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

TL;DR

Using the instantaneous Bethe-Salpeter (Salpeter) framework, this work investigates $S$-$D$ mixing in vector quarkonia by testing eight relativistic wave function representations for the $1^{--}$ state. It shows that tensor forces alone are insufficient and that a $S$-$P$-$D$ mixed structure emerges naturally when relativistic components are included, with the eight representations reduced to four independent radial functions. Among the eight representations, $\varphi_2$ best reproduces charmonium and bottomonium spectra and dileptonic widths, yielding mixing angles $ \theta_{\psi(2S)}=(6.67^{+1.71}_{-1.50})^\circ$, $ \theta_{\psi(1D)}=(7.92^{+1.70}_{-1.43})^\circ$, and $ \theta_{\psi(2D)}=(26.3^{+3.3}_{-3.4})^\circ$ for charmonium, and $ \theta_{\Upsilon(1D)}=(1.78^{+0.32}_{-0.25})^\circ$, $ \theta_{\Upsilon(2D)}=(5.44^{+1.10}_{-0.76})^\circ$ for bottomonium. It also predicts dileptonic widths for the yet-unobserved $\Upsilon(1D)$ and $\Upsilon(2D)$ states: $\Gamma(\Upsilon(1D)\to e^+e^-) = 2.29^{+0.86}_{-0.69}$ eV and $\Gamma(\Upsilon(2D)\to e^+e^-) = 10.5^{+4.2}_{-3.1}$ eV, demonstrating the model’s predictive power and the importance of relativistic multi-wave mixing in vector quarkonia.

Abstract

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 $2S-1D$ 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 $Υ(1D, 2D)$ 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 $Υ(1D)$ and $Υ(2D)$ 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.

Figures (1)

• Figure 1: Radial wave functions of the wave function $\varphi_{2}$ for charmonium, where $q\equiv |\vec{q}|$.