Consistency Relation for Fixed Point Dynamics

We gain insight on the fixed point dynamics of $d$ dimensional quantum field theories by exploiting the critical behavior of the $d-\epsilon$ sister theories. To this end we first derive a self-consistent relation between the $d-\epsilon$ 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 $d-\epsilon$ scaling exponent must be multi-valued in $\epsilon$. 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 $1/N^2$ contributions.