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

TL;DR

This work extends the dilaton-driven approach to the a-theorem from 4d to general d, revealing that even d requires a Wess-Zumino term while odd d involves only Weyl invariants; it uncovers a unified description in terms of GJMS operators 𝓟_k through the building blocks 𝓦_k, producing a compact flat-space action whose anomaly flow appears in the 𝓦_{d/2} term. The authors compute explicit dilaton actions up to eight derivatives and verify them against 1-loop dilaton amplitudes for a free massive scalar in d up to 10, confirming that the a-anomaly Δa is correctly captured in several dimensions (e.g., Δa for d=4,6,8) and matching zeta-function results for a_scalar,d. In 8d there is a notable contamination of the O(p^8) amplitudes by a Weyl-invariant, but the anomaly signal can still be extracted from higher-point data, with Δa > 0 and Γ_{8d} = 2Γ_1 + Γ_2 remaining positive. The study also maps out higher-derivative structures, reports initial 10- and 12-derivative explorations, and discusses potential need for new curved-space GJMS-type operators, signaling rich connections between RG flows, conformal geometry, and anomaly theorems across dimensions.

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.