RG flows in d dimensions, the dilaton effective action, and the a-theorem

TL;DR

This paper explores the structure of the dilaton effective action in various dimensions, analyzing its components, and investigates implications for the a-theorem, especially in 8 dimensions, using explicit calculations with a free scalar.

Contribution

It provides explicit forms of the dilaton effective action in d dimensions up to 8 derivatives, unifies Weyl-invariants and anomalies using GJMS-operators, and examines challenges in proving the a-theorem in 8d.

Findings

Explicit dilaton actions in d=3 to 10 confirmed.

Identified 8-derivative Weyl-invariant contaminates amplitudes in 8d.

Extracted the 8d a-anomaly from scattering amplitudes matching zeta-function results.

Abstract

Motivated by the recent dilaton-based proof of the 4d a-theorem, we study the dilaton effective action for RG flows in d dimensions. When d is even, the action consists of a Wess-Zumino (WZ) term, whose Weyl-variation encodes the trace-anomaly, plus all Weyl-invariants. For d odd, the action consists of Weyl-invariants only. We present explicit results for the flat-space limit of the dilaton effective action in d-dimensions up to and including 8-derivative terms. GJMS-operators from conformal geometry motivate a form of the action that unifies the Weyl-invariants and anomaly-terms into a compact general-d structure. A new feature in 8d is the presence of an 8-derivative Weyl-invariant that pollutes the O(p^8)-contribution from the WZ action to the dilaton scattering amplitudes; this may challenge a dilaton-based proof of an a-theorem in 8d. We use the example of a free massive scalar for two purposes: 1) it allows us to confirm the structure of the d-dimensional dilaton effective action explicitly; we carry out this check for d=3,4,5,...,10; and 2) in 8d we demonstrate how the flow (a_UV - a_IR) can be extracted systematically from the O(p^8)-amplitudes despite the contamination from the 8-derivative Weyl-invariant. This computation gives a value for the a-anomaly of the 8d free conformal scalar that is shown to match the value obtained from zeta-function regularization of the log-term in the free energy.