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Triple parton scatterings in proton-nucleus collisions at high energies

A generic expression to compute triple parton scattering (TPS) cross sections in high-energy proton-nucleus (pA) collisions is derived as a function of the corresponding single-parton cross sections and an effective parameter encoding the transverse parton profile of the proton. The TPS cross sections are enhanced by a factor of $9\,A\approx 2000$ in pPb compared to those in proton-nucleon collisions at the same center-of-mass energy. Estimates for triple charm ($c\overline{c}$) and bottom ($b\overline{b}$) production in pPb collisions at LHC and FCC energies are presented based on next-to-next-to-leading order (NNLO) calculations for $c\overline{c}, b\overline{b}$ single-parton cross sections. At $\sqrt{s_{NN}} = 8.8$ TeV, about 10% of the pPb events have three $c\overline{c}$ pairs produced in separate partonic interactions. At $\sqrt{s_{NN}} = 63$ TeV, the pPb cross sections for triple-J$/ψ$ and triple-$b\overline{b}$ are ${\cal O}$(1--10 mb). In the most energetic cosmic-ray collisions observed on earth, TPS $c\overline{c}$-pair cross sections equal the total p-Air inelastic cross section.