Impact of the resistive wall impedance on beam dynamics in the Future Circular $e^+ e^-$ Collider

The Future Circular Collider study, which aims at designing post-LHC particle accelerator options, is entering in the final stage, which foresees a conceptual design report containing the basic requirements for a hadron and a lepton collider, as well as options for an electron-proton machine. Due to the high beam intensities of these accelerators, collective effects have to be carefully analyzed. Among them, the finite conductivity of the beam vacuum chamber represents a major source of impedance for the electron-positron collider. By using numerical and analytical tools, a parametric study of longitudinal and transverse instabilities caused by the resistive wall is performed in this paper for the case of the Future Circular Collider lepton machine, by taking into account also the effects of coating, used to fight the electron cloud build up. It will be proved that under certain assumptions the coupling impedance of a two layer system does not depend on the conductivity of the coating and this property represents an important characteristic for the choice of the material itself. The results and findings of this study have an impact on the machine design in several aspects. In particular the quite low threshold of single bunch instabilities with respect to the nominal beam current and the not negligible power losses due to the resistive wall are shown, together with the necessity of a new feedback system to counteract the fast transverse coupled bunch instability. The importance of a round vacuum chamber to avoid the quadrupolar tune shift is also discussed. Finally the crucial importance of the beam pipe material coating and thickness choice for the above results is underlined.