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The framework to compute the cross sections for the production of particles with high mass and/or large transverse momentum in double- (DPS), triple- (TPS), and in general $n$-parton scatterings, from the corresponding single-parton ($\sigma_{\rm SPS}$) values in high-energy proton-proton, proton-nucleus, and nucleus-nucleus is reviewed. The basic parameter of the factorized $n$-parton scattering ansatz is an effective cross section $\sigma_{\rm eff}$ encoding all unknowns about the underlying generalized $n$-parton distribution in the proton (nucleon). In its simplest and most economical form, the $\sigma_{\rm eff}$ parameter can be derived from the transverse parton profile of the colliding protons and/or nucleus, using a Glauber approach. Numerical examples for the cross sections and yields expected for the concurrent DPS or TPS production of heavy-quarks, quarkonia, and/or gauge bosons in proton and nuclear collisions at LHC and Future Circular Collider (FCC) energies are provided. The obtained cross sections are based on perturbative QCD predictions for $\sigma_{\rm SPS}$ at next-to-leading-order (NLO) or next-to-NLO (NNLO) accuracy including, when needed, nuclear modifications of the corresponding parton densities.

Double, Triple, and n-Parton Scatterings in High-Energy Proton and Nuclear Collisions