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Ultra-relativistic hollow electron beams can drive plasma wakefields ($\sim$GV/m) suitable for positron acceleration. Stable propagation of hollow electronbeams for long distances in plasmas is required to accelerate positrons to highenergies by these plasma wakefields. In this work, we show by quasi-statickinetic simulations using the code WAKE that an ultra-relativisticazimuthally-symmetric hollow electron beam propagates in a plasma by developingfish-bone like structure and shifting its bulk, differentially along its length(rear part fastest), towards its axis due to the decrease in the betatron timeperiod of beam electrons from the beam-front to beam-rear. Hollow electronbeams with small radius collapse into their axis due to the pull by thesecondary wakefields generated by some of the beam electrons reaching the axis.Hollow beams with sufficiently large radius, however, can propagate stably inplasmas for several meters and be used for positron acceleration.

Evolution of ultra-relativistic hollow-electron-beam wakefield drivers during their propagation in plasmas