# Exact Correlators from Conformal Ward Identities in Momentum Space and Perturbative Realizations

## TL;DR

This paper derives exact conformal correlators in momentum space using Ward identities, compares them with perturbative anomaly actions, and explores their implications for conformal symmetry breaking, ghost states, and cosmology.

## Contribution

It provides an exact solution to conformal Ward identities in momentum space and compares these with perturbative anomaly actions, revealing the nature of conformal symmetry breaking and associated ghost phenomena.

## Key findings

- Exact 3-point functions match between CWI solutions and perturbation theory.
- Massless scalar exchanges are signatures of the conformal anomaly.
- Both nonlocal and Wess-Zumino actions predict ghost states in the spectrum.

## Abstract

The general solution of the conformal Ward identities (CWI's) in momentum space, and their matching to perturbation theory, allows to uncover some specific characteristics of the breaking of conformal symmetry, induced by the anomaly. It allows to compare perturbative features of the 1-particle irreducible (1PI, nonlocal) anomaly action with the prediction of a similar (but exact) nonlocal action identified by the CWI's. The two predictions can be exactly matched at the level of 3-point functions. The analysis of the $TJJ$ and $TTT$ shows that both approaches - based either on 1PI or on the exact solutions of the CWI's - predict massless (dynamical) scalar exchanges in 3-point functions as the signature of the conformal anomaly. In a local formulation such 1PI actions exhibit a ghost in the spectrum which may induce ghost condensation. We also discuss alternative approaches, which take to Wess-Zumino forms of the action with an asymptotic dilaton, which should be considered phenomenological alternatives to the exact nonlocal action. If derived by a Weyl gauging, they also include a ghost in the spectrum. The two formulations, nonlocal and of WZ type, can be unified under the assumption that they describe the same anomaly phenomenon at two separate (UV/IR) ends of the renormalization group flow, possibly separated by a vacuum rearrangement at an intermediate scale. A similar analysis is presented for an $\mathcal{N}=1$ supersymmetric Yang-Mills theory. We comment on the possibile cosmological implications of such quasi Nambu-Goldstone modes as ultralight dilatons and axions.