Identifying the $\Lambda_b(6146)^0$ and $\Lambda_b(6152)^0$ as $D$-wave bottom baryons

We study the $\Lambda_b(6146)^0$ and $\Lambda_b(6152)^0$ recently observed by LHCb using the method of QCD sum rules within the framework of heavy quark effective theory. Our results suggest that they can be interpreted as $D$-wave bottom baryons of $J^P = 3/2^+$ and $5/2^+$ respectively, both of which contain two $\lambda$-mode excitations. We also investigate other possible assignments containing $\rho$-mode excitations. We predict masses of their strangeness partners to be $m_{\Xi_b(3/2^+)} = 6.26^{+0.11}_{-0.14}$ GeV and $m_{\Xi_b(5/2^+)} = 6.26^{+0.11}_{-0.14}$ GeV with the mass splitting $\Delta M = 4.5^{+1.9}_{-1.5}$ MeV, and propose to search for them in future LHCb and CMS experiments.


I. INTRODUCTION
In the past few years important experimental progresses were made in the field of bottom baryons. All the S-wave singly bottom baryons, except the Ω * b of J P = 3/2 + , have been well observed in experiments [1]. However, no excited bottom baryon were established until the LHCb Collaboration discovered the Λ b (5912) 0 and Λ b (5920) 0 in 2012 [2], which were later confirmed by the CDF Collaboration [3]. At that time, they are the only two excited bottom baryons well observed in experiments, while in the past two years the LHCb and CMS Collaborations continuously observed as many as nine excited bottom baryons: • In 2018 the LHCb Collaboration reported their discoveries of two excited bottom baryons, the Σ b (6097) ± in the Λ 0 b π ± invariant mass spectrum and the Ξ b (6227) − in both the Λ 0 b K − and Ξ 0 b π − invariant mass spectra [4,5].
In this paper we shall use the same approach to study D-wave bottom baryons. Some of these studies have been done in our previous papers [51,52], but at that time: a) we did not construct all the bottom baryon interpolating fields, and b) we did not complete all the sum rule calculations. In the present study we shall finish these two steps and systematically study D-wave bottom baryons of the SU (3) flavor3 F . The obtained results will be used to examine whether the Λ b (6146) 0 and Λ b (6152) 0 can be interpreted as D-wave bottom baryons. Before doing this, we note that this assignment has been discussed and supported by several theoretical studies, using the chiral quark model [79], the quark pair creation model [80,81], and QCD sum rules [82], etc.
This paper is organized as follows. In Sec. II, we construct all the interpolating fields for D-wave bottom baryons of the SU (3) flavor3 F , which are used to perform QCD sum rule analyses in Sec. III. The obtained sum rule equations are further used to perform numerical analyses in Sec. IV. In Sec. V we discuss the results and conclude this paper.

II. INTERPOLATING FIELDS FOR THE D-WAVE BOTTOM BARYON
The D-waves heavy baryons have been systematically classified in Ref. [83], and their interpolating fields have been partly constructed in Refs. [51,52]. In this section we further construct all the D-wave heavy baryon interpolating fields of the SU (3) flavor3 F . Note that some of them are different from those given in Refs. [51,52], since we have explicitly used several projection operators in the present study.
First we briefly introduce our notations. A D-wave bottom baryon consists of one bottom quark and two light up/down/strange quarks. We use l ρ to denote the orbital angular momentum between the two light quarks, and l λ to denote the orbital angular momentum between the bottom quark and the two-light-quark system. There can be ρρ-mode excited D-wave bottom baryons (l ρ = 2 and l λ = 0 into L = 2), λλ-mode ones (l ρ = 0 and l λ = 2 into L = 2), and ρλ-mode ones (l ρ = 1 and l λ = 1 into L = 2). Altogether its internal symmetries are as follows: • Color structure of the two light quarks is antisymmetric (3 C ).
• Flavor structure of the two light quarks is either antisymmetric (3 F ) or symmetric (6 F ).
• Spin structure of the two light quarks is either antisymmetric (s l = 0) or symmetric (s l = 1).
• Orbital structure of the two light quarks is either antisymmetric (l ρ = 1) or symmetric (l ρ = 0/2). • Totally, the two light quarks are antisymmetric due to the Pauli principle.
Accordingly, we categorize D-wave bottom baryons into twelve multiplets, five of which belong to the SU (3) fla-vor3 F representation, as shown in Fig. 1. We denote them as [F (lavor), 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 two bottom baryons, whose total angular momentum are j = j l ⊗ s b = j l ± 1/2, with s b the spin of the bottom quark. We use the notation J α1···α j−1/2 j,P,F,j l ,s l ,ρ/λ to denote the D-wave bottom baryon interpolating field, and separately construct them for the [3 F ρλ] multiplets. Note that Eqs. (6), (7), (10), (11), and (18) are the same as those given in Refs. [51,52] except some overall factors; Eqs. (12), (13), and (17) are different since we have explicitly used some projection operators in the present study; Eqs. (15) and (16) were not constructed in Refs. [51,52].
As an example, we use the bottom baryon doublet [Ξ b (3 F ), 3, 1, ρλ] to perform QCD sum rule analyses, through the field From this field, we obtain: Sum rules for other multiplets are listed in Appendix A, and we refer to Refs. [49,50] for detailed analyses.
Their variations are shown in Fig. 2 as functions of the Borel mass T , where their T dependence is weak and acceptable inside the Borel window 0.638 GeV< T < 0.685 GeV. Then we use Eqs. (21)(22) to further obtain where m Ξ b (5/2 + ) and m Ξ b (7/2 + ) are the masses of the Ξ b (5/2 + ) and Ξ b (7/2 + ) belonging to the [Ξ b (3 F ), 3, 1, ρλ] multiplet, with ∆m [Ξ b (3F ),3,1,ρλ] their mass splitting. The variation of m Ξ b (5/2 + ) is shown in the left panel of Fig. 3 as a function of the Borel mass T , where its T dependence is also weak and acceptable inside the Borel window 0.638 GeV< T < 0.685 GeV.
Secondly, we change the threshold value ω c and redo the above procedures. The variation of m Ξ b (5/2 + ) is shown in the right panel of Fig. 3 as a function of the threshold value ω c . We find that there exist nonvanishing Borel windows as long as ω c ≥ 4.4 GeV, and the ω c dependence is weak and acceptable in the region 4.4 GeV< ω c < 4.8 GeV.
Hence, we fix our working regions to be 4.4 GeV< ω c < 4.8 GeV and 0.638 GeV< T < 0.685 GeV, and obtain: where the central values correspond to ω c = 4.6 GeV and T = 0.662 GeV, and the uncertainties are due to the threshold value ω c , the Borel mass T , the strange and bottom quark masses, and various quark and gluon condensates.
Following the same procedures, we study the bottom baryon doublet [Λ b (3 F ), 3, 1, ρλ], which contains the Λ b (5/2 + ) and Λ b (7/2 + ). They are the partner states of the Ξ b (5/2 + ) and Ξ b (7/2 + ) belonging to the [Ξ b (3 F ), 3, 1, ρλ] multiplet, and their masses are extracted to be  Fig. 5 as functions of the threshold value ω c . We summarize all the above results in Table I, which will be discussed in the next section.

V. SUMMARY AND DISCUSSIONS
In this paper we apply the method of QCD sum rules within the heavy quark effective theory to study D-wave bottom baryons of the SU (3) Table I. Before discussing these results, we note that there is considerable uncertainty in our results for the absolute value of the mass, because it depends significantly on the bottom quark mass, as shown in Eq. (21); however, the mass difference within the same doublet does not depend much on the bottom quark mass, so it is produced quite well with much less (theoretical) uncertainty and gives more useful information.
The advantage of this assignment is: the lower state Λ b (5/2 + ) would dominantly decay only into the Pwave Σ * b π channel, while the higher state Λ b (7/2 + ) would decay both into the F -wave Σ b π channel and the F -wave Σ * b π channel, which behaviors are consistent with the Λ b (6146) 0 and Λ b (6152) 0 observed by LHCb [7]. Note that GeV, so that the F -wave decay widths might not be suppressed too much. Anyway, we still need to explicitly study their decay properties to verify this possibility.
• The sum rule results extracted from the [3 F , 2, 0, ρρ] multiplet is a bit strange, because the Borel windows become larger as the threshold value ω c decreases, which behavior has already been found in Fig. 9 of Ref. [51]. Hence, we do not use them to draw any conclusion.
Summarizing the above analyses, our results obtained using the method of QCD sum rules within the heavy quark effective theory support to interpret the Λ b (6146) 0 and Λ b (6152) 0 as D-wave bottom baryons of J P = 3/2 + and 5/2 + , respectively. They both contain two λ-mode excitations, and belong to the bottom baryon doublet [3 F , 2, 0, λλ]. This doublet contains two other bottom baryons, Ξ b (3/2 + ) and Ξ b (5/2 + ), whose masses are extracted to be However, this assignment faces a serious problem, which has already been discussed in Refs. [79][80][81]: the lower state Λ b (3/2 + ) would decay both into the P -wave Σ b π channel and the P -wave Σ * b π channel, while the higher state Λ b (5/2 + ) would dominantly decay only into the Pwave Σ * b π channel, which behaviors are just opposite to the Λ b (6146) 0 and Λ b (6152) 0 observed by LHCb [7].
To solve this problem, we investigate another possible assignment, that is to interpret the Λ b (6146) 0 and Λ b (6152) 0 as D-wave bottom baryons of J P = 5/2 + and 7/2 + respectively, both of which belong to the [3 F , 3, 1, ρλ] multiplet. The advantage of this assignment is: the lower state Λ b (5/2 + ) would dominantly decay only into the P -wave Σ * b π channel, while the higher state Λ b (7/2 + ) would decay both into the F -wave Σ b π channel and the F -wave Σ * b π channel, which behaviors seem to be consistent with the Λ b (6146) 0 and Λ b (6152) 0 observed by LHCb [7]. However, this assignment faces another serious problem: the masses of the Λ b (5/2 + ) and Λ b (7/2 + ) as well as their mass splitting are calculated in the present study to be significantly larger than, although not too far from, those of the Λ b (6146) 0 and Λ b (6152) 0 measured by LHCb and CMS [7,8].
There exist many possible assignments for the Λ b (6072) 0 observed by CMS and LHCb [8,9], such as the Λ b (2S) state, while another possible assignment is to interpret it as the D-wave Λ b state. To verify this, one good choice is to further examine whether it has a nearby partner state in future LHCb and CMS experiments.