Technical naturalness of a flat χ-direction in the two-field cosmic-string scenario

Determine whether a hierarchy of scales that renders the potential along the real scalar field χ significantly flatter than along the complex scalar field φ can be realized in a technically natural manner consistent with observational bounds within the two-field model defined by Vφ(φ) = −μ^2|φ|^2 − λ|φ|^4 + λ6|φ|^6, Vχ(χ) = (m^2/2)χ^2 − (λ3/3)χ^3 + (λχ/4)χ^4, and Vint(φ, χ) = (ξ χ^2 − η χ)|φ|^2, in order to delay the roll-over toward the false minimum and postpone the onset of string-induced tunneling.

Background

To connect string-induced vacuum decay with potential gravitational-wave observations, the paper introduces a two-field extension in which a real scalar χ couples to the complex tunneling field φ. In this setup, cosmic strings form when the U(1) symmetry in the φ sector breaks, after which evolution in the χ direction can render the string vacuum metastable and trigger delayed decay via O(2)×O(2) bubbles.

Explaining or matching pulsar timing array signals motivates delaying the onset of tunneling. The authors argue this requires the χ direction to be considerably flatter than the φ direction so that roll-over in χ is slow. They explicitly state that it is an open model-building question whether such a flat hierarchy can be achieved in a technically natural way while remaining consistent with observational constraints.