Quantum irreversibility in arbitrary dimension

TL;DR

This work generalizes quantum irreversibility from four dimensions to arbitrary dimension $n$ using the trace anomaly in external gravity and introduces the pondered Euler density $ ilde{G}_n$, which is linear in the conformal factor and enforces $a=a'$. It shows that the total RG flow of $a$ equals the invariant area under the beta-function graph between fixed points, expressed via a one-form $oldsymbol{ extomega}=-doldsymbol{eta}(oldsymbol{ extlambda}) f(oldsymbol{ extlambda})$, and verifies this structure in the six-dimensional $oldsymbol{}$ theory up to four loops. The analysis extends to general even $n$ and, by dimensional continuation, provides a predictive odd-dimensional formula even when the external-gravity trace anomaly vanishes. The results connect a scheme-independent RG interpolation to a geometric monotonicity principle and outline a program to test quantum irreversibility in higher-derivative theories.

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.