Quasi-Strong-Strong Beam-Beam Modeling of Bootstrapping Injection in FCC-ee

The FCC-ee is designed to operate with exceptionally strong beam–beam interactions, making continuous injection a critical and non-trivial aspect of its operation. During the injection process, an unavoidable charge imbalance between the two colliding beams leads to asymmetric beam–beam forces, potentially compromising transverse stability. In this paper, we introduce a quasi-strong-strong (QSS) beam–beam scheme, implemented in the SAD simulation framework. The method preserves a self-consistent beam–beam lens by coupling paired weak–strong simulations, while avoiding the computational cost of full strong–strong tracking. The injection process is modeled as a gradual increase of the stored bunch population, allowing the isolated study of beam–beam–driven optics deformation under charge imbalance. Using the QSS approach, we investigate the feasibility of bootstrapping injection in the Z, W, and H operating modes of FCC-ee. Stable injection paths up to the nominal bunch population are identified in the W and H modes. In contrast, in the explored parameter region, the Z mode exhibits saturation of the stored population below the nominal value.