J an 2 02 0 Excited Ω b baryons and fine structure of strong interaction

Hua-Xing Chen, Er-Liang Cui, Atsushi Hosaka, Qiang Mao, and Hui-Min Yang School of Physics, Beihang University, Beijing 100191, China School of Physics, Southeast University, Nanjing 210094, China College of Science, Northwest A&F University, Yangling 712100, China Research Center for Nuclear Physics (RCNP), Osaka University, Ibaraki 567-0047, Japan Advanced Science Research Center, Japan Atomic Energy Agency (JAEA), Tokai 319-1195, Japan Department of Electrical and Electronic Engineering, Suzhou University, Suzhou 234000, China

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 this letter we shall apply these sum rule results to study the four excited Ω b baryons recently observed by LHCb [12]. We shall find that all of them can be well interpreted as P -wave Ω 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 Ω b baryons.
A global picture from the heavy quark effective theory -First let us briefly introduce our notations. A P -wave Ω b baryon consists of one bottom quark and two strange quarks. Its orbital excitation can be either between the two strange quarks (l ρ = 1) or between the bottom quark and the two-strange-quark system (l λ = 1), so there are ρ-mode excited Ω b baryons (l ρ = 1 and l λ = 0) and λmode ones (l ρ = 0 and l λ = 1). Altogether its internal symmetries are as follows: • Color structure of the two strange quarks is antisymmetric (3 C ).
• Flavor structure of the two strange quarks is sym- metric, that is the SU (3) flavor 6 F .
• Spin structure of the two strange quarks is either antisymmetric (s l = 0) or symmetric (s l = 1).
• Totally, the two strange quarks should be antisymmetric due to the Pauli principle.
Accordingly, we can categorize P -wave Ω b baryons into four multiplets, as shown in Fig. 1. We denote them as [6 F , j l , s l , ρ/λ], where j l is the total angular momentum of the light components (j l = l λ ⊗ l ρ ⊗ s l ). Each multiplet contains one or two Ω b baryons, denoted as [Ω b (j P ), j l , s l , ρ/λ], where j P are their total spin-parity quantum numbers (j = j l ⊗ s b = |j l ± 1/2| with s b the bottom quark spin). Note that there are other four multiplets with the SU (3) flavor3 F , and we refer to Refs. [14,45,46] for more discussions.
Mass spectrum from QCD sum rules within HQET --We have systematically constructed all the P -wave heavy baryon interpolating fields in Ref. [45], and applied them to study the mass spectrum of P -wave bottom baryons in Refs. [46,48,49] using the method of QCD sum rules within HQET. In this framework the Ω b baryon belonging to the multiplet [F, j l , s l , ρ/λ] has the mass: ,j l ,s l ,ρ/λ is the sum rule result evaluated at the leading order, and δm Ω b (j P ),j l ,s l ,ρ/λ is the sum rule result evaluated at the O(1/m b ) order.
We clearly see from Eq. (5) that the Ω 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 [64] in Ref. [46], while the pole mass m b = 4.78±0.06 GeV [4] and the 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 theoretical uncertainty in our results for absolute values of the masses, which prevents us from touching the nature of the four excited Ω 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 theoretical uncertainty and give more useful information.
Besides, we can extract even (much) more useful information from strong decay properties of P -wave Ω b baryons. Before doing this, we slightly modify one of the free parameters in QCD sum rules, the threshold value ω c , to get a better description of the four excited Ω b baryons' masses measured by LHCb [12]. The obtained results are summarized in Table I.
Decay property from light-cone sum rules within HQET-We have systematically studied various strong decay properties of P -wave heavy baryons in Refs. [47][48][49] 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 Pwave Ω b baryons is decaying into Ξ b K (given their largest mass to be the mass of the Ω 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 Ω b baryons into ground-state heavy baryons and light pseudoscalar mesons can not happen, except The above value is evaluated through using the mass of [Ω b (1/2 − ), 0, 1, λ] given in Table I.
• All the decays of P -wave Ω b baryons into groundstate heavy baryons and light vector mesons (as intermediate states) can not happen.
Recently, we have systematically studied D-wave decays of P -wave heavy baryons into ground-state heavy baryons and light pseudoscalar mesons [63]. The results suggest: • All the D-wave decays of P -wave Ω b baryons into ground-state heavy baryons and light pseudoscalar mesons can not happen, except The former one is evaluated through  In the third and fourth columns we show the results for the S-and D-wave decays of P -wave Ω b baryons into ΞcK (both Ξ 0 b K − and Ξ − bK 0 ), respectively. A.M.F. means that these channels are forbidden due to the conservation of angular momentum; K.F. 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.

Multiplets
Baryon using the mass of the Ω b (6350) − measured by LHCb [12]. The latter is not calculated because the width of its corresponding S-wave decay mode is already too large.
We summarize the above decay properties in Table II.
Excited Ω b baryons in the heavy quark effective theory -Based on Tables I and II, we can well understand the four excited Ω b baryons observed by LHCb [12] as P -wave Ω b baryons. There are altogether seven P -wave Ω b baryons, belonging to four multiplets: , Our results suggest: • The width of [Ω b (1/2 − ), 0, 1, λ] is too large for it to be observed in experiments.
• The natural widths of the Ω b (6330) − and Ω 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 [Ω b (1/2 − ), 1, 1, λ] and [Ω b (3/2 − ), 1, 1, λ] 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 Ξ c K are both forbidden, but this might be related to some constrain(s) from their internal flavor symmetries.
• The natural width of the Ω b (6316) − was also measured by LHCb to be "consistent with zero" [12]. We can explain it as either 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 ρ-mode excitation is lower than the λ-mode, [6 F (Ω b ), 1, 1, λ], consistent with our previous results for their corresponding multiplets with the SU (3) flavor3 F [45,46], but in contrast to the quark model expectation [5,19].
• The reason is quite straightforward within the framework of HQET why the Ω b (6316) − , Ω b (6330) − , and Ω b (6340) − have natural widths "consistent with zero" but they can still be observed in the Ξ 0 b K − mass spectrum [12]: the HQET is an effective theory, so the three J = 1/2 − Ω b states can mix together and the three J = 3/2 − ones can also mix together, making it possible to observe them in the Ξ 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.
Summary -We have systematically studied mass spectra and strong decay properties of P -wave Ω 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 theoretical 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 Ω b baryons are both useful information, based on which we can well understand the four excited Ω b baryons recently discovered by LHCb [12] as P -wave Ω b baryons.
Our results suggest: the Ω b (6350) − is a P -wave Ω b baryon with J P = 3/2 − and λ-mode excitation, and it has a J P = 5/2 − partner whose mass is 10.0 +4.6 −3.8 MeV larger; the Ω b (6330) − and Ω b (6340) − are partner states both with λ-mode excitation, and they have J P = 1/2 − and 3/2 − , respectively; the Ω b (6316) − is a P -wave Ω b baryon of either J P = 1/2 − or 3/2 − , with ρ-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 Ω b baryons have also been extracted, as discussed above.
To end this letter, we conclude that the beautiful fine structure of the four excited Ω b baryons observed by LHCb [12] is directly related to the rich internal structure of P -wave Ω b baryons. Recalling that the development of quantum theory 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 theory.