Universal effective couplings of the three-dimensional $n$-vector model and field theory

Abstract

We calculate the universal ratios $R_{2k}$ of renormalized coupling constants $g_{2k}$ entering the critical equation of state for the generalized Heisenberg (three-dimensional $n$-vector) model. Renormalization group (RG) expansions of $R_8$ and $R_{10}$ for arbitrary $n$ are found in the four-loop and three-loop approximations respectively. Universal octic coupling $R_8^{\ast }$ is estimated for physical values of spin dimensionality $n=0,1,2,3$ and for $n=4,...64$ to get an idea about asymptotic behavior of $R_8^{\ast }$. Its numerical values are obtained by means of the resummation of the RG series and within the pseudo-$\epsilon$ expansion approach. Regarding $R_{10}$ our calculations show that three-loop RG and pseudo-$\epsilon$ expansions possess big and rapidly growing coefficients for physical values of $n$ what prevents getting fair numerical estimates.