Elimination of witness beam hosing in the linear collider regime

Beam driven plasma wakefield acceleration (PWFA) has shown great potential to be the basis for future linear colliders. Recent progress has shown PWFA can acheieve high acceleration gradients with  high energy transfer efficiency while maintaining low energy spread. In the regime relevant for linear colliders, the witness beam emittance is $\sim$ 100 nm and the charge is $\sim$ 1 nC. With these parameters, the ion collapse will be drastic and lead to emittance growth, however An et al. (2017) [1] showed this emittance growth is controllable. Mehrling et al. (2018) [2] showed hosing of the witness beam is suppressed in this regime due to the BNS mechanism. Hildebrand et al. (2018) [3] showed that the on-axis ion column formed by a high density drive beam can fully eliminate the witness beam hosing and realign it with the drive beam. This will occur for unmatched drive beams that oscillate to small sizes but can also be controlled by matching the drive beam with a small spot size.  We will explore the emittance growth of the witness beam and decay time of its centroid as a function of the drive beam spot size and subsequent ion peak formed in the witness beam region.