Conformal symmetry breaking and degeneracy of high-lying unflavored mesons

We show that though conformal symmetry can be broken by the dilaton, such can happen without breaking the conformal degeneracy patterns in the spectra. We departure from R^1XS^3 slicing of AdS_5 noticing that the inverse radius, R, of S^3 relates to the temperature of the deconfinement phase transition and has to satisfy, \hbar c/R >> \Lambda_{QCD}. We then focus on the eigenvalue problem of the S^3 conformal Laplacian, given by 1/R^2 (K^2+1), with K^2 standing for the Casimir invariant of the so(4) algebra. Such a spectrum is characterized by a (K+1)^2 fold degeneracy of its levels, with K\in [0,\infty). We then break the conformal S^3 metric as, d\tilde{s}^2=e^{-b\chi} ((1+b^2/4) d\chi^2 +\sin^2\chi (d\theta ^2 +\sin^2\theta d\varphi ^2)), and attribute the symmetry breaking scale, b\hbar^2c^2/R^2, to the dilaton. We show that such a metric deformation is equivalent to a breaking of the conformal curvature of S^3 by a term proportional to b\cot \chi, and that the perturbed conformal Laplacian is equivalent to (\tilde{K}^2 +c_K), with c_K a representation constant, and \tilde{K}^2 being again an so(4) Casimir invariant, but this time in a representation unitarily inequivalent to the 4D rotational. In effect, the spectra before and after the symmetry breaking are determined each by eigenvalues of a Casimir invariant of an so(4) algebra, a reason for which the degeneracies remain unaltered though the conformal group symmetry breaks at the level of the representation of its algebra. We fit the S^3 radius and the \hbar^2c^2b/R^2 scale to the high-lying excitations in the spectra of the unflavored mesons, and observe the correct tendency of the \hbar c /R=373 MeV value to notably exceed \Lambda_{QCD}. The size of the symmetry breaking scale is calculated as \hbar c \sqrt{b}/R=673.7 MeV.Comment: Presented at the "XIII Mexican Workshop on Particles and Fields", Leon, Guanajuato, Mexico, October 201

La simetría conforme en los espectros de los mesones ligeros

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 foot- print 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 experimental- ly 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 can affect the shape of the metric of the compactified Minkowski space-time, R1 × S3, one of the possible conformally invariant topologies embedded by AdS5 boundary, through a deformation of the S3 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 S3 ball (unbroken con- formal 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 represen- tation unitarily-inequivalent to the one of the conformally invariant S3 surface. We further- more demonstrate that the Casimir invariant of the so(4) algebra describing the free motion on the deformed metric is equivalent to a perturbation of the free geodesic motion on S3 by a harmonic potential there and given by a cotangent function of the second polar angle parametrizing S3. In solving the eigenvalue problem of the so(4) Casimir invariant on the deformed metric, we find same degeneracy patterns as on the undeformed. 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