Abstract
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 $\ensuremath{\pi}$-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 $\ensuremath{\pi}$ 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.
Highlights
Two colliding beams exert a force on each other which is focusing for beams of opposite polarity and defocusing for those of equal polarity, as for the two proton beams at the Large Hadron Collider (LHC)
Solutions of the linearized Vlasov equation suggest that in the case of one bunch per beam with equal parameters two dipole coherent modes of oscillation appear: the s mode, whose frequency is equal to the unperturbed betatron tune, and the p mode with a tune shift of 1.21–1.33 times the beam-beam parameter j [1]
The coherent mode frequencies are compared with those predicted from the linearized Vlasov equation, and, for round beams, we study the dependence of the coherent-mode frequencies on the intensity ratio and its consequences for Landau damping and emittance growth
Summary
Two colliding beams exert a force on each other which is focusing for beams of opposite polarity and defocusing for those of equal polarity, as for the two proton beams at the Large Hadron Collider (LHC). In the case of nearly equal horizontal and vertical squared beam sizes, Mxix ഠ Myiy, Eq (2) is ill defined. The results for initially round beams and equal transverse tunes are qualitatively and quantitatively the same as obtained using the exact expression, Eq (2) This seems to indicate that small variations in the relative transverse beam sizes do not much affect the coherent modes. The betatron phase advance between the long-range collision points on one side of the interaction region is small. We lump all npar parasitic collisions into a single kick 90± with npar times enhanced strength, applied behind the IP This may slightly overestimate analytical predictions, for the vertical motion of flat beams. To simulate parasitic (long-range) collisions, the same possible phase differences between bunches [11]. The actual number of parasitic The long-range beam-beam kick is
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