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We calculate cross sections for $g g \to Q \bar Q Q \bar Q$ in thehigh-energy approximation in the mixed (longitudinal momentum fraction, impactparameter) and momentum space representations. Besides the total cross sectionas a function of subsystem energy also differential distributions (in quarkrapidity, transverse momentum, $Q Q$, $Q \bar Q$ invariant mass) are presented.The elementary cross section is used to calculate production of $(c \bar c) (c\bar c)$ in single-parton scattering (SPS) in proton-proton collisions. Wepresent integrated cross section as a function of proton-proton center of massenergy as well as differential distribution in $M_{(c \bar c)(c \bar c)}$. Theresults are compared with corresponding results for double-parton scattering(DPS) discussed recently in the literature. We find that the considered SPScontribution to $(c \bar c)(c \bar c)$ production is at high energy ($\sqrt{s}>$ 5 TeV) much smaller than that for DPS contribution.

Production of two $c \bar c$ pairs in gluon-gluon scattering in high energy proton-proton collisions