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

We calculate cross sections for $gg\to Q\bar{Q}Q\bar{Q}$ in the high-energy approximation in the mixed (longitudinal momentum fraction, impact parameter) and momentum space representations. Besides the total cross section as a function of subsystem energy also differential distributions (in quark rapidity, transverse momentum, $QQ$, $Q\bar{Q}$ 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. We present integrated cross section as a function of proton-proton center of mass energy as well as differential distribution in $M_{(c\bar{c})(c\bar{c})}$. The results are compared with corresponding results for double-parton scattering (DPS) discussed recently in the literature. We find that the considered SPS contribution to $(c\bar{c})(c\bar{c})$ production is at high energy ($\sqrt{s}

$ 5 TeV) much smaller than that for DPS contribution.