Renormalized coupling constants for the three-dimensional scalar $\lambda\phi^4$ field theory and pseudo-$\epsilon$-expansion
M. A. Nikitinaa,
A. I. Sokolovab a St. Petersburg State University, St. Petersburg, Russia
b National Research University ITMO, St. Petersburg, Russia
Abstract:
The renormalized coupling constants
$g_{2k}$ that enter the equation of state and determine nonlinear susceptibilities of the system have universal values
$g_{2k}^*$ at the Curie point. We use the pseudo-
$\epsilon$-expansion approach to calculate them together with the ratios
$R_{2k}^{}=g_{2k}^{}/ g_4^{k-1}$ for the three-dimensional scalar
$\lambda\phi^4$ field theory. We derive pseudo-
$\epsilon$-expansions for
$g_6^*$,
$g_8^*$,
$R_6^*$, and
$R_8^*$ in the five-loop approximation and present numerical estimates for
$R_6^*$ and
$R_8^*$. The higher-order coefficients of the pseudo-
$\epsilon$-expansions for
$g_6^*$ and
$R_6^*$ are so small that simple Padé approximants turn out to suffice for very good numerical results. Using them gives
$R_6^*= 1.650$, while the recent lattice calculation gave
$R_6^*=1.649(2)$. The pseudo-
$\epsilon$-expansions of
$g_8^*$ and
$R_8^*$ are less favorable from the numerical standpoint. Nevertheless, Padé–Borel summation of the series for
$R_8^*$ gives the estimate
$R_8^*=0.890$, differing only slightly from the values
$R_8^*=0.871$ and
$R_8^*=0.857$ extracted from the results of lattice and field theory calculations.
Keywords:
nonlinear susceptibility, effective coupling constant, Ising model, renormalization group, pseudo-$\epsilon$-expansion. Received: 09.12.2015
DOI:
10.4213/tmf9117
Fulltext:
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