Hourglass effects for asymmetric colliders

Expressions are presented for the geometrical reduction factor of the luminosity and the geometrical beam-beam aggravating factor for the general asymmetric case, for tri-Gaussian bunches colliding head-on. With these formulas the author attempts a (limited) analytic understanding of the multiparticle tracking simulations carried out for the proposed SLAC (Stanford Linear Accelerator Center)/LBL (Lawrence Berkeley Laboratory)/LLNL (Lawrence Livermore National Laboratory) B factory when parasitic crossings are ignored. It is concluded that (a) the geometrical reduction in luminosity is approximately 6% relative to the zero-bunch-length (nominal) value; (b) only the vertical beam-beam parameter of the LER is significantly altered by the hourglass effect: the geometrical enhancement of the central positron´s vertical beam-beam parameter is approximately 10% relative to the nominal value, and (c) the positrons at the head or tail of the bunch have vertical beam-beam parameters much larger than nominal.<>