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.

在2012年发现希格斯玻色子后，提出了一些未来的圆形对撞机（希格斯工厂），例如FCC-ee和CEPC。在这些e \（^ + \） e \（^-\）在对撞机中，束流冲击可以极大地影响平衡束长度和能量散布。如果在碰撞点处的色散函数不为零，则束射束也将增加横向发射率。在本文中，我们考虑了碰撞过程中变化的电磁场，并考虑了沙漏效应，交叉角和蟹腰，得出了平衡梁参数的解析公式。通过考虑一些示例，我们说明了束流束缚的重要性，包括束长度的增加以及在碰撞点处的杂散函数的隐含公差。