BeamBeam3d Tests

Poisson Solver

Two bunch coherent modes

This simulation ran one proton bunch, one antiproton bunch through a ring with one interaction point. The parameters of the beam are:

The initial offset of the proton beam was set to 3e-6 m. The turn by turn dipole moment of the simulated beams was analyzed by FFT to get the tunes of the bunch modes. The sigma mode is at the same tune as a non-interacting beam bunch has the two bunches moving in phase with each other. The pi mode is shifted upwards (because the beams have opposite charge) corresponds to the bunches oscillating out of phase.

Calculations by Yokoya (Part. Accel. 27, 83 (1990)) show that the amount of the shift is approximately 1.3 times the beam-beam parameter :

Here, is the number of protons in the bunch, is the classical radius of the proton , is the Lorentz factor. For this example, had a value of 0.00727, which agrees very well with the shift from 0.579 to 0.588.

Four bunch coherent modes

The next calculation shows the simulated tune spectrum with two bunches in one beam interacting with two bunches in the opposing beam. For this simulation, the bunches in the proton beam have 10% greater intensity than the bunches in the antiproton beam to break a four-fold degeneracy of tune modes.

The tunes for the coupled modes for this system can be computed in a linearized beam-beam model as the phase of the eigenvalues of the one-turn transfer matrix that includes the beam-beam kick:

The tune spectrum for the combination y1-y2+y3-y4 is shown below. This mode deemphasizes the sigma mode showing the collective pi modes. The sigma mode is at 0.579, (the unperturbed bare tune); the broad peak around 0.585 are two nearly degenerate pi modes at 0.586 and 0.587 which correspond to direct bunch-to-bunch interaction, and the pi mode at 0.598 which corresponds to the interaction of bunches in the same beam interacting indirecting through bunches in the opposing beam.