1. 12.12.26toki@tensorrcnp1 理論から見たテンソル力 Hiroshi Toki (RCNP, Osaka University) In collaboration with T. Myo (Osaka IT) Y. Ogawa (RCNP) K. Horii (RCNP) K. Ikeda (RIKEN)

2. 12.12.26toki@tensorrcnp2 Pion is a pseudo-scalar particle (Nambu) Pion Tensor spin-spin Difficult to handle Low momentum High momentum

3. 12.12.26toki@tensorrcnp3 運動量（ q ２ ） テンソル力 湯川力（中心力）

4. 12.12.26toki@tensorrcnp4 Deuteron (1 + ) NN interaction S=1 and L=0 or 2

5. 12.12.26toki@tensorrcnp5 Deuteron (1 + ) 80% of attraction comes from tensor D-wave component moves very fast K.Ikeda, T.Myo, K.Kato,and H.Toki Lecture Notes in Phys.818 (2010) 165

6. 12.12.26toki@tensorrcnp6 BE = 5MeV for 4 He

7. 12.12.26toki@tensorrcnp7 How to handle tensor interaction in heavy nuclei Transition from relative S-wave to D-wave provides large attraction In shell model, this is achieved by taking 2p-2h state (Myo et al.) Tensor optimized shell model (TOSM) Transition is important small

8. 12.12.26toki@tensorrcnp8 TOSM (+UCOM) with AV8’ T V LS E VCVC VTVT Few body Calculation (Kamda et al (2001)) (Myo Toki Ikeda) PTP 121 (2009)

9. 12.12.26toki@tensorrcnp9 Configurations in TOSM protonneutron Gaussian expansion nlj particle states hole states (harmonic oscillator basis) Application to Hypernuclei by Umeya to investigate  N-  N coupling C0C0 C1C1 C2C2 C3C3

10. 12.12.26toki@tensorrcnp10 4-8 He with TOSM+UCOM Excitation energies in MeV Bound state app. No continuum No V NNN Excitation energy spectra are reproduced well T.Myo,A.Umeya,H.Toki,K.Ikeda PRC84 (2011) 034315

11. 12.12.26toki@tensorrcnp11 We cannot treat the tensor interaction in HF space. Y.Ogawa H.Toki Annals of Physics 326 (2011) 2039 Extension of Hartree-Fock theory with TOSM TOSM ansatz We improve Brueckner-Hartree-Fock theory

12. 12.12.26toki@tensorrcnp12 Comparison of BHF and EBHF EBHF 1 2 BHF Hartree-Fock equations look very similar

13. 12.12.26toki@tensorrcnp13 TOSM wave function 80~90% Shell model High momentum Component Momentum Distribution Shell model Tensor state Matrix element

14. 12.12.26toki@tensorrcnp14 Variational calculation of few body system with NN interaction C. Pieper and R. B. Wiringa, Annu. Rev. Nucl. Part. Sci.51(2001) VMC+GFMC V NNN Fujita-Miyazawa Relativistic Pion is keyHeavy nuclei (Super model)

15. 12.12.26toki@tensorrcnp15 Importance of delta Importance of delta We treat delta explicitly for three body interaction. Two body potential including delta → Argonne delta model potential (AV28) N NN NN N Δ ΔΔ Δ Δ Δ NN channel NΔ channel ΔΔ channel R.B. Wiringa et al, PRC 29, 1207(1984) AV14 & AV28 Three body interaction

16. 12.12.26toki@tensorrcnp16 Effect of delta in deuteron Effect of delta in deuteron Deuteron 1 + AV14 AV28 L・SL・S 0.36 0.86 L2L2 3.07 3.63 (L ・ S) 2 -4.03 -4.14 P NN [ 3 S 1 ] % 93.96 93.22 P NN [ 3 D 1 ] 6.04 6.23 P ΔΔ [ 3 S 1 ] 0.04 P ΔΔ [ 3 D 1 ] 0.02 P ΔΔ [ 7 D 1 ] 0.43 P ΔΔ [ 7 G 1 ] 0.04 -18.80 -36.12 -2.17-2.33 AV14AV28 Kinetic +Mass Central Energy Tensor -25.99(NN) 21.37+3.16

17. 12.12.26toki@tensorrcnp17 Prior studies for nuclei with delta Prior studies for nuclei with delta No calculations for Too many states are necessary. A=3 1. Hannover group (Germany) Bonn potential + single delta 2. Los Alamos group (Fadeev calculations) AV28 potential → Not enough binding for 3 H Approximation : double Δ up to L=2 Wave function in deuteron ←about 0.04 % TOSM Sauer, PHYS. REV. C68, 024005 (2003) A. Picklesimer etal. Phys. Rev. C46 (1992)

18. 12.12.26toki@tensorrcnp18 Results in 1 Even channel Results in 1 Even channel Delta contribution is about 1/5~1/10 in odd channels 1 E channel L=even, S=even, T=1 J=0 d=1.70 fm no NΔ Energy [MeV] 5.7 8.2 Kinetic 10.51(NN=9.23) 9.51 (NN=9.30) Central 0.90(NN=0.70) -0.12 (NN=-0.12) Tensor -7.92 -1.75 L・SL・S ------ L2L2 (L ・ S) 2 ------ P NN [ 1 S 0 ] % 99.37 99.89 P ΔΔ [ 1 D 0 ] 0.01 0.007 P ΔΔ [ 5 D 0 ] 0.12 0.10 P NΔ [ 5 D 0 ] 0.49 ------ Odd channels

19. 12.12.26toki@tensorrcnp19 テンソル力は高い運動量成分を持ち込む 低い運動量成分は１０％くらい小さくなる 高い運動量成分は新たな行列要素を作る テンソル力がデルタ成分を持ち込む テンソル力は陽子中性子間で働く 不安定核ではテンソル力の寄与は大きく変 化 新しい実験

20. 12.12.26toki@tensorrcnp20 Delta with and without 7 G 1 Delta with and without 7 G 1 Deuteron 1 + FullNo G-wave L・SL・S 0.86 0.66 L2L2 3.63 2.82 (L ・ S) 2 -4.14 -3.21 P NN [ 3 S 1 ] % 93.22 94.29 P NN [ 3 D 1 ] 6.23 5.28 P ΔΔ [ 3 S 1 ] 0.04 0.03 P ΔΔ [ 3 D 1 ] 0.02 0.01 P ΔΔ [ 7 D 1 ] 0.43 0.37 P ΔΔ [ 7 G 1 ] 0.04 ------- -28.91 -36.12 -2.3 -1.7 FullNo G-wave Kinetic Central Energy Tensor -25.99(NN) -21.43(NN) 7.05 Horii et al.

21. 12.12.26toki@tensorrcnp21 Unitary Correlation Operator Method H. Feldmeier, T. Neff, R. Roth, J. Schnack, NPA632(1998)61 short-range correlator Bare Hamiltonian Shift operator depending on the relative distance r (UCOM)

22. 12.12.26toki@tensorrcnp22 4 He with UCOM