$S-P-D$ Mixing in Vector Quarkonia from the Salpeter Equation with Optimized Wave Function Representations

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 $\Upsilon(1D, 2D)$ in the absence of experimental data. To more accurately describe $S-D$ mixing and its relativistic effects, this paper systematically compares eight possible relativistic wave function representations ($\varphi_1$ to $\varphi_8$) 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 $\varphi_2$ 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 $\Upsilon(1D)$ and $\Upsilon(2D)$ to be $(1.78^{+0.32}_{-0.25})^\circ$ and $(5.44^{+1.10}_{-0.76})^\circ$, with dileptonic decay widths of $2.29^{+0.86}_{-0.69}$ eV and $10.5^{+4.2}_{-3.1}$ eV, respectively.