Mariana Kirchbach, Adrian Pallares-Rivera, Cliffor Compean, Alfredo Raya
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 S^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) 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, as reported by the Crystal Barrel collaboration, 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.
View original:
http://arxiv.org/abs/1202.0545
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