Abstract
This paper discusses novel coherent beam-beam instability in collisions with a large crossing angle. The instability appears in the correlated head-tail motion of two colliding beams. A cross-wake force, which is localized at the collision point, is introduced to represent the head-tail correlation between colliding beams. A mode-coupling theory based on this localized cross-wake force enables us to explain the correlated heal-tail instability. The use of a collision scheme with a large crossing angle is becoming popular in the design of electron--positron colliders. An example thereof is the SuperKEKB project, in which a collision with a large crossing angle is performed to boost the luminosity to $0.8\ifmmode\times\else\texttimes\fi{}{10}^{36}\text{ }\text{ }{\mathrm{cm}}^{\ensuremath{-}2}\text{ }{\mathrm{s}}^{\ensuremath{-}1}$. Future circular colliders will also be designed with a large crossing angle. Strong-strong simulations, which have shown the first coherent head-tail instability, can limit the performance of proposed future colliders. The mechanism whereby this instability occurs is mode coupling due to the cross-wake force. This instability may affect all collider designs based on the crab waist scheme.
Highlights
An ordinary transverse wake force characterizes a transverse momentum kick of a particle at z according to the dipole moment density at z0 in a bunch: Z ∞ΔpxðzÞ 1⁄4 − Wðz − z0Þρxðz0Þdz0: ð1Þ zBased on the causality, the head part of the dipole moment affects the tail part of the particles, Wðz − z0Þ ≠ 0 for z0 > z.px, which is normalized by the total momentum, is dimensionless
The use of a collision scheme with a large crossing angle is becoming popular in the design of electron–positron colliders
Future circular colliders will be designed with a large crossing angle
Summary
An ordinary transverse wake force characterizes a transverse momentum kick of a particle at z according to the dipole moment density at z0 in a bunch: Z. The wake force is independent of the sign of z − z0, that is, there is no causal property, unlike the ordinary wake force Another important point is that the cross-wake force is localized at the point at which the two beams interact, and this localized wake force is used as the basis on which the mode-coupling theory is developed. The present mode-coupling theory for the cross-wake force explains the instability observed in the simulations. Understanding this instability is indispensable for future accelerator designs.
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