Identifying the $\Xi_{b}(6227)$ and $\Sigma_{b}(6097)$ as $P$-wave bottom baryons of $J^P = 3/2^-$

We use the method of QCD sum rules within the framework of heavy quark effective theory to study the mass spectrum of the $\Sigma_{b}(6097)^{\pm}$ and $\Xi_{b}(6227)^{-}$, and use the method of light-cone sum rules still within the heavy quark effective theory to study their decay properties. Our results suggest that they can be well interpreted as $P$-wave bottom baryons with the spin-parity $J^P = 3/2^-$. They belong to the baryon doublet $[\mathbf{6}_F, 2, 1, \lambda]$, where the total and spin angular momenta of the light degree of freedom are $j_l = 2$ and $s_l = 1$, and the orbital angular momentum is between the bottom quark and the two-light-quark system ($\lambda$-type). This doublet contains six bottom baryons, and we predict masses (mass differences) and decay widths of the other four states to be $M_{\Omega_b(3/2^-)} = 6.46 \pm 0.12 {~\rm GeV}$, $\Gamma_{\Omega_b(3/2^-)} = 58{^{+65}_{-33}} {~\rm MeV}$, $M_{\Sigma_b(5/2^-)}-M_{\Sigma_b(3/2^-)}= 13 \pm 5 {~\rm MeV}$, $M_{\Xi_b^{\prime}(5/2^-)}-M_{\Xi_b^{\prime}(3/2^-)} = 12 \pm 5 {~\rm MeV}$, and $M_{\Omega_b(5/2^-)}-M_{\Omega_b(3/2^-)} = 11 \pm 5 {~\rm MeV}$. We propose to search for them in further LHCb experiments.

In this paper we shall update our previous QCD sum rule analyses [22,23]. We shall further study decay properties of P -wave bottom baryons to check which baryon multiplet is preferred. To do this we shall use the method of light-cone sum rules within HQET. Note that similar light-cone sum rule studies were performed in Refs. [46,47] to interpret the Σ b (6097) and Ξ b (6227) as P -wave bottom baryons with J P = 3/2 − , but those studies are in full QCD and not in HQET. In Ref. [48] we have systematically investigated the S-wave decay properties of P -wave charmed baryons, and in this paper we just need to replace the charm quark by the bottom one. We shall also investigate their D-wave decay properties. We shall find that the baryon doublet [6 F , 2, 1, λ] can well explain both the masses and decay widths of the Σ b (6097) and Ξ b (6227) [1,2]. This paper is organized as follows. In Sec. II, we study the mass spectrum of P -wave bottom baryons using the method of QCD sum rules within HQET. In Sec. III and Sec. IV we further study their S-and D-wave decay properties using the method of light-cone sum rules still within HQET. A short summary is given in Sec. V.

II. MASS SPECTRUM OF P -WAVE BOTTOM BARYONS
The mass spectrum of P -wave heavy baryons have been systematically investigated in Refs. [22,23] using the method of QCD sum rules within HQET. In this section we update these calculations, and at the same time evaluate the parameters which are needed when calculating their decay widths.
Firstly, let us briefly explain our notations. A bottom baryon contains one bottom quark and two light quarks, and the symmetries between the two light quarks are (A = antisymmetric and S = symmetric): • The two light quarks have the antisymmetric color structure3 C (A).
• The spin of the two light quarks can be either s l = 0 (A) or s l = 1 (S).
• The SU (3) flavor representation of the two light quarks can be either3 F (A) or 6 F (S).
• The total symmetry of the two light quarks should be antisymmetric due to the Pauli principle.
Accordingly, all the P -wave bottom baryons can be categorized into eight baryon multiplets, denoted as [F (flavor), j l , s l , ρ/λ], where j l is the total angular momentum of the light components, i.e., j l = l λ ⊗ l ρ ⊗ s l . Every multiplet contains several bottom baryons with the total angular momentum j = j l ⊗ s Q = |j l ± 1/2|, where s Q = 1/2 is the spin of the bottom quark. We show the above categorization in Fig. 1. All the P -wave charmed baryon interpolating fields have been systematically constructed in Ref. [22], and the bottom baryon fields can be easily obtained by replacing the charm quark field by the bottom one. We use J α1···α j−1/2 j,P,F,j l ,s l ,ρ/λ (x) to denote one of these fields, which couples to the bottom baryon B through 0|J α1···α j−1/2 j,P,F,j l ,s l ,ρ/λ (x)|B = f F,j l ,s l ,ρ/λ u α1···α j−1/2 (x) . ( Here f F,j l ,s l ,ρ/λ is the decay constant, P is the parity of the bottom baryon, and u α1···αj is the relevant spinor. The two-point correlation function at the hadron level can be written as F,j l ,s l ,ρ/λ Λ F,j l ,s l ,ρ/λ − ω + higher states . Here ω is the external off-shell energy ω = v · k; S[· · · ] denotes symmetrization and subtracting the trace terms in the sets (α 1 · · · α j−1/2 ) and (β 1 · · · β j−1/2 ); Λ F,j l ,s l ,ρ/λ ≡ Λ |j l −1/2|,P,F,j l ,s l ,ρ/λ = Λ j l +1/2,P,F,j l ,s l ,ρ/λ is the sum rule result evaluated at the leading order

(A)
where m b is the bottom quark mass. We also need to consider the sum rule result at the O(1/m b ) order so that the mass of the bottom baryon B can be written as: It depends on two free parameters, the threshold value ω c and the Borel mass T . We refer interested readers to Refs. [22,23,48,49] for detailed explanations of the above equations. Note that the notations used in this paper are the same as those used in Ref. [48], but a bit different from those used in Refs. [22,23].
In the present study we update the QCD sum rule results of Ref. [23], and reinvestigate the baryon multiplets [6 F , 0, 1, λ], [6 F , 1, 0, ρ] and [6 F , 2, 1, λ]. As we have found in Ref. [23], there are four baryon multiplets of the flavor 6 F representation, but the other one [6 F , 1, 1, λ] do not give useful QCD sum rule results, so we shall not discuss it in the present study.
From Eq. (6) we clearly see that the baryon mass depends significantly on the bottom quark mass, whose uncertainty is not so small [3]. This suggests that there are large theoretical uncertainties in our results for the absolute values of the heavy baryon masses, while the mass differences within the same doublet are produced quite well with much less theoretical uncertainties, because they do not depend much on the bottom quark mass [22,23,44,45].
Moreover, the baryon mass also moderately depends on the threshold value ω c , whose uncertainty has been included in our calculation. In the present study we slightly modify the threshold value ω c in order to better describe the masses of the Σ b (6097) ± and Ξ b (6227) − measured by LHCb [1,2], so that their decay widths can be better evaluated. We have also finetuned ω c in the same multiplet to satisfy the relation 15 GeV, so that the masses of their Ω c partner states can be extracted. The obtained results are listed in Table I.

III. S-WAVE DECAY PROPERTIES OF P -WAVE BOTTOM BARYONS
In the previous section we studied the mass spectrum of P -wave bottom baryons, and found that the masses of the Σ b (6097) ± and Ξ b (6227) − [1,2] are consistent with those of the P -wave bottom baryons belonging to the baryon multiplets [6 F , 1, 0, ρ], [6 F , 0, 1, λ], and [6 F , 2, 1, λ]. However, the uncertainties of our sum rule calculations are not so small, preventing us to differentiate these multiplets, while their decay properties do depend significantly on their internal structures [50][51][52].
In this section we further study their decay properties to check which baryon multiplet is preferred. We shall study their S-wave decays into ground-state bottom baryons accompanied by a pseudoscalar meson (π or K), including both the two-body and three-body decays which are kinematically allowed. Note that their S-wave decays into ground-state bottom baryons accompanied by a vector meson (ρ or K * ) are kinematically forbidden. We have slightly modified the threshold value ωc compared to those from Ref. [23] to better describe the masses of the Σ b (6097) ± and Ξ b (6227) − measured by LHCb [1,2], and finetuned ωc in the same multiplet to satisfy the relation ωc(Ωc) − ωc(Ξ c ) = ωc(Ξ c ) − ωc(Σc) = 0.15 GeV. Note that in the last column we have explicitly taken into account the isospin factors, which come from definitions of interpolating fields, i.e.
To do this we use the method of light-cone sum rules within HQET. Actually, the decays of P -wave charmed baryons have been systematically investigated in Ref. [48] using the same approach. In this paper we just need to replace the charm quark by the bottom one, and reinvestigate the following decay channels: We can calculate their decay widths through the following Lagrangians where X b , and P denotes the excited bottom baryon, ground-state bottom baryon, and pseudoscalar meson, respectively.
At the hadronic level, b π − has the following pole terms from the double dispersion relation: while it can also be calculated at the quark and gluon level using the method of operator product expansion. The sum rules for the charmed baryon decay Σ 0 c (1/2 − ) → Σ + c (1/2 + )π − have been calculated in Ref. [48], and the sum rules for the bottom baryon decay The explicit forms of the light-cone distribution amplitudes contained in this equation can be found in Refs. [53][54][55][56][57][58][59][60], where their values can also be found. In this paper we work at the renormalization scale 2 GeV, and use the following values for various condensates [3,[61][62][63][64][65][66][67][68]: Then we perform the Wick rotations and double Borel transformation to Eq. (23) with ω and ω replaced by T 1 and T 2 : where the uncertainties come from the Borel mass, the parameters of the Σ 0 b , the parameters of the Σ − b  Fig. 2(a) as a function of the Borel mass T .
In the following subsections we shall follow the same procedures to separately study the three baryon multiplets, There are altogether six non-vanishing decay channels: (l), (o), (s), (t), (u) and (v). We use light-cone sum rules with HQET to separately investigate them, and the relevant coupling constants are extracted to be For completeness, we show them as functions of the Borel mass T in Fig. 2. Using the above values, we can further extract their decay widths to be is kinematically forbidden when using the mass value M Ω b (3/2 − ) = 6.42±0.11 GeV listed in Table I, while the same channel for the Ω b (3/2 − ) belonging to [6 F , 2, 1, λ] is kinematically allowed when using the other value M Ω b (3/2 − ) = 6.46 ± 0.12 GeV, although the obtained decay width is quite small.
For completeness, we show them as functions of the Borel mass T in Fig. 3. Using the above values, we can further extract their decay widths to be   to separately investigate them, and the relevant coupling constants are extracted to be  For completeness, we show them as functions of the Borel mass T in Fig. 4. Using the above values, we can further extract their decay widths to be The large uncertainties of the above results mainly come from the light-cone distribution amplitudes of pseudoscalar mesons.

IV. D-WAVE DECAY PROPERTIES OF P -WAVE BOTTOM BARYONS
In the previous section we studied the S-wave decay properties of P -wave bottom baryons, and the results are summarized in Table II. From this table we find that none of the three bottom baryon multiplets, [6 F , 0, 1, λ],  accompanied by a pseudoscalar meson (π or K), i.e., We can calculate their decay widths through the following Lagrangians We use light-cone sum rules with HQET to separately investigate them, and the obtained sum rule equations are given in Appendix B. The relevant coupling constants are extracted to be For completeness, we show them as functions of the Borel mass T in Fig. 5. Using the above values, we can further extract their decay widths to be The above results are summarized in Table III.

V. SUMMARY AND DISCUSSIONS
In this paper we have investigated the Ξ b (6227) − and Σ b (6097) ± newly observed by LHCb [1,2]. We use the method of QCD sum rules within the framework of heavy quark effective theory to study their mass spectrum, and use the method of light-cone sum rules still within the heavy quark effective theory to study their decay properties. Based on our previous studies [22,23,48], we have investigated three P -wave bottom baryon multiplets, [6 F , 0, 1, λ], [6 F , 1, 0, ρ] and [6 F , 2, 1, λ], and their masses and decay widths are extracted and summarized in Tables I, II, and III.  Especially, the masses and decay widths of the Σ b (3/2 − ) and Ξ b (3/2 − ) belonging to [6 F , 2, 1, λ] are extracted to be M Σ b (3/2 − ) = 6.10 ± 0.12 GeV , which are consistent with the experimental parameters of the Ξ b (6227) − and Σ b (6097) ± [1,2]. Their non-vanishing decay channels are extracted to be Especially, the branching ratio is also consistent with the one measured by LHCb [1] (see Eq. (2)). Its uncertainty come from the Borel mass, the parameters of the Λ 0 b , the parameters of the Ξ 0 b , the parameters of the Σ − b [ 3 2 − ], and various quark masses and condensates listed in Eq. (24). Summarizing the above results, we conclude that the Σ b (6097) ± and Ξ b (6227) − can be well interpreted as P -wave bottom baryons with the spin-parity J P = 3/2 − , which belong to the baryon doublet [6 F , 2, 1, λ]. We predict the mass and decay width of their Ω b (3/2 − ) partner state to be with the following decay channels We propose to search for the above four P -wave bottom baryons in further LHCb experiments.  [48,49] for detailed discussions. The flavor3F bottom baryons of J P = 1/2 + composes one baryon multiplet where the spin of the two light quarks is s l = 0, and the flavor 6F bottom baryons of J P = 1/2 + and 3/2 + compose another baryon multiplet with s l = 1. See Refs. [48,49]