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 2 nd June Ricotti in Tokai village ´

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

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

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

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

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

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

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.

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.

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

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

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

Short-range part; referring to Av18, fitted with a few range Gaussians. Long-range part; Akaishi-san’s effective NN interaction for ppnK - (ρ max =9ρ 0 ) [fm] [MeV] Av18-like Av18 Akaishi Respect the repulsive-core part Important in ppK - NN potential

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

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

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

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

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

3. Result of ppK - Kamimura Gauss, N=9, r 1 =0.1 fm, r N =9.0 fm P-wave int. : non-perturbative Self consistency a; range parameter [fm] a=1.00 fm a=0.90 fm a=0.80 fm a=0.70 fm a=0.67 fm 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. 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 - Property [fm] [MeV] [fm] The total binding energy of ppK - is 42 – 76 MeV. cf) It doesn’t exceed 53 MeV in the previous AMD study. The total binding energy of ppK - is 42 – 76 MeV. cf) It doesn’t exceed 53 MeV in the previous AMD study.

3. Result of ppK - Property [fm] [MeV] [fm] 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. 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.

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. 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

Future plans Double counting problem (claimed by Prof. Akaishi and Prof. Morimatsu) + … ++ ……= K N K N KK N N πππ ΣΣΣ K N K N t matrix K N N If we solve the three body system, ppK -, with this … 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… In the, the KN pair interacts again and again, coupling to the Σπ pair.

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.