  The beam hosing instability is analyzed theoretically for a witness beam in
the quasi-linear regime of plasma accelerators. In this regime, the hosing
instability saturates, even for a monoenergetic bunch, at a level much less
than standard scalings predict. Analytic expressions are derived for the
saturation distance and amplitude and are in agreement with numerical results.
Saturation is due to the natural head-to-tail variations in the focusing force,
including the self-consistent transverse beam loading.
