A map for the thick beam-beam interaction

We give a closed-form expression for the thick beam-beam interaction for a small disruption parameter, as typical in electron-positron storage rings. The dependence on transverse angle and position of the particle trajectory as well as the longitudinal position of collision and the waist-modified shape of the beam distribution are included. Large incident angles, as are present for beam-halo particles or for large crossing-angle geometry, are accurately represented. The closed-form expression is well approximated by polynomials times the complex error function. Comparisons with multi-slice representations show even the first order terms are more accurate than a five slice representation, saving a factor of 5 in computation time