Intrabunch Motion with Both Impedance and Beam-Beam Using the Circulant Matrix Approach

In high-intensity high-brightness circular colliders such as the CERN LHC, coherent beam-beam effects and impedance cannot be treated independently. Coherent beam-beam dipole modes can couple with higher order head-tail modes and lead to the transverse mode coupling instability of colliding beams. This mechanism has been analysed in detail in the past through the eigenvalues, which describe the evolution of the beam oscillation mode-frequency shifts. In this contribution, the transverse mode coupling instability of colliding beams is studied using the eigenvectors, which describe the evolution of the intrabunch motion. As this instability exhibits several mode couplings and mode decouplings, the evolution of the intrabunch motion reveals quite some interesting features (such as a propagation of the traveling-wave not only from the head to the tail but also from the tail to the head and similar intrabunch signals for some mode coupling and mode decoupling), which are compared to past predictions in the presence of impedance only.