Transverse equilibria and luminosity enhancement in linear collider beam-beam collisions.

Abstract

It has been observed in simulations of the beam-beam interaction in linear colliders that a near-equilibrium pinched state of the colliding beams develops when the disruption parameter is large (D\ensuremath{\gg}1) [P. Chen and K. Yokoya, Phys. Rev. D 38, 987 (1988); in Frontiers of Particle Beams: Intensity Limitations, edited by M. Dienes, Lecture Notes in Physics Vol. 400 (Springer-Verlag, Berlin, 1992), p. 415]. In this state the beam transverse density distributions are peaked at center, with long tails. We present here an analytical model of the Maxwell-Vlasov equilibrium approached by the beams, that of a generalized Bennett pinch [W. H. Bennett, Phys. Rev. 45, 890 (1934); 98, 1584 (1955)] which develops through collisionless damping due to the strong nonlinearity of the beam-beam interaction. In order to calculate the equilibrium pinched beam size, an estimation of the rms emittance growth is made which takes into account the effects of the initial linear rise of the focusing forces, and of phase-space mismatch during the beam-beam collision. This pinched beam size is used to derive the luminosity enhancement factors whose scaling as a function of D and thermal factor A=${\mathrm{\ensuremath{\sigma}}}_{\mathit{z}}$/${\mathrm{\ensuremath{\beta}}}^{\mathrm{*}}$ is in agreement with the simulation results, and explain the previously noted cubic relationship between round and flat beam enhancement factors. The implications for calculation of differential luminosity, beamstrahlung spectra, and associated coherent beam-beam radiation effects, are discussed.