# Nonlinearly realized conformal invariance in scale invariant field theories

## TL;DR

This paper investigates how promoting scale-invariant theories to conformal theories through nonlinear realization affects their symmetry properties, showing that conformal invariance cannot be spontaneously broken to scale invariance in unitary theories and clarifying the structure of conformal primaries.

## Contribution

It introduces a framework for understanding the nonlinear realization of conformal invariance in scale-invariant theories and demonstrates the implications for symmetry breaking and operator structure.

## Key findings

- Conformal invariance cannot be spontaneously broken to scale invariance in unitary theories.
- Scale invariant unitary theories in two dimensions are also conformal.
- Promoted theories contain only conformal primaries, no descendants.

## Abstract

Implications are explored of promoting non-conformal scale-invariant theories to conformal theories by nonlinearly realizing the missing symmetry. Properties of the associated Nambu-Goldstone mode imply that conformal invariance cannot be spontaneously broken to scale invariance in unitary theories and that, as well known, scale invariant unitary theories in two dimensions are also conformal. The promoted theories have only conformal primaries and no descendants. The (non-)decoupling of the Nambu-Goldstone mode is explicitly shown in examples of scale invariant theories that are actually (not) conformal.