Quantum irreversibility in arbitrary dimension

TL;DR

This paper generalizes the concept of quantum irreversibility and trace anomaly relations from four dimensions to arbitrary dimensions, introducing a new Euler density and deriving universal formulas for RG flow and anomaly coefficients.

Contribution

It introduces the pondered Euler density in arbitrary dimensions and establishes a universal relation between trace-anomaly coefficients, extending quantum irreversibility concepts beyond four dimensions.

Findings

The pondered Euler density relates anomaly coefficients a and a' as a=a' in all even dimensions.

Derived a formula expressing the RG flow of a as the invariant area under the beta function graph.

Validated the formula in six dimensions and checked predictions up to the fourth loop in phi^3-theory.

Abstract

Some recent ideas are generalized from four dimensions to the general dimension n. In quantum field theory, two terms of the trace anomaly in external gravity, the Euler density G_n and Box^{n/2-1}R, are relevant to the problem of quantum irreversibility. By adding the divergence of a gauge-invariant current, G_n can be extended to a new notion of Euler density, linear in the conformal factor. We call it pondered Euler density. This notion relates the trace-anomaly coefficients a and a' of G_n and Box^{n/2-1}R in a universal way (a=a') and gives a formula expressing the total RG flow of a as the invariant area of the graph of the beta function between the fixed points. I illustrate these facts in detail for n=6 and check the prediction to the fourth-loop order in the phi^3-theory. The formula of quantum irreversibility for general n even can be extended to n odd by dimensional continuation. Although the trace anomaly in external gravity is zero in odd dimensions, I show that the odd-dimensional formula has a predictive content.