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Two of the conditions that have been suggested to determine the lowerboundary of the conformal window in asymptotically free gauge theories are thelinear condition, $\gamma_{\bar\psi\psi,IR}=1$, and the quadratic condition,$\gamma_{\bar\psi\psi,IR}(2-\gamma_{\bar\psi\psi,IR})=1$, where$\gamma_{\bar\psi\psi,IR}$ is the anomalous dimension of the operator$\bar\psi\psi$ at an infrared fixed point in a theory. We compare theseconditions as applied to an ${\cal N}=1$ supersymmetric gauge theory with gaugegroup $G$ and $N_f$ pairs of massless chiral superfields $\Phi$ and $\tilde\Phi$ transforming according to the respective representations ${\cal R}$ and$\bar {\cal R}$ of $G$. We use the fact that $\gamma_{\bar\psi\psi,IR}$ and thevalue $N_f = N_{f,cr}$ at the lower boundary of the conformal window are bothknown exactly for this theory. In contrast to the case with anon-supersymmetric gauge theory, here we find that in higher-ordercalculations, the linear condition provides a more accurate determination of$N_{f,cr}$ than the quadratic condition when both are calculated to the samefinite order of truncation in a scheme-independent expansion.

A Comparative Study of Criticality Conditions for Anomalous Dimensions using Exact Results in an ${\cal N}=1$ Supersymmetric Gauge Theory