Renormalization Group in $2+\epsilon$ Dimensions and $\epsilon\to2$: A simple model analysis

Using a simple solvable model, i.e., Higgs--Yukawa system with an infinite number of flavors, we explicitly demonstrate how a dimensional continuation of the $\beta$ function in two dimensional MS scheme {\it fails\/} to reproduce the correct behavior of the $\beta$ function in four dimensions. The mapping between coupling constants in two dimensional MS scheme and a conventional scheme in the cutoff regularization, in which the dimensional continuation of the $\beta$ function is smooth, becomes singular when the dimension of spacetime approaches to four. The existence of a non-trivial fixed point in $2+\epsilon$ dimensions continued to four dimensions $\epsilon\to2$ in the two dimensional MS scheme is spurious and the asymptotic safety cannot be imposed to this model in four dimensions.