Hyperfine meson splittings: chiral symmetry versus transverse gluon exchange

Meson spin splittings are examined within an effective Coulomb gauge QCD Hamiltonian incorporating chiral symmetry and a transverse hyperfine interaction necessary for heavy quarks. For light and heavy quarkonium systems the pseudoscalar-vector meson spectrum is generated by approximate Bardeen-Cooper-Schrieffer, random-phase approximation diagonalizations. This relativistic formulation includes both $S$ and $D$ waves for the vector mesons which generates a set of coupled integral equations. A smooth transition from the heavy to the light quark regime is found with chiral symmetry dominating the $\pi \text{−}\rho$ mass difference. A reasonable description of the observed meson spin splittings and chiral quantities, such as the quark condensate and the $\pi$ mass, is obtained. Similar comparisons with Tamm-Dancoff diagonalizations, which violate chiral symmetry, are deficient for light pseudoscalar mesons, indicating the need to simultaneously include both chiral symmetry and a hyperfine interaction. The ${\eta }_b$ mass is predicted to be around $9400\mathrm{MeV}$, consistent with other theoretical expectations and above the unconfirmed $9300\mathrm{MeV}$ candidate. Finally, for comparison with lattice results, the $J$ reliability parameter is also evaluated.