Consistent Perturbative Fixed Point Calculations in QCD and Supersymmetric QCD

We suggest how to consistently calculate the anomalous dimension $\gamma_*$ of the $\bar{\psi}\psi$ operator in finite order perturbation theory at an infrared fixed point for asymptotically free theories. If the $n+1$ loop beta function and $n$ loop anomalous dimension are known then $\gamma_*$ can be calculated exactly and fully scheme independently through $O(\Delta_f^n )$ where $\Delta_f = \bar{N_f} - N_f$ and $N_f$ is the number of flavors and $\bar{N}_f$ is the number of flavors above which asymptotic freedom is lost. For a supersymmetric theory the calculation preserves supersymmetry order by order in $\Delta_f$. We then compute $\gamma_*$ through $O(\Delta_f^2)$ for supersymmetric QCD in the $\overline{\text{DR}}$ scheme and find that it matches the exact known result. We find that $\gamma_*$ is astonishingly well described in perturbation theory already at the few loops level throughout the entire conformal window. We finally compute $\gamma_*$ through $O(\Delta_f^3)$ for QCD and a variety of other non-supersymmetric fermionic gauge theories. Small values of $\gamma_*$ are observed for a large range of flavors.