Conformal symmetry algebra of the quark potential and degeneracies in the hadron spectra

The essence of the potential algebra concept [3] is that quantum mechanical free motions of scalar particles on curved surfaces of given isometry algebras can be mapped on 1D Schroedinger equations with particular potentials. As long as the Laplace-Beltrami operator on a curved surface is proportional to one of the Casimir invariants of the isometry algebra, free motion on the surface is described by means of the eigenvalue problem of that very Casimir operator and the excitation modes are classified according to the irreps of the algebra of interest. In consequence, also the spectra of the equivalent Schroedinger operators are classified according to the same irreps. We here use the potential algebra concept as a guidance in the search for an interaction describing conformal degeneracies. For this purpose we subject the so(4) isometry algebra of the S^3 ball to a particular non-unitary similarity transformation and obtain a deformed isometry copy to S^3 such that free motion on the copy is equivalent to a cotangent perturbed motion on S^3, and to the 1D Schroedinger operator with the trigonometric Rosen-Morse potential as well. The latter presents itself especially well suited for quark-system studies insofar as its Taylor series decomposition begins with a Cornell-type potential and in accord with lattice QCD predictions. We fit the strength of the cotangent potential to the spectra of the unflavored high-lying mesons and obtain a value compatible with the light dilaton mass. We conclude that while the conformal group symmetry of QCD following from AdS_5/CFT_4 may be broken by the dilaton mass, it still may be preserved as a symmetry algebra of the potential, thus explaining the observed conformal degeneracies in the unflavored hadron spectra, both baryons and mesons.Comment: Invited talk presented at "Beauty in Physics:Theory and Experiment", May 14-May 18, Cocoyoc, Mexico, dedicated to the 70th birthday of Francesco Yachell

Breaking Pseudo-Rotational Symmetry through H+2{\bf H}^2_+ Metric Deformation in the Eckart Potential Problem

The peculiarity of the Eckart potential problem on ${\bf H}^2_+$ (the upper sheet of the two-sheeted two-dimensional hyperboloid), to preserve the $(2l+1)$-fold degeneracy of the states typical for the geodesic motion there, is usually explained in casting the respective Hamiltonian in terms of the Casimir invariant of an so(2,1) algebra, referred to as potential algebra. In general, there are many possible similarity transformations of the symmetry algebras of the free motions on curved surfaces towards potential algebras, which are not all necessarily unitary. In the literature, a transformation of the symmetry algebra of the geodesic motion on ${\bf H}^2_+$ towards the potential algebra of Eckart's Hamiltonian has been constructed for the prime purpose to prove that the Eckart interaction belongs to the class of Natanzon potentials. We here take a different path and search for a transformation which connects the $(2l+1)$ dimensional representation space of the pseudo-rotational so(2,1) algebra, spanned by the rank-$l$ pseudo-spherical harmonics, to the representation space of equal dimension of the potential algebra and find a transformation of the scaling type. Our case is that in so doing one is producing a deformed isometry copy to ${\bf H}^2_+$ such that the free motion on the copy is equivalent to a motion on ${\bf H}^2_+$, perturbed by a $\coth$ interaction. In this way, we link the so(2,1) potential algebra concept of the Eckart Hamiltonian to a subtle type of pseudo-rotational symmetry breaking through ${\bf H}^2_+$ metric deformation.Comment: misprints are correcte

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

Lorentz Group Projector Technique for Decomposing Reducible Representations and Applications to High Spins

The momentum-independent Casimir operators of the homogeneous spin-Lorentz group are employed in the construction of covariant projector operators, which can decompose anyone of its reducible finite-dimensional representation spaces into irreducible components. One of the benefits from such operators is that any one of the finite-dimensional carrier spaces of the Lorentz group representations can be equipped with Lorentz vector indices because any such space can be embedded in a Lorentz tensor of a properly-designed rank and then be unambiguously found by a projector. In particular, all the carrier spaces of the single-spin-valued Lorentz group representations, which so far have been described as 2 ( 2 j + 1 ) column vectors, can now be described in terms of Lorentz tensors for bosons or Lorentz tensors with the Dirac spinor component, for fermions. This approach facilitates the construct of covariant interactions of high spins with external fields in so far as they can be obtained by simple contractions of the relevant S O ( 1 , 3 ) indices. Examples of Lorentz group projector operators for spins varying from 1 / 2 –2 and belonging to distinct product spaces are explicitly worked out. The decomposition of multiple-spin-valued product spaces into irreducible sectors suggests that not only the highest spin, but all the spins contained in an irreducible carrier space could correspond to physical degrees of freedom