Pseudo-$ϵ$ Expansion and Renormalized Coupling Constants at Criticality

Abstract

Universal values of dimensional effective coupling constants $g_{2k}$ that determine nonlinear susceptibilities ${\chi }_{2k}$ and enter the scaling equation of state are calculated for $n$-vector field theory within the pseudo-$ϵ$ expansion approach. Pseudo-$ϵ$ expansions for $g_6$ and $g_8$ at criticality are derived for arbitrary $n$. Analogous series for ratios $R_6=g_6/g_4^2$ and $R_8=g_8/g_4^3$ figuring in the equation of state are also found and the pseudo-$ϵ$ expansion for Wilson fixed point location $g_4^{\ast }$ descending from the six-loop RG expansion for $\beta$-function is reported. Numerical results are presented for $0\le n\le 64$ with main attention paid to physically important cases $n=0,1,2,3$. Pseudo-$ϵ$ expansions for quartic and sextic couplings have rapidly diminishing coefficients, so Padé resummation turns out to be sufficient to yield high-precision numerical estimates. Moreover, direct summation of these series with optimal truncation gives the values of $g_4^{\ast }$ and $R_6^{\ast }$ almost as accurate as those provided by Padé technique. Pseudo-$ϵ$ expansion estimates for $g_8^{\ast }$ and $R_8^{\ast }$ are found to be much worse than that for the lower-order couplings independently on the resummation method employed. Numerical effectiveness of the pseudo-$ϵ$ expansion approach in two dimensions is also studied. Pseudo-$ϵ$ expansion for $g_4^{\ast }$ originating from the five-loop RG series for $\beta$-function of 2D $\lambda {\varphi }^4$ field theory is used to get numerical estimates for $n$ ranging from 0 to 64. The approach discussed gives accurate enough values of $g_4^{\ast }$ down to $n=2$ and leads to fair estimates for Ising and polymer ($n=0$) models.

Comments: 24 pages, 11 tables