C. B. Schroeder, E. Esarey, C. Benedetti, and W. P. Leemans {Phys. Rev. ST Accel. Beams 13, 101301 (2010) and 15, 051301 (2012)} have proposed a set of parameters for a TeV-scale collider based on plasma wake field accelerator principles. In particul
ar, it is suggested that the luminosities greater than 10^34 cm-2s-1 are attainable for an electron-positron collider. In this comment we dispute this set of parameters on the basis of first principles. The interactions of accelerating beam with plasma impose fundamental limitations on beam properties and, thus, on attainable luminosity values.
Transverse beam stability is strongly affected by the beam space charge. Usually it is analyzed with the rigid-beam model. However this model is only valid when a bare (not affected by the space charge) tune spread is small compared to the space char
ge tune shift. This condition specifies a relatively small area of parameters which, however, is the most interesting for practical applications. The Landau damping rate and the beam Schottky spectra are computed assuming that validity condition is satisfied. The results are applied to a round Gaussian beam. The stability thresholds are described by simple fits for the cases of chromatic and octupole tune spreads.
