Consistent Perturbative Fixed Point Calculations in QCD and SQCD

We suggest how to consistently calculate the anomalous dimension $γ_*$ of the $\barψψ$ 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 $γ_*$ can be calculated exactly and fully scheme independently through $O(Δ_f^n )$ where $Δ_f = \bar{N_f} - N_f$ 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 $Δ_f$. We then compute $γ_*$ through $O(Δ_f^2)$ for supersymmetric QCD in the $\overline{\text{DR}}$ scheme and find that it matches the exact known result. We find that $γ_*$ is astonishingly well described in perturbation theory already at the few loops level throughout the entire conformal window. We finally compute $γ_*$ through $O(Δ_f^3)$ for QCD and a variety of other non-supersymmetric fermionic gauge theories. Small values of $γ_*$ are observed for a large range of flavors.

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