Excited $\varOmega _b$ baryons and fine structure of strong interaction

Abstract

The heavy baryon system bounded by the strong interaction has a rich internal structure, so its mass spectra can have the fine structure similar to the line spectra of atom bounded by the electromagnetic interaction. We systematically study the internal structure of P-wave $\varOmega _b$ baryons and calculate their D-wave decay properties. The present study, together with our previous studies on their mass spectra and S-wave decay properties, suggest that all the four excited $\varOmega _b$ baryons recently discovered by LHCb can be well explained as P-wave $\varOmega _b$ baryons, and their beautiful fine structure is directly related to the rich internal structure of P-wave $\varOmega _b$ baryons.

1 Introduction

The electromagnetic interaction holds the electrons and protons together inside a single atom, leading to the gross, fine, and hyperfine structures of the line spectra. The strong interaction occurring between quarks and gluons is similar in some aspects, and it is interesting to investigate whether the hadron spectra also have the fine structure. An ideal platform to study this is the heavy baryon system containing one charm or bottom quark, which is interesting in a theoretical point of view [1,2,3]: the light quarks and gluons circle around the nearly static heavy quark, so that the whole system behaves as the QCD analogue of the hydrogen bounded by the electromagnetic interaction. This system has a rich internal structure, so its mass spectra can have the fine structure similar to hydrogen spectra [4,5,6,7].

In the past years important progress has been made in this field, and many heavy baryons were observed in experiments [4, 8,9,10,11,12]. Especially, in 2017 the LHCb Collaboration discovered as many as five excited $\varOmega _c$ states, $\varOmega _c^0(3000)$, $\varOmega _c^0(3050)$, $\varOmega _c^0(3066)$, $\varOmega _c^0(3090)$, and $\varOmega _c^0(3119)$, simultaneously in the $\varXi _c^+ K^-$ mass spectrum [8]. Very recently, they further discovered four excited $\varOmega _b$ states at the same time in the $\varXi _b^0 K^-$ mass spectrum [12]:

$$\begin{aligned} \varOmega _b(6316)^- : M&= 6315.64 \pm 0.31 \pm 0.07 \pm 0.50~\mathrm{MeV} \, , \nonumber \\ \varGamma&< 2.8~\mathrm{MeV} \, , \end{aligned}$$

(1)

$$\begin{aligned} \varOmega _b(6330)^-: M&= 6330.30 \pm 0.28 \pm 0.07 \pm 0.50~\mathrm{MeV} \, , \nonumber \\ \varGamma&< 3.1 ~\mathrm{MeV} \, , \end{aligned}$$

(2)

$$\begin{aligned} \varOmega _b(6340)^-: M&= 6339.71 \pm 0.26 \pm 0.05 \pm 0.50~\mathrm{MeV} \, , \nonumber \\ \varGamma&< 1.5~\mathrm{MeV} \, , \end{aligned}$$

(3)

$$\begin{aligned} \varOmega _b(6350)^-: M&= 6349.88 \pm 0.35 \pm 0.05 \pm 0.50~\mathrm{MeV} \, , \nonumber \\ \varGamma&= 1.4^{+1.0}_{-0.8} \pm 0.1~\mathrm{MeV} \, . \end{aligned}$$

(4)

These excited $\varOmega _c$ and $\varOmega _b$ states are good candidates of P-wave charmed and bottom baryons. To understand them, many phenomenological methods and models have been applied, such as various quark models [13,14,15,16,17,18,19,20,21,22,23,24,25,26], the chiral perturbation theory [27, 28], the molecular model [29,30,31,32,33,34,35,36], Lattice QCD [37, 38], and QCD sum rules [39,40,41,42,43,44,45,46,47,48,49,50,51], etc. We refer to reviews [7, 52,53,54] and references therein for their recent progress.

Fig. 1

Categorization of P-wave $\varOmega _b$ baryons

In Refs. [47, 48] we systematically studied mass spectra of P-wave heavy baryons using QCD sum rules [55, 56] within the the framework of heavy quark effective theory (HQET) [57,58,59]. Later in Refs. [49,50,51] we systematically studied their S-wave decays into ground-state heavy baryons together with light pseudoscalar and vector mesons, using light-cone sum rules [60,61,62,63,64] still within HQET. Recently, we have applied the same method to systematically study their D-wave decays into ground-state heavy baryons and light pseudoscalar mesons [65]. Hence, we have performed a rather complete study on both the mass spectra and strong decay properties of P-wave heavy baryons within the framework of HQET.

In this letter we shall apply these sum rule results to study the four excited $\varOmega _b$ baryons recently observed by LHCb [12]. We shall find that all of them can be well interpreted as P-wave $\varOmega _b$ baryons, so that both their mass spectra and decay properties can be well explained. Especially, their beautiful fine structure can be well explained in the framework of HQET, that is directly related to the rich internal structure of P-wave $\varOmega _b$ baryons.

2 A global picture from the heavy quark effective theory

First let us briefly introduce our notations. A P-wave $\varOmega _b$ baryon consists of one bottom quark and two strange quarks. Its orbital excitation can be either between the two strange quarks ($l_\rho = 1$) or between the bottom quark and the two-strange-quark system ($l_\lambda = 1$), so there are $\rho$-mode excited $\varOmega _b$ baryons ($l_\rho = 1$ and $l_\lambda = 0$) and $\lambda$-mode ones ($l_\rho = 0$ and $l_\lambda = 1$). Altogether its internal symmetries are as follows:

• Color structure of the two strange quarks is antisymmetric (${\bar{\mathbf {3}}}_C$).

• Flavor structure of the two strange quarks is symmetric, that is the SU(3) flavor $\mathbf {6}_F$.

• Spin structure of the two strange quarks is either antisymmetric ($s_l = 0$) or symmetric ($s_l = 1$).

• Orbital structure of the two strange quarks is either antisymmetric ($l_\rho = 1$) or symmetric ($l_\rho = 0$).

• Totally, the two strange quarks should be antisymmetric due to the Pauli principle.

Accordingly, we can categorize P-wave $\varOmega _b$ baryons into four multiplets, as shown in Fig. 1. We denote them as ${[}\mathbf {6}_F, j_l, s_l, \rho /\lambda ]$, where $j_l$ is the total angular momentum of the light components ($j_l = l_\lambda \otimes l_\rho \otimes s_l$). Each multiplet contains one or two $\varOmega _b$ baryons, denoted as $[\varOmega _b(j^P), j_l, s_l, \rho /\lambda ]$, where $j^P$ are their total spin-parity quantum numbers ($j = j_l \otimes s_b = | j_l \pm 1/2 |$ with $s_b$ the bottom quark spin). Note that there are other four multiplets with the SU(3) flavor ${\bar{\mathbf {3}}}_F$, and we refer to Refs. [14, 47, 48] for more discussions.

Table 1 Mass spectra of P-wave $\varOmega _b$ baryons belonging to the bottom baryon multiplets $[\mathbf {6}_F, 1, 0, \rho ]$, $[\mathbf {6}_F, 0, 1, \lambda ]$, $[\mathbf {6}_F, 1, 1, \lambda ]$, and $[\mathbf {6}_F, 2, 1, \lambda ]$. There is considerable uncertainty in our results for absolute values of the masses due to their (almost linear) dependence on the bottom quark mass, but the mass differences within the same doublet do not depend much on the bottom quark mass, so they are produced quite well with much less uncertainty

3 Mass spectrum from QCD sum rules within HQET

We have systematically constructed all the P-wave heavy baryon interpolating fields in Ref. [47], and applied them to study the mass spectrum of P-wave bottom baryons in Ref. [48] using the method of QCD sum rules within HQET. In this framework the $\varOmega _b$ baryon belonging to the multiplet $[F, j_l, s_l, \rho /\lambda ]$ has the mass:

$$\begin{aligned} m_{\varOmega _b(j^P),j_l,s_l,\rho /\lambda } = m_b + \overline{\varLambda }_{\varOmega _b,j_l,s_l,\rho /\lambda } + \delta m_{\varOmega _b(j^P),j_l,s_l,\rho /\lambda }, \end{aligned}$$

(5)

where $m_b$ is the bottom quark mass, $\overline{\varLambda }_{\varOmega _b,j_l,s_l,\rho /\lambda } = \overline{\varLambda }_{\varOmega _b(|j_l-1/2|),j_l,s_l,\rho /\lambda } = \overline{\varLambda }_{\varOmega _b(j_l+1/2),j_l,s_l,\rho /\lambda }$ is the sum rule result evaluated at the leading order, and $\delta m_{\varOmega _b(j^P),j_l,s_l,\rho /\lambda }$ is the sum rule result evaluated at the ${\mathscr {O}}(1/m_b)$ order:

$$\begin{aligned}&\delta m_{\varOmega _b(j^P),j_l,s_l,\rho /\lambda } \nonumber \\&\quad = -\frac{1}{4m_{b}}\left( K_{\varOmega _b,j_l,s_l,\rho /\lambda } + d_{j,j_{l}}C_{mag}\varSigma _{\varOmega _b,j_l,s_l,\rho /\lambda }\right) . \end{aligned}$$

(6)

Here $C_{mag} = [ \alpha _s(m_b) / \alpha _s(\mu ) ]^{3/\beta _0}$ with $\beta _0 = 11 - 2 n_f /3$, and the coefficient $d_{j,j_{l}}$ is

$$\begin{aligned} d_{j_{l}-1/2,j_{l}} = 2j_{l}+2\, ,~~~~~ d_{j_{l}+1/2,j_{l}} = -2j_{l} \, . \end{aligned}$$

(7)

Hence, the $\varSigma _{\varOmega _b,j_l,s_l,\rho /\lambda }$ term is directly related to the mass splitting within the same multiplet. This term is usually positive, so the mass splitting within the same multiplet is also positive.

We clearly see from Eq. (5) that the $\varOmega _b$ mass depends significantly (almost linearly) on the bottom quark mass, for which we used the 1S mass $m_b = 4.66 ^{+0.04}_{-0.03}$ GeV [66] in Ref. [48], while the pole mass $m_b = 4.78 \pm 0.06$ GeV [4] and the $\overline{\mathrm{MS}}$ mass $m_b = 4.18 ^{+0.04}_{-0.03}$ GeV [4] are used in some other QCD sum rule studies. This suggests that there is considerable uncertainty in our results for absolute values of the masses, which prevents us from touching the nature of the four excited $\varOmega _b$ baryons observed by LHCb [12]. However, the mass differences within the same doublet do not depend much on the bottom quark mass, so they are produced quite well with much less uncertainty and give more useful information.

Besides, we can extract even (much) more useful information from strong decay properties of P-wave $\varOmega _b$ baryons. Before doing this, we slightly modify one of the free parameters in QCD sum rules, the threshold value $\omega _c$, to get a better description of the four excited $\varOmega _b$ baryons’ masses measured by LHCb [12]. The obtained results are summarized in Table 1.

Table 2 Strong decay properties of P-wave $\varOmega _b$ baryons belonging to the bottom baryon multiplets $[\mathbf {6}_F, 1, 0, \rho ]$, $[\mathbf {6}_F, 0, 1, \lambda ]$, $[\mathbf {6}_F, 1, 1, \lambda ]$, and $[\mathbf {6}_F, 2, 1, \lambda ]$. In the third and fourth columns we show the results for the S- and D-wave decays of P-wave $\varOmega _b$ baryons into $\varXi _c K$ (both $\varXi _b^0 K^-$ and $\varXi _b^- \bar{K}^0$), respectively. AMF means that these channels are forbidden due to the conservation of angular momentum; KF means that these channels are kinematically forbidden; 0 means that decay widths of these channels are calculated to be zero; – means that this channel is not calculated

4 Decay property from light-cone sum rules within HQET

We have systematically studied various strong decay properties of P-wave heavy baryons in Refs. [49,50,51] using light-cone sum rules within HQET. There are indeed a lot of decay processes that can happen. However, in the present case the only possible strong decay mode for P-wave $\varOmega _b$ baryons is decaying into $\varXi _b K$ (given their largest mass to be the mass of the $\varOmega _b(6350)^-$, so that all the other strong decay modes are kinematically forbidden). Actually, we can draw even stronger conclusions:

• All the S-wave decays of P-wave $\varOmega _b$ baryons into ground-state heavy baryons and light pseudoscalar mesons can not happen (some of them are forbidden due to the conservation of angular momentum, while some due to the selection rules of the light components of the baryons), except

  $$\begin{aligned} \varGamma \left( [\varOmega _b(1/2^-), 0, 1, \lambda ] \rightarrow \varXi _b K\right) = 2800^{+3600}_{-1800}~{\mathrm{MeV}}. \end{aligned}$$

  (8)

  The above value is evaluated through the Lagrangian

  $$\begin{aligned} {\mathscr {L}} = g {\bar{\varOmega }_{b}}(1/2^{-}) \varXi _b \,K, \end{aligned}$$

  using the mass of $[\varOmega _b(1/2^-), 0, 1, \lambda ]$ given in Table 1.

• All the decays of P-wave $\varOmega _b$ baryons into ground-state heavy baryons and light vector mesons (as intermediate states) are kinematically forbidden.

Recently, we have systematically studied D-wave decays of P-wave heavy baryons into ground-state heavy baryons and light pseudoscalar mesons [65]. The results indicate:

• All the D-wave decays of P-wave $\varOmega _b$ baryons into ground-state heavy baryons and light pseudoscalar mesons can not happen, except (a) $[\varOmega _b(3/2^-), 2, 1, \lambda ] \rightarrow \varXi _b K$ and (b) $[\varOmega _b(1/2^-), 0, 1, \lambda ] \rightarrow \varXi _b K$. The former channel (a) is calculated to be

  $$\begin{aligned} \varGamma \left( [\varOmega _b(3/2^-), 2, 1, \lambda ] \rightarrow \varXi _b K\right) = 4.7^{+6.1}_{-2.9}~\mathrm{MeV}, \end{aligned}$$

  (9)

  using the mass of the $\varOmega _b(6350)^-$ measured by LHCb [12] through the Lagrangian

  $$\begin{aligned} {\mathscr {L}}^\prime = g^\prime ~{\bar{\varOmega }_b}^\mu (3/2^-) \gamma ^\nu \gamma _5 \varXi _b ~ \partial _\mu \partial _\nu K, \end{aligned}$$

  (10)

  The latter channel (b) is too large due to the S-wave nature of the decay mode.

We summarize the above decay properties in Table 2.

5 Excited $\varOmega _b$ baryons in the heavy quark effective theory

Based on Tables 1 and 2, we can well understand the four excited $\varOmega _b$ baryons observed by LHCb [12] as P-wave $\varOmega _b$ baryons. There are altogether seven P-wave $\varOmega _b$ baryons, belonging to four multiplets:

$$\begin{aligned} \varOmega _b(1/2^-), \varOmega _b(3/2^-)\in & {} [\mathbf{6}_F, 1, 0, \rho ] \, , \\ \varOmega _b(1/2^-) \qquad \,\in & {} [\mathbf{6}_F, 0, 1, \lambda ], \\ \varOmega _b(1/2^-), \varOmega _b(3/2^-)\in & {} [\mathbf{6}_F, 1, 1, \lambda ] \, , \\ \varOmega _b(3/2^-), \varOmega _b(5/2^-)\in & {} [\mathbf{6}_F, 2, 1, \lambda ]. \end{aligned}$$

Our results suggest:

• The width of $[\varOmega _b(1/2^-), 0, 1, \lambda ]$ is too large for it to be observed in experiments.

• Only the natural width of the $\varOmega _b(6350)^-$ was measured by LHCb to be “$2.5\sigma$ from zero”, that is $\varGamma _{\varOmega _b(6350)^-} = 1.4^{+1.0}_{-0.8}\pm 0.1$ MeV [12]. Its best candidate is $[\varOmega _b(3/2^-), 2, 1, \lambda ]$, whose width is calculated to be $\varGamma _{[\varOmega _b(3/2^-), 2, 1, \lambda ]} = 4.7^{+6.1}_{-2.9}$ MeV, quite narrow because this is a D-wave decay mode. The $\varOmega _b(6350)^-$ is the partner state of the $\varSigma _b(6097)^\pm$ [11] and $\varXi _b(6227)^-$ [10], and it has another partner state, $[\varOmega _b(5/2^-), 2, 1, \lambda ]$, whose mass is $10.0^{+4.6}_{-3.8}$ MeV larger.

• The natural widths of the $\varOmega _b(6330)^-$ and $\varOmega _b(6340)^-$ were both measured by LHCb to be “consistent with zero”, and their mass difference was measured to be about 9.4 MeV [12]. Their best candidates are $[\varOmega _b(1/2^-), 1, 1, \lambda ]$ and $[\varOmega _b(3/2^-), 1, 1, \lambda ]$ respectively, whose widths are both calculated to be zero and mass difference to be $6.3^{+2.3}_{-2.1}$ MeV. We are not sure about the reason why their decays into $\varXi _c K$ are both forbidden, but this might be related to some constrain(s) from their internal (flavor) symmetries.

• The natural width of the $\varOmega _b(6316)^-$ was also measured by LHCb to be “consistent with zero” [12]. We can explain it as either $[\varOmega _b(1/2^-), 1, 0, \rho ]$ or $[\varOmega _b(3/2^-), 1, 0, \rho ]$. It can be further separated into two states with the mass splitting $2.3^{+1.0}_{-0.9}$ MeV. We would like to note here that this $\rho$-mode excitation is lower than the $\lambda$-mode, $[\mathbf{6}_F(\varOmega _b), 1, 1, \lambda ]$, consistent with our previous results for their corresponding multiplets with the SU(3) flavor ${\bar{\mathbf {3}}}_F$ [47, 48], but in contrast to the quark model expectation [5, 19]. However, this may be simply because that the mass differences between different multiplets have a considerable uncertainty in our framework, similar to absolute values of the masses, but unlike the mass differences within the same multiplet.

• The reason is quite straightforward within the framework of HQET why the $\varOmega _b(6316)^-$, $\varOmega _b(6330)^-$, and $\varOmega _b(6340)^-$ have natural widths “consistent with zero” but they can still be observed in the $\varXi _b^0 K^-$ mass spectrum [12]: the HQET is an effective theory, so the three $J=1/2^-$ $\varOmega _b$ states can mix together and the three $J=3/2^-$ ones can also mix together, making it possible to observe them in the $\varXi _b K$ mass spectrum; while the HQET works quite well for the bottom system, so this mixing is not large and some of them still have very narrow widths.

6 Summary

In the present study we have systematically studied the internal structure of P-wave $\varOmega _b$ baryons and calculated their D-wave decay properties. Together with our previous studies on their mass spectra and S-wave decay properties [47,48,49, 51], we have systematically studied mass spectra and strong decay properties of P-wave $\varOmega _b$ baryons using the methods of QCD sum rules and light-cone sum rules within the framework of heavy quark effective theory. Although there is considerable uncertainty in our results for absolute values of the masses due to their (almost linear) dependence on the bottom quark mass, the mass differences within the same doublet as well as strong decay properties of P-wave $\varOmega _b$ baryons are both useful information, based on which we can well understand the four excited $\varOmega _b$ baryons recently discovered by LHCb [12] as P-wave $\varOmega _b$ baryons.

Our results suggest: the $\varOmega _b(6350)^-$ is a P-wave $\varOmega _b$ baryon with $J^P = 3/2^-$ and $\lambda$-mode excitation, and it has a $J^P = 5/2^-$ partner whose mass is $10.0^{+4.6}_{-3.8}$ MeV larger; the $\varOmega _b(6330)^-$ and $\varOmega _b(6340)^-$ are partner states both with $\lambda$-mode excitation, and they have $J^P = 1/2^-$ and $3/2^-$, respectively; the $\varOmega _b(6316)^-$ is a P-wave $\varOmega _b$ baryon of either $J^P = 1/2^-$ or $3/2^-$, with $\rho$-mode excitation, and it can be further separated into two states with the mass splitting $2.3^{+1.0}_{-0.9}$ MeV. The internal quantum numbers (and so internal structures) of these four excited $\varOmega _b$ baryons have also been extracted, as discussed above.

The above conclusions are drawn by combining our systematical studies of mass spectra (as well as mass splittings with the same multiplets) and decay properties of P-wave $\varOmega _b$ baryons. We would like to note that these are just possible explanations, and there exist many other possibilities for the four excited $\varOmega _b$ baryons observed by LHCb [12], so further experimental and theoretical studies are demanded to fully understand them. However, their beautiful fine structure is in any case directly related to the rich internal structure of P-wave $\varOmega _b$ baryons. Recalling that the development of quantum physics is sometimes closely related to the better understanding of the gross, fine, and hyperfine structures of atom (hydrogen) spectra, one naturally guesses that the currently undergoing studies on heavy baryons would not only improve our understandings on their internal structures, but also enrich our knowledge of the quantum physics.