2 research outputs found
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
The Trigonometric Rosen-Morse Potential in the Supersymmetric Quantum Mechanics and its Exact Solutions
The analytic solutions of the one-dimensional Schroedinger equation for the
trigonometric Rosen-Morse potential reported in the literature rely upon the
Jacobi polynomials with complex indices and complex arguments. We first draw
attention to the fact that the complex Jacobi polynomials have non-trivial
orthogonality properties which make them uncomfortable for physics
applications. Instead we here solve above equation in terms of real orthogonal
polynomials. The new solutions are used in the construction of the
quantum-mechanic superpotential.Comment: 16 pages 7 figures 1 tabl