# Description of the \(\Xi _c\) and \(\Xi _b\) states as molecular states

## Abstract

In this work we study several \(\Xi _c\) and \(\Xi _b\) states dynamically generated from the meson–baryon interaction in coupled channels, using an extension of the local hidden gauge approach in the Bethe–Salpeter equation. These molecular states appear as poles of the scattering amplitudes, and several of them can be identified with the experimentally observed \(\Xi _c\) states, including the \(\Xi _c(2790)\), \(\Xi _c(2930)\), \(\Xi _c(2970)\), \(\Xi _c(3055)\) and \(\Xi _c(3080)\). Also, for the recently reported \(\Xi _b(6227)\) state, we find two poles with masses and widths remarkably close to the experimental data, for both the \(J^P=1/2^-\) and \(J^P=3/2^-\) sectors.

## 1 Introduction

Heavy baryons containing one *c* or *b* quark have been the subject of intense study. Starting from early quark models [1], work along this line has been rather extensive and fruitful [2, 3, 4, 5, 6, 7, 8]. QCD lattice has also contributed to this area [9, 10, 11] and dynamical models building molecular states in coupled meson–baryon channels [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24] have also brought their share to this intense research. There are also many review papers on the subject to which we refer the reader [25, 26, 27, 28, 29, 30, 31, 32].

In the present work we study in detail the \(\Xi _c\) and \(\Xi _b\) states from the molecular point of view. There are many \(\Xi _c\) states reported in the PDG [33] corresponding to excited states. One of the \(\Xi _c\) states, \(\Xi _c(2930)\), first reported by the BaBar Collaboration [34], was recently confirmed with more statistics in two experiments by the Belle Collaboration [35, 36]. On the other hand, for \(\Xi _b\), apart from the \(J^P=1/2^+\) ground states, \(\Xi _b\), \(\Xi ^\prime _b\), and the \(J^P=3/2^+\) \(\Xi _b^*\), there are no states reported in the PDG [33]. Yet, the LHCb Collaboration has recently reported one such state, the \(\Xi _b(6227)\) [37], which we shall also investigate in the present work.

Recent studies of such states using QCD sum rules can be found in Refs. [38, 39, 40, 41], where also reference to works on this particular issue is done, mostly on quark models. As to molecular states of this type we refer to the work of Ref. [15].

*SU*(4) of the chiral Lagrangians. The interesting result from this work was that two states could be interpreted as \(1/2^-\) resonances and the mass and width were well reproduced. This is a non trivial achievement since in other approaches mostly masses are studied and not widths. Some quark models go one step forward and using the \(^3P_0\) model also evaluate widths, as in Ref. [7]. The fact that the widths obtained are quite different than in the molecular model is a positive sign that the study of the widths, and partial decay widths of these resonances, carry valuable information concerning their nature.

Charm sector channels with \(J^P=1/2^-\) and respective thresholds

Channel | \(\Xi _c\pi \) | \(\Xi _c^\prime \pi \) | \(\Lambda _c{\bar{K}}\) | \(\Sigma _c{\bar{K}}\) | \(\Lambda D\) | \(\Xi _c\eta \) | \(\Sigma D\) | \(\Xi _c^\prime \eta \) | \(\Omega _c K\) | \(\Xi D_s\) |
---|---|---|---|---|---|---|---|---|---|---|

Threshold (MeV) | 2607 | 2716 | 2782 | 2949 | 2983 | 3017 | 3060 | 3126 | 3191 | 3287 |

The work of Ref. [44], with vector meson exchange in an extension to *SU*(4) of the chiral Lagrangians, got a boost from Ref. [45], where it was shown that the relevant matrix elements of the interaction can be obtained considering the exchange of light vector mesons in an extension of the local hidden gauge approach [46, 47, 48, 49, 50], where the heavy quarks were mere spectators, such that there was no need to invoke *SU*(4) and one could make a mapping of the *SU*(3) results where the local hidden gauge approach was developed. Like in Ref. [44], in Ref. [45] the same two states were obtained with similar widths, and in addition there was another state reproduced with \(3/2^-\), which was not addressed in [44]. Similar results were then obtained in Ref. [51] with a continuation of the work of Ref. [15] with parameters adjusted to input from the experiment of Ref. [42].

The former results stimulated further work along these lines with predictions for \(\Omega _b\) states in Ref. [52], for which there are not yet experimental counterparts. Encouraged by the success in the \(\Omega _c\) states, in the present work we follow this line of research to study \(\Xi _c\) and \(\Xi _b\) states. In the first case there are several states to compare with our predictions, and in the second case only one excited state, such that many of the states found will be predictions to be tested with future experiments.

Related to these works is the study of the \(\Xi _{cc}\) molecular states in Ref. [53], stimulated by the new measurement of the \(\Xi _{cc}\) by the LHCb Collaboration, with a mass of \(3621\,\mathrm{MeV}\) [54]. This value is higher than that previously measured by the SELEX Collaboration [55, 56]. However, this first measurement by SELEX was not confirmed by the FOCUS [57], Belle [58], BABAR [59] and the LHCb [60] Collaborations. Using the value of the new measurement of the LHCb Collaboration [54], molecular \(\Xi _{cc}\) states were studied in Ref. [53], where excited bound states were found above \(4000\,\mathrm{MeV}\) and broad \(\Xi _{cc} \pi \) and \(\Xi ^*_{cc} \pi \) resonances were found around 3837 and 3918 MeV, respectively.

With all this recent experimental activity there is much motivation to make predictions with different models which can serve as potential guide for experimental set ups and finally to deepen our understanding of the nature of the baryon resonances.

## 2 Formalism

*PB*), vector meson–baryon \((1/2^+)\) (

*VB*) and pseudoscalar meson–baryon \((3/2^+)\) (\(PB^*\)). One should mention that in this theory, these three sectors do not decay into each other, because that would require the exchange of pseudoscalar mesons, and those transitions are momentum-dependent and small compared to the ones with a vector meson exchange [45, 61]. Analyzing the spin-parity of each sector, we find that for the states that arise from

*PB*we have \(J^P = 0^- \otimes 1/2^+ = 1/2^-\), for

*VB*we have degenerate states \(J^P =1^- \otimes 1/2^+ = 1/2^-, \ 3/2^- \) and for \(PB^*\) we have \(J^P =0^- \otimes 3/2^+ = 3/2^- \). In Tables 1, 2, 3 we show, for the charm sector, the channels chosen for the

*PB*,

*VB*and \(PB^*\) sectors. To get the channels for the beauty sector, one needs only to substitute the

*c*quark by a

*b*quark, and we show the results in Tables 4, 5, 6.

Charm sector channels with \(J^P=1/2^-, \ 3/2^-\) and respective thresholds

Channel | \(\Lambda D^*\) | \(\Lambda _c{\bar{K}}^*\) | \(\Sigma D^*\) | \(\Xi _c\rho \) | \(\Xi _c\omega \) | \(\Sigma _c{\bar{K}}^*\) |
---|---|---|---|---|---|---|

Threshold (MeV) | 3124 | 3182 | 3202 | 3245 | 3252 | 3349 |

Charm sector channels with \(J^P=3/2^-\) and respective thresholds

Channel | \(\Xi _c^*\pi \) | \(\Sigma _c^*{\bar{K}}\) | \(\Xi _c^*\eta \) | \(\Sigma ^*D\) | \(\Omega _c^*K\) |
---|---|---|---|---|---|

Threshold (MeV) | 2784 | 3014 | 3194 | 3252 | 3262 |

Beauty sector channels with \(J^P=1/2^-\) and respective thresholds

Channel | \(\Xi _b\pi \) | \(\Xi _b^\prime \pi \) | \(\Lambda _b{\bar{K}}\) | \(\Sigma _b{\bar{K}}\) | \(\Lambda \bar{B}\) | \(\Xi _b\eta \) | \(\Sigma {\bar{B}}\) | \(\Xi _b^\prime \eta \) | \(\Omega _b K\) | \(\Xi B_s\) |
---|---|---|---|---|---|---|---|---|---|---|

Threshold (MeV) | 5931 | 6073 | 6115 | 6309 | 6395 | 6341 | 6473 | 6483 | 6542 | 6685 |

In this work, the kernel will be calculated using an extension of the local hidden gauge approach (LHG) [46, 47, 48, 49, 50], which produces Feynman diagrams of the type shown in Fig. 1, that is, the initial meson baryon pair goes into the final pair through the exchange of a vector meson in the \(t-\)channel.

*PB*case, the meson-meson interaction (the upper vertex in Fig. 1) is given by the

*VPP*Lagrangian:

*SU*(4) pseudoscalar meson and vector meson flavor matrices, respectively, and \(\langle \cdots \rangle \) is the trace over the

*SU*(4) matrices. Note that the original \({\mathcal {L}}_{VPP}\) interaction obeys

*SU*(3) flavor symmetry, but just like in Ref. [45], we extend it to

*SU*(4) to take into account the

*c*(

*b*) quark. The meson matrices are

Beauty sector channels with \(J^P=1/2^-, \ 3/2^-\) and respective thresholds

Channel | \(\Lambda {\bar{B}}^*\) | \(\Lambda _b{\bar{K}}^*\) | \(\Sigma {\bar{B}}^*\) | \(\Xi _b\rho \) | \(\Xi _b\omega \) | \(\Sigma _b{\bar{K}}^*\) |
---|---|---|---|---|---|---|

Threshold (MeV) | 6440 | 6515 | 6518 | 6568 | 6576 | 6709 |

Beauty sector channels with \(J^P=3/2^-\) and respective thresholds

Channel | \(\Xi _b^*\pi \) | \(\Sigma _b^*{\bar{K}}\) | \(\Xi _b^*\eta \) | \(\Sigma ^*{\bar{B}}\) | \(\Omega _b^*K\) |
---|---|---|---|---|---|

Threshold (MeV) | 6091 | 6329 | 6500 | 6664 | 6567 |

The use of *SU*(4) in Eq. (1) is a formality. We shall see later that the dominant terms are due to the exchange of light vectors, where the heavy quark are spectators. Then Eq. (1) automatically projects over *SU*(3). The terms with the exchange of a heavy vector are very suppressed, as we shall see. In principle, in this case one would be using explicitly *SU*(4), however, as seen in Ref. [62], since the matrices \(\phi \) and *V* stand for \(q{\bar{q}}\), Eq. (1) actually only measures the quark overlap of \(\phi \) and *V* and provides a vector structure, hence, the role of *SU*(4) is just a trivial counting of the number of quarks. The lower vertex \(V^\mu BB\) does not rely on *SU*(4) either, as we see below.

*SU*(3) can be described by the following Lagrangian

*B*is the

*SU*(3) baryon matrix, and

*V*the \(3\times 3\) part of

*V*in Eq. (3) containing \(\rho \), \(\omega \), \(K^*\), \(\phi \). Here we do a non-relativistic approximation, which consists in substituting \(\gamma ^{\mu } \rightarrow \gamma ^{0}\). The extension to the charm or bottom sectors is done without relying on

*SU*(4) as explained below. As discussed in Refs. [45, 53], it can be shown that the same interaction in

*SU*(3) of Eq. (4) can be obtained considering an operator at the quark level, such that Eq. (4) becomes

The states described by Eq. (6) are constructed using only *SU*(3) symmetry, taking the heavy quark as a spectator, which implies that all diagonal terms are described through the exchange of light vectors, respecting heavy quark spin symmetry (HQSS) [63].

*PB*sector are constructed using the method outlined in Ref. [64], but with the necessary changes in phases in order to obey the sign notation in Refs. [65, 66], which is consistent with the chiral matrices. Doing this we obtain the following states:

- 1.
\(\big |\Xi _c \pi \big> = \sqrt{\frac{2}{3}}\big |\Xi _c^0 \pi ^+ \big> +\sqrt{\frac{1}{3}}\big |\Xi _c^+ \pi ^0 \big > ,\)

- 2.
\(\big |{\Xi '}_c \pi \big> = \sqrt{\frac{2}{3}}\big |{\Xi '}_c^0 \pi ^+ \big> +\sqrt{\frac{1}{3}}\big |{\Xi '}_c^+ \pi ^0 \big > ,\)

- 3.
\(\big |\Lambda _c \bar{K} \big> = \big |\Lambda _c^+ \bar{K}^0 \big >, \)

- 4.
\(\big |\Sigma _c \bar{K} \big> = -\left( \sqrt{\frac{2}{3}}\big |\Sigma _c^{++} K^- \big> + \sqrt{\frac{1}{3}}\big |\Sigma _c^+ \bar{K}^0 \big >\right) , \)

- 5.
\(\big |\Lambda D \big> = \big |\Lambda ^0 D^+ \big >, \)

- 6.
\(\big |\Xi _c \eta \big> = \big |\Xi _c^+ \eta \big > , \)

- 7.
\(\big |\Sigma D \big> = \sqrt{\frac{2}{3}}\big |\Sigma ^+ D^0 \big> -\sqrt{\frac{1}{3}}\big |\Sigma ^0 D^+ \big > ,\)

- 8.
\(\big |{\Xi '}_c \eta \big> = \big |{\Xi '}_c^+ \eta \big > , \)

- 9.
\(\big |\Omega _c \eta \big> = \big |\Omega _c^0 K^+ \big > , \)

- 10.
\(\big |\Xi D_s\big> = \big |\Xi ^0 D_s^+ \big >.\)

\(D_{ij}\) coefficients for the *PB* states coupling to \(J^P=1/2^-\)

\(J_{baryon}=1/2\) | \(\Xi _c\pi \) | \(\Xi _c^\prime \pi \) | \(\Lambda _c{\bar{K}}\) | \(\Sigma _c{\bar{K}}\) | \(\Lambda D\) | \(\Xi _c\eta \) | \(\Sigma D\) | \(\Xi _c^\prime \eta \) | \(\Omega _c K\) | \(\Xi D_s\) |
---|---|---|---|---|---|---|---|---|---|---|

\(\Xi _c\pi \) | \(-2\) | 0 | \(-\sqrt{\frac{3}{2}}\) | 0 | \(\frac{1}{2\sqrt{2}}\lambda \) | 0 | \(-\frac{1}{2\sqrt{2}}\lambda \) | 0 | 0 | 0 |

\(\Xi _c^\prime \pi \) | \(-2\) | 0 | \(-\frac{1}{\sqrt{2}}\) | \(-\frac{3}{2\sqrt{6}}\lambda \) | 0 | \(-\frac{1}{2\sqrt{6}}\lambda \) | 0 | \(-\sqrt{3}\) | 0 | |

\(\Lambda _c{\bar{K}}\) | \(-1\) | 0 | \(-\frac{1}{\sqrt{3}}\lambda \) | \(\frac{2}{\sqrt{3}}\) | 0 | 0 | 0 | 0 | ||

\(\Sigma _c{\bar{K}}\) | \(-3\) | 0 | 0 | \(-\frac{1}{\sqrt{3}}\lambda \) | \(-2\) | 0 | 0 | |||

\(\Lambda D\) | \(-1\) | \(-\frac{1}{6}\lambda \) | 0 | \(\frac{1}{2\sqrt{3}}\lambda \) | 0 | \(-\frac{\sqrt{6}}{2}\) | ||||

\(\Xi _c\eta \) | 0 | \(-\frac{1}{2}\lambda \) | 0 | 0 | \(\frac{1}{\sqrt{6}}\lambda \) | |||||

\(\Sigma D\) | \(-3\) | \(-\frac{1}{2\sqrt{3}}\lambda \) | 0 | \(-\sqrt{\frac{3}{2}}\) | ||||||

\(\Xi _c^\prime \eta \) | 0 | \(-\frac{2\sqrt{6}}{3}\) | \(-\frac{1}{3\sqrt{2}}\lambda \) | |||||||

\(\Omega _c K\) | \(-2\) | \(-\frac{1}{\sqrt{3}}\lambda \) | ||||||||

\(\Xi D_s\) | \(-2\) |

*VB*sector, the upper vertex of the three vector meson interaction is given by [67]

*VPP*interaction given by Eq. (1). Since from Eq. (4) to Eq. (5) we have made the approximation that \(\gamma ^{\mu } \rightarrow \gamma ^{0}\), this makes Eq. (5) spin independent and as such, we can still use it for the \(V B^*B^*\) vertices. Additionally, we have, for the \(B^*\) baryons, the following spin-flavor states:

- 1.
\(\big |\Xi ^{*+}_c \big> = \big | \frac{1}{\sqrt{2}} c(us+su) \big> \big | \chi _{S}\big >, \)

- 2.
\(\big |\Xi ^{*0}_c \big> = \big | \frac{1}{\sqrt{2}} c(ds+sd) \big> \big | \chi _{S}\big >, \)

- 3.
\(\big |\Omega ^*_c \big> = \big | css \big> \big | \chi _{S}\big >,\)

- 4.
\(\big |\Sigma _c^{* ++} \big> = \big |cuu \big> \big | \chi _{S}\big >, \)

- 5.
\(\big |\Sigma _c^{* +} \big> = \big | \frac{1}{\sqrt{2}} c(ud+du) \big> \big | \chi _{S}\big >,\)

- 6.
\(\big |\Sigma _c^{* 0} \big> = \big |cdd \big> \big | \chi _{S}\big >, \)

- 7.
\(\big |\Sigma ^{* +} \big> = \frac{1}{\sqrt{3}}\big |u(su+us) + suu \big> \big | \chi _{S}\big >, \)

- 8.
\(\big |\Sigma ^{* 0} \big> = \frac{1}{\sqrt{6}} \big |s(du+ud) + d(su+us) +u(sd+ds) \big> \big | \chi _{S}\big >, \)

- 9.
\(\big |\Sigma ^{* -} \big> = \frac{1}{\sqrt{3}} \big |d(sd+ds) + sdd \big> \big | \chi _{S}\big >.\)

*VB*and \(PB^*\) cases are similar to the ones of the

*PB*case.

*VB*interaction we get the same kernel, even though the

*VVV*vertex is described by a different Lagrangian, assuming the three momentum of the vectors are small compared to their masses [67]. Actually the meson baryon chiral lagrangians [69, 70] can be obtained from the local hidden gauge approach neglecting the \(\left( \displaystyle \frac{p}{m_V}\right) ^2\) term in the exchanged vectors [67]. Then, the kernel will be the same as in Eq. (17) with an extra \(\vec {\epsilon }\cdot \vec {\epsilon }\ '\) factor, due to the polarizations of the initial and final vector mesons, which can be factorized in the Bethe–Salpeter equation. This means that the equation is spin independent, and that is why we find degenerate states with \(J^P=1/2^-\) and \(J^P=3/2^-\) with this interaction [61]. Because of this, we can just omit that factor.

*PB*,

*VB*and \(PB^*\) sectors respectively.

\(D_{ij}\) coefficients for the *VB* states coupling to \(J^P=1/2^-\), \(3/2^-\)

\(J_{baryon}=1/2\) | \(\Lambda D^*\) | \(\Lambda _c{\bar{K}}^*\) | \(\Sigma D^*\) | \(\Xi _c\rho \) | \(\Xi _c\omega \) | \(\Sigma _c{\bar{K}}^*\) |
---|---|---|---|---|---|---|

\(\Lambda D^*\) | \(-1\) | \(-\frac{1}{\sqrt{3}}\lambda \) | 0 | \(\frac{1}{2\sqrt{2}}\lambda \) | \(-\frac{1}{2\sqrt{6}}\lambda \) | 0 |

\(\Lambda _c{\bar{K}}^*\) | \(-1\) | 0 | \(-\sqrt{\frac{3}{2}}\) | \(\frac{1}{\sqrt{2}}\) | 0 | |

\(\Sigma D^*\) | \(-3\) | \(-\frac{1}{2\sqrt{2}}\lambda \) | \(-\frac{3}{2\sqrt{6}}\lambda \) | \(-\frac{1}{\sqrt{3}}\lambda \) | ||

\(\Xi _c\rho \) | \(-2\) | 0 | 0 | |||

\(\Xi _c\omega \) | 0 | 0 | ||||

\(\Sigma _c{\bar{K}}^*\) | \(-3\) |

*L*a light meson (see Fig. 2). Then, the propagator will be,

*q*the transferred momentum. Since \(V_H\) is heavy, we can take \(\vec {q}\simeq 0\) and

\(D_{ij}\) coefficients for the \(PB^*\) states coupling to \(J^P=3/2^-\)

\(J_{baryon}=3/2\) | \(\Xi _c^*\pi \) | \(\Sigma _c^*{\bar{K}}\) | \(\Xi _c^*\eta \) | \(\Sigma ^*D\) | \(\Omega _c^*K\) |
---|---|---|---|---|---|

\(\Xi _c^*\pi \) | \(-2\) | \(-\frac{1}{\sqrt{2}}\) | 0 | \(-\frac{1}{\sqrt{6}}\lambda \) | \(-\sqrt{3}\) |

\(\Sigma _c^*{\bar{K}}\) | \(-3\) | \(-2\) | \(\frac{1}{\sqrt{3}}\lambda \) | 0 | |

\(\Xi _c^*\eta \) | 0 | \(-\frac{1}{\sqrt{3}}\lambda \) | \(-\frac{2\sqrt{6}}{3}\) | ||

\(\Sigma ^*D\) | \(-3\) | 0 | |||

\(\Omega _c^*K\) | \(-2\) |

Since the \(\Xi _c\) and \(\Xi _b\) states are heavy quark states, one should comment on how our model deals with HQSS. For that, one should note that, with the exception of the vertices with the \(\lambda \), in all other vertices the heavy quark behaves as a spectator, which guarantees that the dominant terms (in the \(1/m_Q\) counting) obey HQSS rules. In the terms where that does not happen, their influence is scaled down because of the introduction of the \(\lambda \) parameter, which is a small number.

Finally, the same process can be repeated for the beauty sector, where one only needs to substitute the *c* quark by a *b* quark. Then, the \(D_{ij}\) coefficients will be equal to the ones in the charm sector case, the only difference being that now \(\lambda =0.1\) [52], because the heavy vector mesons will now be \(B^*\) and \(B_s^*\) instead of \(D^*\) and \(D_s^*\).

## 3 Results

*V*is the kernel matrix,

*G*is a diagonal matrix where the diagonal terms correspond to the loop functions of each channel, and the cutoff scheme is used here to regularize the loop integration. The cutoff regularization avoids potential pathologies of the dimensional regularization in the charm sector or beauty sector, where the real part of

*G*can become positive below the threshold and artificial poles can be found in the

*T*-matrix, which can lead to the production of the bound states with a repulsive potential [75]. Also, in order to respect the rules of the heavy quark symmetry in bound states, the same cutoff has to be taken for all channels as it was shown in Refs. [76, 77, 78].

Poles in the \(J^P=1/2^-\) sector from pseudoscalar-baryon interaction (all units are in MeV)

\(q_{max}\) | 600 | 650 | 700 | 750 | 800 |
---|---|---|---|---|---|

\(2684.23+i89.72\) | \(2679.71+i76.48\) | \(2673.49+i64.54\) | \(2666.24+i54.01\) | \(2658.68+i44.52\) | |

\(2800.72+i100.03\) | \(2801.80+i86.16\) | \(2803.28+i72.06\) | \(2803.31+i57.77\) | \(2794.76+i31.06\) | |

\(2880.76+i10.31\) | \(2842.47+i10.13\) | \(\varvec{2791.30+i3.63}\) | \(2738.46+i1.36\) | \(2685.56+i0.89\) | |

\(2896.57+i1.34\) | \(2870.10+i10.64\) | \(2850.70+i16.38\) | \(2830.84+i23.17\) | \(2817.77+i40.45\) | |

\(2969.50+i3.30\) | \(2955.62+i5.10\) | \(\varvec{2937.15+i7.31}\) | \(2913.82+i10.03\) | \(2886.31+i13.46\) | |

\(3171.55+i32.48\) | \(3160.12+i37.77\) | \(3148.11+i41.88\) | \(3135.67+i44.96\) | \(3125.96+i47.18\) |

*G*is given by [79]

*l*, will be calculated in the first Riemann sheet for Re(\(\sqrt{s}\)) smaller than the threshold of that channel (\(\sqrt{s}_{th, l}\)), and in the second Riemann sheet for Re(\(\sqrt{s}\)) bigger than \(\sqrt{s}_{th, l}\). To take this into account, we define a new loop function

*q*is given by

*T*-matrix can be expressed as

*j*) for each resonance as in Eq. (27), and then calculate the remaining couplings in relation to this one:

*i*-channel at the origin [81].

### 3.1 Molecular \(\Xi _c\) states generated from meson–baryon states

*PB*states, which will lead us to the states with \(J^P=1/2^-\). The poles that appear in this sector are illustrated in Table 10, where we vary the value of the cutoff \(q_{max}\) from 600 to \(800\,\mathrm{MeV}\).

The coupling constants to various channels and \(g_iG^{II}_i\) for the pole at 2191.30 \(+\) *i*3.63 MeV in the \(J^P=1/2^-\) sector with \(q_{max}=700\,\mathrm{MeV}\) (all units are in MeV)

\(\varvec{2791.30+i3.63}\) | \(\Xi _c\pi \) | \(\Xi _c^\prime \pi \) | \(\Lambda _c{\bar{K}}\) | \(\Sigma _c{\bar{K}}\) | \(\Lambda D\) |
---|---|---|---|---|---|

\(g_i\) | \(-0.01-i0.03\) | \(0.39-i0.44\) | \(-0.09-i0.05\) | \(1.05-i0.47\) | \(1.91-i0.09\) |

\(g_iG^{II}_i\) | \(0.78+i0.53\) | \(-3.98+i14.85\) | \(2.70+i0.73\) | \(-11.27+i4.95\) | \(-7.45+i0.27\) |

\(\Xi _c\eta \) | \(\Sigma D\) | \(\Xi _c^\prime \eta \) | \(\Omega _c K\) | \(\Xi D_s\) | |
---|---|---|---|---|---|

\(g_i\) | \(0.23+i0.03\) | \(\varvec{8.82+i0.38}\) | \(0.49-i0.17\) | \(0.21-i0.26\) | \(5.44+i0.20\) |

\(g_iG^{II}_i\) | \(-2.00-i0.26\) | \(\varvec{-29.16-i1.48}\) | \(-3.53+i1.19\) | \(-1.42+i1.74\) | \(-11.96-i0.49\) |

The widths of pole \(2791.30+i3.63\) decaying to various channels (all units are in MeV)

Channel | \(\Xi _c\pi \) | \(\Xi _c^\prime \pi \) | \(\Lambda _c{\bar{K}}\) |
---|---|---|---|

\(\Gamma _i\) | 0.04 | 8.00 | 0.12 |

The coupling constants to various pseudoscalar-baryon channels and \(g_iG^{II}_i\) for the pole at 2937.15 MeV in the \(J^P=1/2^-\) sector with \(q_{max}=700\,\mathrm{MeV}\) (all units are in MeV)

\(\varvec{2937.15+i7.31}\) | \(\Xi _c\pi \) | \(\Xi _c^\prime \pi \) | \(\Lambda _c{\bar{K}}\) | \(\Sigma _c{\bar{K}}\) | \(\Lambda D\) |
---|---|---|---|---|---|

\(g_i\) | \(-0.29+i0.10\) | \(0.03-i0.32\) | \(0.28-i0.22\) | \(0.27+i0.08\) | \(\varvec{3.96-i0.29}\) |

\(g_iG^{II}_i\) | \(0.83-i9.44\) | \(6.42+i6.63\) | \(0.31+i10.35\) | \(-5.40-i1.98\) | \(\varvec{-27.75+i0.73}\) |

\(\Xi _c\eta \) | \(\Sigma D\) | \(\Xi _c^\prime \eta \) | \(\Omega _c K\) | \(\Xi D_s\) | |
---|---|---|---|---|---|

\(g_i\) | \(-0.07+i0.39\) | \(-2.44+i0.10\) | \(0.09+i0.04\) | \(-0.12-i0.26\) | \(3.55-i0.13\) |

\(g_iG^{II}_i\) | \(1.11-i5.19\) | \(12.15-i0.15\) | \(-0.85-i0.40\) | \(0.98+i2.23\) | \(-9.83+i0.23\) |

*a*th-channel baryon and meson respectively, and \(M_R\) is the mass of the resonance (the real part of the pole). In Table 12, we give the partial decay widths of the pole in Table 11, and it can be clearly seen that the state decays mostly to \(\Xi _c^\prime \pi \), as expected.

Similarly, for the state located at \(2937.15\,\mathrm{MeV}\), we can see that the resonance has a large contribution from the \(\Lambda D\) channel. Also, we have the same open channels as the ones in Table 12. We can see that the coupling constant to the channel \(\Xi _c^\prime \pi \) becomes smaller than before as shown in Table 13. However, the couplings to the channels \(\Xi _c\pi \) and \(\Lambda _c{\bar{K}}\) are bigger, yet, there is more phase space for decay for \(\Xi _c\pi \) and \(\Xi _c^\prime \pi \), but altogether the final widths to these three channels are comparable as one can see in Table 14. The \(\Lambda _c{\bar{K}}\) channel accounts for about 1 / 3 of the total width and this is the channel where the BaBar Collaboration observed the state \(\Xi _c(2930)\) [34].

On the other hand, for the *VB* channels, in Table 15, we obtain four poles for all the cutoffs, and in order to be consistent with the \(J^P=1/2^-\) sector, we stick to the same cutoff \(q_{max}=700\,\mathrm{MeV}\), which leads us to three poles that can be selected as possible candidates for \(\Xi _c(2970)\), \(\Xi _c(3055)\) or \(\Xi _c(3080)\), and \(\Xi _c(3123)\) states.

As shown in Table 16, we present the couplings of the first three poles for \(q_{max}=700\,\mathrm{MeV}\). The first state, \(2973.76\,\mathrm{MeV}\) couples very strongly to \(\Sigma D^*\) and almost nothing to the rest of the channels, thus it can be considered as a \(\Sigma D^*\) bound state. The second state, located at \(3068.21\,\mathrm{MeV}\), couples to both \(\Lambda _c {\bar{K}}^*\) and \(\Xi _c\rho \), with similar values for the coupling as well as \(gG^{II}\). The situation of the third state, \(3109.04\,\mathrm{MeV}\), is similar to what we found in the first state, where it practically only couples to \(\Lambda D^*\), and the product \(gG^{II}\) is also significantly larger than for the rest of the channels. Moreover, we notice that all these three poles are below thresholds, so they do not decay to any of the coupled states shown in Table 16, instead it may decay into the pseudoscalar-baryon ones.

The widths of pole \(2937.15+i7.31\) decaying to various channels (all units are in MeV)

Channel | \(\Xi _c\pi \) | \(\Xi _c^\prime \pi \) | \(\Lambda _c{\bar{K}}\) |
---|---|---|---|

\(\Gamma _i\) | 5.22 | 4.45 | 5.88 |

The poles in the \(J^P=1/2^-\), \(3/2^-\) sector from the vector-baron interaction (all units are in MeV)

\(q_{max}\) | 600 | 650 | 700 | 750 | 800 |
---|---|---|---|---|---|

3055.63 | 3016.46 | \(\varvec{2973.76}\) | 2928.28 | 2880.75 | |

3117.37 | 3094.39 | \(\varvec{3068.21}\) | 3040.89 | 3013.14 | |

3121.75 | 3115.67 | \(\varvec{3109.04}\) | 3100.55 | 3090.16 | |

3234.03+i0.22 | 3204.98 | 3174.50 | 3143.09 | 3111.43 |

The coupling constants to various vector-baryon channels and \(g_iG^{II}_i\) for the poles in the \(J^P=1/2^-,3/2^-\) sector with \(q_{max}=700\,\mathrm{MeV}\) (all units are in MeV)

\(\varvec{2973.76}\) | \(\Lambda D^*\) | \(\Lambda _c{\bar{K}}^*\) | \(\Sigma D^*\) | \(\Xi _c\rho \) | \(\Xi _c\omega \) | \(\Sigma _c{\bar{K}}^*\) |
---|---|---|---|---|---|---|

\(g_i\) | 0 | 0.07 | \(\varvec{9.30}\) | 0.33 | 0.30 | 0.55 |

\(g_iG^{II}_i\) | 0 | \(-0.48\) | \(\varvec{-31.85}\) | \(-2.29\) | \(-2.02\) | \(-2.87\) |

\(\varvec{3068.21}\) | \(\Lambda D^*\) | \(\Lambda _c{\bar{K}}^*\) | \(\Sigma D^*\) | \(\Xi _c\rho \) | \(\Xi _c\omega \) | \(\Sigma _c{\bar{K}}^*\) |
---|---|---|---|---|---|---|

\(g_i\) | 0.37 | \(\varvec{3.08}\) | \(-0.26\) | \(\varvec{3.57}\) | \(-0.85\) | \(-0.04\) |

\(g_iG^{II}_i\) | \(-2.33\) | \(\varvec{-30.22}\) | 1.20 | \(\varvec{-30.89}\) | 7.19 | 0.22 |

\(\varvec{3109.04}\) | \(\Lambda D^*\) | \(\Lambda _c{\bar{K}}^*\) | \(\Sigma D^*\) | \(\Xi _c\rho \) | \(\Xi _c\omega \) | \(\Sigma _c{\bar{K}}^*\) |
---|---|---|---|---|---|---|

\(g_i\) | \(\varvec{3.05}\) | 0.05 | 0.03 | \(-0.51\) | 0.09 | 0.01 |

\(g_iG^{II}_i\) | \(\varvec{-26.23}\) | \(-0.60\) | \(-0.17\) | 5.04 | \(-0.81\) | \(-0.05\) |

The poles in the \(J^P=3/2^-\) sector from the pseudoscalar-baron interaction (all units are in MeV)

\(q_{max}\) | 600 | 650 | 700 | 750 | 800 |
---|---|---|---|---|---|

\(2868.84+i101.02\) | \(2869.69+i87.71\) | \(2870.00+i71.15\) | \(2871.12+i55.04\) | \(2888.93+i43.98\) | |

\(2950.39+i11.19\) | \(2932.11+i15.01\) | \(\varvec{2912.78+i19.94}\) | \(2891.71+i27.88\) | \(2855.31+i26.46\) | |

\(3099.36+i0.55\) | \(3059.03+i0.89\) | \(\varvec{3015.18+i1.37}\) | \(2968.69+i2.98\) | \(2918.23+i7.32\) | |

\(3243.94+i32.64\) | \(3233.36+i38.32\) | \(3222.35+i42.93\) | \(3211.05+i46.56\) | \(3199.61+i49.32\) |

### 3.2 Molecular states for \(\Xi _b\) generated from meson–baryon states

*c*quark replaced by a

*b*quark in each channel. The poles from the

*PB*interaction are given in Table 19, where we obtain six poles for each cutoff. Taking into account the uncertainty caused by the variation of the cutoff, we can associate the pole, \(6220.30\,\mathrm{MeV}\) with \(q_{max}=650\,\mathrm{MeV}\), to the state \(\Xi _b(6227)\) recently observed by the LHCb Collaboration [37]. The newly observed state \(\Xi _b(6227)\) is reported with the values \(6226.9\pm 2.0\pm 0.3\pm 0.2\,\mathrm{MeV}/c^2\) and \(18.1\pm 5.4\pm 1.8\,\mathrm{MeV}/c^2\) for its mass and width, respectively. We can see that the mass obtained is merely a few MeV below the experimental data, and the width (\(25.20\,\mathrm{MeV}\)) is also in very good agreement with the data.

The coupling constants to various pseudoscalar-baryon channels and \(g_iG^{II}_i\) for the poles in the \(J^P=3/2^-\) sector with \(q_{max}=700\,\mathrm{MeV}\) (all units are in MeV)

\(\varvec{2912.78+i19.94}\) | \(\Xi _c^*\pi \) | \(\Sigma _c^*{\bar{K}}\) | \(\Xi _c^*\eta \) | \(\Sigma ^*D\) | \(\Omega _c^*K\) |
---|---|---|---|---|---|

\(g_i\) | \(0.41-i1.28\) | \(\varvec{3.78-i0.47}\) | \(1.87-i0.12\) | \(-1.09-i0.83\) | \(0.16-i0.85\) |

\(g_iG^{II}_i\) | \(11.65+i36.25\) | \(\varvec{-48.53+i2.60}\) | \(-14.80+i0.42\) | \(3.17+i2.60\) | \(-1.34+i6.20\) |

\(\varvec{3015.18+i1.37}\) | \(\Xi _c^*\pi \) | \(\Sigma _c^*{\bar{K}}\) | \(\Xi _c^*\eta \) | \(\Sigma ^*D\) | \(\Omega _c^*K\) |
---|---|---|---|---|---|

\(g_i\) | \(0.03-i0.24\) | \(0.11+i0.07\) | \(0.42+i0.03\) | \(\varvec{8.94-i0.04}\) | \(-0.04-i0.20\) |

\(g_iG^{II}_i\) | \(4.82+i5.04\) | \(-3.19-i1.64\) | \(-4.18-i0.34\) | \(\varvec{-33.36+i0.03}\) | \(0.31+i1.72\) |

The poles in the \(J^P=1/2^-\) sector from the pseudoscalar-baron interaction (all units are in MeV)

\(q_{max}\) | 600 | 650 | 700 | 750 | 800 |
---|---|---|---|---|---|

\(6002.21+i81.90\) | \(5997.45+i69.73\) | \(5991.25+i58.74\) | \(5984.13+i49.02\) | \(5976.61+i40.29\) | |

\(6152.19+i91.66\) | \(6152.17+i78.48\) | \(6152.24+i64.42\) | \(6150.40+i48.36\) | \(6137.15+i27.48\) | |

\(6237.52+i11.30\) | \(\varvec{6220.30+i12.60}\) | \(6201.74+i19.00\) | \(6183.11+i27.85\) | \(6175.03+i40.45\) | |

\(6263.48+i0.07\) | \(6205.08+i2.94\) | \(6141.06+i1.73\) | \(6073.96+i0.17\) | \(6004.35+i0.44\) | |

\(6359.89+i0.82\) | \(6338.97+i1.44\) | \(6312.50+i2.42\) | \(6280.97+i3.77\) | \(6244.54+i5.65\) | |

\(6513.45+i29.56\) | \(6501.26+i33.87\) | \(6488.63+i37.17\) | \(6482.88+i39.75\) | \(6482.86+i41.90\) |

The coupling constants to various pseudoscalar-baryon channels and \(g_iG^{II}_i\) for the poles in the \(J^P=1/2^-\) sector with \(q_{max}=650\,\mathrm{MeV}\) (all units are in MeV)

\(\varvec{6220.30+i12.60}\) | \(\Xi _b\pi \) | \(\Xi _b^\prime \pi \) | \(\Lambda _b{\bar{K}}\) | \(\Sigma _b{\bar{K}}\) | \(\Lambda {\bar{B}}\) |
---|---|---|---|---|---|

\(g_i\) | \(0.01+i0.02\) | \(0.34-i0.91\) | \(0.01-i0.01\) | \(\varvec{3.53-i0.14}\) | \(-1.03+i0.61\) |

\(g_iG^{II}_i\) | \(-0.60+i0.05\) | \(10.08+i25.84\) | \(0.42+i0.15\) | \(\varvec{-44.85-i0.67}\) | \(1.47-i0.78\) |

\(\Xi _b\eta \) | \(\Sigma {\bar{B}}\) | \(\Xi _b^\prime \eta \) | \(\Omega _b K\) | \(\Xi B_s\) | |
---|---|---|---|---|---|

\(g_i\) | \(-0.00+i0.04\) | \(-2.09+i4.72\) | \(1.80+i0.02\) | \(0.09-i0.65\) | \(-1.93+i2.81\) |

\(g_iG^{II}_i\) | \(0.02-i0.38\) | \(2.54-i5.25\) | \(-13.14-i0.55\) | \(-0.74+i4.33\) | \(1.45-i2.02\) |

Moreover, for the couplings of this pole we can look at the results in Table 20, where we can see that the main contribution comes from the \(\Sigma _b {\bar{K}}\) channel^{1}. Also, we see that it has only three open channels, and it should be noted that two of these channels, the \(\Xi _b\pi \) and \(\Lambda _b{\bar{K}}\), are the ones where the state \(\Xi _b(6227)\) has been observed [37]. However, according to our findings, it couples mostly to the \(\Xi _b^\prime \pi \) among the open channels, which suggests that it would be easier to find the state \(\Xi _b(6227)\) in the \(\Xi _b^\prime \pi \) channel instead of the other two, which can be confirmed by future experiments.

*VB*interaction, we observed four poles for each cutoff, but none of these poles for \(q_{max}=650\,\mathrm{MeV}\) can be associated to any known \(\Xi _b\) states with negative parity, since there are not enough data available for \(\Xi _b\) states. Furthermore, almost all of the poles found in this sector are below their respective thresholds, which makes it more plausible that these channels could qualify as bound states rather than resonances (Table 22).

The widths of pole \(6220.30+i12.60\) decaying to various channels (all units are in MeV)

Channel | \(\Xi _b\pi \) | \(\Xi _b^\prime \pi \) | \(\Lambda _b{\bar{K}}\) |
---|---|---|---|

\(\Gamma _i\) | 0.02 | 35.01 | 0.01 |

The poles in the \(J^P=1/2^-\), \(3/2^-\) sector from the vector-baron interaction (all units are in MeV)

\(q_{max}\) | 600 | 650 | 700 | 750 | 800 |
---|---|---|---|---|---|

6342.09 | 6295.86 | 6244.94 | 6190.01 | 6131.80 | |

6425.22 | 6407.58 | 6379.79 | 6351.13 | 6322.26 | |

6434.39 | 6417.24 | 6406.58 | 6393.82 | 6378.88 | |

\(6579.47+i0.05\) | \(6548.79+i0.03\) | 6516.72 | 6484.01 | 6451.25 |

The poles in the \(J^P=3/2^-\) sector from the pseudoscalar-baron interaction (all units are in MeV)

\(q_{max}\) | 600 | 650 | 700 | 750 | 800 |
---|---|---|---|---|---|

\(6169.97+i92.88\) | \(6169.85+i80.29\) | \(6168.88+i66.55\) | \(6166.89+i49.80\) | \(6155.48+i29.00\) | |

\(6258.53+i10.88\) | \(\varvec{6240.21+i14.65}\) | \(6221.49+i19.71\) | \(6203.04+i27.94\) | \(6193.81+i39.67\) | |

\(6474.16+i0.14\) | \(6424.37+i0.20\) | \(6369.22+i0.34\) | \(6309.51+i0.68\) | \(6245.85+i2.08\) | |

\(6538.85+i30.15\) | \(6527.01+i34.64\) | \(6514.86+i38.05\) | \(6502.41+i40.62\) | \(6500.43+i42.56\) |

The coupling constants to various pseudoscalar-baryon channels and \(g_iG^{II}_i\) for the poles in the \(J^P=3/2^-\) sector with \(q_{max}=650\,\mathrm{MeV}\) (all units are in MeV)

\(\varvec{6240.21+i14.65}\) | \(\Xi _b^*\pi \) | \(\Sigma _b^*{\bar{K}}\) | \(\Xi _b^*\eta \) | \(\Sigma ^*{\bar{B}}\) | \(\Omega _b^*K\) |
---|---|---|---|---|---|

\(g_i\) | \(0.23-i0.93\) | \(\varvec{3.39-i0.36}\) | \(1.74-i0.10\) | \(-0.78-i0.36\) | \(0.03-i0.65\) |

\(g_iG^{II}_i\) | \(13.03+i24.31\) | \(\varvec{-43.16+i1.85}\) | \(-12.78+i0.27\) | \(0.63+i0.30\) | \(-0.29+i4.27\) |

Moving on to the \(J^P=3/2^-\) sector, we also get four poles for each cutoff, which are given in Table 23, where we also find a possible candidate for \(\Xi _b(6227)\). The state \(6240.21\,\mathrm{MeV}\) agrees really well with the experimental data [37], as both mass and width are within acceptable ranges. For the couplings as well as \(g_iG^{II}_i\), shown in Table 24, it can be seen that the state at \(6240.21\,\mathrm{MeV}\) couples mostly to \(\Sigma _b^*{\bar{K}}\) and \(\Xi _b^*\eta \), and only slightly to the rest of the channels. However, when we look at the magnitude of \(gG^{II}\), we can see that not only the channel \(\Sigma _b^*{\bar{K}}\) is significantly bigger than the others, also the only open channel \(\Xi _b^*\pi \) is considerably large compared to the other channels. Besides, the decay width of this particular pole to \(\Xi _b^*\pi \) is \(34.38\,\mathrm{MeV}\), which is similar to the value of the width for the pole at 6220.30 MeV decaying to \(\Xi _b^\prime \pi \), in Table 21, because they have almost the same phase space for decay.

## 4 Comparison of the results with other approaches

There are other approaches to excited baryon states which we briefly discuss here in connection to the present work.

In QCD sum rules one introduces a correlation function by means of an integral of the expectation value of some interpolating currents that have the quantum numbers of the hadrons that one wishes to study. The matrix elements are evaluated with the operator product expansion and with a phenomenological model. Then a Borel transformation is used to improve the convergence of the correlation function and the matching between the two ways of evaluating the correlation function. The matrix elements are then related to quark and gluon condensates which are calculated with the help of QCD and empirical information. The uncertainties in the masses of the hadrons obtained are very large. The calculation of the widths is more complicated, since it requires the evaluation of a three point correlation function, and the uncertainty obtained is, in general, large.

In this framework there is work done in [89] about P-wave bottom baryons using the method of QCD sum rule and heavy quark effective theory. The authors obtain several \(\Xi _b\) states of \(1/2^-\), \(3/2^-\), depending on the configuration assumed, with masses around 6200 MeV and errors about 130 MeV. The states correspond to angular excitations of the ground state \(\Xi _b\). Similarly in [38] P-wave charmed baryons are studied using the method of light-cone QCD sum rules, paying particular attention to the widths. Some angular excited states are obtained that could correspond to \(\Xi _c(2930)\), \(\Xi _c(2980)\) and the widths have large uncertainties, depending strongly on the configuration assumed. Again along the same line, there is work in [90] concerning the \(\Xi _c(2930)\). A state around this mass can be obtained, both assuming a radial or an angular excitation of the ground state. However, when the width is contrasted with experiment, the angular excitation leading to \(1/2^-\) state is preferred.

Concerning quark models there is work with the heavy-quark–light-diquark picture in the framework of the QCD-motivated relativistic quark model [4] and nonrelativistic quark models [91, 92, 93, 94, 95] dealing with these states. In all of them, states of \(1/2^-\) and \(3/2^-\) for \(\Xi _c\) are obtained around \(2800\,\mathrm{MeV}\), and around \(6100\,\mathrm{MeV}\) for \(\Xi _b\) (see also [96] concerning the \(\Xi _b(6227)\) state). A discussion of these works is done in [31, 97], where strong and radiative decays of these states are studied. Once again, the values of the widths are strongly dependent on the configurations assumed for the excitation, \(\lambda \)-mode or \(\rho \)-mode and spin assigned. Other scenarios within the quark model and D-wave excitations, leading to positive parity states, are discussed in [98].

We should also note that in our approach, using the s-wave meson baryon interaction as input and the Bethe–Salpeter equation in coupled channels, we only generate ground state molecules with \(1/2^-\), \(3/2^-\). Other quantum numbers cannot be reached with our approach. Excited molecular states can be obtained with more general interactions containing higher partial waves. One example is shown in [22, 99] for the \({\bar{D}}^*\Sigma _c\) and related channels trying to match the pentaquarks signals of [100]. But usually the higher partial waves of the meson–baryon potentials are too weak to bind.

Pentaquarks have also been used to describe excited baryon states [101, 102, 103, 104], mostly stimulated by the LHCb findings of \(P_c(4380)\), \(P_c(4450)\) states [100]. Reviews on this issue can be seen in [30, 105], however, no results for the \(\Xi _c\), \(\Xi _b\) excited states have been reported.

In contrast to the former approaches, our molecular picture has less flexibility. We can change the cut off in the regularization procedure, but this is tuned to some data, for instance one mass. Once this is done, the freedom is lost and one obtains both masses and widths for other states. We should note that since we work in coupled channels, the decay widths into the open channels are automatically implemented and there is no freedom concerning them.

Given the variety of models available to describe excited baryon states, and in particular those studied here, we think that the spectroscopy of these states, determining their masses and widths, and the devoted study of particular decays channels should shed light on the nature of these resonances. One extra way to gain information on these states is to study their production in reactions and decays of other particles. A review on these issues can be found in [106].

## 5 Summary and discussion

Motivated by the experimental findings of \(\Xi _c\) and \(\Xi _b\) states, we use the Bethe–Salpeter coupled channel formalism to study the \(\Xi _{c(b)}\) states dynamically generated from the meson–baryon interaction, considering three types of interactions (*PB*, *VB* and \(PB^*\)), for both the charm and beauty sectors. We search for the pole with different cutoffs in the second Riemann sheet once the scattering matrix is evaluated. Apart from that, the couplings of the poles to various channels are also calculated. With that, we are able the assess the strength at the origin of the wave function and further evaluate the decay widths to the open channels.

The only free parameter in our study is the loop regulator in the meson–baryon loop function, where we employ the cutoff regularization scheme, and we have taken different values for the cutoff in the charm and beauty sectors.

We obtain multiple states for \(\Xi _c\), with some of them agreeing significantly well with the experimental data. For example, the lowest state we observe in the charm sector is the state at \(2791.30\,\mathrm{MeV}\) (with the width \(7.26\,\mathrm{MeV}\)) generated from the pseudoscalar-baryon interaction, which have the same \(J^P\) quantum numbers as the state \(\Xi _c(2790)\) (with width \(8.9\pm 0.6\pm 0.8\,\mathrm{MeV}\)). It can also be seen that there is a very good agreement in their masses and widths. On top of that, we also obtain states at \(2937.15\,\mathrm{MeV}\) (\(2912.78\,\mathrm{MeV}\)), \(2973.76\,\mathrm{MeV}\), \(3068.21\,\mathrm{MeV}\) (\(3015.18\,\mathrm{MeV}\)) and \(3109.04\,\mathrm{MeV}\) (the numbers in the brackets implying the second option), which can be associated to the experimentally observed states \(\Xi _c(2930)\), \(\Xi _c(2970)\), \(\Xi _c(3055)\) and \(\Xi _c(3080)\), respectively. On the other hand, we found two poles, at \(6220.30\,\mathrm{MeV}\) (with a width \(25.20\,\mathrm{MeV}\)) and \(6240.21\,\mathrm{MeV}\) (with width \(29.30\,\mathrm{MeV}\)) in the \(1/2^-\) and \(3/2^-\) sectors, respectively. We can see that both their masses and widths agree well with the recently observed state \(\Xi _b(6227)\) with a width \(18.1\pm 5.4\pm 1.8\,\mathrm{MeV}\).

Overall, the states obtained in this work agree well with some of the already observed states in both the charm and beauty sectors, and it would be interesting to see if further measurements of spin and parity of these states would also agree with our predicted states. Furthermore, with the increased luminosity in future runs, the comparisons of the predictions made here and the experimental measurements will shed light on the nature of these hadrons.

## Footnotes

## Notes

### Acknowledgements

Q. X. Yu acknowledges the support from the National Natural Science Foundation of China (Grant No. 11775024 and 11575023). R. P. and V. R. D. wish to acknowledge the Generalitat Valenciana in the program Santiago Grisolia. This work is partly supported by the Spanish Ministerio de Economia y Competitividad and European FEDER funds under Contracts No. FIS2017-84038-C2-1-P B and No. FIS2017-84038-C2-2-P B, and the Generalitat Valenciana in the program Prometeo II-2014/068, and the project Severo Ochoa of IFIC, SEV-2014-0398.

## References

- 1.S. Capstick, N. Isgur, Phys. Rev. D
**34**, 2809 (1986)ADSCrossRefGoogle Scholar - 2.D. Ebert, R.N. Faustov, V.O. Galkin, Phys. Lett. B
**659**, 612 (2008)ADSCrossRefGoogle Scholar - 3.H. Garcilazo, J. Vijande, A. Valcarce, J. Phys. G
**34**, 961 (2007)CrossRefGoogle Scholar - 4.D. Ebert, R.N. Faustov, V.O. Galkin, Phys. Rev. D
**84**, 014025 (2011)ADSCrossRefGoogle Scholar - 5.P.G. Ortega, D.R. Entem, F. Fernandez, Phys. Lett. B
**718**, 1381 (2013)ADSCrossRefGoogle Scholar - 6.Z. Shah, K. Thakkar, A.K. Rai, P.C. Vinodkumar, Chin. Phys. C
**40**, 123102 (2016)ADSCrossRefGoogle Scholar - 7.E. Santopinto, A. Giachino, J. Ferretti, H. Garca-Tecocoatzi, M.A. Bedolla, M.R. Bijker, E. Ortiz-Pacheco, arXiv:1811.01799 [hep-ph]
- 8.A. Valcarce, H. Garcilazo, J. Vijande, Eur. Phys. J. A
**37**, 217 (2008)ADSCrossRefGoogle Scholar - 9.K.C. Bowler et al., [UKQCD Collaboration], Phys. Rev. D
**54**, 3619 (1996)Google Scholar - 10.T. Burch, C. Hagen, C.B. Lang, M. Limmer, A. Schafer, Phys. Rev. D
**79**, 014504 (2009)ADSCrossRefGoogle Scholar - 11.Z.S. Brown, W. Detmold, S. Meinel, K. Orginos, Phys. Rev. D
**90**, 094507 (2014)ADSCrossRefGoogle Scholar - 12.C. Garcia-Recio, J. Nieves, O. Romanets, L.L. Salcedo, L. Tolos, Phys. Rev. D
**87**, 034032 (2013)ADSCrossRefGoogle Scholar - 13.W.H. Liang, C.W. Xiao, E. Oset, Phys. Rev. D
**89**, 054023 (2014)ADSCrossRefGoogle Scholar - 14.W.H. Liang, T. Uchino, C.W. Xiao, E. Oset, Eur. Phys. J. A
**51**, 16 (2015)ADSCrossRefGoogle Scholar - 15.O. Romanets, L. Tolos, C. Garcia-Recio, J. Nieves, L.L. Salcedo, R.G.E. Timmermans, Phys. Rev. D
**85**, 114032 (2012)ADSCrossRefGoogle Scholar - 16.M.Z. Liu, T.W. Wu, J.J. Xie, M. Pavon Valderrama, L.S. Geng, Phys. Rev. D
**98**, 014014 (2018)ADSCrossRefGoogle Scholar - 17.Y. Huang, Cj Xiao, Q.F. Lü, R. Wang, J. He, L. Geng, Phys. Rev. D
**97**, 094013 (2018)ADSCrossRefGoogle Scholar - 18.R. Chen, A. Hosaka, X. Liu, Phys. Rev. D
**97**, 036016 (2018)ADSCrossRefGoogle Scholar - 19.Y. Dong, A. Faessler, T. Gutsche, S. Kumano, V.E. Lyubovitskij, Phys. Rev. D
**82**, 034035 (2010)ADSCrossRefGoogle Scholar - 20.Y. Dong, A. Faessler, T. Gutsche, V.E. Lyubovitskij, Phys. Rev. D
**90**, 094001 (2014)ADSCrossRefGoogle Scholar - 21.P.G. Ortega, D.R. Entem, F. Fernandez, Phys. Rev. D
**90**, 114013 (2014)ADSCrossRefGoogle Scholar - 22.J. He, Phys. Lett. B
**753**, 547 (2016)ADSCrossRefGoogle Scholar - 23.R. Chen, A. Hosaka, X. Liu, Phys. Rev. D
**96**, 116012 (2017)ADSCrossRefGoogle Scholar - 24.M.Z. Liu, F.Z. Peng, M. Sanchez Sanchez, M.P. Valderrama, Phys. Rev. D
**98**, 114030 (2018)ADSCrossRefGoogle Scholar - 25.J.G. Korner, M. Kramer, D. Pirjol, Prog. Part. Nucl. Phys.
**33**, 787 (1994)ADSCrossRefGoogle Scholar - 26.S. Bianco, F.L. Fabbri, D. Benson, I. Bigi, Riv. Nuovo Cim
**26N7**, 1 (2003)Google Scholar - 27.E. Klempt, J.M. Richard, Rev. Mod. Phys.
**82**, 1095 (2010)ADSCrossRefGoogle Scholar - 28.V. Crede, W. Roberts, Rept. Prog. Phys.
**76**, 076301 (2013)ADSCrossRefGoogle Scholar - 29.H.Y. Cheng, Front. Phys. (Beijing)
**10**, 101406 (2015)CrossRefGoogle Scholar - 30.H.X. Chen, W. Chen, X. Liu, S.L. Zhu, Phys. Rept.
**639**, 1 (2016)ADSCrossRefGoogle Scholar - 31.H.X. Chen, W. Chen, X. Liu, Y.R. Liu, S.L. Zhu, Rept. Prog. Phys.
**80**, 076201 (2017)ADSCrossRefGoogle Scholar - 32.F.K. Guo, C. Hanhart, U.G. Meißner, Q. Wang, Q. Zhao, B.S. Zou, Rev. Mod. Phys.
**90**, 015004 (2018)ADSCrossRefGoogle Scholar - 33.M. Tanabashi et al. [Particle Data Group], Phys. Rev. D
**98**, 030001 (2018)Google Scholar - 34.B. Aubert et al. [BaBar Collaboration], Phys. Rev. D
**77**, 031101 (2008)Google Scholar - 35.
- 36.Y.B. et al. [Belle Collaboration], Eur. Phys. J. C
**78**, 928 (2018)Google Scholar - 37.R. Aaij et al. [LHCb Collaboration], Phys. Rev. Lett.
**121**, 072002 (2018)Google Scholar - 38.H.X. Chen, Q. Mao, W. Chen, A. Hosaka, X. Liu, S.L. Zhu, Phys. Rev. D
**95**, 094008 (2017)ADSCrossRefGoogle Scholar - 39.H.X. Chen, Q. Mao, A. Hosaka, X. Liu, S.L. Zhu, Phys. Rev. D
**94**, 114016 (2016)ADSCrossRefGoogle Scholar - 40.K. Azizi, Y. Sarac, H. Sundu, Phys. Rev. D
**98**, 054002 (2018)ADSCrossRefGoogle Scholar - 41.Z.G. Wang, J.X. Zhang, Eur. Phys. J. C
**78**, 503 (2018)ADSCrossRefGoogle Scholar - 42.R. Aaij et al. [LHCb Collaboration], Phys. Rev. Lett.
**118**, 182001 (2017)Google Scholar - 43.J. Yelton et al., [Belle Collaboration], Phys. Rev. D
**97**, 051102 (2018)Google Scholar - 44.G. Montana, A. Feijoo, A. Ramos, Eur. Phys. J. A
**54**, 64 (2018)ADSCrossRefGoogle Scholar - 45.V.R. Debastiani, J.M. Dias, W.H. Liang, E. Oset, Phys. Rev. D
**97**, 094035 (2018)ADSCrossRefGoogle Scholar - 46.M. Bando, T. Kugo, S. Uehara, K. Yamawaki, T. Yanagida, Phys. Rev. Lett.
**54**, 1215 (1985)ADSCrossRefGoogle Scholar - 47.M. Bando, T. Kugo, K. Yamawaki, Phys. Rept.
**164**, 217 (1988)ADSCrossRefGoogle Scholar - 48.U.G. Meissner, Phys. Rept.
**161**, 213 (1988)ADSCrossRefGoogle Scholar - 49.M. Harada, K. Yamawaki, Phys. Rept.
**381**, 1 (2003)ADSCrossRefGoogle Scholar - 50.H. Nagahiro, L. Roca, A. Hosaka, E. Oset, Phys. Rev. D
**79**, 014015 (2009)ADSCrossRefGoogle Scholar - 51.J. Nieves, R. Pavao, L. Tolos, Eur. Phys. J. C
**78**, 114 (2018)ADSCrossRefGoogle Scholar - 52.W.H. Liang, J.M. Dias, V.R. Debastiani, E. Oset, Nucl. Phys. B
**930**, 524 (2018)ADSCrossRefGoogle Scholar - 53.J.M. Dias, V.R. Debastiani, J.J. Xie, E. Oset, Phys. Rev. D
**98**, 094017 (2018)ADSCrossRefGoogle Scholar - 54.R. Aaij et al. [LHCb Collaboration], Phys. Rev. Lett.
**119**, 112001 (2017)Google Scholar - 55.
- 56.A. Ocherashvili et al. [SELEX Collaboration], Phys. Lett. B
**628**, 18 (2005)Google Scholar - 57.S.P. Ratti, Nucl. Phys. Proc. Suppl.
**115**, 33 (2003)ADSCrossRefGoogle Scholar - 58.B. Aubert et al., [BaBar Collaboration], Phys. Rev. D
**74**, 011103 (2006)Google Scholar - 59.
- 60.R. Aaij et al. [LHCb Collaboration], JHEP
**1312**, 090 (2013)Google Scholar - 61.C.W. Xiao, J. Nieves, E. Oset, Phys. Rev. D
**88**, 056012 (2013)ADSCrossRefGoogle Scholar - 62.S. Sakai, L. Roca, E. Oset, Phys. Rev. D
**96**, 054023 (2017)ADSCrossRefGoogle Scholar - 63.A.V. Manohar, M.B. Wise, Heavy quark physics. Camb. Monogr. Part. Phys. Nucl. Phys. Cosmol.
**10**, 1 (2000)Google Scholar - 64.F.E. Close,
*An Introduction to Quarks and Partons*(Academics Press, Cambridge, 1979)Google Scholar - 65.K. Miyahara, T. Hyodo, M. Oka, J. Nieves, E. Oset, Phys. Rev. C
**95**, 035212 (2017)ADSCrossRefGoogle Scholar - 66.R.P. Pavao, W.H. Liang, J. Nieves, E. Oset, Eur. Phys. J. C
**77**, 265 (2017)ADSCrossRefGoogle Scholar - 67.E. Oset, A. Ramos, Eur. Phys. J. A
**44**, 445 (2010)ADSCrossRefGoogle Scholar - 68.E.J. Garzon, J.J. Xie, E. Oset, Phys. Rev. C
**87**, 055204 (2013)ADSCrossRefGoogle Scholar - 69.G. Ecker, Prog. Part. Nucl. Phys.
**35**, 1 (1995)ADSCrossRefGoogle Scholar - 70.V. Bernard, N. Kaiser, U.G. Meißner, Int. J. Mod. Phys. E
**4**, 193 (1995)ADSCrossRefGoogle Scholar - 71.E. Oset, A. Ramos, C. Bennhold, Phys. Lett. B,
**527**, 12 (2002) 99Google Scholar - 72.T. Mizutani, A. Ramos, Phys. Rev. C
**74**, 065201 (2006)ADSCrossRefGoogle Scholar - 73.J.A. Oller, E. Oset, Phys. Rev. D
**60**, 074023 (1999)ADSCrossRefGoogle Scholar - 74.J.A. Oller, U.G. Meißner, Phys. Lett. B
**500**, 263 (2001)ADSCrossRefGoogle Scholar - 75.J.J. Wu, B.S. Zou, Phys. Lett. B
**709**, 70 (2012)ADSCrossRefGoogle Scholar - 76.A. Ozpineci, C.W. Xiao, E. Oset, Phys. Rev. D
**88**, 034018 (2013)ADSCrossRefGoogle Scholar - 77.J.X. Lu, Y. Zhou, H.X. Chen, J.J. Xie, L.S. Geng, Phys. Rev. D
**92**, 014036 (2015)ADSCrossRefGoogle Scholar - 78.M. Altenbuchinger, L.S. Geng, Phys. Rev. D
**89**, 054008 (2014)ADSCrossRefGoogle Scholar - 79.E. Oset, A. Ramos, Nucl. Phys. A
**635**, 99 (1998)ADSCrossRefGoogle Scholar - 80.E.J. Garzon, E. Oset, Eur. Phys. J. A
**48**, 5 (2012)ADSCrossRefGoogle Scholar - 81.D. Gamermann, J. Nieves, E. Oset, E. Ruiz Arriola, Phys. Rev. D
**81**, 014029 (2010)ADSCrossRefGoogle Scholar - 82.J. Yelton et al., [Belle Collaboration], Phys. Rev. D
**94**, 052011 (2016)Google Scholar - 83.S.E. Csorna, et al., [CLEO Collaboration], Phys. Rev. Lett.
**86**, 4243 (2001)Google Scholar - 84.J.P. Alexander et al. [CLEO Collaboration], Phys. Rev. Lett.
**83**, 3390 (1999)Google Scholar - 85.M. Bayar, R. Pavao, S. Sakai, E. Oset, Phys. Rev. C
**97**, 035203 (2018)ADSCrossRefGoogle Scholar - 86.Y. Kato et al. [Belle Collaboration], Phys. Rev. D
**94**, 032002 (2016)Google Scholar - 87.Y. Fan, J.Z. Li, C. Meng, K.T. Chao, Phys. Rev. D
**85**, 034032 (2012)ADSCrossRefGoogle Scholar - 88.Y. Huang, C.J. Xiao, L.S. Geng, J. He, Phys. Rev. D
**99**, 014008 (2019)ADSCrossRefGoogle Scholar - 89.Q. Mao, H.X. Chen, W. Chen, A. Hosaka, X. Liu, S.L. Zhu, Phys. Rev. D
**92**, 114007 (2015)ADSCrossRefGoogle Scholar - 90.T.M. Aliev, K. Azizi, H. Sundu, Eur. Phys. J. A
**54**, 159 (2018)ADSCrossRefGoogle Scholar - 91.T. Yoshida, E. Hiyama, A. Hosaka, M. Oka, K. Sadato, Phys. Rev. D
**92**, 114029 (2015)ADSCrossRefGoogle Scholar - 92.B. Chen, K.W. Wei, X. Liu, T. Matsuki, Eur. Phys. J. C
**77**, 154 (2017)ADSCrossRefGoogle Scholar - 93.W. Roberts, M. Pervin, Int. J. Mod. Phys. A
**23**, 2817 (2008)ADSCrossRefGoogle Scholar - 94.M. Karliner, J.L. Rosner, Phys. Rev. D
**98**, 074026 (2018)ADSCrossRefGoogle Scholar - 95.K.L. Wang, Q.F. L, X.H. Zhong, Phys. Rev. D
**99**, 014011 (2019)ADSCrossRefGoogle Scholar - 96.B. Chen, K.W. Wei, X. Liu, A. Zhang, Phys. Rev. D
**98**, 031502 (2018)ADSCrossRefGoogle Scholar - 97.K.L. Wang, Y.X. Yao, X.H. Zhong, Q. Zhao, Phys. Rev. D
**96**, 116016 (2017)ADSCrossRefGoogle Scholar - 98.Y. Kato, T. Iijima, arXiv:1810.03748 [hep-ex]
- 99.J. He, Phys. Rev. D
**95**, 074004 (2017)ADSCrossRefGoogle Scholar - 100.R. Aaij et al. [LHCb Collaboration], Phys. Rev. Lett.
**115**, 072001 (2015)Google Scholar - 101.R. Chen, X. Liu, S.L. Zhu, Nucl. Phys. A
**954**, 406 (2016)ADSCrossRefGoogle Scholar - 102.R.F. Lebed, Phys. Lett. B
**749**, 454 (2015)ADSCrossRefGoogle Scholar - 103.R. Zhu, C.F. Qiao, Phys. Lett. B
**756**, 259 (2016)ADSCrossRefGoogle Scholar - 104.R. Ghosh, A. Bhattacharya, B. Chakrabarti, Phys. Part. Nucl. Lett.
**14**, 550 (2017)CrossRefGoogle Scholar - 105.A. Esposito, A. Pilloni, A.D. Polosa, Phys. Rept.
**668**, 1 (2016)ADSCrossRefGoogle Scholar - 106.E. Oset, Int. J. Mod. Phys. E
**25**, 1630001 (2016)ADSCrossRefGoogle Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Funded by SCOAP^{3}