$P$-wave bottom baryons of the $SU(3)$ flavor $\mathbf{6}_F$

We investigate $P$-wave bottom baryons of the $SU(3)$ flavor $\mathbf{6}_F$, and systematically study their $D$-wave decays into ground-state bottom baryons and pseudoscalar mesons. Together with [PRD91, 054034 (2015); PRD92, 114007 (2015); PRD95, 094008 (2017); EPJC80, 80 (2020)], a rather complete study is performed on both mass spectra and decay properties of $P$-wave bottom baryons, using the method of QCD sum rules and light-cone sum rules within the framework of heavy quark effective theory. Among all the possibilities, we find four $\Sigma_b$, four $\Xi^\prime_b$, and six $\Omega_b$ baryons, with limited widths and so capable of being observed. Their masses, mass splittings within the same multiplets, and decay properties are extracted (summarized in Table V) for future experimental searchings.


I. INTRODUCTION
The strong interaction holds quarks and gluons together inside a single hadron. It is similar to the electromagnetic interaction in some aspects, which holds electrons and protons together inside a single atom. The latter leads to the well-known fine structure of line spectra, and it is interesting to investigate whether the former also leads to some fine structure of hadron spectra [5][6][7][8]. An ideal platform to study this is the singly bottom baryon system [9][10][11][12]: light quarks together with gluons circle around the nearly static bottom quark, and the whole system behaves as the QCD analogue of the hydrogen.
In recent years important experimental progresses have been made in the field of singly bottom baryons. Until three years ago there were only two excited bottom baryons, Λ b (5912) 0 and Λ b (5920) 0 , which were observed by LHCb and CDF in 2012 [13,14]. However, in the past three years the LHCb and CMS Collaborations discovered as many as nine excited bottom baryons: • In 2018 the LHCb Collaboration observed the Σ b (6097) ± in the Λ 0 b π ± invariant mass spectrum, and the Ξ b (6227) − in the Λ 0 b K − and Ξ 0 b π − invariant mass spectra [15,16]: • In 2019 the LHCb Collaboration observed the Λ b (6146) 0 and Λ b (6152) 0 in the Λ 0 b π + π − invariant mass distribution [18]. Later in 2020 the CMS Collaboration confirmed them, and further observed a broad excess of events in the Λ 0 b π + π − mass distribution in the region of 6040-6100 MeV [19], whose mass and width were later measured by LHCb to be [20]: Λ b (6072) 0 : M = 6072.3 ± 2.9 ± 0.6 ± 0.2 MeV , Γ = 72 ± 11 ± 2 MeV .
This paper is organized as follows.

II. P -WAVE BOTTOM BARYONS
At the beginning we briefly introduce our notations. A singly bottom baryon is composed by one bottom quark and two light up/down/strange quarks, with the following internal structures: • According to the Pauli principle, the total symmetry of the two light quarks is antisymmetric.
• The color structure of the two light quarks is antisymmetric (3 C ).
• The spin structure of the two light quarks is either symmetric (s l ≡ s qq = 1) or antisymmetric (s l = 0).
• The orbital structure of the two light quarks is either symmetric (λ-type with l ρ = 0 and l λ = 1, meaning that the orbital excitation is between the bottom quark and the two-light-quark system) or antisymmetric (ρ-type with l ρ = 1 and l λ = 0, meaning that the orbital excitation is between the two light quarks).
Accordingly, we categorize P -wave bottom baryons into eight baryon multiplets, four of which belong to the SU (3) flavor 6 F representation, as shown in Fig. 1. We use [F (flavor), j l , s l , ρ/λ] to denote them, where j l is the total angular momentum of the light components, satisfying j l = l λ ⊗ l ρ ⊗ s l . Each multiplet contains one or two bottom baryons with the total angular momenta j = j l ⊗ s b = |j l ± 1/2|, which have similar masses according to the heavy quark effective theory. In Refs. [1,2] we have systematically studied the mass spectrum of P -wave bottom baryons, and the results are reanalysed in the present study, as summarized in Table I. Some of them are used as input parameters when studying decay properties of P -wave bottom baryons. Especially, we use the following mass values when calculating their decay widths: • In Ref. [97] we found that the Ω b (6316) − can be explained as a P -wave Ω b baryon of either J P = 1/2 − or 3/2 − , belonging to the [6 F , 1, 0, ρ] doublet. Hence, we use the following mass values for this doublet, taken from the LHCb experiment [17] as well as their mass sum rules: Note that the above interpretations are just possible assignments, and there exist many other possibilities We use the following parameters for ground-state bottom baryons, pseudoscalar and vector mesons [8]: III. D-WAVE DECAY PROPERTIES In the present study we shall investigate P -wave bottom baryons of the SU (3) flavor 6 F , and study their D-wave decays into ground-state bottom baryons together with pseudoscalar mesons (π or K). Their Swave decay properties have been systematically studied in Refs. [3,4], and we shall use the same method of lightcone sum rules within the heavy quark effective theory to investigate the following decay channels (the coefficients at right hand sides are isospin factors): We shall calculate their decay widths through: b , and P denote the P -wave bottom baryon, ground-state bottom baryon, and pseudoscalar meson, respectively.
As an example, we shall study the D-wave decay of and [6 F , 2, 1, λ], will be separately investigated in the following subsections.
In this subsection we study the D-wave decay of the To do this we need to calculate the three-point correlation function at both hadron and quark-gluon levels. In this expression k ′ = k+q, ω = v·k, and ω ′ = v·k ′ . The two interpolating fields J α At the hadron level, we write G α where · · · denotes other possible decay amplitudes. At the quark-gluon level, we calculate G α Then we perform double Borel transformations to both Eqs. (51) and (52), and obtain: In the above expression, ω and ω ′ have been transformed to T 1 and T 2 ; we work at the symmetric point x k k! . We refer to Refs. [77,78,[99][100][101][102][103][104] for explicit forms of the light-cone distribution amplitudes contained in the above sum rule equations, and more examples can be found in Appendix A.
In the present study we use the following values for various quark and gluon parameters at the renormalization scale 2 GeV [8,[105][106][107][108][109][110][111][112]: After fixing ω c = 1.665 GeV to be the average of the threshold values of the Ω − b (3/2 − ) and Ξ 0 b mass sum rules, we calculate the coupling constant Here the uncertainties are due to the Borel mass, the parameters of Ξ 0 b , the parameters of Ω − b (3/2 − ), and various quark and gluon parameters listed in Eq. (54), respectively. Some of these parameters can be found in Sec. II. Fig. 4(d) as a function of the Borel mass T , where we find that its Borel mass dependance is moderate and acceptable.
The D-wave decay of

and its amplitude is
, and the finial state K − , respectively. This amplitude can be used to further calculate its width through In the following subsections we shall follow the same procedures to separately investigate the four bottom baryon multiplets, There are six bottom baryons contained in the and We study their D-wave decays into ground-state bottom baryons and pseudoscalar mesons, and find twelve non-zero coupling constants: We show some of these coupling constants as functions of the Borel mass T in Fig. 2. Based on them, we further find six D-wave decay channels that are kinematically allowed: We summarize these D-wave decay widths in Table II, where possible experimental candidates are given for comparisons. In Refs. [1,2] we have studied the mass spectrum of P -wave bottom baryons, and the results are reanalysed and summarized in this table. In Refs. [3,4] we have studied S-wave decay properties of P -wave bottom baryons into ground-state bottom baryons together with pseudoscalar mesons or vector mesons, and the results are also reanalysed and summarized in this table.
There are three bottom baryons contained in the and We study their D-wave decays into ground-state bottom baryons and pseudoscalar mesons, but find all the coupling constants to be zero. For completeness, we summarize these results in Table III, together with their mass spectrum, S-wave decay properties, and possible experimental candidates.
There are six bottom baryons contained in the and We study their D-wave decays into ground-state bottom baryons and pseudoscalar mesons, and find twelve non-zero coupling constants: We show some of these coupling constants as functions of the Borel mass T in Fig. 3. Based on them, we further find six D-wave decay channels that are kinematically allowed: We summarize these D-wave decay widths in Table IV, together with their mass spectrum, S-wave decay properties, and possible experimental candidates.
ground-state bottom baryons and pseudoscalar mesons, and find twelve non-zero coupling constants: We show some of these coupling constants as functions of the Borel mass T in Fig. 4. Based on them, we further find eight D-wave decay channels that are kinematically allowed: We summarize these D-wave decay widths in Table V, together with their mass spectrum, S-wave decay properties, and possible experimental candidates.

IV. SUMMARY AND DISCUSSIONS
To summarize this paper, we have investigated the P -wave bottom baryons belonging to the SU (3) flavor 6 F representation, and studied their D-wave decays into ground-state bottom baryons and pseudoscalar mesons. Together with Refs. [1][2][3][4], we have performed a rather complete study on both mass spectra and decay properties of P -wave bottom baryons using the method of QCD sum rules and light-cone sum rules within the framework of heavy quark effective theory.
Accordingly to the heavy quark effective theory, we categorize the P -wave bottom baryons of the SU (3) flavor 6 F into four multiplets: [6 F , 1, 0, ρ], [6 F , 0, 1, λ], [6 F , 1, 1, λ], and [6 F , 2, 1, λ]. In this paper we have studied their D-wave decay properties, and the results are separately summarized in Tables II/III/IV/V. Besides, in Refs. [1,2] we have studied the mass spectrum of Pwave bottom baryons, and the results are reanalysed and summarized in these tables; in Refs. [3,4] we have studied S-wave decay properties of P -wave bottom baryons into ground-state bottom baryons together with pseudoscalar mesons or vector mesons, and the results are also reanalysed and summarized in these tables.
Before drawing our conclusions, we note that there are considerable (theoretical) uncertainties in our results for absolute values of the bottom baryon masses due to their significant dependence on the bottom quark mass [1,2]; however, their mass splittings within the same doublets do not depend much on this, so they are produced quite well with much less (theoretical) uncertainties and give more useful information; moreover, we can extract even (much) more useful information from S-and D-wave strong decay properties of P -wave bottom baryons. Based on the results summarized in Tables II/III/IV/V, we can well understand P -wave bottom baryons as a whole: • The [6 F , 1, 1, λ] doublet contains six bottom baryons: and Their total widths are all calculated to be less than 100 MeV.
• The [6 F , 2, 1, λ] doublet contains six bottom baryons: and Their total widths are all calculated to be less than 100 MeV. Hence, among all the possible P -wave bottom baryons of the flavor 6 F , we find altogether four Σ b , four Ξ ′ b , and six Ω b baryons, with limited widths (< 100 MeV) and so capable of being observed. Their masses, mass splittings within the same multiplets, and decay properties are summarized in Table VI. Their possible experimental candidates are also given in this table for comparisons. We suggest the LHCb and CMS Collaborations to search for these excited bottom baryons, but note that it still depends on the production rates whether these baryons can be observed or not. Especially, it is interesting to further investigate the Λ b (6072) 0 , i.e., the broad excess of events in the Λ 0 b π + π − mass distribution in the region of 6040-6100 MeV [19,20].
In the present study the ρ-mode doublet [6 F , 1, 0, ρ] is found to be lower than the two λ-mode doublets [6 F , 1, 1, λ] and [6 F , 2, 1, λ], which behaviour is consistent with our previous results for their corresponding doublets of the SU (3) flavor3 F [1,2], but in contrast to the quark model expectation [6,25]. However, this may be simply because that the mass differences between different multiplets have considerable uncertainties in our framework, similar to the absolute values of baryon masses, but unlike the mass differences within the same multiplet. We propose to verify whether the ρ-mode doublet [6 F , 1, 0, ρ] exist or not by investigating: a) the spin-parity quantum number of the Ω b (6316) − , b) whether it can be separated into two states almost degenerate, and c) whether its Σ b and Ξ ′ b partners states can be observed.  6F , 1, 0, ρ] doublet. The sixth column "D-wave width" is newly obtained in the present study. Besides, in Refs. [1,2] we studied the mass spectrum of P -wave bottom baryons, and the results are reanalysed and summarized in the second and third columns; in Refs. [3,4] we studied the S-wave decay properties of P -wave bottom baryons into ground-state bottom baryons together with pseudoscalar mesons or vector mesons, and the results are reanalysed and summarized in the fifth column. Possible experimental candidates are given in the last column for comparisons.

The sum rule equation for the Ω
The sum rule equation

The sum rule equation for the Σ
TABLE VI: Among all the possible P -wave bottom baryons of the flavor 6F , we find altogether four Σ b , four Ξ ′ b , and six Ω b baryons, with limited widths (< 100 MeV) and so capable of being observed. Their masses, mass splittings within the same multiplets, and decay properties are extracted for future experimental searchings. We note that there are considerable uncertainties in our results for absolute values of the bottom baryon masses due to their significant dependence on the bottom quark mass [1,2]; however, their mass splittings within the same doublets do not depend much on this, so they are produced quite well with much less uncertainties and give more useful information; moreover, we can extract even more useful information from S-and D-wave strong decay properties of P -wave bottom baryons.