On Ambiguities and Divergences in Perturbative Renormalization Group Functions

There is an ambiguity in choosing field-strength renormalization factors in the $\overline{\text{MS}}$ scheme starting from the 3-loop order in perturbation theory. More concerning, trivially choosing Hermitian factors has been shown to produce divergent renormalization group functions, which are commonly understood to be finite quantities. We demonstrate that the divergences of the RG functions are such that they vanish in the RG equation due to the Ward identity associated with the flavor symmetry. It turns out that any such divergences can be removed using the renormalization ambiguity and that the use of the flavor-improved $\beta$-function is preferred. We show how our observations resolve the issue of divergences appearing in previous calculations of the 3-loop SM Yukawa $\beta$-functions and provide the first calculation of the flavor-improved 3-loop SM $\beta$-functions in the gaugeless limit.