Possible interpretation of the newly observed $\Omega _b$ states

Abstract

Inspired by the newly observed $\Omega _b$ states by the LHCb Collaboration, we investigate the two-body strong decays of the low-lying $\lambda$-mode $\Omega _b$ baryons up to $N=2$ shell using the chiral quark model within the j-j coupling scheme. Our results indicate that: (i) the newly observed states $\Omega _b(6316)^-$ and $\Omega _b(6330)^-$ are good candidates of the light spin $j=1$ states, while the spin-parity $J^P=1/2^-$ and $J^P=3/2^-$ cannot be distinguished. The other two states, $\Omega _b(6340)$ and $\Omega _b(6350)$ mostly correspond to the light spin $j=2$ states, while the spin-parity $J^P=3/2^-$ and $J^P=5/2^-$ cannot be distinguished as well. (ii) The 2S states with spin-parity $J^P=1/2^+$ and $J^P=3/2^+$, respectively, might be narrow states with a width of $\Gamma <2$ MeV. (iii) The 1D states are not broad and the total decay widths vary from several to dozens of MeV, which have a good potential to be observed in future experiments.

1 Introduction

Since the heavy quark provides a “flavour tag” that may give insight into the systematics of hadron resonances and the mechanism of confinement, the studies of hadrons containing heavy quarks (c or b) are important topics in hadron physics. It is evident that heavy-quark hadrons play an important role in our understanding of QCD. Since the first observation of the beauty baryon $\Lambda _b$ at the CERN $p\bar{p}$ collider [1], many singly heavy baryons were observed by various world wide experimental facilities in the past three decades [2]. In recent years, there have been specially fruitful in the observation of the heavy baryon states [3,4,5,6,7,8,9]. The rapidly growth of the experimental progress attracts a a great deal of attention from the hadron physics community.

Table 1 The predicted masses (MeV) of the $\Omega ^-_b$ states within various phenomenal models and methods. $|J^P,j\rangle ~(nl)$ denotes the state within the j-j coupling scheme

Very recently, in the $\Xi ^0_bK^-$ mass spectrum obtained using pp collisions the LHCb Collaboration reported four narrow peaks [10],

$$\begin{aligned} m[\Omega _b(6316)^-]= & {} 6315.64\pm 0.31\pm 0.07\pm 0.50~\text {MeV},\nonumber \\&\Gamma [\Omega _b(6316)^-]<2.8(4.2)~~\text {MeV}, \end{aligned}$$

(1)

$$\begin{aligned} m[\Omega _b(6330)^-]= & {} 6330.30\pm 0.28\pm 0.07\pm 0.50~\text {MeV},\nonumber \\&\Gamma [\Omega _b(6330)^-]<3.1(4.7)~~\text {MeV}, \end{aligned}$$

(2)

$$\begin{aligned} m[\Omega _b(6340)^-]= & {} 6339.71\pm 0.26\pm 0.05\pm 0.50~\text {MeV},\nonumber \\&\Gamma [\Omega _b(6340)^-]<1.5(1.8)~~\text {MeV}, \end{aligned}$$

(3)

$$\begin{aligned} m[\Omega _b(6350)^-]= & {} 6349.88\pm 0.35\pm 0.05\pm 0.50~\text {MeV},\nonumber \\&\Gamma [\Omega _b(6316)^-]<2.8(3.2)~~\text {MeV}, \end{aligned}$$

(4)

where the uncertainties are statistical, systematic and the limited knowledge of the $\Xi ^0_b$ mass, respectively. The natural widths $\Gamma$ correspond to $90\%$ ($95\%$) confidence level upper limits.

Before the LHCb’s measurement [10], there exist many theoretical calculations of the mass spectrum of the excited $\Omega _b$ states with various quark models [11,12,13,14,15,16,17,18], Regge phenomenology [19], QCD two-point sum rule method [20, 21], molecular model [22], etc. We collect the theoretical predictions in Table 1. From the table, the four peaks observed by the LHCb Collaboration [10] are in the predicted mass region of the 1P wave excitations [11,12,13,14, 16,17,18]. In addition, the possibility as 2S excitations cannot be excluded absolutely based simply on the predicted masses [15, 17]. There are also some discussions of the strong decay properties of the excited $\Omega ^-_b$ states. In the framework of the QCD sum rule method [23], the 1P- and 2S-wave states with spin-1/2 and spin-3/2 are predicted to be narrow resonances. Within an harmonic oscillator based model, the authors in Ref. [14] obtained that the 1P-wave states should be narrow resonances as well. With a chiral quark model our group predicted that in the L-S coupling scheme the 1P-wave states with spin-parity $J^P=3/2^-$ and $J^P=5/2^-$ should be narrow states with a width of a few MeV, while two $J^P=1/2^-$ 1P-wave states $| ^2P_{1/2}\rangle$ and $| ^4P_{1/2}\rangle$ are most likely to be broad states with a width of $\Gamma \sim$ (50–100) MeV [24]; however, in the j-j coupling scheme only the $j=0$ P-wave state with $J^P=1/2^-$ is a broad state, the other four 1P-wave states should be narrow states [25]. Recently, Liang et al. [26] assigned the observed four peaks as the 1P wave states within the quark pair creation model. The similar results were obtained in Refs. [27, 28]. In addition, inspired by the LHCb’s measurement [10], the possible interpretations of the peaks in the higher energy as the meson-baryon molecular states were presented as well in Ref. [29].

We notice that for the singly heavy flavour quark systems, proper consideration of the heavy quark symmetry is necessary, thus, the states may more favor the j-j coupling scheme. With this coupling scheme, we have studied the strong decays of the 1P-wave singly bottom heavy baryons in our previous work [25]. In the present work, we further conduct a systematical study of the strong decay properties of the low-lying $\lambda$-mode $\Omega _b$ states in the $j-j$ coupling scheme, and attempt to decode the inner structures of the four peaks observed by the LHCb Collaboration [10]. Our theoretical results suggest that both the $\Omega _b(6316)^-$ and $\Omega _b(6330)^-$ are good candidates of the light spin $j=1$ states, while the spin-parity $J^P=1/2^-$ and $J^P=3/2^-$ cannot be distinguished due to the limited data. The other two states, $\Omega _b(6340)^-$ and $\Omega _b(6350)^-$ are good candidates of the light spin $j=2$ states, while the spin-parity $J^P=3/2^-$ and $J^P=5/2^-$ cannot be distinguished as well. In addition, we notice that the light spin $j=0$ states with the spin-parity $J^P=1/2^-$ are most likely to be a moderate state with a width of $\Gamma \sim 100$ MeV, and the $\Xi _b K$ decay channel almost saturates its total decay widths. For the 2S-wave $\Omega _b$ states with $J^P=1/2^+$ and $J^P=3/2^+$, they might be narrow states with a width of $\Gamma <2$ MeV. We notice that the 1D states are not broad and the total decay widths vary from several to dozens of MeV, which have a good potential to be observed in future experiments.

This paper is organized as follows. In Sect. II we give a brief introduction of the strong decay model and the relationship of the singly-heavy baryon spectrum between the j-j coupling scheme and the L-S coupling scheme. We discuss the strong decays of the low-lying 1P-, 2S-, and 1D-wave $\Omega _b$ states within the j-j coupling scheme in Sect. III and summarize our results in Sect. IV.

2 The model

In this work we study the strong decay properties with the chiral quark model. This model has been successfully applied to study the strong decays of baryons and heavy-light mesons [24, 25, 30,31,32,33,34,35,36,37,38,39,40,41,42]. In the chiral quark model, the effective low energy quark-pseudoscalar-meson coupling in the SU(3) flavor basis at tree level is described by [43]

$$\begin{aligned} H_m=\sum _j \frac{1}{f_m}\bar{\psi }_j\gamma ^{j}_{\mu }\gamma ^{j}_{5}\psi _j\vec {\tau }\cdot \partial ^{\mu }\vec {\phi }_m, \end{aligned}$$

(5)

where $\psi _j$ corresponds to the jth quark field in a baryon. $f_m$ stands for the pseudoscalar meson decay constant. $\phi _m$ denotes the pseudoscalar meson octet.

To match the nonrelativistic harmonic oscillator spatial wave function $^N\Psi _{LL_z}$ in the calculations, we adopt a nonrelativistic form of the quark-pseudoscalar coupling and obtain [44,45,46]

$$\begin{aligned} H^{nr}_{m}= & {} \sum _j\Big \{\frac{\omega _m}{E_f+M_f}{\varvec{\sigma }}_j\cdot \mathbf{P} _f+ \frac{\omega _m}{E_i+M_i}{\varvec{\sigma }}_j \cdot \mathbf{P} _i \nonumber \\&-{\varvec{\sigma }}_j \cdot \mathbf{q} +\frac{\omega _m}{2\mu _q}{\varvec{\sigma }}_j\cdot \mathbf{p} '_j\Big \}I_j e^{-i\mathbf {q}\cdot \mathbf {r}_j}, \end{aligned}$$

(6)

where $(E_i,~M_i,~\mathbf {p}_i)$ and $(E_f,~M_f,~\mathbf {p}_f)$ represent the energy, mass and three-vector momentum of the initial and final baryon, respectively. ($\omega _m,~\mathbf {q}$) are the energy and three-vector momentum of the final light pseudoscalar meson. $\mathbf {p}'_j=\mathbf {p}_j-(m_j/M)\mathbf {P}_{\text {c.m.}}$ is the internal momentum of the jth quark in the baryon rest frame. $\mu _q$ stands for a reduced mass expressed as $1/\mu _q=1/m_j+1/m'_j$, and $\mathbf {\sigma }_j$ denotes the Pauli spin vector on the jth quark. The isospin operator $I_j$ associated with K meson is given by

$$\begin{aligned} I_{j}={\left\{ \begin{array}{ll} a^{\dagger }_j(u)a_j(s) &{} \text {for}~K^{-},\\ a^{\dagger }_j(d)a_j(s) &{} \text {for}~\bar{K}^0. \end{array}\right. } \end{aligned}$$

(7)

Here, $a_j^{\dagger }(u,d)$ and $a_j(s)$ are the creation and annihilation operator for the u,  d and s quarks on jth quark.

Then the partial decay width for the emission of a light pseudoscalar meson is

$$\begin{aligned} \Gamma =\left( \frac{\delta }{f_m}\right) ^2\frac{(E_f +M_f)|\mathbf {q}|}{4\pi M_i}\frac{1}{2J_i+1}\sum _{J_{iz}J_{fz}}|\mathcal {M}_{J_{iz},J_{fz}}|^2, \end{aligned}$$

(8)

where $J_{iz}$ and $J_{fz}$ are the third components of the total angular momenta of the initial and final baryons, respectively. $\delta$ denotes a global parameter accounting for the strength of the quark-meson couplings. $\mathcal {M}_{J_{iz},J_{fz}}$ stands for the transition amplitude.

According to our previous works [25, 30] together with the nature of the newly observed four peaks [10], the physical states of the singly heavy baryons seem more favor the j-j coupling scheme. Thus, in the present work we study the strong decay properties within this coupling scheme.

In the L-S coupling scheme, the states are constructed by [16]

$$\begin{aligned} \left| ^{2S+1}L_{J}\right\rangle = \left| \left[ \left( l_{\rho } l_{\lambda }\right) _L\left( s_{\rho }s_Q\right) _S\right] _{J^P}\right\rangle , \end{aligned}$$

(9)

where $l_{\rho }$ and $l_{\lambda }$ are the quantum numbers of the orbital angular momenta $\mathbf {l}_{\rho }$ and $\mathbf {l}_{\lambda }$ for the $\rho$- and $\lambda$-mode oscillators, respectively. L corresponds to the quantum number of the total orbital angular momentum $\mathbf {L}=\mathbf {l}_{\rho }+\mathbf {l}_{\lambda }$, which determine the parity of a state by $P=(-1)^{l_{\rho }+l_{\lambda }}$. $s_{\rho }$ and $s_Q$ stand for the quantum numbers of the total spin of the two light quarks $\mathbf {s}_{\rho }$ and the spin of the heavy quark $\mathbf {s}_Q$, respectively. S is the quantum number of the total spin angular momentum $\mathbf {S}=\mathbf {s}_{\rho }+\mathbf {s}_Q$. According to the quark model classification, there are five $\lambda$-mode 1P-wave states, six $\lambda$-mode 1D-wave states and two 2S-wave states, and their corresponding quantum numbers have been collected in Table 2.

Table 2 The classifications of the low-lying $\lambda$-mode 1P-, 1D- and 2S-wave states within the L-S coupling scheme

Under the heavy-quark symmetry limit [47], the states within the j-j coupling scheme are constructed by

$$\begin{aligned} \left| J^P,j\right\rangle = \left| \left\{ \left[ \left( \ell _\rho \ell _\lambda \right) _L s_{\rho }\right] _js_Q\right\} _{J^P}\right\rangle . \end{aligned}$$

(10)

Here, the total orbital angular momentum $\mathbf {L}=\mathbf {l}_{\rho }+\mathbf {l}_{\lambda }$ couples to the total spin of the two light quarks $\mathbf {s}_{\rho }$, and then gets $\mathbf {j}=\mathbf {L}+\mathbf {s}_{\rho }$. $\mathbf {j}$ couples to the spin of the heavy quark $\mathbf {s}_{Q}$, and gives the total angular momentum $\mathbf {J}=\mathbf {j}+\mathbf {s}_Q$. Similarly, with this coupling scheme the low-lying $\lambda$-mode 1P-, 1D- and 2S-wave states are listed in Table 3.

The states within the j-j coupling scheme can be expressed as linear combinations of the states within the L-S coupling scheme by [16]

$$\begin{aligned} |\{[(l_\rho l_\lambda )_Ls_{\rho }]_js_Q\}_J\rangle= & {} (-1)^{L+s_\rho +1/2+J}\sqrt{2j+1} \sum _{S}\sqrt{2S+1} \nonumber \\&\begin{Bmatrix}L&s_\rho&j\\s_Q&J&S \end{Bmatrix} |[(l_\rho l_\lambda )_L(s_{\rho }s_Q)_S]_J\rangle . \end{aligned}$$

(11)

This relationship between the two different coupling schemes indicates that the states may well be mixed with the same spin-parity $J^P$ within the L-S coupling scheme for the singly heavy baryons. The mixing angles can be estimated by Eq. (11).

In this work, 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 K is taken as $f_{K}=160$ MeV. The harmonic oscillator space-wave functions are adopted to describe the spatial wave function of the initial and final baryons, and the harmonic oscillator parameter $\alpha _{\rho }$ in the wave functions for ds/us and ss systems are taken as $\alpha _{\rho }=420$ and 440 MeV, respectively. Another harmonic oscillator parameter $\alpha _{\lambda }$ is determined by $\alpha _{\lambda }=(\frac{3m'}{2m+m'})^{1/4}\alpha _{\rho }$, where $m'$ denotes the heavy quark mass and m denote the light quark mass. The masses of the final baryons and mesons are taken from the PDG [48]. For the global parameter $\delta$, we fix its value the same as our previous works [25, 31, 39], i.e., $\delta =0.557$.

Table 3 The classifications of the low-lying $\lambda$-mode 1P-, 1D- and 2S-wave states within the j-j coupling scheme

3 Results and analysis

Inspired by the newly observed four peaks by the LHCb Collaboration [10], we carry out a systematic study of the strong decay behaviors of the singly bottom baryons $\Omega _b$ up to $N=2$ shell within the j-j coupling scheme in the framework of the chiral quark model. Our results and theoretical predictions are presented as follows.

3.1 1P states

The predicted masses of the $\lambda$-mode 1P-wave baryons $\Omega _b$ are about 6.3 GeV (see Table 1), which are in good agreement with the masses of the newly observed four $\Omega _b$ states [10]. In addition, via evaluation the OZI-allowed two-body decays, some people obtained that the 1P $\Omega _b$ states have a narrow decay widths [14, 23], which are in accordance with the nature of the newly observed $\Omega _b$ states. To pin down the inner structures of the four peaks, it is crucial to study the decay properties of those 1P states carefully.

3.1.1 $J^P=1/2^-$ states

There are two $J^P=1/2^-$ states $|J^P=\frac{1}{2}^-,0\rangle$ and $|J^P=\frac{1}{2}^-,1\rangle$ within the j-j coupling scheme (see Table 3). They can be expressed as mixed states between $|^2P_{\lambda }\frac{1}{2}^-\rangle$ and $|^4P_{\lambda }\frac{1}{2}^-\rangle$:

$$\begin{aligned} \Bigg |J^P= & {} \frac{1}{2}^-,0 \Bigg \rangle =-\sqrt{\frac{1}{3}}\Bigg | {}^2P_{\lambda }\frac{1}{2}^-\Bigg \rangle +\sqrt{\frac{2}{3}}\Bigg | {}^4P_{\lambda }\frac{1}{2}^-\Bigg \rangle , \end{aligned}$$

(12)

$$\begin{aligned} \Bigg |J^P= & {} \frac{1}{2}^-,1\Bigg \rangle =\sqrt{\frac{2}{3}}\Bigg |{}^2P_{\lambda }\frac{1}{2}^-\Bigg \rangle +\sqrt{\frac{1}{3}}\Bigg |{}^4P_{\lambda }\frac{1}{2}^-\Bigg \rangle . \end{aligned}$$

(13)

Their masses are predicted to be above the threshold of $\Xi _b K$. Considering the uncertainties of the predicted mass, we plot the variation of the decay width of the $\Xi _b K$ decay mode as a function of the mass of the state $|J^P=1/2^-,0\rangle$ in Fig. 1. The decay width increases with the mass increasing in the range of (6290–6350) MeV, and varies in the region of $\Gamma \sim (33-151)$ MeV. The decay properties of this state is inconsistent with the natures of the four $\Omega _b$ states, thus, the four $\Omega _b$ states as the state $|J^P=1/2^-,0\rangle$ should be excluded. Since the predicted width of $|J^P=1/2^-,0\rangle$ is not broad, this state might be observed in the $\Xi _b K$ channel when enough data are accumulated in experiments.

Fig. 1

Partial decay widths of the $\Xi _b K$ channel for the $J^P = 1/2^-$, $J^P = 3/2^-$ and $J^P = 5/2^{-} \Omega _b$ states as a function of the mass

For the other $J^P=1/2^-$ state $|J^P=1/2^-,1\rangle$, the decay mode $\Xi _b K$ is forbidden in the heavy quark symmetry limit (see Table 4), thus, this state should be very narrow state. We notice that the widths of two peaks, $\Omega _b(6316)$ and $\Omega _b(6330)$, observed in the $\Xi ^0_b K^-$ mass spectra [10] are $0.00^{+0.7}_{-0.0}$ MeV and $0.00^{+0.4}_{-0.0}$ MeV, respectively, which are good candidates of this state.

Considering the heavy-quark symmetry is only an approximation, the mixing angle between two states within the L-S coupling scheme should be slightly different from that determined in the j-j coupling scheme. Namely, the mixing angle $\phi$ for the physical states as mixed states between $|^2P_{\lambda }J\rangle$ and $|^4P_{\lambda }J\rangle$ within the L-S coupling scheme, i.e.,

$$\begin{aligned} \left( \begin{array}{c}| P~{J^P}\rangle _1\\ | P~{J^P}\rangle _2 \end{array}\right) =\left( \begin{array}{cc} \cos \phi &{} \sin \phi \\ -\sin \phi &{}\cos \phi \end{array}\right) \left( \begin{array}{c} |^2P_{\lambda }J^P \rangle \\ |^4P_{\lambda }J^P\rangle \end{array}\right) , \end{aligned}$$

(14)

may rang from the value of the L-S coupling scheme ($\phi =0^\circ$) to that of the j-j coupling scheme.

Table 4 Partial decay widths (MeV) of the 1P states within the j-j coupling scheme in the $\Omega _b$ family. The value inside of the braces denotes the reference mass (MeV) of the corresponding state

Fig. 2

Partial decay widths for the $J^P = 1/2^-$ and $J^P = 3/2^-$ $\Omega _b$ states as a function of the mixing angle

For the P-wave states with $J^P=1/2^-$, the mixing angle obtained in the heavy quark limit within the j-j coupling scheme is $\phi \simeq 35^\circ$, namely $\cos \phi =\sqrt{2/3}$ and $\sin \phi =\sqrt{1/3}$. Thus according to the mixing scheme defined in Eq. (14), in Fig. 2 we plot the strong decay widths of the $J^P=1/2^-$ states as a function of the mixing angle $\phi$ in the region of $(0^\circ ,35^\circ )$ by fixing the mass of the resonance at $M=6316$ MeV. It is found that the decay widths are strongly dependent on the mixing angle. The decay width of $|P \frac{1}{2}^-\rangle _2$ is about $\Gamma \sim (83-126)$ MeV with the mixing angle varied in the region what we considered. For the the other mixed state $|P \frac{1}{2}^-\rangle _1$, the decay width is close to zero with the mixing angle tending to $\phi \simeq 35^\circ$ which is determined in the j-j coupling scheme. This may be evidence that the heavy quark symmetry is a good approximation for the physical states of the singly bottom baryons, thus, the properties of the newly observed $\Omega _b$ states, $\Omega _b(6316)$ and $\Omega _b(6330)$, can be naturally understood.

3.1.2 $J^P=3/2^-$ states

Within the j-j coupling scheme, there are two $J^P=3/2^-$ P-wave states $|J^P=\frac{3}{2}^-,1\rangle$ and $|J^P=\frac{3}{2}^-,2\rangle$, which can be expressed as mixed states in the L-S coupling scheme by Eq. (11):

$$\begin{aligned}&\left| J^P=\frac{3}{2}^-,1\right\rangle =-\sqrt{\frac{1}{6}}\left| ^2P_{\lambda }\frac{3}{2}^-\right\rangle +\sqrt{\frac{5}{6}}\left| ^4P_{\lambda }\frac{3}{2}^-\right\rangle , \end{aligned}$$

(15)

$$\begin{aligned}&\left| J^P=\frac{3}{2}^-,2\right\rangle =\sqrt{\frac{5}{6}}\left| ^2P_{\lambda }\frac{3}{2}^-\right\rangle +\sqrt{\frac{1}{6}}\left| ^4P_{\lambda }\frac{3}{2}^-\right\rangle . \end{aligned}$$

(16)

For the two $J^P=3/2^-$ states, the $\Xi _b K$ decay mode is the only OZI-allowed two-body strong decay channel. The decay mode $\Xi _b K$ is forbidden for the state $|J^P=\frac{3}{2}^-,1\rangle$ in the heavy-quark symmetry limit, thus, this state should be very narrow state. Combining the natures of the newly observed states $\Omega _b(6316)$ and $\Omega _b(6330)$, we obtain that these two $\Omega _b$ resonances are good candidates of the $|J^P=\frac{3}{2}^-,1\rangle$ state as well.

For the other $J^P=3/2^-$ state $|J^P=\frac{3}{2}^-,2\rangle$, we also plot the decay width as a function of the mass in the range of $M=(6290-6350)$ MeV in Fig. 1. It is found that the decay width is very narrow as well due to the D-wave suppression. Fixing the mass of $|J^P=\frac{3}{2}^-,2\rangle$ at $M=6340$ MeV, we obtain

$$\begin{aligned} \Gamma \left[ |J^P=\frac{3}{2}^-,2\rangle \rightarrow \Xi _bK \right] \simeq 2.2~\text {MeV}. \end{aligned}$$

(17)

This theoretical width is well comparable with the observations of the two resonances $\Omega _b(6340)$ and $\Omega _b(6350)$ [10], thus the two resonances are good candidates of the $J^P=3/2^-$ state $|J^P=\frac{3}{2}^-,2\rangle$.

The mixing angle for the $J^P=3/2^-$ physical states may be slightly different from the value $\phi =24^\circ$ determined by the j-j coupling scheme in the heavy-quark symmetry limit. Similarly we fix the masses at $M=6330$ MeV and plot the variation of the strong decay widths of the $J^P=3/2^{-}$ states as a function of the mixing angle in Fig. 2. From the figure, it is found that the decay widths for the two $J^P=3/2^-$ states are slightly depend on the mixing angle and less than $\Gamma <1.3$ MeV in the range of ($0^\circ$, $24^\circ$).

3.1.3 $J^P=5/2^-$ states

There is only one P-wave state with $J^P=5/2^-$. There is no differences between the j-j coupling scheme and L-S coupling scheme, namely

$$\begin{aligned} \left| J^P=\frac{5}{2}^-,2\right\rangle =\left| ^4P_{\lambda }\frac{5}{2}^-\right\rangle . \end{aligned}$$

(18)

According to the Table 1, the theoretical masses of the $J^P=5/2^-$ states are about 6.3 GeV. Thus, we plot the decay width of this state as a function of the mass in Fig. 1 in the region of (6290–6350) MeV. It is found that this state is most likely narrow ($\Gamma <3.5$ MeV) because of the D-wave suppression within the mass range what we considered, and comparable with the nature of the newly observed $\Omega _b$ states, $\Omega _b(6340)$ and $\Omega _b(6350)$. Fixing the mass of $|J^P=\frac{5}{2}^-,2\rangle$ at $M=6350$ MeV, we obtain

$$\begin{aligned} \Gamma \left[ |J^P=\frac{5}{2}^-,2\rangle \rightarrow \Xi _bK \right] \simeq 3.4~\text {MeV}. \end{aligned}$$

(19)

So far, we have calculated the strong decay properties of the five $\lambda$-mode P-wave $\Omega _b$ states within the j-j coupling scheme. Our results show that excepting the state $|1/2^-,0\rangle$ being a middle width state with a width of $\Gamma \sim 100$ MeV, the other four states are most likely to be narrow states, which are good candidates for the newly observed four $\Omega _b$ states. The newly observed resonances $\Omega _b(6316)$ and $\Omega _b(6330)$ can be interpreted as the light spin $j=1$ states, while their spin-parity $J^P=1/2^-$ and $J^P=3/2^-$ cannot be distinguished. The other two states $\Omega _b(6340)$ and $\Omega _b(6350)$ may be the light spin $j=2$ states, but the spin-parity $J^P=3/2^-$ and $J^P=5/2^-$ cannot be distinguished.

3.2 2S states

According to the predicted masses of the 2S states in the $\Omega _b$ family (see Table 1), the possibility as the candidates of the newly observed resonances cannot be excluded completely. It is necessary to study the strong decay properties of the 2S states.

There are two 2S states with spin-parity $J^P=1/2^+$ and $j^P=3/2^+$, respectively. For these two 2S-wave states, there is no differences between the j-j coupling scheme and L-S coupling scheme. Adopting the predicted masses from the Ref. [13], we calculate the partial decay widths of the two 2S states. Our results have been listed in Table 5. From the table, we get that the two 2S states may be narrow state with a comparable total decay width of $\Gamma \sim 1$ MeV.

Table 5 Partial decay widths (MeV) for the 2S-wave $\Omega _b$ states, whose masses (MeV) are taken the quark model predictions of Ref. [13]

Considering the uncertainty of the predicted masses, we plot the strong decay properties of the two states as functions of the masses in the range of (6.35,6.55) MeV, as shown in Fig. 3. The total decay widths of the 2S states are about $\Gamma <2$ MeV within the masses we considered. Meanwhile, we notice that the partial decay width of the $\Xi _bK$ mode has an oscillation as the mass increases. That’s because the 2S-wave function has a node, and then the overlap between the initial and final states is a quadratic function of the momentum of the final state. As the mass increases, the tension between phase space and overlap makes the variation in Fig. 3.

The decay properties of the 2S states were calculated in Ref. [26] within quark pair creation model as well. The authors predicted that the total decay widths of the two 2S states were about $\Gamma \sim 50$ MeV, which were incomparable with our predictions.

Fig. 3

Partial decay widths of the 2S states with $J^P = 1/2^+$ and $J^P = 3/2^+$, respectively, as a function of the masses

Table 6 Partial widths (MeV) of strong decays for the $\lambda$-mode D-wave $\Omega _b$ baryons, whose masses (MeV) are taken the quark model predictions of Ref. [13]

It should be pointed out that although some people obtain the theoretical masses of the 2S states in the $\Omega _b$ family are about 6.3–6.4 GeV [15, 17] and the possibility as the new observed $\Omega _b$ states cannot be excluded absolutely, however, the mass splitting between the 1P and 2S is about 100-150 MeV. We notice that the splitting between the four newly observed resonances ranges from several to 10 MeV. If the four resonances are conventional baryons and two of them are interpreted as the 2S states, while how to understand the other two resonances will be a great challenge.

3.3 1D states

The 1D-wave $\Omega _b$ states are most likely to be observed in forthcoming experiments as well. In Ref. [34] we have studied the strong decays of the 1D-wave $\Omega _b$ states in the L-S coupling scheme. Since the heavy-quark symmetry indicates the the singly-heavy baryons may more favor the j-j coupling scheme, in present work we further give our predictions in the j-j coupling scheme.

For the $\lambda$-mode 1D states, there are six states within the j-j coupling scheme and the corresponding relations to the states in the L-S coupling scheme are

$$\begin{aligned} \Bigg |J^P =\frac{1}{2}^+,1\Bigg \rangle= & {} \Big |{}^4D_{\lambda \lambda }\frac{1}{2}^+\Big \rangle , \end{aligned}$$

(20)

$$\begin{aligned} \Bigg |J^P =\frac{3}{2}^+,1\Bigg \rangle= & {} -\sqrt{\frac{1}{2}}\Big |{}^2D_{\lambda \lambda }\frac{3}{2}^+ \Big \rangle + \sqrt{\frac{1}{2}}\Big |{}^4D_{\lambda \lambda }\frac{3}{2}^+ \Big \rangle , \end{aligned}$$

(21)

$$\begin{aligned} \Bigg |J^P=\frac{3}{2}^+,2\Bigg \rangle= & {} \sqrt{\frac{1}{2}}\Big |{}^2D_{\lambda \lambda }\frac{3}{2}^+ \Big \rangle + \sqrt{\frac{1}{2}}\Big |{}^4D_{\lambda \lambda }\frac{3}{2}^+ \Big \rangle , \end{aligned}$$

(22)

$$\begin{aligned} \Bigg |J^P=\frac{5}{2}^+,2\Bigg \rangle= & {} -\frac{\sqrt{2}}{3}\Big |{}^2D_{\lambda \lambda }\frac{5}{2}^+ \Big \rangle + \frac{\sqrt{7}}{3}\Big |{}^4D_{\lambda \lambda }\frac{5}{2}^+ \Big \rangle , \end{aligned}$$

(23)

$$\begin{aligned} \Bigg |J^P=\frac{5}{2}^+,3\Bigg \rangle= & {} \frac{\sqrt{7}}{3}\Big |{}^2D_{\lambda \lambda }\frac{5}{2}^+ \Big \rangle + \frac{\sqrt{2}}{3}\Big |{}^4D_{\lambda \lambda }\frac{5}{2}^+ \Big \rangle , \end{aligned}$$

(24)

$$\begin{aligned} \Bigg |J^P=\frac{7}{2}^+,3\Bigg \rangle= & {} \Big |{}^4D_{\lambda \lambda }\frac{7}{2}^+ \Big \rangle . \end{aligned}$$

(25)

For the D-wave $\Omega _b$ states, within the masses adopted from Ref. [13] we calculate the strong decays of the 1D states and collect in Table 6.

From Table 6, it is seen that the total decay widths of $|J=1/2^+,1\rangle$ and $|J=3/2^+,1\rangle$ are about $\Gamma \sim 28$ MeV, and both mainly decay into $\Xi _bK$. The strong decays of the states $|J=5/2^+,3\rangle$ and $|J=7/2^+,3\rangle$ are governed by the $\Xi _bK$ channel as well, and the total decay widths of both states are about $\Gamma \sim 10$ MeV. However, the decay mode $\Xi _bK$ is forbidden for the states $|J=3/2^+,2\rangle$ and $|J=5/2^+,2\rangle$. The two states might be narrow state with the total decay widths of $\Gamma \sim 11$ MeV and $\Gamma \sim 6$ MeV, respectively. Meanwhile, we notice that the strong decays of the state $|J=3/2^+,2\rangle$ is dominated by the $\Xi '_bK$ channel, while that of the state $|J=5/2^+,2\rangle$ is the $\Xi ^*_bK$ channel.

Comparing with the predictions in Ref. [26], the main decay channel for each 1D states is consistent with each other. However the total decay widths are difference, especially for the states $|J=1/2^+,1\rangle$ and $|J=3/2^+,1\rangle$. The predicted total decay widths for the two states in Ref. [26] are about three times larger than ours.

4 Summary

In this paper, we carry out a systematic study of the OZI allowed two-body strong decays of the low-lying $\lambda$-mode $\Omega _b$ resonances up to the $N=2$ shell in the framework of a chiral quark model within the j-j coupling scheme. For the newly observed four $\Omega _b$ states, we give a possible interpretation in theory. Meanwhile, we give the predictions for the decay properties of the 2S- and 1D- wave states, and hope to provide helpful information for searching these missing $\Omega _b$ states in the future.

Our theoretical results show that the newly observed states $\Omega _b(6316)$ and $\Omega _b(6330)$ are most likely to be explained as the 1P wave states with the light spin $j=1$, while the spin-parity $J^P=1/2^-$ and $J^P=3/2^-$ cannot be distinguished. The other two newly observed states $\Omega _b(6340)$ and $\Omega _b(6350)$ may correspond to the 1P-wave states with the light spin $j=2$ excepting the spin-parity $J^P=3/2^-$ and $J^p=5/2^-$ cannot being distinguished. In addition, we notice that the state $|1/2^-,0\rangle (1P)$ might be a moderate width state with a total decay width of $\Gamma \sim 100$ MeV, and the $\Xi _{bK}$ channel mostly saturates its total decay width.

For the two 2S states with the spin-parity $J^P=1/2^+$ and $J^P=3/2^+$, they might be narrow states with a total width of $\Gamma <2$ MeV. Considering the large mass splitting between 1P- and 2S-wave states together with the small mass splitting along the observed four peaks, it is unnatural to explain two of the four peaks as 2S-wave states within the conventional baryon picture.

For the 1D states, the total decay widths are not broad and vary from several to dozens of MeV. In addition, we notice that the strong decays of the states $|J=1/2^+,1\rangle$, $|J=3/2^+,1\rangle$, $|J=5/2^+,3\rangle$, and $|J=7/2^+,3\rangle$ are governed by the $\Xi _bK$ channel, while this channel is forbidden for the states $|J=3/2^+,2\rangle$ and $|J=5/2^+,2\rangle$. The state $|J=3/2^+,2\rangle$ dominantly decays to the $\Xi '_bK$ channel, and the state $|J=5/2^+,2\rangle$ are governed by the $\Xi ^{*}_{bK}$ channel.