Beam-Beam Limit in $e^{+}{e}^{-}$ Circular Colliders

Beam-beam effects limit the luminosity of circular colliders. Once the bunch population exceeds a threshold, the luminosity increases at a slower rate. This phenomenon is called the beam-beam limit. Onset of the beam-beam limit has been analyzed with various simulation methods based on the weak-strong and strong-strong models. We have observed that an incoherent phenomenon is mainly concerned in the beam-beam limit. The simulation have shown that equilibrium distributions of the two colliding beams are distorted from Gaussians when the luminosity is limited. The beam-beam limit is estimated to be $\xi \sim 0.1$ for a B factory with damping time of several thousand turns.

Figure 1

Evolution of luminosity, vertical rms beam size, and kurtosis ($⟨y^4⟩/3⟨y^2⟩$) obtained by the strong-strong simulation.

Figure 2

Beam sizes and the beam-beam parameter function of the nominal beam-beam parameter obtained by the strong-strong simulations with the particle-in-cell method and Gaussian approximation. Horizontal and vertical beam sizes are depicted in (a) and (b). Effective beam-beam parameters are depicted in (c).

Figure 3

Evolution of the vertical distribution of electron and positron beams. The distributions during the sudden change and after 35 000 turns are depicted in (a) and (b), respectively.

Figure 4

Evolution of the beam-beam parameter and vertical size of electron and positron beams. Evolutions of $\xi$ for the weak-strong (W.S.) and strong-strong (S.S.) simulation are depicted in (a), where the final distribution of the strong-strong simulation is used as a fixed distribution in the weak-strong simulation. Evolution of the beam size for the weak-strong simulation is depicted in (b). Evolution of the size of the positron beam that interacts with fixed a Gaussian electron beam is depicted in (c).