Decay rate of PQ-ball

Q-balls are non-topological solitons that arise in theories with a complex scalar field possessing a conserved global U(1) charge. Their stability is ensured by this charge, making them potentially significant in cosmology. In this paper, we investigate Q-ball-like objects in scenarios where the scalar field acquires a finite vacuum expectation value, spontaneously breaking the global U(1) symmetry. A well-motivated example is the Peccei-Quinn field, where the U(1) symmetry is identified as the Peccei-Quinn symmetry, and hence we refer to such objects as PQ-balls. We first discuss the existence of stable PQ-ball solutions in a finite-density plasma and argue that they become unstable in vacuum. Using detailed numerical simulations under spherical symmetry, we confirm their formation, compute their decay rate, and derive an analytical formula for it. Our results have important implications for axion cosmology, particularly in the context of the kinetic misalignment mechanism.