Critical behaviour of the O(n)-$\phi^{4}$ model with an antisymmetric order parameter

Critical behaviour of the O(n)-symmetric $\phi^{4}$-model with an antisymmetric tensor order parameter is studied by means of the field-theoretic renormalization group (RG) in the leading order of the $\varepsilon=4-d$-expansion (one-loop approximation). For $n=2$ and 3 the model is equivalent to the scalar and the O(3)-symmetric vector models, for $n\ge4$ it involves two independent interaction terms and two coupling constants. It is shown that for $n>4$ the RG equations have no infrared (IR) attractive fixed points and their solutions (RG flows) leave the stability region of the model. This means that fluctuations of the order parameter change the nature of the phase transition from the second-order type (suggested by the mean-field theory) to the first-order one. For $n=4$, the IR attractive fixed point exists and the IR behaviour is non-universal: if the coupling constants belong to the basin of attraction for the IR point, the phase transition is of the second order and the IR critical scaling regime realizes. The corresponding critical exponents $\nu$ and $\eta$ are presented in the order $\varepsilon$ and $\varepsilon^{2}$, respectively. Otherwise the RG flows pass outside the stability region and the first-order transition takes place.