Beam-beam collisions with an arbitrary crossing angle: analytical tune shifts, tracking algorithm without Lorentz boost, crab-crossing

The beam-beam collision with a finite crossing angle has become a reality since the DAΦNE and KEKB colliders were put in operation. Nowadays, designs or/and upgrade options for a number of colliders and particle factories are based on lattices having a crossing angle. Recently new innovative schemes were proposed, such as crab-crossing for KEKB and very large crossing angle for DAΦNE-II. Hence it is clear that the beam physics related to the crossing angle has become critically important. In this paper we would like to continue discussing these questions, already raised in ICFA Beam Dynamics Letter 34 by Y. Cai.