for g*_6, g*_8, R*_6, R*_8 derived in five-loop approximation.

Numerical estimates R*_6 = 1.650 and R*_8 = 0.890 closely match lattice results.

Simple Pade approximants and Pade-Borel resummation yield accurate results.

Abstract

Renormalized coupling constants g_{2k} that enter the critical equation of state and determine nonlinear susceptibilities of the system possess universal values g*_{2k} at the Curie point. They are calculated, along with the ratios R_{2k} = g_{2k}/g_4^{k-1}, for the three-dimensional scalar \lambda\phi^4 field theory within the pseudo-\epsilon-expansion approach. Pseudo-\epsilon-expansions for g*_6, g*_8, R*_6, and R*_8 are derived in the five-loop approximation, numerical estimates are presented for R*_6 and R*_8. The higher-order coefficients of the pseudo-\epsilon-expansions for the sextic coupling are so small that simple Pade approximants turn out to be sufficient to yield very good numerical results. Their use gives R*_6 = 1.650 while the most recent lattice estimate is R*_6 = 1.649(2). For the octic coupling pseudo-\epsilon-expansions are less favorable from the numerical point of view. Nevertheless, Pade-Borel resummation leads in this case to R*_8 = 0.890, the number differing only slightly from the values R*_8 = 0.871, R*_8 = 0.857 extracted from the lattice and field-theoretical calculations.