Effect of Beamstrahlung on Bunch Length and Emittance in Future Circular e+e- Colliders

In future circular e+e− colliders, beamstrahlung may limit the beam lifetime at high energies, and increase the energy spread and bunch length at low energies. If the dispersion or slope of the dispersion is not zero at the collision point, beamstrahlung will also affect the transverse emittance. In this paper, we first examine the beamstrahlung properties, and show that for the proposed FCC-ee, the radiation is fairly well modelled by the classical formulae describing synchrotron radiation in bending magnets. We then derive a set of equations determining the equilibrium emittances in the presence of a nonzero dispersion at the collision point. An example case from FCC-ee will serve as an illustration.