Coherent Dipole Modes for Multiple Interaction Regions

In the Large Hadron Collider (LHC) two proton beams of similar strength will collide at several interaction points. For a single interaction point it is known that the head-on collision of two equally strong beams with the same betatron tune, excites two coherent dipole modes whose frequencies are different from the frequencies of oscillation of individual particles in the beam. Because of this frequency difference Landau damping does not act on the dipole modes and the beams can be unstable. In this paper we extend these studies to several interaction points and explore the possibility of cancellation of the dipole coherent modes by carefully adjusting the phase difference between the beams from one collision to the next. We also study the collision of the two beams with LHC optics V 6.1. Special attention should be paid to coherent resonances that are excited due to local phase advance correlations. It will be shown also that a tune split of 0.03 between the two beams suppresses these coherent dipole modes.