A previously unexplored experimental scheme is presented for generating low-divergence, ultra-dense, relativistic, electron-positron beams using 400 GeV/c protons available at facilities such as HiRadMat and AWAKE at CERN. Preliminary Monte-Carlo and Particle-in-cell simulations demonstrate the possibility of generating beams containing $10^{13}-10^{14}$ electron-positron pairs at sufficiently high densities to drive collisionless beam-plasma instabilities, which are expected to play an important role in magnetic field generation and the related radiation signatures of relativistic astrophysical phenomena. The pair beams are quasi-neutral, with size exceeding several skin-depths in all dimensions, allowing for the first time the examination of the effect of competition between transverse and longitudinal instability modes on the growth of magnetic fields. Furthermore, the presented scheme allows for the possibility of controlling the relative density of hadrons to electron-positron pairs in the beam, making it possible to explore the parameter spaces for different astrophysical environments.
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