Modeling the electron positron beam-beam interaction with non-Gaussian beam distributions

A new program to simulate the beam-beam interaction between asymmetric {dollar}esp+{dollar} and {dollar}esp-{dollar} beams has been developed. Beam bunch distributions are expanded in terms of orthogonal basis functions constructed from solutions to the two-dimensional quantum mechanical harmonic oscillator. Including all solutions corresponding to oscillator energies up to the {dollar}Nsp{lcub}rm th{rcub}{dollar} level yields a basis which spans the set of functions composed of a product of a Gaussian times a Hermite polynomial of order N or lower. A consistent and economical description of non-Gaussian beam shapes is thus made possible. In addition, the use of continuous density functions isolates the effects of statistical fluctuations which arise when beam bunches are modeled by tracking particles. The beam dynamics are encapsulated in matrices which operate on the expansion coefficients of the bunches. These matrices are computed once for each beam with any given set of basis functions and for any particular accelerator. The evolution of a beam distribution is computed by matrix multiplication.