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クオーク模型によるエキゾチックハドロン構造研究

クオーク模型によるエキゾチックハドロン構造研究. E. Hiyama (RIKEN). 今までハイパー核の構造計算をしてきました・・。. Single Λ hypernuclei. P-shell double Λ hypernuclei （ A=7-10) ： α+x+Λ+Λ model E. Hiyama, M. Kamimura, T. Motoba, T. Tamada and Y. Yamamoto, Phys. Rev. C66, 024007 (2002). でも、基本的に、私の計算法は、 Hamltonian が与えられるのでしたが、

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クオーク模型によるエキゾチックハドロン構造研究

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  1. クオーク模型によるエキゾチックハドロン構造研究クオーク模型によるエキゾチックハドロン構造研究 E. Hiyama (RIKEN)

  2. 今までハイパー核の構造計算をしてきました・・。今までハイパー核の構造計算をしてきました・・。

  3. Single Λhypernuclei

  4. P-shell double Λ hypernuclei （A=7-10)：α+x+Λ+Λ model E. Hiyama, M. Kamimura, T. Motoba, T. Tamada and Y. Yamamoto, Phys. Rev. C66, 024007 (2002)

  5. でも、基本的に、私の計算法は、Hamltonianが与えられるのでしたが、でも、基本的に、私の計算法は、Hamltonianが与えられるのでしたが、 なんでも計算できます。 ということで・・。

  6. (1) Pentaquark system E. Hiyama, M. Kamimura, A. Hosaka, H. Toki and M. Yahiro, Phys. Lett.B 633,237(2006). u u s d d Θ+ Non-relativistic 4- and 5-body constituent quark model (2) Tetra quark u u X(3872) C C

  7. So far, experimentally, exotic hadron systems such as X(3872) etc have been observed. One of the theoretical important issue is whether we can explain the experimental value or not. The purpose of our work is to answer this question in the framework of the non-relativistic 4- and 5-body constituent quark model. Before going to study of pentaquark and tetra quark systems, I shall explain our method briefly.

  8. Applied to Gaussian Expansion Method Developed by Kyushu Univ. group Kamimura (1) 3-cluster structure of light nuclei (2) Coulomb 3-body muonic molecular ions appearing in the muon-catalyzed fusion cycles (1987～) (3) 3-nucleon bound states with realistic NN and 3N forces (1988) (4)Metastable antiprotonic helium atom (He++p+e)(1995～) E. Hiyama, M. Kamimura and Y. Kino, Prog. Part. Nucl. Phys. 51 (2003), 223.

  9. (1) Pentaquark system u u s d d (2) Tetra quark u u C C

  10. u u Θ+ T.Nakano et al. (LEPS collaboration) Phys. Rev. Lett. 91 (2003), 012002. Mass: 1540MeV Γ‹25 MeV S=+1 s d d Θ+ (1540 MeV) About 100 MeV N+K threshold (1433 MeV) The ground state of N -- 938 MeV The ground state of K -- 495 MeV

  11. Theoretically, it is requested to evaluate accurately the mass and decay width of this five quark system. For this purpose, we need to impose any proper boundary condition to NK scattering channel of this system. u u s d d

  12. To study this pentaquark system is very interesting from view points of few-body physics. Because pentaquark system gives us to develop our method to 5-body problem to treat both resonance states and continuum scattering states. Very recently, I developed my method to such a calculation together with M. Kamimura H. Toki A. Hosaka M. Yahiro

  13. I succeeded to perform 5-body calculation imposing proper boundary condition to NK scattering channel of this system. N K I shall discuss the structure of pentaquark, Θ+, within the framework of non-relativistic 5-body constituent quark model. q q q q s E. Hiyama et al., Phys. Lett.B 633,237 (2006).

  14. Hamiltonian N. Isgur and G. Karl, Phys. Rev. D 20, 1191 (1971)

  15. N+K scattering channel Model space of 5-quark system: We employ precise 5-body basis functions that are appropriate for describing the q-q and q-q correlations and for obtaining energies of 5-quark states accurately.

  16. ΨJM(qqqqq)=ΦJM(C=1) +ΦJM (C=2) +ΦJM (C=3) +ΦJM (C=4) +ΦJM (C=5) ΦJM (C) =∑Aα(C)Φα,JM(qqqqq) α Φα,JM(qqqqq)=Aqqqq{[(color)(c)α(isospin)(c)α (spin)(C)α(spatial)(c)α]JM} (spatial)(c)α=φnl(c)(rc)ψνλ(c)(ρc)φ(c)kjχNL(c)(Rc)

  17. (spatial)(c)α=φnl(c)(rc)ψνλ(c)(ρc)φ(c)kj(Sc)χNL(c)(Rc)(spatial)(c)α=φnl(c)(rc)ψνλ(c)(ρc)φ(c)kj(Sc)χNL(c)(Rc) ^ 2 φnlm(c)=rle-(r/r ) Ylm(rc), rn=r1an-1(n=1～nmax) n 2 Ψνλμ(ρc)=ρλe-(ρ/ρ) Yλμ(ρc), ρμ=ρ1αμ-1 (μ=1～μmax) ^ μ ^ 2 Φkjm’ (c)(Sc)=Sje-(S/S )Yjm’(Sc) , Sk=S1a’k-1 (k=1～kmax) k χNL(c)(Rc)=RLe-(R/R )YLM(Rc), RN=R1AN-1(N=1～Nmax) Geometric progression For many reasons, Gaussian basis functions are good Basis functions for few-body systems. (H-E)Ψ=0 By the diagonalization of Hamiltonian, we obtain N eigenstates for each Jπ.

  18. Here, we use 15,000 basis functions. Then, we obtained 15,000 eigenfunction for each Jπ. We investigate two spin parity states, J=1/2- and J=1/2+ of Isospin I=0. ・・・・ ・・・・ ・・・・ ・・・・ ½-2 ½+2 ½-1 ½+1 I=0,1/2- I=0,1/2+

  19. We finally solve 5-body problem under scattering boundary condition for the N+K channel in order to examine whether 5 quark system are resonance states or non-resonance continuum states. So far in the literature calculations, N+K channel was neglected due to very difficulty of this calculation. But, this difficult calculation was performed by our method for the first time. N K q q q q s

  20. R. Jaffe and F. Wilczek, Phys. Rev. Lett. 91, 232003, (2003). Important to describe the pentaquark system

  21. s Jaffe and Wilczek N+K threshold

  22. s ½+ ½- qq correlation is twice stronger than the qq correlation. C=2 and 3 channels are appropriate Jacobian coordinate to describe the qq correlation. N+K threshold

  23. N+K scattering channel

  24. Results before doing the scattering calculation Bound-state approximation Large contribution from N+K channel! N+K threshold Do these states correspond to resonance states or Discrete non-resonance continuum states?

  25. useful method: real scaling method often used in atomic physics In this method, we artificially scale the range parameters of our Gaussian basis functions by multiplying a factor α: rn→αrn in rlexp(-r/r ) for exmple 0.8 <α<1.5 2 n and repeat the diagonalization of Hamiltonian for many value of α. ← resonance state Non-resonance continuum state α: range parameter of Gaussian basis function [schematic illustration of the real scaling] What is the result in our pentaquark calculation?

  26. Real scaling calculation N+K threshold No resonance in 0 – 500 MeV above the N-K threshold

  27. Real scaling calculation N+K threshold No resonance in 0 – 500 MeV above the N-K threshold

  28. In order to see in more detail whether each discrete states survive as a resonance or melt into the non- resonance NK continuum state, we finally solve the scattering problem using Kohn-type coupled-channel variational method. This method is useful for the scattering between composed particle and were used in the 3q-3q scattering in the study of NN interaction. (For example, M. Oka and K. Yazaki, Prog. Thor. Phys. 66, 556 (1981).) Phase shift of J=1/2 – and 1/2 +

  29. Scattering phase shifts N+K threshold No resonance in 0 – 500 MeV above the N-K threshold

  30. I did the same calculation using a linear type confining potential, recently. The result that we had no resonance in the reported energy region of Θ+ was qualitatively was the same as the present one. I also calculated the other spin parity states such as 3/2+ and 3/2- . But, we had no resonance states in the observed energy region.

  31. (1) Pentaquark system u u Θ+ s d d Non-relativistic 4- and 5-body constituent quark (2) Tetra quark u u X(3872) C C

  32. X(3872):3871.2±0.5 MeV 1++ Γ<2.3MeV Phys. Rev. Lett. 91, 262001 (2003) Belle Group 3872.4 ρ+J/Ψ（T=1) X(3871.2) D0 + D 3871.2 ω+ηC (T=1) 3763 u u 3756 ρ+ηC(T=1) C C 3644MeV η+J/Ψ (T=0)

  33. Theoretical interesting issue: • Can we explain the observed mass and width of X(3872) as a tetra quark state? Or X(3872) is qq meson state? (2) Is there any manifestly exotic tetra quark meson? For example, X+(ccud),X-(ccdu) are the manifestly exotic hadrons which cannot be constructed by qq.

  34. X+(ccud) X0(ccqq) X-(ccdu) Tetra quark At first, let’s search for manifestly Exotic tetra-quark hadron ( I=1 state) around the energy region of DD* threshold. u u C C It is possible to find X± states which are manifestly exotic mesons. If X(3872) is I=1

  35. π、ρ I=1 state u u C C J/Ψ+ρ 3872 X(3871.2) 3871 DD* J/Ψ、ηC 3756 ηC+ρ u u C 3255 MeV C J/Ψ+π The qq interaction employed in the pentaquark does not reproduce the experimental data. Therefore, I employed another qq interaction. D D

  36. Proposed by J Vijande etc, J. Phys. G 31(2005) 481 Vqq=Vcon+VOGE+VqqC Goldstone-boson exchange interaction

  37. Cal. Exp. J/Ψ：　　3097 MeV 3097 MeV ηC : 2989 2990 D* : 2017 2009 D : 1897 1868 ρ: 773 771 π : 149 138

  38. I=1 state 3914 (3875) DD* 3868 (3872) J/Ψ+ρ ηC+ρ 3761 (3756) J/Ψ+π 3244 MeV (Exp.3255) How is the results of 4- quark system using this potential?

  39. All the 15 Jacobi-coordinate channels

  40. D D* π、ρ u u u u C C C C J/Ψ、ηC C=2 C=1 Scattering channel

  41. 15 C=1 ΨJM(qqqq)=∑Φ(rc,Rc,ρc) Here, we use 13,000 basis functions. Then, we obtained 13,000 eigenfunction for 1+. ・・・・ ・・・・ 1+2 1+1 I=1,1+

  42. Results before doing the scattering calculation Bound state approximation I=1 Are there are bound states or resonance state? Or All states are non-resonance states? 46MeV D+D* threshold 0MeV J/Ψ+ρ

  43. Real scaling I=1 no bound sate and no resonance state D+D* 46 0MeV J/Ψ+ρ Scaling parameter of Gaussian basis function

  44. In order to confirm it, I solved the scattering problem under the scattering boundary condition for the J/Ψ+ρ channel and D+D* channel with the full 15 channel coupled. • In the next figure, I show 2 types of phase shifts. • For the J/Ψ+ρ incoming and the J/Ψ+ρ outgoing • For the D+D* incoming and D+D* outgoing • in the presence of the 15 channel coupled.

  45. No resonance No bound state δ（DD*→DD*) δ(J/Ψ+ρ→J/Ψ＋ρ) Phase shift (degree) (MeV) D+D* threshold J/Ψ+ρ

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