Triple parton scatterings in high-energy proton-proton collisions

A generic expression to compute triple parton scattering (TPS) cross sections in high-energy proton-proton (pp) collisions is presented as a function of the corresponding single-parton cross sections and the transverse parton distribution in the proton encoded in an effective parameter $\sigma_{\rm eff,TPS}$. The value of $\sigma_{\rm eff,TPS}$ is closely related to the similar effective cross section that characterizes double-parton scatterings, and amounts to $\sigma_{\rm eff,TPS} = 12.5 \pm 4.5$ mb. Estimates for triple charm ($\rm c\overline{c}$) and bottom ($\rm b\overline{b}$) production in pp collisions at LHC and FCC energies are presented based on next-to-next-to-leading order perturbative calculations for single $\rm c\overline{c},\rm b\overline{b}$ cross sections. At $\sqrt{s}\approx$ 100 TeV, about 15% of the pp collisions produce three $\rm c\overline{c}$ pairs from three different parton-parton scatterings.