Revisiting metastable cosmic string breaking

The study focuses on cosmic strings, linear defects resulting from nontrivial fundamental groups of vacuum manifolds in cosmological phase transitions. These strings, particularly those arising from U(1) symmetry breaking, are significant in new physics searches. Gravitational waves (GWs) are a promising observational channel; while infinitely long strings are stable, loops and interactions produce a stochastic GW background. Recent pulsar timing array (PTA) experiments have reported a stochastic GW signal consistent with metastable cosmic strings, whose metastability suppresses the low-frequency GW spectrum. This paper revisits the string breaking rate estimation, crucial for the interpretation of PTA data, as the conventional approximation (Preskill-Vilenkin) assumes a large hierarchy between breaking scales—an assumption potentially invalid for the parameters favored by observations. The authors re-analyze the breaking rate using Ansätze on the string unwinding process, incorporating the finite sizes of the string and monopoles, aiming for a robust estimate even in regimes where the conventional approximation fails.

The paper draws heavily on existing literature regarding cosmic strings, their formation in cosmological phase transitions, and their gravitational wave emission. It references works on the cosmological evolution of monopoles connected by strings, signals of inflationary models with cosmic strings, and the analysis of gravitational wave backgrounds from cosmic strings using LISA. The authors cite the recent results from multiple PTA collaborations (NANOGrav, EPTA, and Parkes PTA), which have observed a stochastic gravitational wave background. Importantly, the paper engages with previous theoretical work on the gravitational wave spectrum from metastable cosmic strings, pointing out limitations in the existing Preskill-Vilenkin approximation, which ignores finite sizes of the monopoles and strings. Finally, the authors acknowledge the work of Shifman and Yung, whose Ansätze for the string unwinding process they utilize for their analysis.

The study employs an SU(2) gauge theory with an adjoint scalar field and a doublet scalar field, creating a model with hierarchical symmetry breaking. The Lagrangian includes gauge field terms and a Higgs potential with hierarchical vacuum expectation values (VEVs) (V >> v). The analysis involves describing monopoles ('t Hooft-Polyakov monopoles) formed at the first phase transition (SU(2) → U(1)) and Abrikosov-Nielsen-Olesen (ANO) strings formed at the second transition (U(1) → nothing). The authors focus on the process where monopoles and antimonopoles are created inside the string, leading to its breaking. Two Ansätze for the string unwinding process are employed: a primitive Ansatz and an improved Ansatz. The primitive Ansatz assumes a simple unwinding of the string and accounts for the size of the string and monopole. The improved Ansatz offers a refinement by considering separate radial variations for different parts of the configuration, and further allows for better handling of the finite sizes of the monopole and string. The study uses Euclidean path integral methods, and the bounce action (tunneling factor) is numerically evaluated for both Ansätze, comparing to the Preskill-Vilenkin result for comparison and validation. The thin-wall approximation is also explored and compared to the numerical results. The numerical procedure involves minimizing the string tension and solving the Euclidean equations of motion for the unwinding parameter to find the bounce solutions.

The paper's key findings center on the improved estimation of the cosmic string breaking rate, taking into account the finite sizes of the monopoles and strings. For a large hierarchy between the symmetry-breaking scales (V >> v), the authors find that their estimate of the bounce action using the primitive Ansatz agrees with the conventional Preskill-Vilenkin approximation. However, for regimes where the hierarchy is not large (m_w/m_γ = O(10) or less), a regime relevant to interpretations of the PTA data, the conventional approximation is unreliable. The improved Ansatz yielded a larger bounce action than the primitive Ansatz, though the authors point to potential improvement in numerical techniques in this case. The authors find that the thin-wall approximation accurately approximates the full bounce solution when V >> v, but underestimates the bounce action for a mild hierarchy. A major finding is the derivation of a robust lower limit on the tunneling factor, even in regimes where the conventional estimate is not reliable. The numerical results, considering both Ansätze, provide constraints on model parameters consistent with PTA observations, even in the non-hierarchical regime, where the Preskill-Vilenkin approximation is unreliable. The study also analyzes how the bounce action depends on various mass parameters, finding that it decreases with increasing m_h1/m_w (doublet Higgs mass over SU(2) gauge boson mass).

The findings address the limitations of the conventional infinitely thin string approximation in estimating the cosmic string breaking rate, particularly in the context of recent PTA observations. The authors’ analysis provides a more accurate method for estimating the breaking rate, considering finite size effects. The results have significant implications for understanding the origin of the observed stochastic gravitational wave background. The fact that the results are consistent with the PTA data even when the hierarchy between symmetry-breaking scales is not large provides strong support for the proposed model of metastable cosmic strings and shows that this refined method is reliable for interpreting gravitational wave observations from cosmological sources. The authors note that the improved ansatz leads to a larger bounce action but suggest there is potential for improvement in the numerical procedure. Therefore, even though the results in this paper still have some limitations, it shows that even the most basic models of string formation and decay can be improved by accurate analysis methods.

This paper offers a significant improvement in the theoretical understanding of metastable cosmic string breaking. By incorporating finite size effects, the authors provide a more robust and reliable method for estimating the breaking rate. The results demonstrate the limitations of the conventional infinitely thin string approximation and offer insights into interpreting recent PTA data. Future work should focus on further refining the numerical methods, extending the analysis to different symmetry breaking patterns, and addressing the formation of the metastable string network at phase transitions.

The study focuses on a specific symmetry breaking pattern (SU(2) → U(1) → nothing) and may not be directly applicable to all models of metastable cosmic strings. The authors acknowledge potential improvements in the numerical techniques, particularly in handling the improved Ansatz. The analysis assumes that the string configuration with β = 0 is a stable and the lowest energy configuration even for m_w/m_γ = O(1), which remains to be fully demonstrated. Furthermore, the paper does not explicitly address the cosmological formation of the string network and how the assumptions of the model affect the final number of strings in the universe.