Approximate Physical Treatment of the Beam-Beam Effects

When the beam-beam force is approximated as a periodic δ-function non-linear potential kicks, it is expected to excite a continuum(density of rational numbers) of resonances. The totality of resonance within a narrow "width" of the betatron oscillation tune contribute to a diffusion-like amplitude growth as if the kicks are random. A semi-quantitative formulation is derived and applied to electron colliding beams.