## 4 Influences of Non-Gaussian Transverse Distributions
### Axisymmetric Non-Gaussian Transverse Distribution
Similar to the evolution of the normalized transverse emittance shown in Figure 3b, the halo parameters of different initial beam profiles see a rapid absolute increase at the early stage and then evolve slowly with the beam propagation. However, unlike the emittance, the final beam halo parameters strongly depend on the initial distributions. Beams with a high initial halo parameter, i.e., sharply peaked with a low \(p\)-factor, results in a higher final halo parameter. These effects are true for both charge groups. The difference due to different witness beam charge is shown in the early-stage evolution of the halo parameter. The low-charge beams see a much larger initial growth of the halo parameters than high-charge beams. This is related to the weak plasma focusing force outside the plasma bubble. It is also due to the same reason, the halo parameter of the low-charge cases drops after reaching the peak as halo electrons with large transverse momentum leave the simulation window and are lost. To better understand the evolution of the beam halo parameter, the slice distributions of the halo parameters are analysed, as shown in Figure 6c. Here, results of the high-charge cases are shown for example. As shown in Figure 1, the plasma bubble does not cover the whole witness beam. The witness electrons inside and outside the bubble experience different strengths of the plasma focusing force. This then generates different beam slice halo parameter evolution. For particles at the rear of the witness beam, their halo parameters are generally consistent along \(\xi\), while for beam electrons at the head, the slice halo parameter varies with the longitudinal location of each transverse slice. Since the halo parameter is an invariant under the linear transverse focusing force according to Equation (A2), we expect that the slice halo parameters are generally \"preserved\" in the bubble's field. However, it is shown in Figure 6c that the halo parameters of witness electrons inside the bubble still see a small increase after the acceleration, which is due to the minor non-linearity in the loaded transverse wakefield. Figure 6: _Cont._  
For head slices around \(\xi_{0e}+2\sigma_{ze}\), i.e., near the leading edge of the witness beam, we can see that local slice halo parameters for beams with different initial spatial profiles converge after the acceleration. This suggests that after reaching full phase-mixing in the non-linear transverse wakefield, these beams have the similar transverse distributions at the head. It should be noted that at the front tip of the high-charge beam, these witness electrons are decelerated and slip backward since the bunch is too long to be accommodated by the accelerating phase. This causes the head erosion, then leads to the reduction of the local slice halo parameter. For beam slices at around \(\xi_{0e}+\sigma_{ze}\), one can see a larger increase of the slice halo parameter in Figure 6c. In this region, the plasma bubble starts to form but the local bubble radius is smaller than that of the witness beam. Although a majority of those witness electrons stay inside the bubble and are focused by the strong bubble-regime wakefield, there is still a small portion of the witness charge falling outside the bubble. The dense sheath of plasma electrons at the edge of the bubble shields the plasma ion charge, leading to the reduction of the plasma focusing strength outside. This huge disparity in the plasma focusing strength leads to a larger increase of the local halo parameter than at other positions along the witness bunch. It is also the main contribution for the whole beam halo parameter increase in the first metre. For the 120-pC beams, we observe a similar evolution of the slice halo parameters. However, as the plasma bubble takes longer to form than in the high-charge case, the initial halo parameter growth is much more significant. According to the results shown above, whole-beam statistics such as the emittance are easily dominated by particles in the halo.