# Conformality Loss, Walking, and 4D Complex Conformal Field Theories at Weak Coupling

## TL;DR

This paper constructs weakly coupled 4D gauge theories demonstrating conformality loss via fixed point merging, exhibiting walking dynamics, phase transitions, and a light dilaton-like scalar, advancing understanding of conformal windows.

## Contribution

It provides explicit, controllable examples of 4D gauge theories showing conformality loss through fixed point merging and walking behavior at weak coupling.

## Key findings

- Controlled weakly coupled models demonstrating conformality loss.
- Identification of a scalar particle resembling a dilaton in the spectrum.
- Observation of a first-order phase transition ending the walking regime.

## Abstract

Four-dimensional gauge theories with matter can have regions in parameter space, often dubbed conformal windows, where they flow in the infrared to non-trivial conformal field theories. It has been conjectured that conformality can be lost because of merging of two nearby fixed points that move into the complex plane, and that a walking dynamics governed by scaling dimensions of operators defined at such complex fixed points can occur. We find controlled, parametrically weakly coupled, and ultraviolet-complete 4D gauge theories that explicitly realize this scenario. We show how the walking dynamics is controlled by the coupling of a double-trace operator that crosses marginality. The walking regime ends when the renormalization group flow of this coupling leads to a (weak) first-order phase transition with Coleman-Weinberg symmetry breaking. A light dilaton-like scalar particle appears in the spectrum, but it is not parametrically lighter than the other excitations.