Models to study multi bunch coupling through head-on and long-range beam-beam interactions

In the LHC almost 6000 bunches will collide in four interaction regions where they experience head-on as well as clustered long range interactions. These lead to a coupling between all bunches and coherent beam-beam effects. For two colliding bunches this is well understood. However, for a larger number of bunches colliding with different collision patterns, it results in a complex spectrum of oscillation frequencieswith consequences for beam measurements and Landau damping. To study the coherent beam-beam modes, three complementary models have been developped and will be described in this report. Two of these methods rely on self-consistent multi-bunch and multi-particle tracking while the third is an analytic model based on a complex matrix algorithms. The three methods together provide useful information about the beambeam coupling of multi bunch beams and together provide a deeper insight into the underlying physics.