Four- and Five-Body Scattering Calculations of
Exotic Hadron Systems
1 Introduction: importance of the wave function of multi-quarks
From the viewpoint of Elementary Particle Physics, the lattice QCD calculation is one of the most standard approaches to investigate hadrons including multi-quarks. However, it is rather difficult to extract the “wave function” of hadrons in the lattice QCD calculation because it is based on the path-integral, where all contributions of possible states are summed up and only vacuum expectation values can be obtained.
Needless to say, the wave function is one of the most important quantities in quantum physics, and, of course, also in Quark-Hadron Physics. In particular for multi-quark systems, the analysis with the quark wave function is necessary to clarify whether the multi-quark hadron is an exotic resonance state or a two-hadron scattering state. Thus, to extract the state information (the wave function) of hadrons, we need a reliable calculational method for multi-quark systems instead of lattice QCD. One of the attractive methods is the constituent quark-model calculation, since it seems workable up to several hundred MeV excitation, and its precise calculation for multi-quark systems can be done with the Gaussian Expansion Method (GEM).
In Nuclear Physics, a precise calculational method for bound and scattering states of various few-body systems using GEM has been developed by two of the present authors (E.H. and M.K.) and their collaborators, and has successfully been applied to light nuclei, light hypernuclei, exotic atoms/molecules and so on.
Using GEM within the framework of a constituent quark model, Hiyama et al. investigated for the first time scattering and resonance states of the five-quark system under the explicit NK scattering boundary condition. It was made clear that there appears no resonance state around the reported energy of .
In this paper, we perform the similar precise calculation of the five-quark system with and using the linear confinement potential, which is more appropriate as indicated by lattice QCD calculations. Furthermore, we apply the same method to the four-quark system in connection with X(3872) and discuss about possibility of any bound/resonance state near the threshold.
2 Five-body quark-model calculation for
We study the five-quark system by solving the five-body Schrödinger equation including the NK scattering channel explicitly, as was done in Ref.2). We here take a standard non-relativistic quark-model Hamiltonian with the linear-type confining potential and the color-magnetic potential :
Using the parameters listed in Table I, we can well reproduce masses of ordinary baryons and mesons as shown in Table II and III, and well reproduce or predict hadron properties, e.g., (exp. 2.78nm), (exp. 1.91nm), (exp. 0.61nm), , (exp. 2.02nm) for the magnetic moment. Note here that the form of the linear-type confining potential is appropriate as an approximation of the Y-type linear three-quark potential indicated by lattice QCD, although the adopted value of the string tension 0.56GeV/fm is smaller than the standard value 0.89GeV/fm.
|adopted value||330MeV||500MeV||0.56 GeV/fm||0.5fm||572MeV||240MeV|
|calculated mass [MeV]||939||1235||1064.8||1129||1324||1539.5||1462|
|empirical mass [MeV]||939||1232||1115.7||1191||1318||1672.5||1440|
|calculated mass [MeV]||759||759.3||864.4||457.9||152|
|empirical value [MeV]||782.7||775.5||892||496||138|
Now, we perform an almost precise quark-model calculation with GEM for multi-quark systems and clarify whether each obtained state is a resonance or a continuum scattering state through the phase-shift analysis in the model calculation. For the five-quark system with the energy , the total wave function is given as
The first term is the NK scattering component expressed by
The second term describes five-body degrees of freedom in the interaction-region amplitude which vanishes asymptotically. The amplitude is expanded using a nearly complete set of five-body eigenstates, , constructed by diagonalizing the total Hamiltonian as Here, each of is described as a superposition of -type five-body Gaussian basis functions written in all the Jacobi coordinates as shown in Fig.1. By employing 15,000 five-body basis functions, i.e., , the eigenfunction set forms a nearly complete set in the finite interaction region.
The eigenfunctions stand for discretized continuum states of the five-body system, and are called “pseudostates” in the scattering theory. It is known that most of the pseudostates do not actually represent resonance states but melt into non-resonant continuum states when the scattering boundary condition is imposed to the total wave function.
We actually perform the precise quark-model calculation including all the pseudostate terms in (2.2), and calculate the phase shift in the NK elastic scattering N+KN+K for channels, respectively. For , all the pseudostates located at MeV above the NK threshold melt into non-resonant continuum state when the NK scattering boundary condition is imposed, which means the strong coupling between those pseudostates and the NK scattering state. We show in Fig.2 the calculated phase shift , and find no resonance for MeV above the NK threshold, i.e., GeV in mass region around . We find a sharp resonance for and a broad one for around 1.9GeV in Fig.2, although their energies are too high to be identified with the . For the states, we find no resonance up to 500 MeV above the NK threshold.
Note that these results on the absence of low-lying pentaquark resonances with are consistent with the lattice QCD results in Refs.9) and 10). As for the presence of penta-quark resonance around 1.9GeV, Ref.11) shows a consistent lattice QCD result indicating a penta-quark resonance around 1.8GeV.
To conclude, there is no five-quark resonance with below 1.85GeV. Instead, the quark-model calculation predict a five-quark resonance state of with the mass of about 1.9GeV and the width of 2.68MeV.
3 Four-body quark-model calculation for X(3872)
With GEM, we analyze four-quark systems for in a quark model to investigate X(3872), which may have the tetraquark structure as . Note here that, if X(3872) is an state, it is manifestly exotic and there should exist its isospin partners X ( and ) around 3.87GeV. For the calculation of four-quark charmed systems, we adopt the quark-quark interaction of Ref.13), which effectively includes the exchange effect of Nambu-Goldstone bosons. With 1752MeV, 313MeV and 555MeV, the quark model leads light hadron masses as 772.8MeV, 696.3MeV and 148.7MeV, and the calculated (experimental) masses of charmed mesons are 1897.6(1867)MeV, 2017.1(2008)MeV, 3096.5(3097)MeV and 2989.1(2980)MeV.
For the four-quark calculation, we employ all the 18 sets of the Jacobi coordinates of the four-body system. We show in Fig.4 the three important sets describing the and correlations (), the and ones () and the and ones (). The four-body Gaussian basis functions are prepared in the Jacobi sets and the four-body pseudostates are obtained by diagonalizing the four-quark Hamiltonian using nearly 13,000 Gaussian basis functions. For the phase shift calculation, we consider the scattering channels and J/ + for , and and J/ + for .
For the states, we find a very sharp resonance, dominantly having the component, slightly below the threshold. In this calculation, however, the J/ threshold energy is much lower than the experimental value by 86 MeV. We find that if the quark-quark interaction is artificially adjusted so as to reproduce the experimental value of the J/ threshold, neither bound nor resonance state appears. In fact, the result seems rather sensitive to the quark-quark interaction, and we need better interaction to obtain definite conclusions for states.
For the states, we obtain several pseudostates near the threshold region, but all of them disappear when the scattering boundary condition is switched on. The resultant scattering phase shift by the full coupled-channel calculation is given in Fig.3, which shows no resonance behavior. We thus find no -type tetraquark resonance with in mass region of 3.874.0GeV.
In this way, the almost precise quark-model calculation using the Gaussian Expansion Method (GEM) is a powerful tool to clarify the state properties of multi-quark systems, which is a theoretical search for exotic hadron resonances.
E.H. acknowledges Profs. A. Hosaka, H. Toki and M. Yahiro for useful discussions. The authors are thankful to the Yukawa Institute for Theoretical Physics at Kyoto University, where stimulating discussions were made about the present work during the YKIS2006 on “New Frontiers on QCD”.
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