Prediction of possible exotic states in the ηˉKK system

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Xu Zhang and Ju-Jun Xie. Prediction of possible exotic states in the ηˉKK system[J]. Chinese Physics C. doi: 10.1088/1674-1137/44/5/054104
Xu Zhang and Ju-Jun Xie. Prediction of possible exotic states in the ηˉKK system[J]. Chinese Physics C.  doi: 10.1088/1674-1137/44/5/054104 shu
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Received: 2019-10-28
Revised: 2019-12-20
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Prediction of possible exotic states in the ηˉKK system

    Corresponding author: Xu Zhang, zhangxu@impcas.ac.cn
    Corresponding author: Ju-Jun Xie, xiejujun@impcas.ac.cn
  • 1. Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China
  • 2. School of Nuclear Sciences and Technology, University of Chinese Academy of Sciences, Beijing 101408, China
  • 3. School of Physics and Microelectronics, Zhengzhou University, Zhengzhou, Henan 450001, China

Abstract: We investigate the ηˉKK three-body system in order to look for possible IG(JPC)=0+(1+) exotic states in the framework of the fixed-center approximation of the Faddeev equation. We assume the scattering of η on a clusterized system ˉKK, which is known to generate f1(1285), or a ˉK in a clusterized system ηK, which is shown to generate K1(1270). In the case of η-(ˉKK)f1(1285) scattering, we find evidence of a bound state IG(JPC)=0+(1+) below the ηf1(1285) threshold with a mass of around 1700 MeV and a width of about 180 MeV. Considering ˉK-(ηK)K1(1270) scattering, we obtain a bound state I(JP)=0(1) just below the ˉKK1(1270) threshold with a mass of around 1680 MeV and a width of about 160 MeV.

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    1.   Introduction
    • Exotic states cannot be described by the traditional quark model and may have a more complex structure allowed by QCD such as glueballs, hybrid mesons and multiquark states. The discovery of exotic states and the study of their structure will extend our knowledge of the strong interaction dynamics [1-3].

      A meson with quantum numbers JPC=1+, which is excluded by the traditional quark model in the qˉq picture, is an exotic state [4]. Interestingly, three isovector JPC=1+ exotic candidates, π1(1400), π1(1600), and π1(2015), have been reported by experiments [5]. On the theoretical side, the isovector exotic states are interpreted as hybrid mesons in different theoretical approaches, such as the flux tube model [6-8], ADS/QCD model [9, 10], and Lattice QCD [11-13]. In addition, the hybrid meson decay properties were studied in the framework of the QCD sum rules in Refs. [14-17]. Some studies suggest that the isovector exotic state might be a four-quark state [18] or a molecule/four-quark mixing state [19]. On the other hand, the three-body system can also carry the quantum numbers JPC=1+. In Ref. [20], by retaining the strong interactions of ˉKK which generate the f1(1285) resonance [21, 22], the πˉKK three-body system was investigated in the framework of the fixed-center approximation (FCA) of the Faddeev equation, where π1(1600) could be interpreted as a dynamically generated state in the π-(ˉKK)f1(1285) system.

      In principle, an isoscalar exotic state is also possible, although it has not been experimentally observed [8, 13]. In fact, these isoscalar exotic states were studied with the QCD sum rules using the tetraquark currents [23], where the obtained mass is around 1.82.1 GeV, and the decay width is about 150 MeV.

      In this paper, we study the ηˉKK three-body system in order to look for possible IG(JPC)=0+(1+) exotic states in the FCA approach, which has been used to investigate the interaction of Kd at the threshold [24-26]. A possible state in the three-body system Kpp, according to calculations performed with the FCA approach [27, 28], is supported by the J-PARC experiments [29]. In Ref. [30], the Δ5/2+(2000) puzzle is solved in a study of the π-(Δρ) interaction. In Ref. [31], a peak is found around 1920 MeV, indicating that the NKˉK state with I=1/2 could exists around that energy, which supports the existence of the N resonance with JP=1/2+ around 1920 MeV obtained in Refs. [32-35], where the full Faddeev calculations were performed. Recently, predictions of several heavy flavor resonance states in a three-body system have been reported in the framework of the FCA approach, for example ˉK()B()ˉB() [36], D()B()ˉB() [37], ρBˉB [38], ρDˉD [39, 40], DKK (DKˉK) [41], BDD(BDˉD) [42], and KˉDD [43]. The DDK system was investigated in Ref. [44] with coupled channels by solving the Faddeev equation using the two-body input, and it was found that an isospin 1/2 state is formed at 4140 MeV when Ds0(2317) is formed in the DK subsystem. This result is compatible with Ref. [45], where the system D-Ds0(2317) was studied without the explicit three-body dynamics. In a more recent work [46], where the Gaussian expansion method was used, the existence of DDK states was further confirmed. The above examples show that the results of FCA prove to be reasonable. However, as important as it may be to understand the success of FCA, there are problems in the FCA calculations of the ϕKˉK system [47] (more details about the limits of FCA can also be found in this reference), in which ϕ(2175) can be reproduced by the full Faddeev calculations [48].

      There are two possible scattering cases in the ηˉKK three-body system since the ˉKK and ηK systems lead to the formation of two dynamically generated resonances, f1(1285) and K1(1270). Based on the two-body ηˉK, ηK and ˉKK scattering amplitudes obtained from the chiral unitary approach [21, 49, 50], we perform an analysis of the η-(ˉKK)f1(1285) and ˉK-(ηK)K1(1270) scattering amplitudes, which allows to predict the possible exotic states with quantum numbers IG(JPC)=0+(1+).

      The paper is organized as follows. In Sec. 2, we present the FCA formalism and ingredients to analyze the η-(ˉKK)f1(1285) and ˉK-(ηK)K1(1270) systems. In Sec. 3, numerical results and a discussion are presented. Finally, a short summary is given in Sec. 4.

    2.   Formalism and ingredients

      2.1.   Fixed-center approximation formalism

    • In the framework of FCA, we consider ˉKK(ηK) as a cluster, and η(ˉK) interacts with the components of the cluster. The total three-body scattering amplitude T can be simplified as the sum of two partition functions T1 and T2, by summing all diagrams in Fig. 1, starting with the interaction of particle 3 with particle 1(2) of the cluster. The FCA equations can be written in terms of T1 and T2, which give the total scattering amplitude T, and read [25, 26, 51]

      Figure 1.  Diagrammatic representation of FCA for the Faddeev equations.

      T1=t1+t1G0T2,

      (1)

      T2=t2+t2G0T1,

      (2)

      T=T1+T2,

      (3)

      where the amplitudes t1 and t2 represent the unitary scattering amplitudes with coupled channels for the interactions of particle 3 with particle 1 and 2, respectively. The function G0 in the above equations is the propagator for particle 3 between the particle 1 and 2 components of the cluster, which we discuss below.

      We calculate the total scattering amplitude T in the low energy regime, close to the threshold of the ηˉKK system or below, where FCA is a good approximation. The on-shell approximation for the three particles is also used.

      Following the field normalization of Refs. [52, 53], we can write the S matrix for the single scattering term [Fig. 1(a) and 1(e)] as

      S(1)=S(1)1+S(1)2=(2π)4V2δ4(k3+kclsk3kcls)×12w312w3(it12w112w1+it22w212w2),

      (4)

      where V stands for the volume of a box in which the states are normalized to unity, while the momentum k(k) and the on-shell energy w(w) refer to the initial (final) particles.

      The double scattering contributions are obtained from Fig. 1(b) and 1(f). The expression for the S matrix for double scattering [S(2)2=S(2)1] is given by

      S(2)=it1t2(2π)4V2δ4(k3+kclsk3kcls)×12w312w312w112w112w212w2×d3q(2π)3Fcls(q)1q02|q|2m23+iϵ,

      (5)

      where Fcls(q) is the form factor of the cluster which is a bound state of particles 1 and 2. The information about the bound state is encoded in the form factor Fcls(q) in Eq. (5), which is the Fourier transform of the cluster wave function. The variable q0 is the energy carried by particle 3 in the center-of-mass frame of particle 3 and the cluster, and is given by

      q0(s)=s+m23m2cls2s,

      (6)

      where s is the invariant mass squared of the ηˉKK system.

      For the form factor Fcls(q), we take the following expression for s wave bound states only, as discussed in Refs. [52-54]:

      Fcls(q)=1N|p|<Λ,|pq|<Λd3p12w1(p)12w2(p)×1mclsw1(p)w2(p)12w1(pq)12w2(pq)×1mclsw1(pq)w2(pq),

      (7)

      where the normalization factor N is

      N=|p|<Λd3p(12w1(p)12w2(p)1mclsw1(p)w2(p))2,

      with mcls the mass of the cluster. Note that the width of K should also be included in Fcls(q) [30]. However, as shown below, the masses of f1(1285) and K1(1270) are below the threshold of ˉKK and ηK, and the effect of the width of K is small and can be neglected.

      Similarly, the full S matrix for the scattering of particle 3 on the cluster is given by

      S=iT(2π)4V2δ4(k3+kclsk3kcls)×12w312w312wcls12wcls.

      (8)

      By comparing Eqs. (4), (5), and (8), we see that it is necessary to introduce a weight in t1 and t2 so that Eqs. (4) and (5) include the factors that appear in Eq. (8). This is achieved by,

      ˜t1=t12wcls2w12wcls2w1,˜t2=t22wcls2w22wcls2w2.

      Eq. (3) can then be solved to give

      T=˜t1+˜t2+2˜t1˜t2G01˜t1˜t2G20,

      (9)

      where G0 depends on the invariant mass of the ηˉKK system, and is given by

      G0(s)=12mclsd3q(2π)3Fcls(q)q02|q|2m23+iϵ.

      (10)
    • 2.2.   Single scattering contribution

    • It is worth noting that the argument of the total scattering amplitude T can be regarded as a function of the total invariant mass s of the three-body system, while the arguments of two-body scattering amplitudes t1 and t2 depend on the two-body invariant masses s1 and s2. s1 and s2 are the invariant masses squared of the external particle 3 with momentum k3, and particle 1 (2) inside the cluster with momentum k1 (k2), which are given by

      s1=m23+m21+(sm23m2cls)(m2cls+m21m22)2m2cls,s2=m23+m22+(sm23m2cls)(m2cls+m22m21)2m2cls,

      where ml (l=1,2,3) are the masses of the corresponding particles in the three-body system.

      It is worth mentioning that in order to evaluate the two-body scattering amplitudes t1 and t2, the isospin of the cluster should be considered. For the case of the η-(ˉKK)f1(1285) system, the cluster ˉKK has isospin IˉKK=0. Therefore, we have

      |ˉKKI=0=12|(12,12)12|(12,12),

      (11)

      where the kets on the right-hand side indicate the Iz components of particles ˉK and K, |(IˉKz,IKz). For the case of the total isospin Iη(ˉKK)=0, the single scattering amplitude is written as [20]

      η(ˉKK)|t|η(ˉKK)=(00|12((12,12)|(12,12)|))(t31+t32)(|0012(|(12,12)|(12,12)))=(12(1212,12)|12(1212,12)|)t31(12|(1212,12)12|(1212,12))+(12(1212,12)|12(1212,12)|)t32(12|(1212,12)12|(1212,12)),

      (12)

      where the notation for the states in the last term is |(IηˉKIzηˉK,IzK) for t31 and |(IηKIzηK,IzˉK) for t32. This leads to the following amplitudes for single scattering [Fig. 1(a) and 1(e)] in the η-(ˉKK)f1(1285) system,

      t1=tI=1/2ηˉKηˉK,t2=tI=1/2ηKηK.

      (13)

      In the ˉK-(ηK)K1(1270) system, the cluster ηK can only have isospin IηK=1/2. Therefore, for the total isospin IˉK(ηK)=0, the scattering amplitude is written as [20]

      ˉK(ηK)|t|ˉK(ηK)=12(1212|(12,12)|1212|(12,12)|)(t31+t32)12(|1212|(12,12)|1212|(12,12))=(12(1212,12)|12(1212,12)|)t31(12|(1212,12)12|(1212,12))+12(12(00,0)+12(00,0)|)t3212(12|(00,0)+12|(00,0)).

      (14)

      This leads to the following amplitudes for single scattering in the ˉK-(ηK)K1(1270) system,

      t1=tI=1/2ˉKηˉKη,t2=tI=0ˉKKˉKK.

      (15)

      We see that only the transition ˉKKˉKK with I=0 gives a contribution, since the total isospin IˉK(ηK)=0 and the η meson has isospin zero.

    • 2.3.   Unitarized ηK and ˉKK interactions

    • An important ingredient in the calculations of the total scattering amplitude for the ηˉKK system using FCA are the two-body ηK, ηK, and ˉKK unitarized s wave interactions from the chiral unitary approach. These two-body scattering amplitudes are studied with the dimensional regularization procedure, and they depend on the subtraction constants aηK, aηK and aˉKK, and also on the regularization scale μ. Note that there is only one parameter for the dimensional regularization procedure, since any change in μ is reabsorbed by the change in a(μ) through a(μ)a(μ)=ln(μ2μ2), so that the scattering amplitude is scale independent. In this work, we use the parameters from Refs. [21, 49, 50]: aηK=1.38 and μ=mK for IηK=1/2; aηK=1.85 and μ=1000 MeV for IηK=1/2; aˉKK=1.85 and μ=1000 MeV for IˉKK=0. With these parameters, we get the masses of f1(1285) and K1(1270) at their estimated values.

      In Figs. 2(a) and (b), we show the numerical results for |tI=0ˉKKˉKK|2 and |tI=1/2ηKηK|2, respectively, where we see clear peaks for the f1(1285) and K1(1270) states.

      Figure 2.  (a) Modulus squared of tI=0ˉKKˉKK as a function of the invariant mass MˉKK of the ˉKK subsystem. (b) Modulus squared of tI=1/2ηKηK as a function of the invariant mass MηK of the ηK subsystem.

    • 2.4.   Form factors Fcls(q) and propagator G0(s)

    • To connect with the dimensional regularization procedure, we choose the cutoff Λ such that the two-body loop function at the threshold coincides in both methods. Thus, we take Λ=990 MeV such that f1(1285) is as obtained in Refs. [55, 56], while for K1(1270), we take Λ=1000 MeV. The cutoff is tuned to get a pole at 1288i74 for the K1(1270) state.

      In Figs. 3 and 4, we show the respective form-factors for f1(1285) and K1(1270), where we take mcls=1281.3 MeV for f1(1285) and 1284 MeV for K1(1270), as obtained in Ref. [49]. In FCA, we keep the wave function of the cluster unchanged in the presence of the third particle. In order to estimate the uncertainties of FCA due to this "frozen" condition, we admit that the wave function of the cluster could be modified by the presence of the third particle. To do so, we perform calculations with different cutoffs. The results, shown in Figs. 3 and 4, are obtained with Λ=890, 990 and 1090 MeV for f1(1285), while for K1(1270) we take Λ=900, 1000 and 1100 MeV.

      Figure 3.  Forms-factor Eq. (7) as a function of q=|q| for the cutoff Λ=890 (dashed), 990 (solid), and 1090 MeV (dotted) for f1(1285) as the ˉKK bound state.

      Figure 4.  As in Fig. 3, but for K1(1270) as the ηK bound state. The dashed, solid and dotted curves are for Λ=900, 1000, and 1100 MeV, respectively.

      In Fig. 5, we show the real (solid line) and imaginary (dashed line) parts of G0 as a function of the invariant mass of the η-(ˉKK)f1(1285) system for Λ=890, 990 and 1090 MeV.

      Figure 5.  (color online) Real (solid line) and imaginary (dashed line) parts of G0 for the η-(ˉKK)f1(1285) system and Λ=890 (blue), 990 (red) and 1090 MeV (green).

      The results for G0 of the ˉK-(ηK)K1(1270) system are shown in Fig. 6, where the real (solid line) and imaginary (dashed line) parts are for Λ=900, 1000 and 1100 MeV.

      From Figs. 5 and 6, it can be seen that the imaginary part of G0(s) is not sensitive to the value of the cutoff, while the real part slightly changes with the cutoff.

      Figure 6.  (color online) Real (solid line) and imaginary (dashed line) parts of G0 for the ˉK-(ηK)K1(1270) system and Λ=900 (blue), 1000 (red) and 1100 MeV (green).

    3.   Numerical results and discussion
    • For the numerical evaluation of the three-body amplitude, we need the two-body interaction amplitudes of ηˉK, ηK, and ˉKK, which were investigated in the chiral dynamics and unitary coupled channels approach in Refs. [21, 49, 50]. The total scattering amplitude T can then be calculated, and the peaks or bumps in the modulus squared |T|2 associated to resonances.

      In Fig. 7, we show the modulus squared |T|2 for the η-(ˉKK)f1(1285) scattering with the total isospin I=0. A clear bump structure can be seen below the ηf1(1285) threshold with a mass of around 1700 MeV and a width of about 180 MeV. Furthermore, taking s=1700 MeV, we get s1=927 MeV and s2=1315 MeV. At this energy, the interactions of ηˉK and ηK are strong.

      Figure 7.  Modulus squared of the total amplitude T for the η-(ˉKK)f1(1285) system. The dashed, solid and dotted curves are for Λ=890,990 , and 1090 MeV, respectively.

      In Fig. 8, we show |T|2 for the ˉK-(ηK)K1(1270) system. A strong resonant structure around 1680 MeV with a width of about 160 MeV is clearly seen, which indicates that the ˉK-(ηK)K1(1270) state could be formed. The mass of this state is below the ˉK and K1(1270) mass threshold. The strength of |T|2 at the peak is much higher than in Fig. 7 for the ηf1(1285)ηf1(1285) scattering. Thus, it is clear that the preferred configuration is ˉKK1(1270). However, ˉK keeps interacting with K, and could sometimes also form f1(1285).

      Figure 8.  Modulus squared of the total amplitude T for the ˉK-(ηK)K1(1270) system. The dashed, solid and dotted curves are for Λ=900,1000, and 1100 MeV, respectively.

      From Figs. 7 and 8, it can be seen that the peak positions and widths for the η-(ˉKK)f1(1285) and ˉK-(ηK)K1(1270) systems are quite stable for small variations of the cutoff parameter Λ. This gives confidence that the η-(ˉKK)f1(1285) and ˉK-(ηK)K1(1270) bound states can be formed. In fact, the ηf1(1285) configuration could mix with ˉKK1(1270). However, since the strength of ˉK-(ηK)K1(1270) scattering is much higher than of η-(ˉKK)f1(1285) scattering, the interference between the two configurations should be small. As both configurations peak around a similar energy, it is expected that the peak of any mixture of states is also around this energy.

      The ηˉKK bound state with quantum numbers I(JP)=0(1) has a dominant ˉKK1(1270) component. Since K1(1270) mostly decays into Kππ [49], the dominant decay mode of the proposed state should be ˉKKππ, and we hope that future experimental measurements could test our predictions.

      We should mention that two K1(1270) states were obtained in Ref. [49]. The one with a mass of 1284 MeV couples more strongly to the ηK and Kρ channels, while the other with a mass of 1195 MeV mainly couples to the πK channel, and couples very weakly to ηK. Thus, one could expect that the lower mass K1(1270) state of Ref. [49] does not affect our calculations.

    4.   Summary
    • In this work, we used FCA of the Faddeev equation to look for possible IG(JPC)=0+(1+) exotic states generated from the ηˉKK three-body interactions. We first selected a cluster ˉKK, which is known to generate f1(1285) with I=0, and then allowed the η meson to interact with ˉK and K. In the modulus squared of the η-(ˉKK)f1(1285) scattering amplitude, we find evidence of a bound state below the ηf1(1285) threshold with a mass of around 1700 MeV and a width of about 180 MeV. In the case of ˉK scattering with the cluster ηK, which was shown to generate K1(1270) with I=1/2, we obtained a bound state I(JP)=0(1) just below the ˉKK1(1270) threshold with a mass of around 1680 MeV and a width of about 160 MeV. In addition, the simplicity of the present approach also allows a transparent interpretation of the results, which are not easy to see when the full Faddeev equation is used. In the present study, it is easily recognized that ˉKK1(1270) is the dominant state, and that the ˉKK subsystem can still couple to the f1(1285) resonance. Yet, one may think that we should rely on the full Faddeev calculations where all scattering processes can be summed up to infinite order, as pointed out, for example, in Refs. [57, 58] in the study of the KdπΣn reaction. Such calculations are welcome and we intend to address this issue in a future study.

      The predictions of existence of possible exotic states have been made in the framework of the flux tube model [8], Lattice QCD [13] and QCD sum rule [23]. The results obtained here provide a different theoretical approach for a particular investigation of these exotic states.

      We would like to thank Prof. Li-Sheng Geng for useful discussions.

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