A study of the Coherent Beam-Beam Effect in the Framework of the Vlasov Perturbation Theory

A number of factors which can influence coherent beam-beam oscillations are studied on the basis of the Vlasov equation:difference in the intensities and single-particle tunes in the beams;difference in the phase advances between interaction points;long-range interactions; synchro-betatron coupling due to betatron phase advance variation in the vicinity of IP chromatic tune modulation and crossing angle. The synchro-betatron coupling appears to have a principal stabilizing effect:at synchrotron tune values in the neighborhood of half the beam-beam parameter it provides Landau damping of the discrete spectral lines by overlapping sidebands; the damping rate being higher with negative chromaticity. At smaller synchrotron tune values a specific mode of the head-tail damping sets in which is not sensitive to chromaticity. Application to LHC shows that with the design values of parameters the coherent beam-beam oscillations should be Landau damped, probably with the help of negative chromaticity of a moderate absolute value.