Quantum-Corrected Q-balls in the Friedberg-Lee-Sirlin Model

We study the real-time quantum dynamics of Q-balls in the Friedberg-Lee-Sirlin model within the inhomogeneous Hartree approximation. The mean fields are evolved self-consistently with the leading quantum two-point functions, which are implemented numerically through a stochastic ensemble representation. After introducing a renormalized formulation and a classical-limit scaling, we simulate single-Q-ball configurations in $3+1$ dimensions and compare their quantum-corrected evolution with the corresponding classical dynamics. We find a clear separation between a classical regime, where quantum fluctuations remain small and the evolution closely follows the classical solution, and a quantum regime, where the fluctuation sector carries a sizable fraction of the Noether charge. We also observe a periodic exchange of Noether charge between the mean fields and the fluctuation modes within the Hartree approximation. We further investigate the stability of quantum-corrected Q-balls and find an intermediate window in which configurations that are classically stable become unstable once Hartree fluctuations are included. Our results provide a first step toward real-time quantum simulations of Q-balls in renormalizable two-field soliton models.