Comparative study of criticality conditions for anomalous dimensions using exact results in an 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, γψ¯ψ,IR=1, and the quadratic condition, γψ¯ψ,IR(2-γψ¯ψ,IR)=1, where γψ¯ψ,IR is the anomalous dimension of the operator ψ¯ψ at an infrared fixed point in a theory. We compare these conditions as applied to an N=1 supersymmetric gauge theory with gauge group G and Nf pairs of massless chiral superfields φ and φ transforming according to the respective representations R and R¯ of G. We use the fact that γψ¯ψ,IR and the value Nf=Nf,cr at the lower boundary of the conformal window are both known exactly for this theory. In contrast to the case with a nonsupersymmetric gauge theory, here we find that in higher-order calculations, the linear condition provides a more accurate determination of Nf,cr than the quadratic condition when both are calculated to the same finite order of truncation in a scheme-independent expansion.