Search for flow invariants in even and odd dimensions

TL;DR

This paper investigates flow invariants in quantum field theories across various dimensions, deriving sum rules for trace anomalies and revealing dimension-dependent properties of flow invariants.

Contribution

It extends the understanding of flow invariants by deriving new sum rules in six dimensions and analyzing their behavior in odd and even dimensions.

Findings

Recovered known results in four and six dimensions

Derived sum rules for trace anomalies a and a' in six dimensions

Found a non-trivial vanishing relation in three dimensions

Abstract

A flow invariant in quantum field theory is a quantity that does not depend on the flow connecting the UV and IR conformal fixed points. We study the flow invariance of the most general sum rule with correlators of the trace Theta of the stress tensor. In even (four and six) dimensions we recover the results known from the gravitational embedding. We derive the sum rules for the trace anomalies a and a' in six dimensions. In three dimensions, where the gravitational embedding is more difficult to use, we find a non-trivial vanishing relation for the flow integrals of the three- and four-point functions of Theta. Within a class of sum rules containing finitely many terms, we do not find a non-vanishing flow invariant of type a in odd dimensions. We comment on the implications of our results.