  Coherent beam-beam tune shift of unsymmetrical beam-beam interactions was
studied experimentally and numerically in HERA where the lepton beam has a very
large beam-beam parameter (up to $\xi_y=0.272$). Unlike the symmetrical case of
beam-beam interactions, the ratio of the coherent and incoherent beam-beam tune
shift in this unsymmetrical case of beam-beam interactions was found to
decrease monotonically with 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 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.
