Phenomenology of $J^{PC} = 3^{--}$ tensor mesons

We study the strong and radiative decays of the anti-quark-quark ground state $J^{PC} = 3^{--}$ ($n^{2 S + 1} L_J = 1^3 D_3$) nonet {$\rho_{3} (1690)$, $K_{3}^{\ast} (1780)$, $\phi_{3} (1850)$, $\omega_{3} (1670)$} in the framework of an effective quantum field theory approach, based on the $SU_\mathrm{V}(3)$-flavor-symmetry. The effective model is fitted to experimental data listed by the Particle Data Group. We predict numerous experimentally unknown decay widths and branching ratios. An overall agreement of theory (fit and predictions) with experimental data confirms the $\bar{q} q$ nature of the states and qualitatively validates the effective approach. Naturally, experimental clarification as well as advanced theoretical description is needed for trustworthy quantitative predictions, which is observed from some of the decay channels. Besides conventional spin-$3$ mesons, theoretical predictions for ratios of strong and radiative decays of a hypothetical glueball state $G_3 (4200)$ with $J^{PC} = 3^{--}$ are also presented.