Q-balls from thermal balls during a first-order phase transition: a numerical study

We numerically study the Q-ball formation triggered by a cosmological first-order phase transition within the Friedberg-Lee-Sirlin model. By performing lattice simulations, we track the nonequilibrium dynamics throughout the transition, providing a precise description of the Q-ball formation mechanism and the resulting mass spectrum. Collapsing false-vacuum regions first form thermal balls, which subsequently cool via dissipative interactions and stabilize into long-lived Q-balls with nonzero spin. We observe a large population of low-mass Q-balls, as well as rare, massive Q-balls that are several times larger than the analytical prediction. The final Q-ball population exhibits a broad mass spectrum spanning over two orders of magnitude, characterized by an exponential tail of number density at large masses. The simulations suggest that the Q-ball abundance is approximately $50%$ higher than predicted by analytical estimates, adjusting the result in the context of Q-balls as dark matter candidates.

💡 Research Summary

This paper presents the first comprehensive numerical investigation of Q‑ball formation during a cosmological first‑order phase transition (FOPT) within the Friedberg‑Lee‑Sirlin (FLS) model. The FLS model contains a complex scalar field χ carrying a conserved global U(1) charge Q and a real scalar field ϕ whose vacuum expectation value determines the effective mass of χ, mχ(ϕ)=g ϕ. In regions where ϕ≈0 (the false vacuum), χ is essentially massless, while in the true vacuum χ becomes massive. This mass hierarchy creates a natural trapping mechanism: χ particles are energetically driven into the false‑vacuum pockets, where they can accumulate large charge densities.

The authors implement three‑dimensional lattice simulations using the open‑source CosmoLattice package on a 512³ cubic grid with periodic boundary conditions. Spatial derivatives are discretized with a second‑order central finite‑difference scheme and time evolution is performed via the leapfrog algorithm with a time step Δt=0.2 Δx for numerical stability. The effective potential for ϕ includes a cubic term, a quartic term, and a temperature‑dependent mass term; the latter receives a mean‑field contribution from χ‑fluctuations. Dissipation from the surrounding plasma is modeled by a friction term Γ ∂tϕ, but in the presented runs Γ is set to zero, effectively neglecting bulk fluid motion.

To trigger the FOPT, the authors nucleate Nb=16 critical bubbles with random, non‑overlapping centers. A net U(1) charge is injected by initializing the canonical momentum of χ with a small asymmetry, thereby mimicking a primordial charge asymmetry. The simulations track the full nonlinear dynamics of bubble expansion, collision, and the subsequent field evolution.

The evolution proceeds in two distinct stages. First, collapsing false‑vacuum regions contract rapidly, forming high‑density “thermal balls” in which χ particles of both signs coexist. The kinetic pressure of these particles temporarily halts further contraction. Second, dissipative effects (encoded in the friction term) drain kinetic energy and allow opposite‑sign charges to annihilate, cooling the thermal ball. The remaining net charge stabilizes the configuration, producing a long‑lived Q‑ball. In many cases the final object is not perfectly spherical but exhibits a dumbbell‑like shape, indicating substantial angular momentum—these are identified as spinning Q‑balls.

A single‑bubble test shows the effective radius R(t) of the false‑vacuum pocket undergoing an initial rapid drop (thermal‑ball formation), followed by an approximately linear decrease during cooling, and finally reaching a constant value once the Q‑ball stabilizes (t ≳ 770 ω*⁻¹).

In the multi‑bubble scenario, the authors perform 30 independent realizations, each yielding a total of 3 352 Q‑balls at the final simulation time t = 1000 ω*⁻¹. Visualizations reveal a mixture of spherical low‑mass Q‑balls and a minority of larger, often spinning objects. The mass spectrum is quantified using the dimensionless variable (\tilde{M}=M,\omega_*,\eta^2). For (\tilde{M}>200) the distribution follows an exponential law
(P(\tilde{M})\propto \exp