The trace anomaly in external gravity is the sum of three terms at criticality: the square of the Weyl tensor, the Euler density and gR, with coefficients, properly normalized, called c, a, and a$, the latter being ambiguously defined by an additive constant. Considerations about unitarity and positivity properties of the induced actions allow us to show that the total RG flows of a and a$ are equal and therefore the a$-ambiguity can be consistently removed through the identification a$=a. The picture that emerges clarifies several long-standing issues. The interplay between unitarity and renormalization implies that the flux of the renormalization group is irreversible. A monotonically decreasing a-function interpolating between the appropriate values is naturally provided by a$. The total a-flow is expressed non-perturbatively as the invariant (i.e., scheme-independent) area of the graph of the beta function between the fixed points. We test this prediction to the fourth loop order in perturbation theory, in QCD with N f 11ร2 N c and in supersymmetric QCD. There is agreement also in the absence of an interacting fixed point (QED and . 4 -theory). Arguments for the positivity of a are also discussed.
1999 Academic Press
1. Introduction
Anomalies are often calculable to high orders in perturbation theory with a relatively moderate effort. Sometimes they are calculable exactly to all orders.
Much of the present knowledge about the low-energy limit of asymptotically free quantum field theories comes from anomalies, via the Adler Bardeen theorem . Conserved axial currents have one-loop exact anomalies and the 't Hooft anomaly matching conditions [2] put constraints on the low-energy limit of the theory.
A second class of anomalies, related to the stress tensor, called central charges, does not satisfy the Adler Bardeen theorem. Nevertheless, they obey various positivity constraints, which also put restrictions on the low-energy limit of the theory.
Other remarkable positivity constraints are those obeyed by the spectrum of anomalous dimensions of the quantum conformal algebra, i.e., the algebra generated by the operator product expansion of the stress tensor. Applications of the Nachtmann theorem [3] reveal non-trivial properties of strongly coupled conformal field theories , especially in the presence of supersymmetry, where the algebraic structure simplifies considerably.
Furthermore, in supersymmetric theories, the two classes of anomalies mentioned above, axial and trace, are related to each other, and the Adler Bardeen theorem can be used to compute the exact IR values of the central charges in the conformal