  Beam-beam interactions pose substantial challenges in the design and
operation of circular colliders, significantly affecting their performance. In
particular, the weak-strong simulation approach is pivotal for investigating
single-particle dynamics during the collider design phase. This paper evaluates
the limitations of existing models in weak-strong simulations, noting that
while they accurately account for energy changes due to slingshot effects, they
fail to incorporate longitudinal coordinate changes ($z$-variation). To address
this gap, we introduce two novel transformations that enhance Hirata's original
framework by including both $z$-variation and slingshot effect-induced energy
changes. Through rigorous mathematical analysis and extensive weak-strong
simulation studies, we validate the efficacy of these enhancements in achieving
a more precise simulation of beam-beam interactions. Our results reveal that
although $z$-variation constitutes a higher-order effect and does not
substantially affect the emittance growth rate within the specific design
parameters of the Electron-Ion Collider (EIC), the refined model offers
improved accuracy, particularly in scenarios involving the interaction between
beam-beam effects and other random diffusion processes, as well as in
simulations incorporating realistic lattice models.
