Spectra of light and heavy mesons with $J \le 5$ in a relativistic Bethe-Salpeter approach

TL;DR

The paper extends a relativistic Dyson-Schwinger/Bethe-Salpeter framework to high-spin ($J=4,5$) mesons, introducing new Dirac tensor bases and performing exploratory spectra within a rainbow-ladder truncation. By employing effective couplings of Maris–Tandy and Qin–Chang types, the authors solve the quark propagator DSE and the meson BSE, mapping out ground-state masses and Regge trajectories across light, strange, charm, and bottom sectors. They find that natural-parity sequences with $J^{PC}=1^{--},2^{++},3^{--},4^{++},5^{--}$ are reasonably reproduced, predicting new high-spin states such as $4^{++}$ and $5^{--}$ in multiple flavor channels, while channels off this trajectory highlight the need for beyond-RL interactions. Regge behaviour emerges in this model due to the intermediate-distance part of the effective potential, but the authors caution that the observed linearity is not fundamental confinement. Overall, the work demonstrates the feasibility of studying high-spin mesons within a consistent functional framework and clarifies where improved interactions are necessary for a complete spectral description.

Abstract

We extend the range of application of the relativistic Dyson-Schwinger/Bethe-Salpeter approach from previously discussed mesons with total angular momentum $J \le 3$ to the ones with $J=4,5$. On a technical level, the new element is the general Dirac tensor representations for the latter which, to our knowledge, are presented here for the first time. As a first application, we provide an exploratory spectrum for these mesons in a rainbow-ladder truncation of Dyson-Schwinger and Bethe-Salpeter equations. We discuss the merits and limitations of this truncation and explore the shape of the heavy-quark potential corresponding to the underlying effective running coupling. With our predictions for the masses of ground state mesons with quantum numbers $J^{P,C}=3^{--}, 4^{++}, 5^{--}$ we identify Regge trajectories in channels where the interaction model can be trusted on a semi-quantitative level. In other channels, discrepancies with experiments confirm the well-known need to go beyond rainbow-ladder in the Dyson-Schwinger/Bethe-Salpeter approach by using more sophisticated interactions.

Figures (6)

• Figure 1: Left panel: Interaction potentials obtained via Fourier transformation of effective running couplings. The MT potential has a (positive) maximum around $r=1.6$ fm and approaches zero from the positive side while the QC potential approaches zero monotonously. Right panel: The contributions of the infrared and ultraviolet parts of the effective running coupling to the MT potential.
• Figure 2: Spectrum of $q\bar{q}$ states obtained from a rainbow-ladder truncation with the Maris-Tandy and Qin-Chang models compared to the experimental spectrum in the isovector channel ParticleDataGroup:2024cfk.
• Figure 3: Spectrum of states with one light and one strange (anti-)quark obtained from a rainbow-ladder truncation with the Maris-Tandy model compared to the experimental kaon spectrum ParticleDataGroup:2024cfk.
• Figure 5: Charmonium spectrum obtained from a rainbow-ladder truncation with the Maris-Tandy model and $\eta = 1.16$ compared to experimental data ParticleDataGroup:2024cfk. Note that potential tetraquark candidates such as the $\eta_{c1}(3872)$ are not included.
• Figure 7: Chew-Frautschi plots for different quark contents.