Positivity constraints on anomalies in supersymmetric gauge theories

Abstract

The relation between the trace and $R$-current anomalies in supersymmetric theories implies that the $\mathrm{U}{(1)}_{R}{F}^{2},$ $\mathrm{U}{(1)}_{R},$ and $\mathrm{U}{(1)}_{R}^{3}$ anomalies which are matched in studies of $N=1$ 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 $N=1$ 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 ${a}_{\mathrm{UV}}\ensuremath{-}{a}_{\mathrm{IR}}$ is always positive, as conjectured by Cardy.