An interplay between beam–beam and beam coupling impedance effects in the Future Circular e$^+$e$^-$ Collider

In order to reach high luminosity, the Future Circular e$^+$e$^-$ Collider will need very intense beams with small emittances and small beta functions in the interaction points. This is achieved with a large Piwinski angle combined with the crab waist collision scheme. Under these conditions, the luminosity and the beam–beam tune shifts are strongly influenced by the bunch length. On the other hand, in this machine, the beamstrahlung effect is dominant too, leading to an increase of bunch length and energy spread. Moreover, due to the extreme beam parameters, new important beam–beam instabilities have been found, such as the so-called coherent X-Z instability. Finally, the bunch length and energy spread are also affected by collective effects. In this paper, we study the beam–beam interaction, by focusing on the X-Z instability for FCC-ee, in the lowest energy configuration (45.6 Gev, Z-resonance), by taking into account, self-consistently, the combined effects of beamstrahlung and the machine impedance model that has been evaluated so far. Finally, we also discuss some possible mitigation methods.