Strong decays of the $\Xi_b(6227)$ as a $\Sigma_b\bar{K}$ molecule

We study the strong decays of the newly observed $\Xi_b(6227)$ assuming that it is a pure $\Sigma_b\bar{K}$ molecular state. Considering four possible spin-parity assingments $J^P=1/2^{\pm}$ and $3/2^{\pm}$, the partial decay widths of the $\Sigma_b\bar{K}$ molecular state into $\Lambda_b\bar{K}$, $\Xi_b\pi$, and $\Xi_b^{'}\pi$ final states through hadronic loops are evaluated with the help of the effective Lagrangians. In comparison with the LHCb data, the spin-party $1/2^-$ assignment is preferred while those of $J^P=1/2^{+}$ and $J^P=3/2^{\pm}$ are disfavored. In addition, we show that the two allowed decay modes $\Lambda_b\bar{K}$ and $\Xi_b\pi$ of the $\Xi_b(6227)$, being a $S$-wave $\Sigma_b\bar{K}$ molecular state, have almost equal branching ratios, consistent with the data.


I. INTRODUCTION
During the past decade, many new narrow baryon resonances containing a b quark, such as Λ b (5912), Λ b (5920), Σ b (5815), Σ * b , and Ξ b , were discovered experimentally [1][2][3]. Very recently, a bottom baryon Ξ b (6227) was reported by the LHCb Collaboration in both the Λ 0 b K − and Ξ 0 b π − final states from pp collisions [4]. Its mass and width are, respectively, M = 6226.9 ± 2.0(stat) ± 0.3(syst) ± 0.2(Λ 0 b ) MeV Γ = 18.1 ± 5.4(stat) ± 1.8(syst) MeV. (1) Its spin-parity, however, remains unknown. Before the discovery of the Ξ * b state, there were already a few theoretical studies on the existence of such a state. For instance, different quark models predicted that there exists a dsb three quark excited state with a mass about 6200 MeV [5][6][7]. The mass spectra of bottom baryons around 6200 MeV were investigated in a Faddeev equation approach [8]. A molecule Ξ b state with a narrow width and a mass around 6200 MeV was predicted in the unitarized coupled channels approach [9]. In Ref [10] the authors also found that theKΣ b interaction is strong enough to form a bound state both in isospin 3/2 and 1/2. [35] Following the discovery of the Ξ * b , several theoretical studies have been performed [11][12][13]. In the heavy quark-light diquark model, based on the analysis of the mass spectrum Although the studies of Refs. [11][12][13] seem to indicate that this state is a conventional three-quark state, the Ξ * b might still be aKΣ b hadronic molecule state, because the mass gap between the Ξ * b and the ground Ξ b , about 440 MeV, is large enough to excite a light quark-antiquark pair to form a molecular state. Indeed, it is shown in Refs. [9,10] that the interaction between aK meson and a Σ b baryon is strong enough to form a bound state with a mass about 6200 MeV. One way to distinguish the two scenarios is to study the two-body strong decay widths of the Ξ * b baryon. In the present paper we consider the following strong decay modes, with the following spin-parity assignments: J P = 1/2 ± and 3/2 ± , using an effective Lagrangian approach and assuming that the Ξ b (6227) is a hadronic molecule state ofK and Σ b .
This work is organized as follows. The theoretical formalism is explained in Sec. II. The predicted partial decay widths are presented in Sec. III, followed by a short summary in the last section.
With the effective Lagrangian in Eq. (2) and Eq. (3) and the Feynman diagram shown in Fig 1, we can obtain the self energy of the Ξ * b , where with k 0 , m Ξ * b denoting the four momenta and mass of the Ξ * b , respectively, k 1 , mK, and m Σ b are the fourmomenta, the mass of theK meson, and the mass of the Σ b baryon, respectively. The coupling constant g Ξ * b Σ bK is determined by the compositeness condition [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30]. It implies that the renormalization constant of the hadron wave function is set to zero, i.e., where the is the transverse part of the self-energy op- The strong decay modes of the Here, we only calculate the decay widths in the chan- The decay widths in the other channels can be obtained by isospin symmetry as The sum of these partial decay widths is the total decay width of the Ξ * b . Fig. 2 shows the hadronic decay of the Σ bK molecular state into Λ 0 b K − , Ξ 0 b π − , and Ξ 0 ′ b π − mediated by the exchange of the ρ andK * mesons. In Refs. [31,32], the couplings of the vector meson with the charm baryons are obtained using the hiddengauge formalism and assuming SU(4) flavor symmetry: where the coupling constant g = 6.6 is taken from Ref. [31]. The symbol V µ represents the vector fields of the 16-plet of b π − decay processes. We also show the definitions of the kinematics (k 0 , k 1 , k 2 , p 1 , p 2 , and q) used in the calculation. the ρ meson, and B represents the 20-plet of the proton, where the indices i, j, k of B i jk denote the quark content of the baryon fields with the identification 1 ↔ u, 2 ↔ d, 3 ↔ s, and 4 ↔ c. Following the strategy of Ref. [32], the Lagrangian for the bottom hadrons can be easily obtained by replacing the charm baryons in Eq. (11) with the bottom ones.
To compute the ρKK and πKK * vertices, we also need the following effective Lagrangians where with τ being the usual Pauli matrices and ρ µ the field operators of the ρ meson. In the S U(3) f limit, the coupling constant satisfies g KKρ = 2.9, which is determined by the Kawarabayashi-SuzukiRiazuddin-Fayyazuddin relation [33]. The coupling g K * Kπ is fixed from the strong decay width of K * → Kπ. With the help of Eq. (13), the two-body decay width Γ(K * + → K 0 π + ) is related to g K * Kπ as where P πK * is the three-momentum of the π in the rest frame of the K * . Using the experimental strong decay width we obtain g K * Kπ = 4.61 [34].
Putting all the pieces together, we obtain the following amplitudes, where K = ( √ 3, √ 2) and H = ( , respectively. Once the amplitudes are determined, the corresponding partial decay widths can be obtained, which read, where J is the total angular momentum of the Ξ * b state, the | p 1 | is the three-momenta of the decay products in the center of mass frame, the overline indicates the sum over the polarization vectors of the final hadrons, and MB denotes the decay channel of MB, i.e.,Λ bK , Ξ b π, Ξ ′ b π.

III. RESULTS AND DISCUSSIONS
In this work, we study the strong decays of the Ξ * b to the two-body final states Λ bK , Ξ b π, and Ξ ′ b π assuming that it is aKΣ b molecular state. In order to obtain the two body decay width through the triangle diagrams shown in Fig. 2, we first compute the coupling constant g Ξ * b Σ bK . With a value of the cutoff Λ = 0.9 − 1.1 GeV, the corresponding coupling constants are listed in Table. I. The Λ dependence of these coupling constants are shown in Fig 3. We  note that they decrease slowly with the increase of the cut-off, and the coupling constant is almost independent of Λ for the J P = 1/2 − case, where the Ξ * b is an S −waveKΣ b molecular state. It is consistent with the conclusion in Refs. [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30] that for an S −wave loosely bound state the effective coupling strength of the bound state to its components is insensitive to its inner structure. We show the dependence of the total decay width on the cutoff Λ in Fig. 4. In the present calculation, we vary Λ from 0.9 to 1.1 GeV. In this Λ range, the total decay width increases for the cases of J P = 1/2 ± and J P = 3/2 + , while decreases for the J P = 3/2 − case. For the case of J P = 1/2 + , the predicted total decay width is much smaller than the experimental total width, which disfavors such a spin-parity assignment for the Ξ * b in theKΣ b molecular picture. The J P = 3/2 + assignment is not favored by our study as well. In this case theKΣ b molecular state should be in P−wave. Hence, the J P = 3/2 + assumption for Ξ * b (6227) can be excluded. The J P = 3/2 − case is disfavored due to the smallest width predicted, and a D−waveKΣ b molecular with J P = 3/2 − is difficult to form through long range meson exchanges. In other words, the J P = 1/2 + and J P = 3/2 ± Σ bK molecular assumptions for the Ξ * b are excluded. Hence, only the assignment as an S −wavē KΣ b molecular state is possible for the Ξ * b based on the total decay width experimentally measured.
In Fig. 4, we show the partial decay widths of the Ξ * b → Λ bK , Ξ b π, and Ξ ′ b π as a function of the cutoff parameter Λ. The two-body decays in the Ξ * b →KΛ b , Ξ b π, and Ξ ′ b π channels are insensitive to the cut-off parameter Λ. The transition Ξ * b →KΛ b and Ξ * b → Ξ b π are the main decay channels in the J p = 1/2 ± case, the sum of which almost saturates the total width. For the J P = 3/2 ± assignment, the transi- π (green dash-dotted lines), and the total decay width(black solid lines) with different spin-parity assignments for the Ξ * b as a function of the parameter Λ. The oycn bands denote the experimental total width [4]. tion Ξ * b → Λ b π provides an dominant contribution to the total decay width. Since the phase space is small compared with the other two channels, the Ξ ′ b π two-body decay width is the smallest in the four spin-parity assignments.
In Table II, we list the decay widths of the Ξ * b (6227) in the J P = 1/2 assignment with Λ = 0.9 − 1.1 GeV. The ratio of the partial decay widths in the Λ bK and Ξ b π channels are shown in the eleventh line. For comparison, we show the results of the quark models [11][12][13] as well. In Ref [12] and Ref [13], although the Ξ * b state can decay into bothKΛ b and πΞ b channels, the partial decay width of theKΛ b mode is much smaller than that of the πΞ b mode. It contradicts with the experimental data, where the ratio of the partial decay widths of the Ξ * b intoKΛ b and πΞ b channels is about 0.5-1.5. In Ref. [11] the new Ξ * b state can be well interpreted as a conventional three quark state in comparison with the experimental total width and the ratio of the partial decay widths of the Ξ * b intoKΛ b and πΞ b . One should note that the difference between the results of Ref. [11] and Refs. [12,13] mainly originates from the different choices on fixing some parameters accounting for the strength of the quark-meson couplings and/or the harmonic oscillator (SHO) wave functions.
In Fig. 5, we show the ratio of the partial decay widths into Λ bK and Ξ b π. For the J P = 1/2 + case, the ratio is about 1.06 but the total decay width is much smaller than the experimental value. The ratio for the J P = 3/2 ± cases are obviously inconsistent with the experimental results. Therefore, our results, in terms of not only the total decay width but also the ratio of the partial decay widths into Λ bK and Ξ b π, support the Ξ * b state as a S −waveKΣ b molecular state.

IV. SUMMARY
We studied the strong decays of the newly observed Ξ * b baryon intoKΛ 0 b , Ξ b π, and π − Ξ ′ b with different spin-parity assignments and assuming that it is aKΣ b molecular structure. With the couplings constants between the Ξ * b and its components determined by the composition condition, we calculated the partial decay widths intoKΛ 0 b , Ξ b π, and π − Ξ ′ b final states through triangle diagrams in an effective Lagrangian approach. In such a picture, the decays Ξ * b →KΛ 0 b , Ξ * b → πΞ b , and Ξ * b → πΞ ′ b occur by exchanging ρ and K * mesons. We found that both the total decay width and the ratio of the partial decay widths into Λ bK and Ξ b π can be reproduced with the assumption that the Ξ * b is an S −waveKΣ b bound state with J P = 1/2 − , while the P− and D−waveKΣ b assignments are excluded. Our study shows that the experimental measurement of the spin-parity of the Ξ * b will be able to tell whether it is a molecular state or a conventional three-quark state. II: Partial decay widths of Ξ * b → Λ bK , Ξ b π, Ξ ′ b π, and the total decay width Γ total with Λ = 0.9 − 1.1 MeV, in comparison with the results of the quark model [11][12][13] and the total width obtained from the LHCb experiments [4]. The relative decay ratio of the Λ bK and Ξ b π channels are also shown in the eleventh row. All masses and widths are in units of MeV.