1. Tensor Optimized Few-body Model for s-shell nuclei Kaori Horii, Hiroshi Toki (RCNP, Osaka univ.) Takayuki Myo, (Osaka Institute of Technology) Kiyomi Ikeda (RIKEN Nishina Center) 1 ２３．８．２０１１ APFB11

2. Introduction 2 ＊ D-wave component is small, the probability is 5% ＊ The dominant attraction of 80 % is caused by the tensor interaction. EnergyKineticCentralTensorLSP(L=2) -2.24 19.88 -4.46 -16.64 -1.02 5.77 ★ Deuteron result with AV8’ potential Tensor Optimized Shell Model (TOSM) Myo, Toki, Ikeda Realistic NN interaction ⇒ strong short range repulsion and tensor interaction We expect to develop medium and heavy nuclei using realistic NN int. ★ Method for treating the strong tensor int.

3. Motivation ＊ Where is the difference between TOSM and the rigorous calculation? ＊ What should be done to improve the description of nuclei ? Few-body technique (relative coordinates) by introducing the spirit of TOSM approximation. → Tensor Optimized Few-body Model (TOFM) TOSM +UCOM (only S-wave relative motion) 4 He calculation with NN int (AV8’). 3 SEnergyKineticCentralTensorLS TOSM -22.30 90.50-55.71-54.55-2.53 Few body -25.92102.35-55.22-68.32-4.71 T.Myo et al (2009) H.Kamada et al (2001)

4. Tensor Optimized Few-body Model (TOFM) 4 TOFM (Few-body) TOSM (Shell-Model) ******* ＊ D-wave state contains a single Y 2 function in the relative coordinates. ( No double and triple Y 2, No Y 1 functions) ＊ The most essential Y 2 function is in the relative coordinate x 1 ＊ small variational model space

5. Tensor Optimized Few-body Model (TOFM) 5 TOFM (Few-body) TOSM (Shell-Model) ******* ＊ D-wave state contains a single Y 2 function in the relative coordinates. ( No double and triple Y 2, No Y 1 functions) ＊ The most essential Y 2 function is in the relative coordinate x 1 ＊ small variational model space Deuteron like state (S=1,T=0) TOFM wave function describes the deuteron like state in nuclei.

6. 6 Jacobi coordinate x 1,x 2,x 3 global vector Tensor Optimized Few-body Model (TOFM) For 4 He Total J=0, S-wave (L=0,S=0), D-wave(L=2,S=2) ＊ Correlated gaussian basis with the global vector Few Body System 42(2008 )33-72 Y.Suzuki, et al TOFM approx. ＊ Variational calculation on the basis of the Stochastic Variational Method (SVM) Gaussian range matrix A are given as random parameters

7. 7 EnergyKineticCentralTensorLS TOFM -7.5446.68-22.30-29.96-1.91 Suzuki -7.7647.57-22.49-30.84-2.00 Energy [MeV] TOFM results compare almost perfectly with the rigorous few-body calculation. ★ Numerical result for 3 H with AV8’ potential Few Body System 42(2008 )33-72 Y.Suzuki, et al Kinetic LS Energy Central Tensor

8. 8 Energy [MeV] ★ Numerical result for 4 He with AV8’ potential (w/o Coul.) EnergyKineticCentralTensorLS TOFM -24.0595.37-54.58-60.79-4.05 Suzuki -25.92102.35-55.22-68.32-4.71 Few Body System 42(2008 )33-72 Y.Suzuki, et al The missing strength come from the kinetic and tensor matrix elements. The inclusion of double Y2 functions brings the total energy very close. Kinetic LS Energy Central Tensor

9. 9 ★ 4 He ~ Comparison with Tensor Optimized Shell Model (TOSM) ~ EnergyKineticCentralTensorLS TOFM-24.05 95.37-54.58-60.78-4.05 TOSM-22.30 90.50-55.71-54.55-2.53 ＊ The energy values are better than the TOSM result. ＊ The large differences come from the kinetic and tensor components. ＊ TOSM calculation can be improved by taking more suitable UCOM correlation function in the short range correlation. TOFM (Few-body) TOSM (Shell-Model) T. Myo et al, PTP121,No3(2009)511,

10. 10 Correlation function with TOFM w.f. S-wave and D-wave Correlation functions for 2 H, 3 H, and 4 He S D ＊ The dip structure below 1 fm reflects the presence of the strong short range repulsion. ＊ The magnitude of the peak reflects the size of nuclei. ＊ d-wave components are found significant and similar among the three nuclei. 4 He (1.52 fm) 3 H (1.78 fm) 2 H (1.95 fm) H.Kamada et al (2001) PRC64, 044001

11. 11 Correlation function with TOFM w.f. 4 He (1.52 fm) 3 H (1.78 fm) 2 H (1.95 fm) S-wave and D-wave Correlation functions for 2 H, 3 H, and 4 He we normalize the correlation functions to those of 4 He. S D ＊ The same short range behavior below 1 fm for both S and D ＊ The size of nuclei determine the correlation function at large distance. ＊ This is an important finding for the study of heavier nuclei. H.Kamada et al (2001) PRC64, 044001

12. Correlation function for S-wave AV8’ (with tensor) MT-V (w/o tensor) S ＊ The short range behavior of the correlation function depends on the properties of interaction. ＊ It is interesting to modify the correlation function in the UCOM. To improve the TOSM (shell-model) results ⇒ the short range correlation is treated by the UCOM ⇒ UCOM function form is obtained with the central MT-V[1] int. Difference of S-wave Correlation function for 4 He between AV8’ and MT-V [1]R.A.Malfliet and j.A..Tjon, NPA127, 61(1969)

13. Summary & Outlook 13 ＊ We have formulated a tensor-optimized few-body model (TOFM) in the spirit of the TOSM ＊ TOFM app. has a small variational model space (single Y 2 in D-wave) ＊ The good reproduction of the rigorous results indicates that nuclei like to have deuteron configuration. ＊ The short range behaviors of correlation function are very similar for the s-shell nuclei. ＊ The correlation function in UCOM should be studied further in the TOSM flamework. ＊ The present study is very encouraging to extend our study for nuclei with A>5 in TOFM flamework. EX) 5 He structure, 4 He+n Phase Shift, 8 Be( 4 He+ 4 He RGM)…

14. 14