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Multiquark States in the Inherent Nodal Structure Analysis Approach

Multiquark States in the Inherent Nodal Structure Analysis Approach. Yu-xin Liu Department of Physics, Peking University, Beijing 100871. Outline I. Introduction II. The INS Analysis Approach III. Application to Penta-quark System

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Multiquark States in the Inherent Nodal Structure Analysis Approach

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  1. Multiquark States in the Inherent Nodal Structure Analysis Approach Yu-xin Liu Department of Physics, Peking University, Beijing 100871

  2. Outline I. Introduction II. The INS Analysis Approach III. Application to Penta-quark System IV. Application to Six-quark System V. Remarks References: P R L 82 (1999) 61; P L B 544 (2002) 280; P R C 67(2003) 055207 (nucl-th/0212069) hep-ph/0401197

  3. I. Introduction • Multi-quark systems are Appropriate to investigate the quark behavior in short distance to explore exotic states of QCD • Many six-quark cluster states e.g., H, d’, d*, (ΩΩ), (ΩΞ), (ΩΞ*), … … have been predictedin many QCD approaches: Lattice QCD (e.g., Nucl. Phys. B-Proc. Sup. 73 (1999) 255) QCD Sum Rules(e.g., Nucl. Phys. A 580 (1994) 445) Bag Model(e.g., Phys. Rev. Lett. 38 (1978) 195, Sov. J. Nucl. Phys. 45 (1987) 445) Quark Delocalization and Color Screening Model (e.g., Phys. Rev. Lett. 69 (1992) 776) SU(3) Chiral Quark Model (e.g., Phys. Rev. C61 (2000) 065204) • No dibaryons have been observed in experiment after more than 25 years efforts.

  4. An exciting point:It was claimed thatPenta-quark state + was observedinLEPS, DIANA, CLAS, SAPHIR, HERMES, ZEUS, … • Many theoretical investigations have been accomplishedin Chiral soliton model (ZPA 359(1997) 305) Diquark-antiquark model(e.g., PRL 91(’03) 232003, etc. ), Skyrme model(PLB 575 (2003) 234) Diquark-triquark cluster model (PLB 575 (2003) 249) Chiral Q model(PLB 575(‘03)18, PLB 577(‘03) 242, hep-ph/0310040) QCD Sum Rules (e.g., PRL 91 (2003) 2320020, etc.) Large Nc QCD(e.g., hep-ph/0309150 ) Lattice QCD(e.g., hep-lat/0309090, hep-lat/0310014 ), ………… • The parity has not been fixed commonly(model dependent). The narrow width has not been reproduced. • To neglecthandling thecomplicated interactions in QCD, ……, we propose a model independent approach ---- INS Analysis.

  5. The INSAnalyzing Approach • 1. General Point of View • Penta/Six - quark clusters involve flavors u, d and ( s ) • Intrinsic space {color, flavor, spin} holds symmetry • Coordinate space holds symmetry • Geometric symmetry INSaccessible • Penta/Six - quark clusters must have symmetry • all the quantum Numbers

  6. 2. Inherent Nodal Structure Analysis • Starting Point: The lessnodal surfacesthe wavefunction • contains, the lowerenergy the state has, • e.g., infinite square well Dynamical Nodal Surface Inherent Nodal Surface • Nodal Surface

  7. Inherent Nodal Surface • Ψ eigenstate, A a geometric configuration, • A may be invariant to a specific operation , i.e., • A A A (1) • The representation of the operation on is a matrix, • Eq.(1) appears as a set of homogeneous linear equations. • In some cases, there exists solution (A) = 0 • Inherent Nodal Surface (INS) which is imposed by the • inherent geometric configuration and independent • of dynamicsexists. Then, the inherent nodeless states • are accessible to the geometric configuration.

  8. An Example of Six-body System • A 6-body system has several regular geometric shapes, • for example: the regular octahedron (OCTA) • the regular pentagon pyramid (PENTA) • the triangular pyramid • the regular hexagon • For OCTA , it is invariant to the operations: • (2) • (3) • (4) • (5)

  9. Denoting the OCTA as A and the basis of the representation • of the rotation, space inversion and permutation as , • for the , we have • AAA(6) • The Solution A depends on the , and . • We obtain then the INS accessible and for each , • and all the quantum numbers further. • Since , S-wave nodeless state is the lowest • state in energy, then the P-wave nodeless one.

  10. Application to Penta-quark System 1. Intrinsic States Since , the orbital symmetry and the flavor-spin symmetry has the following relation

  11. The explicit quantum numbers and configurations

  12. 2. Accessibility of the spatial configurations

  13. The accessibility of the ETH and square configurations to the (L) wave-functions

  14. 3. Possible low-lying penta-quark states Consistent with the results in chiral soliton model, general framework of QCD, Chiral quark model, diquark-triquark cluster model, …… ……

  15. IV. Application to Six-quark System 1. Intrinsic States Since [2 2 2], the orbital symmetry and the flavor-spin symmetry has the following relation

  16. The strangeness, isospin and spin of the states listed above and the baryon-baryon and hidden color channel correspondence

  17. Accessible Orbital Symmetries • Solving the sets of linear equations in Eq.(6) at geometric • configurations OCTA and C-PENTA, we obtain the nodeless • accessible orbital symmetries as • for S-wave ( ) states, • for P-wave ( ) states, • The accessibilities of the states are listed in the following tables.

  18. The accessibility of the S-wave nodeless components

  19. (continued)

  20. The accessibility of P-wave nodeless components

  21. (continued)

  22. 3. Possible low-lying S-wave dibaryon states The configuration with large nodeless accessibility: s=-6, (T, S)=(0, 0) 3 s=-5, (T, S)=(1/2, 1), (1/2, 0) 4, 3 s=-4, (T, S)=(0, 1), (1, 0), (1, 1), (1, 2) 8, 7, 5, 6 s=-1, -2, -3, many configurations s=0, (T, S)=(0, 1), (1, 0), (1, 2), (2, 1) 4, 4, 4, 4 Pauli principle, L+T+S=odd decay to two free baryons low-lying stable S-wave dibaryons: (s, T, S)=(-6, 0, 0) , (-5, ½, 1), (-5, ½, 0), (-4, 1, 1) ?

  23. 4. Possible low-lying P-wave dibaryon states P-wave resonance may have narrow width, but higher energy P-wave accessible, but S-wave inaccessible configurations being taken as P-wave dibaryon states (s, T, S)=(-6, 0, 1), (-4, 0, 0), (-2, 0, 3), (0, 0, 0), (0, 0, 2),(0,2,0), (0,1,3), (0,3,1), (0,3,3) Pauli Principle being taken into account, Possible ones are (s, T, S)=(-6, 0, 1), (-2, 0, 3) Spin-orbital interaction : high J states may have low energy Possible low-lying stable P-wave dibaryons are (s, T, J) = (-6, 0, 2), (-2, 0, 4)

  24. 5. Comparison with other theoretical studies and experimental results The candidates are consistent with the results in Quark-Delocalization and Color-Screening Model (QDCSM) and Chiral SU(3) quark model d* is possible, since the accessibility for (s, T, S)=(0, 0, 3) is 3, if its energy is very low. Consistent with QDCSMresult. d’ is impossible, since the accessibility for L=1, (s, T, S)=(0, 0, 1) is only 1. Inconsistent withBag model and Chiral quark model, but consistent withp-p collision results (PLB 550 (2002) 147, EPJA 18 (2003) 171, 297)

  25. V. Remarks • The inherent nodal structure analysisapproach for few-body system is proposed • The wave-functions of penta/six-quark systems are classified, the quantum numbers and the configurations of the wave-functions are obtained. • The , , , and , and the hidden-color channel states are proposedto bedibaryon states, which may be observed in exp. • The d* is also a possible dibaryon, but the d’ is not. • The parity of the +is proposed to bepositive. • The INS analysisapproach is independent of dynamics. • To obtainnumerical result both the INS analysisand the dynamical calculation are required.

  26. Thanks !!!

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