Metastable Strings and Gravitational Waves in One-Scale Models

TL;DR

The paper investigates metastable cosmic strings as a minimal origin for the nanohertz stochastic gravitational-wave background reported by Pulsar Timing Arrays. It analyzes a one-scale, electroweak-like dark sector with gauge group $SU(2)\times U(1)\to U(1)$ and a single Higgs, computing the Z-string tension via Nielsen–Olesen profiles and the delocalised-flux tension via a CP^1 sigma-model reduction, yielding $\alpha_{\rm str}(\beta)$ and $\alpha_{\infty}=\pi$. The quantum decay of the long string is treated in the thin-defect, semiclassical limit, giving a bounce action $S_B=\pi\kappa$ with $\kappa=8\pi^{2}/(g^{2}\alpha_{\rm str}(\beta))$, and a decay rate $\Gamma\simeq (\mu_{\rm str}/2\pi) e^{-S_B}$. By matching $\sqrt{\kappa}$ to the PTA-preferred range $[7.5,8.5]$, the authors identify regions in parameter space (notably with sizable $\alpha'$) where the one-scale model reproduces the PTA signal without introducing extended Higgs sectors or multi-stage symmetry breaking, demonstrating a robust, minimal mechanism linking microscopic dark-sector physics to gravitational-wave observations.

Abstract

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, and verify that the thin-defect approximation remains valid throughout the phenomenologically favoured region, without invoking extended Higgs sectors or multi-stage symmetry breaking.

Figures (3)

• Figure 1: Left: Profile functions $P(\tilde{r})$ and $V(\tilde{r})$ for the classical string configuration. The functions are plotted for $\beta=0.02,\,0.2$ and 1. These functions are independent of $g$. Right: The blue line shows the dependence of $\alpha_\text{str}$ on $\log\beta$, for the range $-2<\log_{10}\beta<0$, where the string solution can be shown to be metastable. Note also that $\alpha_\text{str}$ depends on $\beta$ only, and has no further dependence on $g$. The grey dotted line shows the analytically approximation for $\alpha_{\rm str}$ valid in the small-$\beta$ limit and the red line $\alpha_{\rm str}=\alpha_{\infty}=\pi$.
• Figure 2: Left: $Z$-string ending on a Nambu monopole with core size $R_{\rm core}$. Right: Euclidean worldsheet of a $Z$-string (width $\delta_s$) with nucleation of a monopole–antimonopole pair, whose closed worldline appears as a circle.
• Figure 3: Regions of parameter space in the $(\beta, \sin^2\theta_w)$ plane. Shaded in pink are points with $\kappa>8.5$; dark green corresponds to $8.5>\kappa>8$; light green to $8>\kappa>7.5$; and blue to $\kappa<7.5$. The grey contour shows the metastable region taken from Fig. 1 of Ref. James:1992wb, based on an analysis of the spectrum of single particle fluctuations around the classical string background. The three panels correspond to different values of the $U(1)$ fine structure constant of $\alpha'=g'^2/4\pi$. An overlap between the metastable region and the $\kappa$-favoured band is visible for $\alpha' \ge 3/4$, showing that the PTA-preferred range can be accommodated for moderately large $U(1)$ gauge couplings.