Longitudinal wakefield implementation in the circulant matrix model

The influence of longitudinal wakefields on the beam dynamics in electron-positron colliders, particularly their role in beam instabilities such as Transverse Mode Coupling Instability (TMCI) and other transverse-longitudinal effects, necessitates a robust approach to accurately model these effects. This work focuses on the implementation of wakefield effects in the Circulant Matrix Model (CMM), a linear model that can facilitate the representation of these instabilities. We study the impact of potential well distortion and synchrotron frequency shifts due to longitudinal wakefields for FCC-ee and implement these effects in the CMM. The implementation is benchmarked against reference multiparticle tracking simulations to validate its accuracy in predicting longitudinal wakefield-driven instabilities. Results enable further studies featuring longitudinal wakefields for collider designs and operating machines.