  Preserving the quality of positron beams in plasma-based accelerators, where
wakefields are generated in electron filaments, is challenging. These
wakefields are characterized by transversely non-linear focusing fields and
non-uniform accelerating fields. However, a nonzero plasma temperature
linearizes the transverse wakefield within the central region of the electron
filament. In this study, we employ 3D particle-in-cell simulations with mesh
refinement to demonstrate that beams with emittances on the order of tens of
nanometers are contained within the linearized region of the transverse
wakefield. This enables emittance preservation to one percent, while positron
beams with the same charge and micrometer emittances, which sample the
non-linear part of the transverse wakefield, experience a relative emittance
growth of ten percent. Additionally, we observe a significant reduction in the
growth rate of the slice energy spread for the tens of nanometers emittance
beams in comparison to the micrometer emittance beams. The utilization of warm
plasmas in conjunction with low-emittance beams opens up new avenues for
enhancing the beam quality across various plasma-based positron acceleration
approaches.
