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We use a non-relativistic model to study the spectroscopy of a tetraquarkcomposed of $[cc][\bar{c}\bar{c}]$ in a diquark-antidiquark configuration. Bynumerically solving the Schr\"{o}dinger equation with a Cornell-inspiredpotential, we separate the four-body problem into three two-body problems.Spin-dependent terms (spin-spin, spin-orbit and tensor) are used to describethe splitting structure of the $c\bar{c}$ spectrum and are also extended to theinteraction between diquarks. Recent experimental data on charmonium states areused to fix the parameters of the model and a satisfactory description of thespectrum is obtained. We find that the spin-dependent interaction is sizable inthe diquark-antidiquark system, despite the heavy diquark mass, and also thatthe diquark has a finite size if treated in the same way as the $c\bar{c}$systems. We find that the lowest $S$-wave $T_{4c}$ tetraquarks might be belowtheir thresholds of spontaneous dissociation into low-lying charmonium pairs,while orbital and radial excitations would be mostly above the correspondingcharmonium pair thresholds. Finally, we repeat the calculations without theconfining part of the potential and obtain bound diquarks and boundtetraquarks. This might be relevant to the study of exotic charmonium in thequark-gluon plasma. The $T_{4c}$ states could be investigated in theforthcoming experiments at the LHC and Belle II.

A non-relativistic model for the $[cc][\bar{c}\bar{c}]$ tetraquark