Beam Blow Up due to Beamstrahlung in Circular $e^+e^-$ Colliders

After the discovery of the Higgs boson in 2012, several future circular colliders — Higgs factories — are proposed, such as FCC-ee and CEPC. At these e$^+$e$^-$ colliders, beamstrahlung can greatly affect the equilibrium bunch length and energy spread. If the dispersion function at the collision point is not zero, beamstrahlung will also increase the transverse emittances. In this paper, we derive analytical formulae for the equilibrium beam parameters, considering the varying electromagnetic field during the collision, and taking into account hourglass effect, crossing angle, and crab waist. We illustrate the importance of beamstrahlung, including the increase of bunch length and the implied tolerance on the spurious dispersion function at the collision point, by considering a few examples.