In recent $e^{+}{e}^{-}$ colliders, a collision scheme with a large crossing angle has been designed and adopted. In two colliding beams, mode coupling between beam-beam oscillation modes is discussed in this paper. The typical $\sigma$ and $\pi$ modes are extended to include longitudinal motion. The tune shifts of the $l=0$ and $l=\pm 1$ modes are presented explicitly. The behavior of beam-beam modes is analyzed for different Piwinski angles. The instability due to the vacuum pipe impedance is discussed in terms of mode coupling between these beam-beam modes. Beam-beam simulations based on the strong-strong model have shown that a vertical beam-beam instability is induced by crosstalk between the beam-beam collision and the impedance. It is shown that the mode coupling theory for colliding beams reproduces the results of the beam-beam simulation. The impact of hourglass effects on the instability is also studied by simulation.

• Received 3 June 2023
• Accepted 19 October 2023

DOI:https://doi.org/10.1103/PhysRevAccelBeams.26.111001

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society