Simulations of coherent beam-beam modes at the Large Hadron Collider

The transverse coherent motion of two colliding proton beams at the Large Hadron Collider is studied by multiparticle tracking. We use the beam-beam force for a Gaussian beam distribution with variable barycenters and rms beam sizes, and optionally include the effect of long-range collisions and external impedance. The simulation yields the coherent and incoherent oscillation frequencies, the emittance growth of either beam, and evidence for the existence or lack of Landau damping. For head-on collisions of beams with equal sizes, we find that the π-mode frequency lies outside of the continuum frequency spread, if the ratio of the beam-beam parameters exceeds 0.6, in accordance with predictions. For smaller ratios of the beam-beam parameters, or if, for equal beam-beam parameters, the beam sizes are widely different, the π mode is Landau damped. When long-range collisions are also included, undamped coherent modes do still exist outside the continuum, both with and without alternating crossing planes at two interaction points. However, separating the tunes of the two beams restores the Landau damping, provided the external impedance is sufficiently small.