Analytical calculation of the smear for long-range beam-beam interactions

The Lie-algebraic method is used to develop generalized Courant-Snyder invariant in the presence of an arbitrary number of beam-beam collisions, head-on or long-range, in a storage ring collider. The invariant is obtained by concatenating nonlinear beam-beam maps in the horizontal plane and to first order in the beam-beam parameter. Tracking evidence is presented to illustrate that with LHC parameters the invariant is indeed preserved and can be used to predict the smear of horizontal emittance observed in tracking simulations. We discuss the limits of applicability of this model for realistic LHC collision schemes.