Forward $J/\psi + J/\psi$ and $J/\psi + \psi^\prime$ production with High Energy Factorization

We calculate the cross sections of associated $J/\psi + \psi^\prime$ and $J/\psi + J/\psi$ production in $pp$ collisions at $\sqrt s = 13$ TeV in the forward kinematic region. The High Energy Factorization ($k_T$-factorization) framework supplemented with the Catani-Ciafaloni-Fiorani-Marchesini evolution of gluon densities in a proton is applied. We demonstrate that latest data on $J/\psi + J/\psi$ production and first experimental data on $J/\psi + \psi^\prime$ events taken very recently by the LHCb Collaboration can be described well by the color singlet terms and contributions from the double parton scattering (DPS) with the standard choice for $\sigma_{\rm eff}$ parameter. The relative production rate $\sigma(J/\psi + \psi^\prime)/\sigma(J/\psi + J/\psi)$ is found to be sensitive to the DPS terms as well as to feeddown contributions.