Renormalized Fokker-Planck equation for the problem of the beam-beam interaction in electron storage rings

It is shown that the beam-beam interaction in electron storage rings is equivalent to an additional source of noise for the particle betatron oscillations. A weak white noise acting upon a nonlinear oscillator causes a fast loss of coherence in its phase. This loss of coherence induces a broadening of the resonances, thus avoiding the problem of the divergent perturbative series which arises in the study of nonintegrable Hamiltonian systems. A “renormalized” Fokker-Planck equation is established which contains new diffusive terms corresponding to the presence of resonances. The solution of this equation is exhibited explicitly in a simplified case. This allows an analytical approach to the problem of the beam-beam instability, which sets an upper limit to the maximum attainable luminosity in storage rings.