We suggest how to consistently calculate the anomalous dimension ${\gamma }_{*}$ of the $\overline{\psi }\psi$ 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 in a Banks-Zaks expansion through $O({\mathrm{\Delta }}_f^n)$, where ${\mathrm{\Delta }}_f={\overline{N}}_f-N_f$, $N_f$ is the number of flavors, and ${\overline{N}}_f$ is the number of flavors above which asymptotic freedom is lost. For a supersymmetric theory, the calculation preserves supersymmetry order by order in ${\mathrm{\Delta }}_f$. We then compute ${\gamma }_{*}$ through $O({\mathrm{\Delta }}_f^2)$ for supersymmetric QCD in the dimensional reduction 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 $O({\mathrm{\Delta }}_f^3)$ for QCD and a variety of other nonsupersymmetric fermionic gauge theories. Small values of ${\gamma }_{*}$ are observed for a large range of flavors.

• Received 3 April 2016

DOI:https://doi.org/10.1103/PhysRevLett.117.071601