This paper proposes a novel mechanism based on the instantaneous Bethe-Salpeter (Salpeter) equation for investigating wave function mixing in vector mesons such as $\psi (3770)$. Conventional theories typically treat $\psi (3770)$ as a $2S-1D$ mixed state; however, considering only tensor forces or relativistic corrections alone often leads to mixing angles that are too small and inconsistent with experimental data. Phenomenological $2S-1D$ mixing requires experimental data as input to determine the mixing angles, resulting in limited theoretical studies on states like $\mathrm{\Upsilon }(1D,2D)$ in the absence of experimental data. To more accurately describe $S-D$ mixing and its relativistic effects, this paper systematically compares four relativistic wave function representations (${\phi }_1$, ${\phi }_2$, ${\phi }_3$, and ${\phi }_4$) by solving the Salpeter equation and calculates the mass spectra and dileptonic decay widths of charmonium and bottomonium. The study finds that the wave function representation ${\phi }_2$ can simultaneously reproduce the experimental data of both charmonium and bottomonium well. Further analysis reveals that, in addition to $S-D$ mixing, the wave functions of vector mesons contain a non-negligible $P$-wave component, meaning they are $S-P-D$ mixed states. We predict the mixing angles for bottomonium $\mathrm{\Upsilon }(1D)$ and $\mathrm{\Upsilon }(2D)$ to be $(1.78_{-0.25}^{+0.32})°$ and $(5.44_{-0.76}^{+1.10})°$, with dileptonic decay widths of ${2.29}_{-0.69}^{+0.86}\mathrm{eV}$ and ${10.5}_{-3.1}^{+4.2}\mathrm{eV}$, respectively.

Close

PrevNext

Reuse & PermissionsView figure in article

• 

• Bethe-Salpeter equation
• Particle interactions
• Particle mixing & oscillations
• Quantum electrodynamics
• Quark model