Numerical Integration of the Vlasov Equation of Two Colliding Beams

Abstract

In a circular collider the motion of particles of one beam is strongly perturbed at the interaction points by the electro-magnetic field associated with the counter-rotating beam. For any two arbitrary initial particle distributions the time evolution of the two beams can be known by solving the coupled system of two Vlasov equations. This collective description is mandatory when the two beams have similar strengths, as in the case of LEP or LHC. The coherent modes excited by this beam-beam interaction can be a strong limitation for the operation of LHC. In this work, the coupled Vlasov equations of two colliding flat beams are solved numerically using a finite difference scheme. The results suggest that, for the collision of beams with equal tunes, the tune shift between the $\sigma$- and $\pi$- coherent dipole mode depends on the unperturbed tune $q$ because of the deformation that the so-called dynamic beta effect induces on the beam distribution. Only when the unperturbed tune $q\rightarrow 0.25$ this tune shift is equal to $Y\times \xi$, with $Y$ the Yokoya factor as predicted from the linearized Vlasov theory. Colliding beams with unequal tunes brings the tunes of the dipole modes back into the continuum, but it also generates a flip-flop asymmetry in the transverse beam size. It will be shown how coherent resonances can excite the amplitude of the coherent modes and induce variations in the beam transverse size (size growth, period-n oscillations) as well as significant deformations of the beam shape