The $1D$-wave bottom-strange baryons and possible interpretation of $\Xi_{b}(6327)^{0}$ and $\Xi_{b}(6333)^{0}$

Establishing and improving hadron spectroscopy always is a key subject in hadron physics. Completing this subject can help us to understand the hadron structure, and then improve our understanding of the dynamics of Quantum Chromodynamics (QCD). As an indispensable part of hadrons, single heavy baryons play an important role since heavy quark symmetry is a good approximation and can provide some qualitative properties, especially in the baryons containing the bottom ($b$) quark. However, differing from the charmed baryons, searching for the bottom states is quite difficult for experiment since higher energy and higher luminance of the beams are required to produce them. Fortunately, experimenters have made important progress in searching for the bottom baryons in recent years [1], which provides us good opportunities to establish an abundant spectrum by decoding the inner structures of these newly observed bottom baryons.

In 2012, two narrow $1P$ $\Lambda_{b}^{0}$ baryons, denoted as $\Lambda_{b}(5912)^{0}$ and $\Lambda_{b}(5920)^{0}$, were firstly observed by the LHCb Collaboration [2] and confirmed by the CDF Collaboration [3] the following year. Later, two bottom baryons, i.e., $\Xi_{b}(6227)^{-}$ [4] and $\Sigma_{b}(6097)^{\pm}$ [5], were found by the LHCb Collaboration. Recently, two $1D$ $\Lambda_{b}^{0}$ candidates, $\Lambda_{b}(6146)^{0}$ and $\Lambda_{b}(6152)^{0}$, were discovered by the LHCb Collaboration in the $\Lambda_{b}^{0}\pi^{+}\pi^{-}$ spectrum [6]. This may be the first time that the low-lying $D$-wave singly bottom baryons are observed in experiment. In addition, four extremely narrow excited $\Omega_{b}$ states, $\Omega_{b}(6316)^{-}$, $\Omega_{b}(6330)^{-}$, $\Omega_{b}(6340)^{-}$ and $\Omega_{b}(6350)^{-}$, were announced by the LHCb Collaboration in the $\Xi_{b}^{0}K^{-}$ mass spectrum[7] in 2020. These observed single bottom baryons have stimulated a wide discussion [19, 18, 21, 20, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 22, 23, 24, 25, 26].

Very recently, the LHCb Collaboration again reported their new discovery of two new excited $\Xi_{b}^{0}$ states in the $\Lambda_{b}^{0}K^{-}\pi^{+}$ mass spectrum using a data sample of $pp$ collisions [27]. The measured masses and decay widths of these two states are

where the natural widths $\Gamma$ correspond to $90\%(95\%)$ confidence level upper limits. It is also found that the $\Xi_{b}(6327)^{0}$ resonance observed in the $\Lambda_{b}^{0}K^{-}\pi^{+}$ final states is dominantly contributed by the intermediate channel $\Sigma_{b}^{+}K^{-}$, while the $\Xi_{b}(6333)^{0}$ resonance is significantly contributed by the intermediate channel $\Sigma_{b}^{*+}K^{-}$. To decode their inner structure and analysis their dynamic mechanism in theory is necessary. Before the LHCb’s measurement [27], there exist many theoretical predictions of the mass spectra of the $\Xi_{b}$ and $\Xi_{b}^{\prime}$ baryons with various models in the literature [23, 11, 28, 24, 25, 10, 26, 29, 30, 31, 32, 33, 34], etc. We collect the some theoretical predictions of the spectrum for the $1D$ $\Xi_{b}$ and $\Xi_{b}^{\prime}$ baryons in Table 1. From the table, the two new exited $\Xi_{b}^{0}$ states observed by the LHCb Collaboration  [27] are in the predicted mass region of the $\lambda$-mode $1D$ $\Xi_{b}$ resonances with spin-parity $J^{P}=3/2^{+}$ and $J^{P}=5/2^{+}$ [16, 29, 30, 31, 32, 33]. Except mass spectrum, decay property is one of the important bases for determining hadron’s properties. However, there are only a few discussions of the strong decays of the $1D$ bottom baryons [16, 17, 22]. It should be pointed out that the predicted strong decay properties of the $\lambda$-mode $1D$ $\Xi_{b}$ resonances with spin-parity $J^{P}=3/2^{+}$ and $J^{P}=5/2^{+}$ in Refs. [16, 17] are in good agreement with the properties of the two new exited $\Xi_{b}^{0}$ states observed by the LHCb Collaboration [27]. Recently, Bijker et al. [35] assigned the newly observed $\Xi_{b}(6327)^{0}$ and $\Xi_{b}(6333)^{0}$ states as the $\lambda$-mode $1D$ $\Xi_{b}$ resonances with spin-parity $J^{P}=3/2^{+}$ and $J^{P}=5/2^{+}$ as well both within the elementary emission model and the quark-pair creation model. The same theoretical results were obtained in Ref. [10] with the method of QCD sum rules.

In the present work, we will further investigate the probable assignments of the new $\Xi_{b}(6327)^{0}$ and $\Xi_{b}(6333)^{0}$ states. Furthermore, considering the powerful detecting ability of LHCb, etc., more and more $1D$ $\Xi_{b}$ and $\Xi^{\prime}_{b}$ baryons are excepted to be observed in the near future. Thus, it is necessary for us to carry out a systematical study of the strong decay properties of the $1D$ $\Xi_{b}$ and $\Xi^{\prime}_{b}$ baryons including both the $\rho$- and $\lambda$-mode excitations. By analyzing the decay properties of the $1D$ $\Xi_{b}$ and $\Xi^{\prime}_{b}$ baryons, we will suggest ideal decay channels to establish missing states in the follow-up experiments. It should be mentioned that proper consideration of the heavy quark symmetry is necessary for the bottom baryons, wherein the states can more favor the $j$-$j$ coupling scheme [36, 19, 30]. Hence, we study the strong decays of the $\rho$- and $\lambda$-mode $1D$ $\Xi_{b}$ and $\Xi^{\prime}_{b}$ baryons within the $j$-$j$ scheme. With this coupling scheme, the classifications of the $1D$ $\Xi_{b}$ and $\Xi^{\prime}_{b}$ states investigated in this work are listed in Table 1.

The paper is organized as follows. In Sec. II, we give a brief review of our theoretical method and the relationship between the $j$-$j$ coupling scheme and the $L$-$S$ coupling scheme. In Sec. III, we investigate the strong decay properties of the $\rho$- and $\lambda$-mode $1D$ $\Xi_{b}$ and $\Xi^{\prime}_{b}$ baryons within the $j$-$j$ scheme, where we attempt to decode the properties of the newly observed $\Xi_{b}(6327)^{0}$ and $\Xi_{b}(6333)^{0}$ states. Finally, we present a short summarization in Sec. IV.

In this work, we systematically investigate the strong decay properties of both the $\rho$- and $\lambda$-modes $1D$ $\Xi_{b}$ and $\Xi^{\prime}_{b}$ baryons within the chiral quark model [37, 38, 39]. In the chiral quark model, the effective low energy quark-pseudoscalar-meson coupling in the SU(3) flavor basis at tree level is given by [37]

where $f_{m}$ represents the pseudoscalar meson decay constant and $\psi_{j}$ denotes the $j$-th quark field in a baryon. $\phi_{m}$ stands for the pseudoscalar meson octet and reads

Considering the harmonic oscillator spatial wave function of baryons in this work being nonrelativistic form, the quark-pseudoscalar-meson coupling is adopt the nonrelativistic form as well and is described by [40, 41, 42]

Here, $\omega_{m}$ and q correspond to the energy and three-vector momentum of the meson, respectively; $E_{i(f)}$, $M_{i(f)}$ and $\textbf{P}_{i(f)}$ represent the energy, mass and three-vector momentum of the initial (final) baryon; The $\mbox{\boldmath$\sigma$\unboldmath}_{j}$ and $\mu_{q}$ denote the Pauli spin vector and the reduced mass of the $j$-th quark in the initial and final baryons, respectively. $\textbf{p}^{\prime}_{j}=\textbf{p}_{j}-(m_{j}/M)\textbf{P}_{c.m.}$ is the internal momentum of the $j$-th quark in the baryon rest frame and $I_{j}$ is the isospin operator associated with the pseudoscalar meson; $\varphi_{m}=e^{(-)i\textbf{q}\cdot\textbf{r}_{j}}$ for absorbing (emitting) a meson.

According to the non-relativistic operator of quark-pseudoscalar-meson coupling, the partial decay amplitudes $M_{J_{iz},J_{fz}}$ of a light pseudoscalar meson emission in a baryon strong decays can be worked out. Here, $J_{iz}$ and $J_{fz}$ stand for the third components of the total angular momenta of the initial and final baryons, respectively. Then, the strong decay width can be calculated by

where $\delta$ is a global parameter accounting for the strength of the quark-meson couplings, which has been determined by experimental data in works [38, 39]. Here, we fix its value the same as that in Refs. [38, 39], i.e. $\delta=0.557$ MeV.

For the single bottom baryons, decoding their inner structure within the $j$-$j$ coupling scheme is considered preferable to that within the $L$-$S$ coupling scheme (listed in Table 2). In the heavy quark symmetry limit [36], the states within the the $j$-$j$ coupling scheme can be expressed as linear combinations of the states within the $L$-$S$ coupling scheme via the following relationship [30]:

In the expression, $l_{\rho}$ and $l_{\lambda}$ correspond to the quantum numbers of the orbital angular momenta for the $\rho$-mode and $\lambda$-mode (see Fig. 1) oscillators, respectively. $L$ ($=|l_{\rho}-l_{\lambda}|,\cdot\cdot\cdot,l_{\rho}+l_{\lambda}$) corresponds to the quantum number of the total orbital angular momentum. $s_{\rho}$ is the quantum numbers of the total spin of the two light quarks and $s_{Q}$ is the spin of the heavy quark. $S$ ($=|s_{\rho}-s_{Q}|,\cdot\cdot\cdot,s_{\rho}+s_{Q}$) denotes the quantum number of the total spin angular momentum.

In the calculation, the standard quark model parameters are adopted. Namely, we set $m_{u}=m_{d}=330$ MeV, $m_{s}=450$ MeV and $m_{b}=5000$ MeV for the constituent quark masses. The decay constants for $\pi$ and $K$ mesons are taken as $f_{\pi}=132$ MeV and $f_{K}=160$ MeV, respectively. The masses of the initial baryons($1D$ $\Xi_{b}$ and $\Xi^{\prime}_{b}$ baryons) are estimated based on the various theoretical predictions listed in Table 1. The masses of the final $S$-wave ground mesons and baryons used in the calculations are adopted from the Particle Data Group [1], and those of the final $P$-wave single-bottom baryons are taken the predictions in Ref. [29]. The spatial wave function ${}^{N}\Psi_{LL_{z}}$ of the baryons is taken the form of non-relativistic harmonic oscillator spatial-wave function. The harmonic oscillator parameter $\alpha_{\rho}$ for $us/ds$ system is taken as $\alpha_{\rho}=420$ MeV [17, 18], and another harmonic oscillator parameter $\alpha_{\lambda}$ is estimated by [38, 17, 18]

We notice that in the simplified case the ratio between oscillator frequencies $\omega_{\lambda}$ and $\omega_{\rho}$ reads

This expression indicates the excitation energy of the $\lambda$-mode is smaller than that of the $\rho$-mode. Thus, the $\rho$-excitation modes are heavier than the $\lambda$-excitation modes for the $1D$ $\Xi_{b}$ and $\Xi^{\prime}_{b}$ baryons. The realistic potential is much more complicated, while the general feature should be similar.

We conduct a systematic investigation of strong decays of $\rho$- and $\lambda$-modes $1D$ $\Xi_{b}$ and $\Xi^{{}^{\prime}}_{b}$ within the $j$-$j$ coupling scheme in the framework of the chiral quark model, emphatically explaining the two newly discovered states by LHCb Collaboration [27] and giving predictions of other missing $1D$-wave states. Our theoretical results are presented as follows.