Covariant study of tensor mesons

We investigate tensor mesons as quark-antiquark bound states in a fully covariant Bethe-Salpeter equation. As a first concrete step we report results for masses of $J^{PC}=2^{++}$ mesons from the chiral limit up to bottomonium and sketch a comparison to experimental data. All covariant structures of the fermion-antifermion system are taken into account and their roles and importance discussed in two different bases. We also review the general construction principle for covariant Bethe-Salpeter amplitudes of mesons with any spin.

Figure 1

Dependence of meson masses on $m_{\pi }$ (the pseudoscalar-meson mass calculated for a given current-quark mass). Vertical dotted lines correspond to positions for light, strange, and charm $\overline{q}q$ states.

Figure 2

Dependence of meson masses on $\omega$. Dotted lines correspond to experimental data [57] where applicable. Figure updated from Ref. 39.

Figure 3

The parabola in the complex $q_{\pm }^2$-plane sampled in the BSE for a meson of mass $M$ together with the typical positions of singularities, marked by “$X$” (see also the text).

Figure 4

Extrapolation of calculated eigenvalues (dots), by polynomials of degree 3 (solid), 4 (dashed), and 5 (dash-dotted), to the value $\lambda (M)=1$, demarcated by the horizontal dotted line. The central value of the final result (box) and the uncertainty (solid horizontal line) are plotted in the graph as determined (see text).