Asymptotic Behavior of Renormalization Constants in Higher Orders of the Perturbation Expansion for the $(4?\epsilon)$-Dimensionally Regularized $O(n)$-Symmetric $\phi^4$ Theory

Higher-order asymptotic expansions for renormalization constants and critical exponents of the $O(n)$-symmetric $\phi^4$ theory are found based on the instanton approach in the minimal subtraction scheme for the $(4-\epsilon)$-dimensional regularization. The exactly known expansion terms differ substantially from their asymptotic values. We find expressions that improve the asymptotic expansions for unknown expansion terms of the renormalization constants.