Renormalization Group in 2 + Dimensions and ->2: A Simple Model Analysis

Using a simple solvable model, i.e., Higgs--Yukawa system with an infinitenumber of flavors, we explicitly demonstrate how a dimensional continuation ofthe $\beta$ function in two dimensional MS scheme {\it fails\/} to reproducethe correct behavior of the $\beta$ function in four dimensions. The mappingbetween coupling constants in two dimensional MS scheme and a conventionalscheme in the cutoff regularization, in which the dimensional continuation ofthe $\beta$ function is smooth, becomes singular when the dimension ofspacetime approaches to four. The existence of a non-trivial fixed point in$2+\epsilon$ dimensions continued to four dimensions $\epsilon\to2$ in the twodimensional MS scheme is spurious and the asymptotic safety cannot be imposedto this model in four dimensions.Comment: 15 pages, PHYZZX. English is improved, some references are adde