  We use a non-relativistic model to study the mass spectroscopy of a
tetraquark composed by $c \, \bar{c} \, c \, \bar{c}$ quarks in the
diquark-antidiquark picture. By numerically solving the Schr\"{o}dinger
equation with a Cornell-inspired potential, we separate the four-body problem
into three two-body problems. Spin-dependent terms (spin-spin, spin-orbit and
tensor) are used to describe the splitting structure of the $c\bar{c}$ spectrum
and are also extended to the interaction between diquarks. Recent experimental
data on charmonium states are used to fix the parameters of the model and a
satisfactory description of the spectrum is obtained. We find that the
spin-dependent interaction is sizable in the diquark-antidiquark system,
despite of the heavy diquark mass, and that the diquark has a finite size if
treated in analogy to the $c\bar{c}$ systems. We find that the lowest $S$-wave
$T_{4c}$ tetraquarks might be below their thresholds of spontaneous
dissociation into low-lying charmonium pairs, while orbital and radial
excitations would be mostly above the corresponding charmonium pair threshold.
These states could be investigated in the forthcoming experiments at LHCb and
Belle II.

High Energy Physics - Phenomenology