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This paper is devoted to the study of exclusive charm and bottom aggregateswhich are suitable, at least partially, for a non-relativistic description.Starting from a simple choice of quarkonia and tetra-quark wave functions westudy their production cross sections, total widths and more interesting decayproperties.  In particular, we compute the production cross sections of the ground statescalar quarkonia getting results compatible with the values appearing in theliterature. We also compute the production cross sections of the lower energystates of the charm and bottom tetra-quarks, their decay widths and the decayrates in two $\mu\bar\mu$ pairs and in the $q\bar q\mu\bar\mu$ channel.  Our results on the tetra-charm are consistent with the data published by theLHCb Collaboration and those on the tetra-bottom show a reasonable agreementwith the indications published by the CMS Collaboration. Our cross sectionanalyses are based on interpolations based on the gluon-gluon luminosity gridspublished by the NNPDF and CTEQ Collaborations. The hard amplitude calculationsare based on QCD in the tree approximation. We think that a careful improvementof our method and a refinement of the experimental data after the foreseenluminosity increase at the LHC will make possible valuable tests of QCD at theintermediate energies.

The heavy tetra-quark states after the discovery of tetra-charm states