“…In particle-beam driven plasma wakefield accelerators (PWA), 2 the maximum energy gain of the witness bunch is restricted by the transformer ratio limit, which roughly states that a witness bunch's energy cannot exceed twice the driver bunch's energy. 13 To achieve unprecedented electron beam energies in the TeV range, TeV proton beams from CERN's Large Hadron Collider (LHC) have been proposed as a driver in a proton driver plasma wakefield acclerator (PDPWA). 14,15 In order to efficiently excite plasma waves that produce large accelerating gradients, the driver beam should be of order one plasma wavelength, k p 2pc=x p , where x p ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4pe 2 n p0 =m e p is the plasma frequency, c is the speed of light, n p0 $ 10 15 cm À3 is the density of the unperturbed plasma, Àe and m e are the electron charge and mass, respectively.…”