A new regime of proton-driven plasma wakefield acceleration is discovered, in which the plasma nonlinearity increases the phase velocity of the excited wave compared to that of the protons. If the beam charge is much larger than minimally necessary to excite a nonlinear wave, there is sufficient freedom in choosing the longitudinal plasma density profile to make the wave speed close to the speed of light. This allows electrons or positrons to be accelerated to about 200 GeV with a 400 GeV proton driver.
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