Asymptotic Behavior of Higher-Order Perturbations: Scaling Functions of the $O(n)$-Symmetric $\phi^4$-Theory in the $(4-\epsilon)$-Expansion

We use an instantonic approach to calculate the asymptotic behavior of higher orders of the $(4-\epsilon)$-expansion for the scaling function of the pair correlator of the $O(n)$-symmetric $\phi^4$-theory in the minimal subtraction scheme. Our results differ substantially from the known exact expression for the $\epsilon^3$ order of the expansion of the scaling function in the small-$\tau$ domain.