Limitations on plasma acceleration due to synchrotron losses

In this letter we consider the effect of synchrotron radiation losses due to the betatron motion of the electron beam in its self-induced magnetic field in a plasma accelerator taking into account the charge neutralization factor. The most favorable case is where the plasma density is smaller than the beam density. The contrary regime is strongly disfavored by the synchrotron radiation loss for beams with characteristics for TeV energies. In both cases we find that upon increasing the plasma density the synchrotron losses kill the acceleration process, so that there are limitations on the maximum allowable plasma density.