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Study of light kaonic nuclei with a Chiral SU(3)-based KN potential

Study of light kaonic nuclei with a Chiral SU(3)-based KN potential. ´. A. Dote (KEK) W. Weise (TU Munich). Introduction ppK - studied with a simple model Simple Correlated Model Test on two nucleons system Result of ppK - Summary and future plan. Nuclear Physics at J-PARC

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Study of light kaonic nuclei with a Chiral SU(3)-based KN potential

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  1. Study of light kaonic nuclei with a Chiral SU(3)-based KN potential ´ A. Dote (KEK) W. Weise (TU Munich) • Introduction • ppK- studied with a simple model • Simple Correlated Model • Test on two nucleons system • Result of ppK- • Summary and future plan Nuclear Physics at J-PARC 2nd June ‘07@ Ricotti in Tokai village

  2. KN interaction Strongly attractive Deeply bound and Dense Kaonic nuclei Decay NN interaction Non-mesonic decay mode KNN → YN, in addition to mesonic decay mode KN → Yπ Repulsive core at short distance Introduction

  3. Strongly attractive Deeply bound and Dense Kaonic nuclei ppK- H. Fujioka et al. @ FINUDA B.E. = 116 MeV, Γ = 67 MeV Introduction KN interaction Chiral SU(3)-based KN potential Decay NN interaction Non-mesonic decay mode KNN → YN, in addition to mesonic decay mode KN → Yπ Av18-like NN potential Repulsive core at short distance

  4. ppK- studied with a simple model and Chiral SU(3)-based KN potential Prof. Akaishi gave advices on the few-body calculation.

  5. Correlations Single-particle motion of nucleons and a kaon NN correlation function nucleon kaon KN correlation 1. Simple Correlated Model Model wave function of ppK- Spatial part NN spin: S=0 NN isopin: TN=1 Total isospin: T=1/2

  6. 1. Simple Correlated Model Energy variation Variational parameters • Included in the spatial part of the wave function • Real parameters • Determined by Simplex method to minimize the total energy Gaussians used for the NN correlation … Kamimura Gauss

  7. Very attractive nucleon isospin=1 nucleon isospin=0 1. Simple Correlated Model Model wave function of ppK- Isospin state Λ（１４０５）：　 ppK-：　 Deuteron+K-：　

  8. 2. Test on 2N system Checked this model in case of pp system. • Variational parameters are determined by the Simplex method. • are fixed to those of Kamimura Gauss.

  9. 2. Test on 2N system NN potential to test Enhanced the long-range attraction of the Av18-like potential slightly so as to make two protons bound.

  10. 2. Test on 2N system Hamiltonian Result obtained by directly diagonalizing the relative Hamiltonian with a lot of Gaussian base. Result Converged

  11. 2. Test on 2N system Relative wave function GDM N=25 Test potential [MeV] SCM N=9 [fm]

  12. 3. Result of ppK- Hamiltonian Coulomb force is neglected.

  13. NN potential Respect the repulsive-core part • Short-range part; referring to Av18, fitted with a few range Gaussians. • Long-range part; Akaishi-san’s effective NN interaction for ppnK- (ρmax=9ρ0) Av18-like Av18 Important in ppK- [MeV] Akaishi [fm]

  14. KN potential S-wave potential P-wave potential 1, Gaussian shape as=ap=a 2, Energy dependent Chiral SU(3) theory : KN scattering amplitude : KN scattering volume 3, P-wave potential including derivative operator.

  15. KN potential B. Borasoy, R. Niβler, and W. Weise, Euro. Phys. J. A 25, 79-96 (2005) S-wave scattering amplitude

  16. R. Brockmann, W. Weise, and L. Taucher, Nucl. Phys. A 308, 365 (1978) ※updated version KN potential P-wave scattering volume

  17. Assume the values of the binding energy of kaon itself “B(K)”. The Hamiltonian is determined. Perform the energy variation by the Simplex method. Then, calculate the binding energy of kaon with the obtained wave function. Finished ! If Yes Check If No Procedure of the present calculation • Self-consistency of kaon’s energy is taken into account.

  18. Procedure of the present calculation Remarks • The imaginary parts are ignored in the current study. • The kaon’s binding energy “B(K)” B(K) = -EK = -(Etotal – Enucl) [pp] in ppK- + K Enucl p+p+K 0 B(K) Etotal [ppK-]

  19. There doesn’t exist any self-consistent solution for the range parameter a < 0.67 fm. This result is the same as that obtained in the previous AMD study reported in YKIS’06 and so on. 3. Result of ppK- Kamimura Gauss, N=9, r1=0.1 fm, rN=9.0 fm P-wave int. : non-perturbative a; range parameter [fm] Self consistency a=0.67 fm a=0.70 fm a=1.00 fm a=0.80 fm a=0.90 fm

  20. 3. Result of ppK- Property [fm] [MeV] The total binding energy of ppK- is 42 – 76 MeV. cf) It doesn’t exceed 53 MeV in the previous AMD study. [MeV] [MeV] [fm]

  21. 3. Result of ppK- Property [fm] [MeV] The relative distance between two nucleons is larger than 1.0 fm. If the size of a nucleon core is 0.5 fm, they don’t touch. This result is the same as that of the previous AMD study. [MeV] [MeV] [fm]

  22. Summary • We are now investigating “prototype of a K cluster” ppK- with a simple model • respecting the NN short-range correlation. • In the present study, we adopt a NN potential which has a strongly repulsive core. (Av18-like) • The present KN potential is based on the Chiral SU(3) theory. It includes • the p-wave interaction in addition to the s-wave interaction. • The model wave function is very simple. The nuclear part is assumed to be purely • L=S=0 and T=1 state. But in this model we introduce a correlation function between • the two nucleons so as to avoid the repulsive core adequately. • Difference from the previous AMD study • The present calculation performs “Variation After Projection” with respect to the total angular momentum and the total isospin. • The p-wave interaction is treated non-perturbatively. • Result • The total binding energy is 42 ～ 76 MeV, when the range parameter changes from 1.00 fm to 0.67 fm. • There exists a lower limit in the range parameter due to the self consistency. • The mean distance between the two nucleons is larger than 1fm. • Essentially, the present result is very similar to the previous one by the AMD study.

  23. Future plans π π π K K K K K K = + … + … + … N N N N Σ Σ Σ N N t matrix In the , the KN pair interacts again and again, coupling to the Σπ pair. If we solve the three body system, ppK-, with this … K N N Double counting problem(claimed by Prof. Akaishi and Prof. Morimatsu) Although it has already been considered that a KN pair interacts infinite times, such a process is incorporated again and again in the three-body calculation…

  24. Future plans • We should directly treat the imaginary part of the KN potential. • This is important to estimate the decay width. • In addition, this will give an influence to the total energy • because the imaginary potential is expected to give a repulsive contribution. • We should determine the range parameter of the KN interaction. • We would like to introduce a correlation between two nucleons • into the AMD calculation so as to investigate larger system.

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