Characterization of the equilibrium configuration for modulated beams in a plasma wakefield accelerator

We analyze the equilibrium configuration for a modulated beam with sharp boundaries exposed to the fields self-generated by the interaction with a plasma. Through a semi-analytical approach, we show the presence of multiple equilibrium configurations and we determine the one more suitable for wakefield excitation. Once pointed out the absence of confinement for the front of the beam and the consequently divergence driven by the emittance, we study the evolution of the equilibrium configuration while propagating in the plasma, discarding all the other time-dependencies. We show the onset of a rigid backward drift of the equilibrium configuration, and we provide an explanation in the increasing length of the first bunch.