Conformal symmetry breaking and degeneracy of high-lying unflavored mesons

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. Our argumentation goes as follows: We departure from the gauge-gravity duality which predicts on the boundaries of the AdS5 geometry a conformal theory, associated with QCD at high temperatures, and consider S1 × S3 slicing. The inverse radius, R, of S3 relates to the temperature of the deconfinement phase transition and has to satisfy, ℏc/R ≫ ΛQCD. On S3, whose isometry group is SO(4), we then focus on the eigenvalue problem of the conformal Laplacian there, given by , with standing for the Casimir invariant of the so(4) algebra. This eigenvalue problem describes the spectrum of a scalar particle, to be associated with a qq̄ system. Such a spectrum is characterized by a (K + l)2-fold degeneracy of its levels, with K ∊ [0, ∞). We then break the conformal S3 metric, ds2 = dχ2 + sin2 χ(dθ2 + sin2θdφ2) -in polar chi;,θ, and azimuthal φ coordinates- according to, ds∼2 = e−bχ((1 + b2/4)dχ2 + sin2 chi;(dθ2 + sin2θdφ2)), and attribute the symmetry breaking scale bℏ2c2/R2 to the dilaton. Next we show that the above metric deformation is equivalent to a breaking of the conformal curvature of S3 by a term proportional to b cot χ, and that the perturbed conformal Laplacian is equivalent to , with cκ a representation constant, and being again an so(4) Casimir invariant, but this time in a representation unitarily nonequivalent to the 4D rotational one. As long as the spectra before and after the symmetry breaking happen to be determined each by eigenvalues of a Casimir invariant of an so(4), no matter whether or not in a representation that generates the orthogonal group SO(4) as a subgroup of the conformal group SO(2,4), the degeneracy patterns remain unaltered though the conformal symmetry breaks at the level of the representation of the algebra. We fit the S3 radius and the ℏ2c2b/R2 scale to the high-lying excitations in the spectra of the unflavored mesons, and observe the correct tendency of the ℏc/R = 373 MeV value to notably exceed ΛQCD. The size of the symmetry breaking scale is calculated as MeV.