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 ψ ( 3770 ) . Conventional theories typically treat ψ ( 3770 ) as a 2 S − 1 D 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 2 S − 1 D mixing requires experimental data as input to determine the mixing angles, resulting in limited theoretical studies on states like ϒ ( 1 D , 2 D ) 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 ( φ 1 , φ 2 , φ 3 , and φ 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 φ 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 ϒ ( 1 D ) and ϒ ( 2 D ) to be ( 1.7 8 − 0.25 + 0.32 ) ° and ( 5.4 4 − 0.76 + 1.10 ) ° , with dileptonic decay widths of 2.29 − 0.69 + 0.86 eV and 10.5 − 3.1 + 4.2 eV , respectively.