Metastable Strings and Gravitational Waves in One-Scale Models

Metastable cosmic strings provide a minimal and predictive origin for the stochastic gravitational-wave background reported by Pulsar Timing Array experiments. We analyse this possibility in electroweak-like dark sectors with a single-stage breaking $SU(2)\times U(1)\!\to\!U(1)$ driven by one Higgs field. In the regime with dark sector Higgs mass below the $Z'$ mass, and for sufficiently small $W'$ mass, the resulting $Z$-string is classically stable but undergoes quantum decay via nucleation of monopole--antimonopole pairs along the string. We compute the corresponding semiclassical bounce action in a thin-defect approximation, treating both the string core and the monopole endpoints as localised defects whose sizes are small relative to their separation in the tunnelling configuration. This yields a decay rate per unit length that depends on the gauge couplings and the mass hierarchy. We delineate the parameter space in which single-scale dark-sector models reproduce the PTA signal, demonstrating the applicability of the thin-defect approximation throughout the phenomenologically favoured region and without invoking extended Higgs sectors or multi-stage symmetry breaking.