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

Four-dimensional gauge theories with matter can have regions in parameter space, often dubbed conformal windows, where they flow in the infrared to nontrivial 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 dilatonlike scalar particle appears in the spectrum, but it is not parametrically lighter than the other excitations.