The relation between the trace and R-current anomalies in supersymmetric theories implies that the U(1) R F 2 , U(1) R and U(1) 3 R 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 U V − a IR , is always positive, as conjectured by Cardy.
Recently a non-perturbative formula for the RG flow between UV and IR fixed points of the coefficient in the trace of the energy momentum tensor of the Euler density has been obtained for N = 1 SUSY gauge theories by relating the trace and R-current anomalies. This result is compared here with an earlier perturbation theory analysis based on a naturally defined metric on the space of couplings for general renormalisable quantum field theories. This approach is specialised to N = 1 supersymmetric theories and extended, using consistency arguments, to obtain the Euler coefficient at fixed points to 4-loops. The result agrees completely, to this order, with the exact formula. emails: dzf@math.mit.edu and ho@damtp.cam.ac.uk
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