Consequences of Weyl consistency conditions

The running of quantum field theories can be studied in detail with the use of a local renormalization group equation. The usual beta-function effects are easy to include, but by introducing spacetime-dependence of the various parameters of the theory one can efficiently incorporate renormalization effects of composite operators as well. An illustration of the power of these methods was presented by Osborn in the early 90s, who used consistency conditions following from the Abelian nature of the Weyl group to rederive Zamolodchikov’s c-theorem in d = 2 spacetime dimensions, and also to obtain a perturbative a-theorem in d = 4. In this work we present an extension of Osborn’s work to d = 6 and to general even d. We compute the full set of Weyl consistency conditions, and we discover among them a candidate for an a-theorem in d = 6, similar to the d = 2, 4 cases studied by Osborn. Additionally, we show that in any even spacetime dimension one finds a consistency condition that may serve as a generalization of the c-theorem, and that the associated candidate c-function involves the coefficient of the Euler term in the trace anomaly. Such a generalization hinges on proving the positivity of a certain “metric” in the space of couplings.