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) that figure 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 renormalization group (RG) expansion for the ß function is reported. Numerical results are presented for 0 ≤ n ≤ 64, with the most attention paid to the 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 values of g(4)(*) and R(6)(*) that are almost as accurate as those provided by the Padé technique. Pseudo-ε expansion estimates for g(8)(*) and R(8)(*) are found to be much worse than those for lower-order couplings independently of the resummation method employed. The 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 the ß function of two-dimensional 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.