Consistency relation for fixed point dynamics

Abstract We gain insight on the fixed point dynamics of d dimensional quantum field theories by exploiting the critical behavior of the $$d-\epsilon $$ d - ϵ sister theories. To this end we first derive a self-consistent relation between the $$d-\epsilon $$ d - ϵ scaling exponents and the associated d dimensional beta functions. We then demonstrate that to account for an interacting fixed point in the original theory the related $$d-\epsilon $$ d - ϵ scaling exponent must be multi-valued in $$\epsilon .$$ ϵ . We elucidate our findings by discussing several examples such as the QCD Banks–Zaks infrared fixed point, QCD at large number of flavors, as well as the O(N) model in four dimensions. For the latter, we show that although the 1/ N corrections prevent the reconstruction of the renormalization group flow, this is possible when adding the $$1/N^2$$ 1 / N 2 contributions.