| Summary: | The luminosity in flat-beam circular colliders is known to “saturate” at some “threshold” beam current above which (because the beam height grows) the luminosity varies (only) linearly with beam current, making both the specific luminosity (luminosity/current) and the beam-beam tune shift parameter ξ_{y} independent of current. The purpose of this paper is to calculate ξ_{y} analytically with the goal of maximizing the luminosity. A zero parameter application of the theory to 13 existing storage ring configurations yields theory/experiment equal to 1.26±0.45 for ξ_{y,max}. Parameter values (especially tunes Q_{x}, Q_{y}, and Q_{s}) expected to maximize ξ_{y} are given. The most favored tune combinations seem not to have been tried so far in colliding beam facilities. The vertical beam growth is ascribed to “parametric pumping” of the vertical betatron amplitude of each individual particle by its own (inexorable) horizontal and longitudinal oscillation. A unique determination of the distribution of all particles then follows from a saturation principle which asserts that the beam height adjusts itself to the value for which the least stable particle (of probable amplitude) is barely stable. The difference equation describing the pumping can be solved by numerical iteration or, because it is (almost) linear, it can be solved analytically, at least for amplitudes small enough that resonances remain isolated. Because of the aliasing (or undersampling) characteristic of accelerators, this equation exhibits an even richer spectrum of resonances than the Mathieu equation, which the present theory generalizes. Contrary to the lore of the field (which motivates the intentional increase of damping decrement δ_{y} using wigglers), the theory presented here predicts the dependence of luminosity on δ_{y} to be quite weak. This is not inconsistent with actual collider performance according to a survey by Rice [D. Rice, Cornell University Report No. CBN 01-09 (2001)] of colliding rings built thus far.
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