Anomalies, Unitarity and Quantum Irreversibility

TL;DR

The paper unifies several long-standing issues about four-dimensional trace anomalies by arguing that the ambiguous $a'$ can be consistently identified with $a$, yielding a clean $a$-theorem. It frames quantum irreversibility as the monotonic decrease of $a$ along RG flows and expresses the total flow as the scheme-independent area under the beta-function graph between fixed points. The analysis hinges on unitarity constraints, positivity of induced actions (Riegert action), and a nonperturbative central-function framework that remains testable through perturbative checks up to four loops in QCD, SQCD, and related theories. It also connects to Casimir energies and introduces the notion of a proper coupling constant, linking scheme dependence to a universal geometric measure of degrees of freedom lost along the flow.

Abstract

The trace anomaly in external gravity is the sum of three terms at criticality: the square of the Weyl tensor, the Euler density and Box R, 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 Nf ~< 11/2 Nc and in supersymmetric QCD. There is agreement also in the absence of an interacting fixed point (QED and phi^4-theory). Arguments for the positivity of a are also discussed.