How to increase the physics output per MW.h for FCC-ee?: Parameter optimization for maximum luminosity

The efficiency of colliders for physics is largely determined by their luminosity, while most of the energy consumed by high-energy e$^+$e$^-$ colliders is proportional to the total beam current. Thus, the energy efficiency is mainly determined by the specific luminosity that needs to be maximized. One of the most effective ways to achieve this is by using the Crab waist collision scheme, which implies a large Piwinski angle (LPA). A distinctive feature of the FCC-ee is the great influence of beamstrahlung (radiation in the field of an opposite bunch) on beam dynamics. At low energies, this manifests itself in a significant increase in the energy spread and bunch length, at high energies, in a limitation of the beam lifetime. The collision of intense bunches with LPA and beamstrahlung can also lead to various kinds of instabilities limiting the luminosity. Here, we discuss the main aspects to consider when optimizing the parameters of the FCC-ee collider at different energies and explain the choice of basic parameters such as RF voltage, lattice functions at IP, bunch intensity, etc. We will also pay attention to open issues requiring further study and identify some key points for the next phase of this project.