Positivity constraints on anomalies in supersymmetric gauge theories

The relation between the trace and $R$-current anomalies in supersymmetric theories implies that the $\mathrm{U}{\mathrm{}\left(1\right)\mathrm{}}_R{F}^2,$ $\mathrm{U}{\mathrm{}\left(1\right)\mathrm{}}_R,$ and $\mathrm{U}{\mathrm{}\left(1\right)\mathrm{}}_R^3$ anomalies which are matched in studies of $N=1\mathrm{}$ Seiberg duality satisfy positivity constraints. Some constraints are rigorous and others conjectured as four-dimensional generalizations of the Zamolodchikov $c$ theorem. These constraints are tested in a large number of $N=1\mathrm{}$ supersymmetric gauge theories in the non-Abelian Coulomb phase, and they are satisfied in all renormalizable models with unique anomaly-free $R$ current, including those with accidental symmetry. Most striking is the fact that the flow of the Euler anomaly coefficient $a_{\mathrm{UV}}-a_{\mathrm{IR}}$ is always positive, as conjectured by Cardy.