Conformal symmetry and color confinement in the infrared regime of QCD are interpreted by means of a conjectured deSitter $dS_4$ geometry of the internal space-time of hadrons, an assumption inspired by the hypothesis on deSitter special relativity. Within such a scenario, the interactions involving the virtual gluon- and constituent quark degrees of freedom of hadrons are deduced from the Green functions of Laplace operators on the $dS_4$ geodesics. Then the conformal symmetry of QCD emerges as a direct consequence of the conformal symmetry of the $dS_4$ space-time, while the color confinement, understood as colorlessness of hadrons, appears as a consequence of the inevitable charge neutrality of the unique closed space-like geodesic on this space, the three dimensional hyper-sphere $S^3$, on which the hadrons constituents are conjectured to reside when near rest frame. Mesons are now modelled as quarkish color-anticolor dipoles, whose free quantum motions on the aforementioned $S^3$ geodesic are perturbed by a potential generated by a gluon--anti-gluon color dipole. The potential predicted presents itself as the color charge analogue to the curved Coulomb potential, i.e. to the electric potential that defines a consistent electrostatic theory on a hyper-spherical surface. The advantage of this method is that it allows to establish a direct relationship of the potential parameters to the fundamental constants of QCD. We apply the model to the description of the spectra of the $a_1$ and $f_1$ mesons, and the pion electric charge form factor, finding fair agreement with data.
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