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Construction of future Muon Collider tangential to the Large Hadron Colliderwill give opportunity to realize mu-p collisions at multi-TeV center of massenergies. Using nominal parameters of high luminosity and high energy upgradesof the LHC, as well as design parameters of muon colliders, it is shown thatL_mu-p of order of 10^33 cm^-2s^-1 is achievable for different options withsqrt(s)_mu-p from 4.58 TeV to 12.7 TeV. Certainly, proposed mu-p colliders havea huge potential for clarifying QCD basics and searches for new physics.

Luminosity and Physics Considerations on HL-LHC and HE-LHC based mu-p Colliders