Flow of the □R Weyl anomaly

An important aspect of Weyl anomalies is that they encode information on the irreversibility of the renormalization group flow. We consider, Delta(b) over bar = (b) over bar (UV) -(b) over bar (IR) the difference of the ultraviolet and infrared value of the square R-term of the Weyl anomaly. The quantity is related to the fourth moment of the trace of the energy momentum tensor correlator for theories which are conformal at both ends. Subtleties arise for nonconformal fixed points as might be the case for infrared fixed points with broken chiral symmetry. Provided that the moment converges, Delta(b) over bar is then automatically positive by unitarity. Written as an integral over the renormalization scale, flow-independence follows since its integrand is a total derivative. Furthermore, using a momentum subtraction scheme (MOM) the 4D Zamolodchikov-metric is shown to be strictly positive beyond perturbation theory and equivalent to the metric of a conformal manifold at both ends of the flow. In this scheme (b) over bar mu can be extended outside the fixed point to a monotonically decreasing function. The ultraviolet finiteness of the fourth moment enables us to define a scheme for the delta L similar to b(0)R(2)-term, for which the R-2-anomaly vanishes along the flow. In the MOM-and the R-2-scheme, (b) over bar mu is shown to satisfy a gradient flow type equation. We verify our findings in free field theories, higher derivative theories and extend Delta(b) over bar and the Euler flow Delta beta(a) for a Caswell-Banks-Zaks fixed point for QCD-like theories to nextto- next-to leading order using a recent < G(2)G(2)>-correlator computation.