Conformal symmetry breaking and degeneracy of high-lying unflavored mesons

TL;DR

This paper investigates how conformal symmetry can be broken in high-lying unflavored mesons without disrupting their spectral degeneracy, using a deformed S^3 metric and fitting the model to experimental data.

Contribution

It introduces a novel approach to symmetry breaking in meson spectra that preserves degeneracy patterns by deforming the conformal metric and analyzing the resulting eigenvalue problem.

Findings

The symmetry breaking scale exceeds b c / R by a significant margin.

The degeneracy of the spectra remains intact despite conformal symmetry breaking.

The model accurately fits the high-lying meson spectra with the proposed deformation.

Abstract

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.