We study relevance of conformal symmetry breaking through the dilaton mass on the high-lying spectra of the unflavored mesons. The conformal symmetry is supposed to leave a footprint in those spectra in consequence of the gauge-gravity duality conjecture in combination with the opening of a conformal window in the infrared as recently observed experimentally through the property of the running coupling constant of QCD to approach a fixed point in the limit of a vanishing momentum transfer. The dilaton mass is well known to affect the metric of the compactified Minkowski space-time, R$^1\times$ S$^3$, one of the possible conformally invariant topologies embedded by AdS$_5$ boundary, through a deformation of the $S^3$ position space by a damping exponential factor.

Towards our purpose, we consider the mesons under investigation as four-dimensional rigid rotators with the quark performing free geodesic motion either on the $S^3$ ball (unbroken conformal symmetry), or, on the deformed metric (symmetry broken by the dilaton mass). We show that so(4) remains an isometry algebra of the deformed manifold though in a representation unitarily-inequivalent to the one of the conformally invariant $S^3$ surface. We furthermore demonstrate that the Casimir invariant of the so(4) algebra describing the motion on the deformed metric is equivalent to a perturbation of the free geodesic motion on $S^3$ by a harmonic potential there and given by a cotangent function of the second polar angle parametrizing $S^3$.

In solving the eigenvalue problem of the so(4) Casimir invariant on the deformed metric, we find there same degeneracy patterns as on $S^3$.

In this manner, a subtle mode of symmetry breaking has been identified in which the violation of the symmetry at the level of the representation function of the algebra can be opaqued by a conservation of the degeneracy patterns of the unbroken symmetry in the spectra.