String-induced vacuum decay

Abstract: False vacuum decay typically proceeds via the nucleation of spherical bubbles of true vacuum, described by $O(4)$ symmetric field configurations in Euclidean time. In this work, we investigate how the presence of cosmic strings can catalyze the decay process. To this end, we consider a complex scalar field charged under a global or local $U(1)$ symmetry. Assuming a non-trivial vacuum manifold, realizable for example in a simple sextic potential, we derive relativistic bounce solutions with $O(2) \times O(2)$ symmetry, corresponding to elongated bubbles seeded by a cosmic string of the same scalar field. Building up on earlier results in the literature, we identify the region of parameter space where vacuum decay predominantly proceeds via this alternative channel, thereby providing an explicit mechanism for the quantum decay of cosmic strings. Finally, we present an initial discussion of the gravitational wave signal associated with this type of vacuum decay and its possible connection to the recently observed stochastic signal in pulsar timing arrays.

• The paper develops a novel mechanism where cosmic strings catalyze false vacuum decay via O(2)xO(2) bubble nucleation, leading to enhanced tunneling rates compared to the Coleman O(4) process.
• It employs a thin-wall approximation with relativistic bounce solutions and provides semi-analytic fits for the normalized bounce action and bubble radius.
• The study reveals significant cosmological implications, including unique gravitational wave signatures from non-spherical bubbles and potential impacts on early Universe dynamics.

String-Induced Vacuum Decay: Mechanism, Dynamics, and Cosmological Implications

Introduction and Motivation

The paper "String-induced vacuum decay" (2510.27579) presents a comprehensive analysis of false vacuum decay catalyzed by cosmic strings, focusing on the nucleation of elongated $O(2)\times O(2)$ symmetric bubbles along the string axis. This mechanism provides an alternative to the conventional Coleman $O(4)$ symmetric bubble nucleation, with significant implications for the dynamics of topological defects and the generation of gravitational waves (GWs) in the early Universe. The study is grounded in a minimal model of a complex scalar field with a sextic potential, charged under either global or local $U(1)$ symmetry, and explores both the theoretical underpinnings and phenomenological consequences of string-induced vacuum decay.

Figure 1: Schematic illustration of the Coleman $O(4)$ bounce (left), and the $O(2)\times O(2)$ bounce in the presence of a cosmic string (right).

Theoretical Framework and Model Construction

The authors construct a minimal setup based on a complex scalar field $\phi$ with a sextic potential:

$$V(\phi) = V_1 + \mu^2 |\phi|^2 - \lambda |\phi|^4 + \lambda_6 |\phi|^6$$

where the parameter regime is chosen to ensure a metastable symmetry-breaking false vacuum at $|\phi|=v_f/\sqrt{2}$, separated from the symmetric true vacuum at $\phi=0$ by a potential barrier. This structure allows for the existence of cosmic strings, which are classically stable due to nontrivial winding, but become metastable in the presence of quantum tunneling.

Figure 2: Sketch of the potential in Eq.~\eqref{eq:V2}, illustrating the metastable false vacuum and the true vacuum separated by a barrier.

The thin-wall approximation is employed to analyze both the string and bounce solutions, with explicit derivations of the wall profile and energy functionals for global and local strings. The validity of the thin-wall regime is carefully assessed, with quantitative constraints on the winding number $n$ and gauge coupling $g$ for reliable application.

Relativistic Bounce Solutions and String-Induced Tunneling

A central result is the derivation of $O(2)\times O(2)$ symmetric bounce solutions, corresponding to bubble nucleation along the string axis. The authors provide a fully relativistic treatment, incorporating Lorentz contraction effects via a $\gamma$-factor in the wall profile ansatz, which is essential for accurately capturing the post-nucleation dynamics.

Figure 3: Radial profiles $R(\varrho)$ of the bounce solution for global (left) and local (right) strings for various parameter values, converging to the $O(4)$-symmetric solution for $x,y \ll 1$.

Numerical solutions are obtained using a shooting algorithm, and semi-analytic fitting functions are provided for the normalized bounce action and radius as functions of the dimensionless parameters $x$ (global string) and $y$ (local string). The bounce action for the string-induced channel is always smaller than the Coleman $O(4)$ action, $b \leq 1$, indicating an enhanced tunneling rate in a broad parameter regime.

Figure 4: Numerical bounce results for global (black) and local (blue) strings, showing the normalized bounce action and radius as functions of $x$ and $y$, respectively.

Phenomenological Implications: Vacuum Decay and Gravitational Waves

Dominance of String-Induced Decay

The nucleation rate for string-induced bubbles is compared to the standard Coleman mechanism. The string-induced channel dominates when $b \lesssim 1/2$, corresponding to $0.4 \lesssim x \leq 1$ (global) and $0.1 \lesssim y \leq 1$ (local), with explicit constraints on model parameters. This regime renders the conventional $O(4)$ decay channel phenomenologically irrelevant.

Quadrupole Moment and GW Emission

A key feature of the $O(2)\times O(2)$ bubbles is their non-vanishing, time-dependent quadrupole moment, which leads to GW emission even from isolated expanding bubbles, in contrast to the spherically symmetric case.

Figure 5: The regularized quadrupole moment $Q^\mathrm{(reg)}$ as a function of bubble radius, demonstrating power-law growth and nonzero GW emission for non-spherical bubbles.

The quadrupole moment scales as $Q^\mathrm{(reg)}(t) \sim R(t)^{\beta_0}$ with $\beta_0 \approx 4$, and the GW contribution from individual bubble expansion is generally suppressed relative to bubble collisions, except in scenarios with rapid percolation or large initial non-sphericity.

GW Spectrum from Metastable String Networks

The GW spectrum from a decaying cosmic string network is computed, showing a suppression at low frequencies due to the finite lifetime of the network. This effect allows evasion of stringent PTA and CMB bounds on the string tension, $G\mu_s$, and provides a potential explanation for the observed stochastic GW background in pulsar timing arrays.

Figure 6: Schematic illustration of the three main GW sources: bubble collisions (blue), expansion of non-spherical bubbles (orange), and loop dynamics in the transient string network (green).

Figure 7: GW spectra from a metastable local string network for different phase transition temperatures, illustrating suppression of the low-frequency spectrum and compatibility with NANOGrav observations.

Model Realization and Cosmological Scenarios

A two-field model is introduced to account for both the formation and quantum decay of cosmic strings, with a real scalar $\chi$ coupled to the tunneling field $\phi$. The evolution proceeds through symmetry breaking, string formation via the Kibble mechanism, and delayed symmetry restoration via string-induced vacuum decay.

Figure 8: The two-field potential $V(\phi, \chi)$, showing the sequence of field evolution: symmetry restoration, string formation, metastable minimum, and final decay.

This framework enables the construction of cosmological scenarios where the timing and dynamics of the phase transition can be tuned to evade observational constraints and potentially address the Hubble tension via energy injection into the dark sector.

Conclusion

The analysis establishes that cosmic strings can catalyze false vacuum decay via $O(2)\times O(2)$ bubble nucleation, with enhanced tunneling rates and distinctive GW signatures. The mechanism is robust for both global and local strings, with explicit parameter regimes identified for dominance over the standard Coleman channel. The GW phenomenology of metastable string networks provides a natural avenue to evade current observational bounds and may offer an explanation for the stochastic GW background observed by PTAs. The two-field model realization further enables the embedding of this mechanism into broader cosmological frameworks, including those addressing the Hubble tension. Future work should focus on refining the treatment of gauge field dynamics, detailed GW signal modeling, and the integration of these mechanisms into early dark energy scenarios.