Three-loop renormalization-group analysis of a complex model with stable fixed point: Critical exponents up toε3andε4

Abstract

The complete analysis of a model with three quartic coupling constants associated with $\mathrm{O}(2N)$-symmetric, cubic, and tetragonal interactions is carried out within the three-loop approximation of the renormalization-group (RG) approach in $D=4\ensuremath{-}2\ensuremath{\epsilon}$ dimensions. Perturbation expansions for RG functions are calculated using dimensional regularization and the minimal subtraction (MS) scheme. It is shown that for $N>~2$ the model does possess a stable fixed point in three-dimensional space of coupling constants, in accordance with predictions made earlier on the base of the lower-order approximations. A numerical estimate for the critical (marginal) value of the order parameter dimensionality ${N}_{c}$ is given using the Pad\'e-Borel summation of the corresponding \ensuremath{\epsilon}-expansion series obtained. It is observed that a twofold degeneracy of the eigenvalue exponents in the one-loop approximation for the unique stable fixed point leads to the substantial decrease of the accuracy expected within three loops and may cause powers of $\sqrt{\ensuremath{\epsilon}}$ to appear in the expansions. The critical exponents \ensuremath{\gamma} and \ensuremath{\eta} are calculated for all fixed points up to ${\ensuremath{\epsilon}}^{3}$ and ${\ensuremath{\epsilon}}^{4},$ respectively, and processed by the Borel summation method modified with a conformal mapping. For the unique stable fixed point the magnetic susceptibility exponent \ensuremath{\gamma} for $N=2$ is found to differ in third order in \ensuremath{\epsilon} from that of an $\mathrm{O}(4)$-symmetric point. Qualitative comparison of the results given by \ensuremath{\epsilon} expansion, three-dimensional RG analysis, nonperturbative RG arguments, and experimental data is performed.