# Analysis of the structure of \(\Xi (1690)\) through its decays

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## Abstract

The mass and pole residue of the first orbitally and radially excited \( \Xi \) state as well as the ground state residue are calculated by means of the two-point QCD sum rules. Using the obtained results for the spectroscopic parameters, the strong coupling constants relevant to the decays \(\Xi (1690)\rightarrow \Sigma K\) and \(\Xi (1690) \rightarrow \Lambda K\) are calculated within the light-cone QCD sum rules and width of these decay channels are estimated. The obtained results for the mass of \( {\widetilde{\Xi }}\) and ratio of the \(Br(\widetilde{\Xi }\rightarrow \Sigma K)/Br(\widetilde{\Xi }\rightarrow \Lambda K)\), with \( \widetilde{\Xi } \) representing the orbitally excited state in \( \Xi \) channel, are in nice agreement with the experimental data of the Belle Collaboration. This allows us to conclude that the \(\Xi (1690)\) state, most probably, has negative parity.

## 1 Introduction

Understanding the spectrum of baryons and looking for new baryonic states constitute one of the main research directions in hadron physics. Impressive developments of experimental techniques allow discovery of many new hadrons. Despite these developments, the spectrum of \(\Xi \) baryon is still not well established. This is due to the absence of high intensity anti-kaon beams and small production rate of the \(\Xi \) resonances. At present time only the ground state octet and decuplet baryons as well as \(\Xi (1320)\) and \(\Xi (1530)\) baryons are well established. Up to present time the quantum numbers of \(\Xi (1690)\), \(\Xi (1820)\) and \(\Xi (1950)\) have not been determined. Theoretically, the spectrum of \(\Xi \) baryon, within different approaches, have been studied intensively (see [1, 2, 3, 4, 5, 6, 7, 8, 9] and references therein).

The main results of these studies are that different phenomenological models explain successfully the nature of \(\Xi (1320)\) and \(\Xi (1530)\) states. However, these approaches predict controversially results for other excitations of \(\Xi \) baryons. In [8] using the nonrelativistic quark model the mass of \(\Xi (1690)\) is calculated and it is obtained that it might be radial excitation of \(\Xi \) with \(J^P=\frac{1}{2}^+\). This result was then supported by the quark model calculations in [5]. However within the relativistic quark model in [9] it was established that the first radial excitation should have mass around \(1840~\mathrm {MeV}\). In [4] the authors suggested that the \(\Xi (1690)\) state might be orbital excitation of \(\Xi \) with \(J^P=\frac{1}{2}^-\). This point of view was supported by calculations performed within Skyrme model [2] and chiral quark model [7]. The controversy results suggests independent analysis for establishing the nature of \(\Xi (1690)\) state.

In the present study, within the light cone QCD sum rules, we estimate the widths of the \(\Xi \rightarrow \Lambda K\) and \(\Xi \rightarrow \Sigma K\) transitions. We suggest that \(\Xi (1690)\) state may be radial (\( \widetilde{\Xi } \)) or orbital (\(\Xi '\)) excitation of \(\Xi \) baryon. For establishing these decays we need to know the residue of \(\Xi (1690)\) as well as the strong coupling constants for these decays. For calculation of the mass and residue of the \(\Xi \) states as the main inputs of the calculations we employ the two point QCD sum rule method.

The paper is arranged as follows. In Sect. 2 the mass and residue of \(\Xi (1690)\) baryon within both scenarios, namely considering \(\Xi (1690)\) as the orbital and radial excitations of \(\Xi \) baryon, are calculated. In Sect. 3 we present the calculations of the strong coupling constants defining the \(\Xi (1690)\rightarrow \Sigma (\Lambda ) K\) transitions within both scenarios. By using the obtained results for the coupling constants we estimate the relevant decay widths and compare our predictions on decay widths with the existing experimental data in this section, as well. We reserve Sect. 1 for the concluding remarks and some lengthy expressions are moved to the Appendix.

## 2 Mass and pole residue of the first orbitally and radially excited \(\Xi \) state

For calculation of the widths of \(\Xi \rightarrow \Sigma K\) and \(\Xi \rightarrow \Lambda K\) decays we need to know the residues of \(\Xi \), \(\Sigma \) and \(\Lambda \) baryons. In present work we consider two possible scenarios about nature of the \(\Xi (1690)\): (a) it is represented as radial excitation of the ground state \(\Xi \). In other words it carries the same quantum numbers as the ground state \(\Xi \), i.e. \(J^P=\frac{1}{2}^+\). (b) The \(\Xi (1690)\) state is considered as first orbital excitation of the ground state \(\Xi \), i.e. it is negative parity baryon with \(J^P=\frac{1}{2}^-\). In the following we will try to answer the question that which scenario is realized in nature? To answer this question we will calculate the mass of \(\Xi (1690)\) state and decay width of the \(\Xi \rightarrow \Sigma K\) and \(\Xi \rightarrow \Lambda K\) transitions and then compare the ratio of these decays as well as the prediction on the mass with existing experimental data. Note that the BABAR Collaboration has measured the mass (\( m=1684.7\pm 1.3^{+2.2}_{-1.6} \)) MeV and width (\( \Gamma =8.1 ^{+3.9+1.0}_{-3.5-0.9}\)) MeV of \(\Xi (1690)\) [10, 11] and Belle Collaboration has measured the mass (\( m=1688\pm 2\)) MeV and width (\( \Gamma =11\pm 4\)) MeV of this state as well as the ratio \(\frac{B(\Xi (1690)^0\rightarrow K^-\Sigma ^+ )}{B(\Xi (1690)^0\rightarrow \bar{K}^0 \Lambda ^0 )}\). The experimental value for this ratio measured by Belle is \(0.50\pm 0.26\) [12].

*a*,

*b*,

*c*are the color indices and \(\beta \) is an arbitrary parameter with \(\beta =-1\) corresponding to the Ioffe current.

*C*is the charge conjugation operator.

*m*, \(\widetilde{m}\) and

*s*, \(\widetilde{s}\) are the masses and spins of the ground and first orbitally excited \(\Xi \) baryons, respectively. Here dots represent the contributions of higher states and continuum.

*I*from these representations to find the following sum rules:

Some input parameters used in the calculations

Parameters | Values |
---|---|

\(m_{s}\) | \(96^{+8}_{-4}~\mathrm {GeV}\) [19] |

\(m_{\Xi }\) | \((1314.86\pm 0.20)~\mathrm {MeV}\) [19] |

\(m_{\Sigma }\) | \((1189.37\pm 0.07)~\mathrm {MeV}\) [19] |

\(m_{\Lambda }\) | \((1115.683\pm 0.006)~\mathrm {MeV}\) [19] |

\(\lambda _{\Sigma }\) | \((0.014\pm 0.03)~\mathrm {GeV^3}\) [20] |

\(\lambda _{\Lambda }\) | \((0.013\pm 0.02)~\mathrm {GeV^3}\) [20] |

\(\langle \bar{u}u \rangle \) | \((-0.24\pm 0.01)^3~\mathrm {GeV}^3\) |

\(\langle \bar{s}s \rangle \) | \(0.8\cdot (-0.24\pm 0.01)^3~\mathrm {GeV}^3\) |

\(\langle \overline{u}g_s\sigma Gu\rangle \) | \(m_{0}^2\langle \bar{u}u\rangle \) |

\(\langle \overline{s}g_s\sigma Gs\rangle \) | \(m_{0}^2\langle \bar{s}s \rangle \) |

\(m_{0}^2 \) | \((0.8\pm 0.1)~\mathrm {GeV}^2\) |

\(\langle \frac{\alpha _sG^2}{\pi }\rangle \) | \((0.012\pm 0.004)~\mathrm {GeV} ^4 \) |

## 3 \({\widetilde{\Xi }}\) and \(\Xi '\) transitions To \(\Lambda K\) and \(\Sigma K\)

In present section we calculate the strong couplings \(g_{\widetilde{\Xi }\Sigma K}\), \(g_{\widetilde{\Xi }\Lambda K}\), \(g_{\Xi '\Sigma K}\) and \(g_{\Xi '\Lambda K}\) defining the \(\widetilde{\Xi }\rightarrow \Sigma K\), \(\widetilde{\Xi }\rightarrow \Lambda K\), \(\Xi ' \rightarrow \Sigma K\) and \(\Xi ' \rightarrow \Lambda K\) transitions.

*a*,

*b*,

*c*are color indices,

*C*is the charge conjugation operator and \(A_{1}^{1}=I\), \(A_{1}^{2}=A_{2}^{1}=\gamma _5\), \(A_{2}^{2}=\beta \). According to the method used, we again calculate the aforesaid correlation function in two representations: hadronic and QCD. Matching these two sides through a dispersion relation leads to the sum rules for the coupling constants under consideration.

The sum rule results for the masses and residues of the first orbitally and radially excited \(\Xi \) baryon as well as residue of the ground state

\(\Xi \) | \(\widetilde{\Xi }\) | \(\Xi '\) | |
---|---|---|---|

\(m~(\mathrm {MeV})\) | \(1685\pm 69\) | \(1685\pm 69\) | |

\(\lambda ~(\mathrm {GeV}^3)\) | \(0.047\pm 0.007\) | \( 0.019\pm 0.004\) | \( 0.055 \pm 0.010\) |

*q*are the momenta of the \(\Xi \), \(\Sigma \) baryons and

*K*meson, respectively. In this expression \( m_{\Sigma }\) is the mass of the \(\Sigma \) baryon. The dots in Eq. (14) stand for contributions of the higher resonances and continuum states.

The sum rules for the coupling constants for \(\Xi ' \rightarrow \Sigma K\) and \(\Xi ' \rightarrow \Lambda K\) transitions can be easily obtained from Eqs. (17) and (18), by replacing \(m_{\widetilde{\Xi }}\rightarrow - m_{\Xi ^{\prime }}\) and \(\lambda _{\widetilde{\Xi }}\rightarrow \lambda _{\Xi ^{\prime }}\).

*K*-meson distribution amplitudes (DAs), can be obtained by using Fierz rearrangement formula

*K*-meson and vacuum states, as well as ones generated by insertion of the gluon field strength tensor \(G_{\lambda \rho }(uv)\) from quark propagators, are determined in terms of the

*K*-meson DAs with definite twists. The DAs are main nonperturbative inputs of light cone QCD sum rules. The

*K*-meson distribution amplitudes are derived in [21, 22, 23] which will be used in our numerical analysis. All of these steps summarized above result in lengthy expression for the OPE side of correlation function. In order not to overwhelm the study with overlong mathematical expressions we prefer not to present them here. Apart from parameters in the distribution amplitudes, the sum rules for the couplings depend also on numerical values of the \(\Sigma \) and \(\Lambda \) baryon’s mass and pole residue, which are given in Table 1. Note that the working region of the Borel mass \(M^2\), threshold \(s_0\) and \(\beta \) parameters for calculations of the relevant couplings are chosen the same as in the residue and mass computations.

The sum rule results for the strong coupling constants and decay widths of the first orbitally and radially excited \(\Xi \) baryon

| \(\Gamma (\mathrm {MeV})\) | |
---|---|---|

\(\widetilde{\Xi } \rightarrow \Sigma K\) | \(1.35\pm 0.37\) | \(32.73\pm 9.16\) |

\(\Xi ^{\prime }\rightarrow \Sigma K\) | \(69.57\pm 19.48\) | \( 3.08\pm 0.86\) |

\(\widetilde{\Xi } \rightarrow \Lambda K\) | \(1.65\pm 0.48\) | \(65.13\pm 18.89\) |

\(\Xi ^{\prime }\rightarrow \Lambda K\) | \(8.41\pm 2.44\) | \(18.23\pm 5.29\) |

## Notes

### Acknowledgements

K. A. thanks Dogus University for the partial financial support through the grant BAP 2015-16-D1-B04.

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