Positivity Constraints on Anomalies in Supersymmetric Gauge Theories

TL;DR

This work tests positivity constraints on supersymmetric anomaly coefficients across a wide class of N=1 SUSY gauge theories, grounding the analysis in the connection between the trace anomaly and U(1)_R current anomalies. Using the all-orders anomaly-free S-current and 't Hooft anomaly matching, the authors derive IR central charges b_IR, c_IR, a_IR and compare them to UV values, demonstrating that in renormalizable models with a unique anomaly-free R-current, b_IR>0, c_IR>0, a_IR>0, and the Euler-flow a_UV−a_IR>0 consistently holds. They analyze various deformations and flows, including mass, Higgs, and Kutasov–Schwimmer-type models, and show that accidental symmetries require corrections but do not generally violate the a-theorem in the renormalizable regime. Nonrenormalizable theories, however, can exhibit negative a_UV−a_IR in interpolating flows, suggesting the a-theorem is not universal beyond renormalizability. Overall, the results provide strong empirical support for a universal four-dimensional a-theorem within SUSY gauge theories and highlight where caution is warranted in nonrenormalizable contexts.

Abstract

The relation between the trace and R-current anomalies in supersymmetric theories implies that the U$(1)_RF^2$, U$(1)_R$ and 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, $a_{UV}-a_{IR}$, is always positive, as conjectured by Cardy.