Equilibrium Beam Invariants of an Electron Storage Ring with Linear X-Y Coupling

In accelerators, it is common that the motion of the horizontal x-plane is coupled to that of the vertical y-plane. Such coupling will induce tune shifts and can cause instabilities. The damping and diffusion rates are also affected, which in turn will lead to a change in the equilibrium in variants. With the perturbative approach which is also used for synchrobetatron coupling [B. Nash, J. Wu, and A. Chao, work in progress], we study the x-y coupled case in this paper. Starting from the one-turn map, we give explicit formulae for the tune shifts, damping and diffusion rates, and the equilibrium invariants. We focus on the cases where the system is near the integer or half integer, and sum or difference resonances where small coupling can cause a large change in the beam distribution.