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

Two of the conditions that have been suggested to determine the lower boundary of the conformal window in asymptotically free gauge theories are the linear 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 these conditions as applied to an $\mathcal{N}=1$ supersymmetric gauge theory with gauge group $G$ and $N_f$ pairs of massless chiral superfields $\mathrm{\Phi }$ and $\tilde{\mathrm{\Phi }}$ transforming according to the respective representations $\mathcal{R}$ and $\bar{\mathcal{R}}$ of $G$. We use the fact that ${\gamma }_{\bar{\psi }\psi ,IR}$ and the value $N_f=N_{f,cr}$ at the lower boundary of the conformal window are both known exactly for this theory. In contrast to the case with a non-supersymmetric gauge theory, here we find that in higher-order calculations, the linear condition provides a more accurate determination of $N_{f,cr}$ than the quadratic condition when both are calculated to the same finite order of truncation in a scheme-independent expansion.