Mass spectrum of $1^{--}$ heavy quarkonium

Over 20 charmoniumlike and bottomoniumlike XYZ states have been observed in the past two decades. The charged states (Z states), which might be good exotic state candidates in the tetraquark or molecule picture, have inspired extensive interests of theorists in revealing their underlying structures Chen et al. (2016). However, distinguishing the real exotic neutral X and Y states from conventional meson states is still a challenging work, and underlying structures of X and Y states are still wildly discussed and debated in the past decade Eichten et al. (2008); Liu et al. (2019).

The exotic states with $J^{PC}=1^{--}$, also known as Y states, are named $\Upsilon$ in the bottomonium region, and $\psi$ in the charmonium region according to the latest PDG (Particle Data Group) naming scheme Workman et al. (2022). It is significant to separate these exotic neutral states from conventional meson picture before treating them in other exotic pictures.

In the bottomonium region, the PDG states $\Upsilon$(9460), $\Upsilon$(10023), $\Upsilon$(10355), and $\Upsilon$(10579) are assigned to be $\Upsilon$(1S) to $\Upsilon$(4S) respectively Workman et al. (2022). Meanwhile, in the charmonium region, the $J/\psi$, $\psi$(3686), $\psi$(3770), $\psi$(4040), $\psi$(4160) are assigned to be $\psi(1S)$, $\psi(2S)$ Workman et al. (2022), $\psi(1D)$ Zhang (2007); Li and Chao (2009); Segovia et al. (2008a); Barnes et al. (2005), $\psi(3S)$ Zhang (2007); Li and Chao (2009); Segovia et al. (2008a); Barnes et al. (2005), and $\psi(2D)$ Zhang (2007); Li and Chao (2009); Segovia et al. (2008a); Barnes et al. (2005) respectively. Theoretical pictures of $c\bar{c}$ bound states including S-D mixings Li and Chao (2009); Shah et al. (2012); Wang et al. (2019); Wang and Liu (2023), hybrid charmonium $c\bar{c}g$ Liu et al. (2012); Luo and Liu (2006); Zhu (2005), compact tetraquark $\left(qc\bar{q}\bar{c}\right)$ Maiani et al. (2005); Drenska et al. (2009); Ebert et al. (2008), and molecule $\left(q\bar{c}\right)\left(\bar{q}c\right)$ Chiu and Hsieh (2006); Ding (2009); Close and Downum (2009); Close et al. (2010) have been proposed for studying the higher excited states, $\psi$(4230), $\psi$(4360), $\psi$(4660), and $Y(4500)$ observed by BESIII recently Ablikim et al. (2022).

Meanwhile, experimental new values of mass and leptonic decay width have been reported for these $J^{PC}=1^{--}$ states, and the understanding of these states has been also improved since many theoretical works have been done. However, theoretical predictions of leptonic widths for higher excited states are still not consistent with the latest experimental data Shah et al. (2012); Godfrey and Moats (2015); Wang et al. (2018); Segovia et al. (2016); Li and Chao (2009); Segovia et al. (2008a); Barnes et al. (2005); Barnes (2004); Mutuk (2018). All established heavy quarkonium states with $J^{PC}=1^{--}$ are listed in Table 1, with experimental data of mass and leptonic decay width from PDG Workman et al. (2022), and also with assignments from cited theoretical works. We briefly review the model of those works here and discuss their results in Sec. III.

Masses and leptonic widths of heavy quarkonium are estimated in a Martin-like potential model where a non-Coulombic power law potential is employed Shah et al. (2012). Different parameters are applied for bottomonium and charmonium mesons.

In Refs Godfrey and Moats (2015); Wang et al. (2018), masses and decay properties of excited bottomonium states are studied in a relativized quark model (Godfrey-Isgur model) developed from Refs. Godfrey and Isgur (1985); Godfrey (1985); Godfrey and Isgur (1986); Godfrey (2004, 2005) where a Cornell-like potential is employed. Masses, radiative transitions, annihilation decays, hadronic transitions and production cross sections of excited bottomonium states are evaluated.

Bottomonium mass spectrum, electromagnetic, strong and hadronic decays are also studied in a non-relativistic quark model Segovia et al. (2016) developed from Vijande et al. (2005) and their previous work Segovia et al. (2008b).

Charmonium spectrum, and electromagnetic decays are estimated in a non-relativistic model with a coulomb potential plus a screened linear potential Li and Chao (2009), and also are studied in a constituent quark model with a screened confinement potential Segovia et al. (2008a). In Ref Barnes et al. (2005), higher charmonium mass spectra are calculated in a non-relativistic model with a Cornell-like potential, and the corresponding leptonic widths are estimated in Ref Barnes (2004).

In this work, we apply a model developed from Ref. Zhao et al. (2021a, 2022) to predict the masses and leptonic decay widths of higher excited $1^{--}$ bottomonium $b\bar{b}$ and charmonium $c\bar{c}$ states. By considering possible S-D mixtures, and comparing the theoretical results with experimental data, we present possible conventional meson interpretation for the higher excited $1^{--}$ heavy quarkonium states. The states which can not be accommodated in the present picture will be studied in our future work by applying exotic pictures.

The paper is organized as follows. In Sec. II, a constituent quark model Zhao et al. (2021a, b, 2022); Xu et al. (2019, 2020) is developed to include a spin-dependent interaction Schoberl (1986) for studying higher orbital excited quarkonium states. In Sec. III, theoretical masses and leptonic decay widths of the $1^{--}$ heavy quarkonium states are calculated and compared with experimental data. Tentative assignments for higher excited heavy quarkonium states are suggested in S-D mixture picture. A summary is given in Sec. IV.

The non-relativistic Hamiltonian for studying the meson system takes the form,

with

where $H_{0}$ is taken from the previous work Zhao et al. (2021a, 2022, b). $\vec{r}$ is the relative coordinate between the two quarks, $M_{ave}$ is the spin-averaged mass taken from experimental data Workman et al. (2022), and $m_{r}$ stands for the reduced quark mass taking the form $m_{1}m_{2}/(m_{1}+m_{2})$. In the work, we employ $m_{c}=1270\ {\rm MeV}$ and $m_{b}=4180\ {\rm MeV}$ Workman et al. (2022). $\vec{L}$, $\vec{S}$, and $\vec{J}$ are the operators of orbital angular momentum, total spin, and total angular momentum, respectively. The tensor operator $S_{12}$ is defined as $S_{12}=(3(\vec{\sigma}_{1}\cdot r)(\vec{\sigma}_{2}\cdot r)-\vec{\sigma}_{1}\cdot\vec{\sigma}_{2})$.

$C_{SS}(r)$, $C_{LS}(r)$, and $C_{T}(r)$ in Eq. (II) are derived by following the Breit-Fermi interaction, that is,

Note that we have employed $V_{V}(r)=-B\,Erf[\sigma r]/r$ and $V_{S}(r)=Ar$, taken from Ref. Schoberl (1986).

$\vec{\sigma}_{i}$ in Eq. (II) are quark spin operators, and the contribution of $\vec{\sigma}_{i}\cdot\vec{\sigma}_{j}$ is $-3$ for $S=0$ and $+1$ for $S=1$ mesons. The matrix elements of $\vec{L}\cdot\vec{S}$ and $S_{12}$ in the $|JMLS\rangle$ basis read

The tensor operator $S_{12}$ has non-vanishing matrix elements between the two orbital parts of spin-triplet states.

The fitting results $M^{cal}$ for the S- and P-wave ground and first radial excited bottomonium and charmonium meson states are listed in Table 2, together with experimental data $M^{exp}$ from PDG Workman et al. (2022). Some typical theoretical mass results from other works for bottomonium mesons Segovia et al. (2016); Godfrey and Moats (2015); Shah et al. (2012) and charmonium mesons Barnes et al. (2005); Li and Chao (2009) are collected for comparison. Our results are fairly compatible with experimental data.

The S-wave and D-wave $1^{--}$ quarkonium leptonic decay width given by the Van Royen-Weisskopf formula Van Royen and Weisskopf (1967), including radiative QCD corrections for S-wave Barbieri et al. (1979), takes the same form with Segovia et al. (2016)

where the fine-structure constant $\alpha\simeq 1/137$. $e_{q}$ is the charge of quarks, $M_{n}$ is the mass of the decaying quarkonium states, $R_{nS}(0)$ and $R_{nD}(0)$ are the radial wave functions of the ${}^{3}S_{1}$ and ${}^{3}D_{1}$ states at the origin respectively. $\alpha_{s}$ is the running strong coupling constant, where $\alpha_{s}(b\bar{b})=0.118$ for bottomonium Shah et al. (2012) and $\alpha_{s}(c\bar{c})=0.26$ for charmonium Li and Chao (2009).

The difference between performing full integration for leptonic width and applying the lowest order approximation is about 50% for light mesons, but is about 10% for charmonium mesons and 4% for bottomonium mesons. Thus the Van’s formula with the first order approximation is reliable to be employed for estimating heavy quarkonium leptonic widths.