6D beam-beam interaction step-by-step

This document describes in detail the numerical method used in different simulation codes for the simulation of beam-beam interactions using the “Synchro Beam Mapping” approach, in order to correctly model the coupling introduced by beam-beam between the longitudinal and the transverse plane. The goal is to provide in a compact, complete and self-consistent manner the set of equations needed for the implementation in a numerical code. The effect of a “crossing angle” in an arbitrary “crossing plane” with respect to the assigned reference frame is taken into account with a suitable coordinate transformation. The employed description of the strong beam allows correctly accounting for the hour-glass effect as well as for linear coupling at the interaction point.