Ξ (1690) as a $K-$ Σ molecular state Open Access

Abstract

1. Introduction

Investigating the internal structure of hadrons is one of the most important subjects in hadron physics. The motivation for the investigation is that we expect the existence of exotic hadrons, which are not able to be classified as |$q q q$| for baryons or |$q \bar {q}$| for mesons. Actually, the fundamental theory of strong interaction, QCD, does not prohibit such exotic systems as long as they are color singlet, and there are indeed several exotic hadron candidates which cannot fit into the classifications by the constituent quark models [1]. In order to clarify the internal structure of exotic hadron candidates and to discover genuine exotic hadrons, great efforts have been continuously made in both experimental and theoretical sides. In this context, it is very encouraging that charged quarkonium-like states and charmonium-pentaquark states were observed in the heavy quark sector by Belle [2] and by LHCb [3], respectively.

In this article we focus on the |$\Xi (1690)$| resonance and theoretically investigate its structure in terms of the |$\bar {K} \Sigma $| component. Historically this resonance was discovered as a threshold enhancement in both the neutral and charged |$\bar {K} \Sigma $| mass spectra in the |$K^{-} p \to \left (\bar {K} \Sigma \right ) K \pi $| reaction at |$4.2\,\hbox {GeV}/c$| [4]. Several experimental and theoretical studies have followed in, e.g., Refs. [5–11] and Refs.[12–20], respectively, and today the |$\Xi (1690)$| resonance is attributed a three-star status in the Particle Data Group table [1]. Its mass and width are |$1690 \pm 10\,\hbox {MeV}$| and |${<}30\,\hbox {MeV}$|⁠, respectively [1], but a relatively narrow width, e.g., |$10 \pm 6\,\hbox {MeV}$| [7] was reported as well. In addition, the small ratio of the |$\Xi (1690)$| branching fractions |$\Gamma (\pi \Xi ) / \Gamma \left (\bar {K} \Sigma \right ) < 0.09$| has been observed [1]. Its spin/parity has been expected to be |$J^{P} = 1/2^{-}$| from the beginning [4], and this was supported by a recent experiment [10]. Then, a difficulty emerges; assuming |$J^{P} = 1/2^{-}$|⁠, |$\Xi (1690)$| couples to the |$\pi \Xi $| channel in |$s$| wave, and hence |$\Xi (1690)$|⁠, as a |$q q q$| state, should inevitably decay to the |$\pi \Xi $| channel to some extent in a naïve quark model, which contradicts the above experimental results and implications. This implies that |$\Xi (1690)$| might have some nontrivial structure other than the usual |$q q q$| state.

In this study, in order to describe |$\Xi (1690)$|⁠, we perform a coupled-channel analysis of the |$s$|-wave |$\bar {K} \Sigma $|⁠, |$\bar {K} \Lambda $|⁠, |$\pi \Xi $|⁠, and |$\eta \Xi $| scatterings. For this purpose we employ the so-called chiral unitary approach [21–25], which is formulated with the meson–baryon coupled-channels scattering equation in an algebraic form based on the combination of the chiral perturbation theory and the unitarization of the scattering amplitude. One of the most remarkable properties of this approach is that a simple driving term, or interaction kernel, provided by the chiral Lagrangian with a small number of free parameters can reproduce experimental observables such as cross sections fairly well. The most important application of the chiral unitary approach is the description of the |$\Lambda (1405)$| resonance [26]. In this article we apply the chiral unitary approach to the strangeness |$S = -2$| sector, where |$\Xi ^{\ast }$| resonances exist. In the chiral unitary approach the |$S = -2$| sector was already studied in Refs. [15, 18, 27]. On the one hand, in Ref. [27] a |$\Xi ^{\ast }$| state below the |$\bar {K} \Lambda $| threshold was discussed and identified with |$\Xi (1620)$|⁠. On the other hand, in Refs. [15, 18] the authors obtained several |$\Xi ^{\ast }$| poles such as |$\Xi (1620)$|⁠, |$\Xi (1690)$|⁠, and |$\Xi (1950)$| together with many hadronic resonances in |$S = 0$| to |$-$|3. In particular, in Ref. [15] they found that in a flavor |$SU(3)$| symmetric world |$\Xi (1690)$| is a member of two degenerated octets, which also contain one of the two-|$\Lambda (1405)$| poles coming from the |$\bar {K} N$| bound state, |$N (1535)$|⁠, and so on. In this study we extend the analyses in Refs. [15, 27] by concentrating on the phenomena near the |$\bar {K} \Sigma $| threshold and on |$\Xi (1690)$|⁠. In the following we will show that the chiral unitary approach in |$S = -2$| can qualitatively reproduce the experimental data of the |$\bar {K}^{0} \Lambda $| and |$K^{-} \Sigma ^{+}$| mass spectra, dynamically generate a |$\Xi (1690)$| pole as an |$s$|-wave |$\bar {K} \Sigma $| molecular state near the |$\bar {K} \Sigma $| threshold, and naturally explain the decay properties of |$\Xi (1690)$|⁠.

2. Formulation

3. Numerical results

Now we solve the scattering equation to obtain the scattering amplitude |$T_{j k} (w)$| and show the numerical results. In the following we mainly consider the neutral charge system since the experimental data on both the |$\bar {K}^{0} \Lambda $| and |$K^{-} \Sigma ^{+}$| mass spectra are available [8].

As we have explained, we have four subtraction constants as the model parameters. First we fix them by using the so-called natural renormalization scheme [34], which can exclude explicit pole contributions from the loop functions. In the natural renormalization scheme, we introduce a certain energy |$w_{\rm m}$| at which we achieve the consistency of the low-energy theorem with respect to the spontaneous breaking of the chiral symmetry. Namely, since we take the interaction |$V$| as the leading-order term of the chiral perturbation theory, we require that the scattering amplitude |$T$| should coincide with the interaction |$V$| at certain “low” energy according to the low-energy theorem. We represent this energy as |$w_{\rm m}$|⁠: |$T_{j k} ( w_{\rm m} ) = V_{j k} ( w_{\rm m} )$| with |$G_{j} ( w_{\rm w} ) = 0$| in every channel |$j$|⁠. According to the discussion in Ref. [34], we fix this matching energy scale as the mass of the “target” baryon of the scatterings, i.e., |$w_{\rm m} = M_{\Lambda }$|⁠, |$M_{\Sigma }$|⁠, or |$M_{\Xi }$|⁠.¹ As a result, we obtain the subtraction constants in the second, third, and fourth columns of Table 2, which we refer to as the parameter sets |$\Lambda $|⁠, |$\Sigma $|⁠, and |$\Xi $|⁠, respectively.² With the parameter set |$\Lambda $|⁠, we find two resonance poles as |$\Xi ^{\ast }$| states with |$J^{P}\,{=}\,1/2^{-}$|⁠; each pole is in the second (unphysical) Riemann sheets of the open channels, whose thresholds are lower than |$\text {Re} (w_{\rm pole})$|⁠. One pole is located at |$1556.5 - 102.9 i \,\hbox {MeV}$|⁠, which corresponds to the pole studied in Refs. [15, 18, 27] as the |$\Xi (1620)$| resonance. We have found that the energy dependence of the Weinberg–Tomozawa interaction in the |$\pi \Xi $| channel is essential to the appearance of |$\Xi (1620)$|⁠. Actually, by taking into account only the |$\pi \Xi $| channel and switching off couplings to other channels, we obtain a resonance pole at a similar position in the |$\pi \Xi $| dynamics. The mechanism is the same as that of the broad |$\Lambda (1405)$| pole in the chiral unitary approach, to which the energy dependence of the Weinberg–Tomozawa interaction in the |$\pi \Sigma $| channel is essential. In addition to the |$\Xi (1620)$| pole, another pole appears at |$1682.6 - 0.8 i\,\hbox {MeV}$| just below the |$\bar {K} \Sigma $| threshold, whose properties are listed in the second column of Table 2. We expect that the latter pole can be identified with the |$\Xi (1690)$| resonance and originates from the |$\bar {K} \Sigma $| bound state. On the other hand, with the parameter sets |$\Sigma $| and |$\Xi $|⁠, we obtain no poles near the |$\bar {K} \Sigma $| threshold as |$\Xi (1690)$|⁠. However, we will not take these parameter sets seriously, since they give larger value of the ratio |$R$| with larger |$\chi ^{2}$| value introduced below and can be excluded by the experimental results.

Let us now concentrate on the |$\Xi ^{\ast }$| state near the |$\bar {K} \Sigma $| threshold in the parameter set |$\Lambda $|⁠. The structure of the |$\Xi ^{\ast }$| state can be investigated with the coupling constants and compositeness. Actually, from Table 2 the coupling constants and compositeness indicate a large |$\bar {K} \Sigma $| component in the |$\Xi ^{\ast }$| state. In particular, the |$\bar {K} \Sigma $| compositeness, |$X_{K^{-} \Sigma ^{+}} + X_{\bar {K}^{0} \Sigma ^{0}}$|⁠, dominates the sum rule (7) with its small imaginary part. This result strongly indicates that the |$\Xi ^{\ast }$| state is indeed a |$\bar {K} \Sigma $| molecular state, on the basis of the similarity to the bound state case; the wave function of the |$\Xi ^{\ast }$| state can be similar to that of a bound state dominated by the |$\bar {K} \Sigma $| channel. We also note that, although the coupling constant approximately satisfies the isospin relation |$g_{K^{-} \Sigma ^{+}} = - \sqrt {2} g_{\bar {K}^{0} \Sigma ^{0}}$|⁠, the |$\bar {K} \Sigma $| compositeness largely breaks the corresponding relation |$X_{K^{-} \Sigma ^{+}} = 2 X_{\bar {K}^{0} \Sigma ^{0}}$|⁠. This is because the |$\Xi ^{\ast }$| state is located very close to the |$K^{-} \Sigma ^{+}$| threshold. This fact will induce further effects of the isospin symmetry breaking on |$\Xi (1690)$|⁠, such as the difference of the |$K^{-} \Sigma ^{+}$| and |$\bar {K}^{0} \Sigma ^{0}$| mass spectra, due to the difference of their thresholds. The dominance of the |$K^{-} \Sigma ^{+}$| compositeness implies a coupled-channels extension of the near-threshold scaling in |$s$| wave discussed in Refs. [35, 36].

In addition to the |$\bar {K} \Sigma $| component, the |$\Xi ^{\ast }$| state has a remarkable property of its small decay width with the small imaginary part of the pole position |${\sim }1\,\hbox {MeV}$|⁠, which can also be seen in Refs. [15, 18]. This decay property can be understood by considering the structure of the coefficient |$C_{j k}$|⁠. Namely, as shown in Table 1, the transitions |$K^{-} \Sigma ^{+}$|⁠, |$\bar {K}^{0} \Sigma ^{0} \leftrightarrow \bar {K}^{0} \Lambda $| are forbidden at the leading order, so the decay of the |$\bar {K} \Sigma $| quasibound state to the |$\bar {K} \Lambda $| channel is highly suppressed. Actually, the decay to |$\bar {K} \Lambda $| takes place through a multiple scattering of |$\bar {K} \Sigma \to \eta \Xi \to \bar {K} \Lambda $|⁠, since the |$\bar {K} \Sigma $|-|$\eta \Xi $| coupling is the strongest among the coupled channels [27]. In addition, the |$\bar {K} \Sigma \leftrightarrow \pi \Xi $| transition is not strong compared to, e.g., the |$\bar {K} N \leftrightarrow \pi \Sigma $| one in the |$\Lambda (1405)$| case; the coefficient |$C_{\bar {K} \Sigma \, \pi \Xi (I=1/2)}$| in the isospin basis is |$- 1/2$| [27], while that of |$\bar {K} N \leftrightarrow \pi \Sigma $| in |$I = 0$| is |$- \sqrt {3/2}$| [22]. As a consequence, the |$\Xi ^{\ast }$| state as a |$\bar {K} \Sigma $| molecule cannot couple strongly to |$\bar {K} \Lambda $| or |$\pi \Xi $| as the decay channels and hence the decay width becomes very small. Moreover, the above argument can also explain the small ratio of the |$\Xi (1690)$| branching fractions |$\Gamma (\pi \Xi ) / \Gamma \left (\bar {K} \Sigma \right ) < 0.09$| [1]. Here we note that higher-order contributions to the interaction, such as the |$s$|- and |$u$|-channel Born terms, can bring tree-level couplings of |$\bar {K} \Sigma \leftrightarrow \bar {K} \Lambda $| and may give a decay width |${\lesssim }10 \,\hbox {MeV}$|⁠.

Fig. 1.

Mass spectra of |$\bar {K}^{0} \Lambda $| (left) and |$K^{-} \Sigma ^{+}$| (right) calculated with Eq. (8). The red solid lines are the theoretical mass spectra with the fitted parameters and |$C = 0.222\,\hbox {ns}^{-1}/\text {GeV}$|⁠, and the blue histograms correspond to the folded results with the size of experimental bins, |$5\,\hbox {MeV}$| and |$4\,\hbox {MeV}$| for |$\bar {K}^{0} \Lambda $| and |$K^{-} \Sigma ^{+}$|⁠, respectively. Experimental data are taken from Ref. [8], and their scale is fixed with the experimental branching fractions (see text).

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We note that the |$\Xi (1690)$| pole in the parameter set Fit is located in the first Riemann sheet of the |$K^{-} \Sigma ^{+}$|⁠, |$\bar {K}^{0} \Sigma ^{0}$|⁠, and |$\eta \Xi ^{0}$| channels and in the second Riemann sheet of the |$\bar {K}^{0} \Lambda $|⁠, |$\pi ^{+} \Xi ^{-}$|⁠, and |$\pi ^{0} \Xi ^{0}$| channels. Therefore, this pole exists above the |$K^{-} \Sigma ^{+}$| threshold (⁠|$1683.05\,\hbox {MeV}$|⁠) but in the first Riemann sheet of this channel, which is connected smoothly from the pole position of parameter set |$\Lambda $|⁠. In this meaning, strictly speaking, the peak seen in the |$\bar {K}^{0} \Lambda $| mass spectrum in Fig. 1 is a cusp at the |$K^{-} \Sigma ^{+}$| threshold rather than the usual Breit–Wigner resonance peak. Other properties of |$\Xi (1690)$| in the parameter set Fit are very similar to those in the parameter set |$\Lambda $|⁠. The |$\pi \Xi $| subtraction constant in the set Fit is larger than “natural” value |$\sim \!\!-2$| [23], but we do not take it seriously since the |$\pi \Xi $| channel contributes negligibly to the |$\Xi (1690)$| resonance. Actually, even changing the subtraction constant |$a_{\pi \Xi } = -0.75$| to |$-2$| in the set Fit, we obtain a similar |$\chi ^{2}$| value.

4. Discussion

Fig. 2.

Loop function |$G$| in the |$K^{-} \Sigma ^{+}$| channel and inverse of the interaction |$V$| in the |$\bar {K} \Sigma (I=1/2)$| channel [see Eq. (12)]. The |$K^{-} \Sigma ^{+}$| threshold is located at |$1683.05\,\hbox {MeV}$|⁠.

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Finally we consider the charged |$S = -2$| system with the channels |$\bar {K}^{0} \Sigma ^{-}$|⁠, |$K^{-} \Sigma ^{0}$|⁠, |$K^{-} \Lambda $|⁠, |$\pi ^{-} \Xi ^{0}$|⁠, |$\pi ^{0} \Xi ^{-}$|⁠, and |$\eta \Xi ^{-}$|⁠. Here we use the parameter set Fit given in Table 2 for the subtraction constants and solve the Bethe–Salpeter equation (1) to obtain the scattering amplitude. As a result, with the set Fit we find a pole near the |$\bar {K} \Sigma $| threshold, which corresponds to the |$\Xi (1690)^{-}$| resonance. The pole is located in the first Riemann sheet of the |$\bar {K}^{0} \Sigma ^{-}$| and |$\eta \Xi ^{-}$| channels and in the second Riemann sheet of the |$K^{-} \Lambda $|⁠, |$K^{-} \Sigma ^{0}$|⁠, |$\pi ^{-} \Xi ^{0}$|⁠, and |$\pi ^{0} \Xi ^{-}$| channels. The properties of |$\Xi (1690)^{-}$| are listed in Table 3. The pole position has a larger imaginary part |$\sim 10\,\hbox {MeV}$| compared to the neutral case, since it exists above the |$K^{-} \Sigma ^{0}$| threshold in its second Riemann sheet and hence the decay |$\Xi (1690)^{-} \to K^{-} \Sigma ^{0}$| is allowed. The coupling constants and compositeness indicate that |$\Xi (1690)^{-}$| has a large |$\bar {K} \Sigma $| component. However, each of |$\bar {K}^{0} \Sigma ^{-}$| and |$K^{-} \Sigma ^{0}$| compositeness has a nonnegligible imaginary part, because the pole exists above the |$K^{-} \Sigma ^{0}$| one in its second Riemann sheet. Nevertheless, the sum |$X_{\bar {K}^{0} \Sigma ^{-}} + X_{K^{-} \Sigma ^{0}}$| is the largest contribution to the sum rule (7) with its small imaginary part, which implies that the charged |$\Xi ^{\ast }$| state is also a |$\bar {K} \Sigma $| molecular state. Moreover, we can expect effects of the isospin symmetry breaking on the charged |$\Xi (1690)$| state in a similar manner to the neutral case due to the |$K^{-} \Sigma ^{0}$|–|$\bar {K}^{0} \Sigma ^{-}$| threshold difference.

5. Conclusion

In this study we have investigated dynamics of |$\bar {K} \Sigma $| and its coupled channels in the chiral unitary approach. It is a great advantage to employ the chiral unitary approach that a simple interaction kernel provided by the leading-order chiral perturbation theory does not contain free parameters and hence only the subtraction constants (or cut-offs) of the loop functions are the model parameters. The subtraction constants are fixed in the natural renormalization scheme, which can exclude explicit pole contributions from the loop functions, or by fitting the |$\bar {K}^{0} \Lambda $| and |$K^{-} \Sigma ^{+}$| mass spectra to the experimental data.

As a result, we have found that, although the |$\bar {K} \Sigma $| interaction from the leading-order chiral perturbation theory alone is slightly insufficient to bring a |$\bar {K} \Sigma $| bound state, multiple scatterings in a meson–baryon coupled-channels approach can dynamically generate a |$\bar {K} \Sigma $| quasibound state near the |$\bar {K} \Sigma $| threshold. The obtained scattering amplitude can qualitatively reproduce the experimental data of the |$\bar {K}^{0} \Lambda $| and |$K^{-} \Sigma ^{+}$| mass spectra and contains a |$\Xi ^{\ast }$| pole as a |$\bar {K} \Sigma $| molecule, which can be identified with the |$\Xi (1690)$| resonance. Due to the small or vanishing couplings of the |$\bar {K} \Sigma $| channel to others, we can naturally explain the decay properties of |$\Xi (1690)$|⁠. We have also pointed out a possibility to observe effects of the isospin symmetry breaking on |$\Xi (1690)$|⁠, such as the difference of the |$K^{-} \Sigma ^{+}$| and |$\bar {K}^{0} \Sigma ^{0}$| mass spectra, due to the difference of their thresholds when the |$\Xi (1690)$| pole exists very close to the one of the |$\bar {K} \Sigma $| thresholds.

Finally we suggest that further experimental studies on the |$\Xi (1690)$| resonance and related |$\bar {K} \Lambda $| and |$\bar {K} \Sigma $| mass spectra are most welcome, since these experiments can constrain more the |$\bar {K} \Sigma $| scattering amplitude and the structure of the |$\Xi (1690)$| resonance. In particular, future studies on multi-strangeness systems at J-PARC, JLab, and other facilities can shed light on the structure of the |$\Xi (1690)$| resonance. We also expect that high-statistics analyses on |$\Xi (1690)$| by Belle, BaBar, and LHCb are promising. On the other hand, for the theoretical support in analyzing the |$\Xi (1690)$| production, structure, and decay processes, we expect that we can utilize the same or a similar approach to the |$\Lambda (1405)$| case, which has been extensively studied in the chiral unitary approach as well as in many other models, effective theories, and lattice QCD simulations. This is because both are dynamically generated resonances in the meson–baryon degrees of freedom and, in particular, originate from the same flavor |$SU(3)$| multiplet in the chiral unitary approach.

Funding

Open Access funding: SCOAP³.

Acknowledgments

The author greatly acknowledges K. Imai for suggesting the author study this topic and for useful discussions. The author also thanks Y. Kato for useful comments on Belle data and J. Nieves on the |$\Xi ^{\ast }$| states in the chiral unitary approach. Stimulating discussions with A. Hosaka and his careful reading of the manuscript are gratefully acknowledged. This work is partly supported by the Grants-in-Aid for Young Scientists from JSPS (No. 15K17649) and for JSPS Fellows (No. 15J06538).

References