Coherent beam-beam tune shift of unsymmetrical beam-beam interactions with large beam-beam parameter.

Coherent beam-beam tune shift of unsymmetrical beam-beam interactions was studied experimentally and numerically in HERA, where the lepton beam has such a large beam-beam parameter (up to xi(y) = 0.272 ) that the single-particle motion is locally unstable at the origin (closed orbit). Unlike the symmetrical case of beam-beam interactions, the ratio of the coherent beam-beam tune shift and the beam-beam parameter in this unsymmetrical case of beam-beam interactions was found to decrease monotonically with an increase of the beam-beam parameter. The results of self-consistent beam-beam simulation, the linearized Vlasov equation, and the rigid-beam model were compared with the experimental measurement. It was found that the coherent beam-beam tune shifts measured in the experiment and calculated in the simulation agree remarkably well but they are much smaller than those calculated by the linearized Vlasov equation with the single-mode approximation or the rigid-beam model. The study indicated that the single-mode approximation in the linearization of the Vlasov equation is not valid in the case of unsymmetrical beam-beam interactions. The rigid-beam model is valid only with a small beam-beam parameter in the case of unsymmetrical beam-beam interactions.