Analytical estimation of the beam–beam interaction limited dynamic apertures and lifetimes in e+e− circular colliders

Physically speaking, the delta function like beam–beam nonlinear forces at interaction points act as a sum of delta function nonlinear multipoles. By applying the general theory established by J. Gao (Nucl. Instr. and Meth. A 451 (2000) 545), in this paper we examine analytically the beam–beam interaction limited dynamic apertures and the corresponding beam lifetimes for both the round and the flat beams. Relations between the beam–beam limited beam lifetimes and the beam–beam tune shifts are established, which show clearly why experimentally one has always a maximum beam–beam tune shift, ξy,max, around 0.045 for e+e− circular colliders, and why one can use round beams to double this value approximately. Comparisons with some machine parameters are given.