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

Abstract

Universal values of dimensional effective coupling constants $g_{2k}$ that determine nonlinear susceptibilities $χ_{2k}$ 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 $g_{4}^*$ descending from the six-loop RG expansion for $β$-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 $g_4^*$ and $R_6^*$ almost as accurate as those provided by Padé technique. Pseudo-$ε$ expansion estimates for $g_8^*$ and $R_8^*$ 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 $g_4^*$ originating from the five-loop RG series for $β$-function of 2D $λϕ^4$ field theory is used to get numerical estimates for $n$ ranging from 0 to 64. The approach discussed gives accurate enough values of $g_{4}^*$ down to $n = 2$ and leads to fair estimates for Ising and polymer ($n = 0$) models.