Large-order asymptotic terms in perturbation theory: The first $(4-\epsilon)$-expansion correction to renormalization constants in the $O(n)$-symmetric theory

We investigate large-order asymptotic terms in the perturbation theory for the $O(n)$ symmetric $\phi^4(4-\epsilon)$-model in the minimal subtraction scheme. Taking the specificity of the $(4-\epsilon)$-minimal-subtraction scheme into account, we calculate corrections to the asymptotic formula for the expansion coefficients of the renormalization constant $Z_g$ and the critical index $\eta$. The resulting corrections essentially improve the asymptotic description of the results in loop calculations.