Renormalization group in 2 + ∈ dimensions and ∈ → 2: A simple model analysis

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