Recently there has been much interest in the pair production of quarkonia (charmonia, bottomonia). There are two main motivations behind these studies: first, these processes may help to differentiate between different proposed production mechanisms via color-octet and color-singlet $Q\bar Q$-pair production.

Second, the production of quarkonium pairs is expected to receive an important contribution from double parton scattering (DPS) processes. There remain a number of open problems, especially with the CMS and ATLAS data.

In the kinematics of these experiments, the leading order of ${\cal{O}}(\alpha_S^4)$ is clearly not sufficient. The double parton scattering (DPS) contribution was claimed to be large or even dominant in some corners of the phase space, when the rapidity distance $\Delta y$ between two $J/\psi$ mesons is large. However the effective cross sections $\sigma_{\rm eff}$ found from empirical analyses are about a factor $2.5$ smaller than the usually accepted $\sigma_{\rm eff} = 15 \, \rm{mb}$.

We will discuss, which single-parton-scattering mechanisms can mimic the behaviour of DPS induced production. Here especially the production of $\chi$-pairs is important.