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)

Pion is a pseudo-scalar particle (Nambu) Pion Tensor spin-spin Difficult to handle Low momentum High momentum

運動量（ q ２ ） テンソル力 湯川力（中心力）

Deuteron (1 + ) NN interaction S=1 and L=0 or 2

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

BE = 5MeV for 4 He

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

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

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

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)

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

Comparison of BHF and EBHF EBHF 1 2 BHF Hartree-Fock equations look very similar

TOSM wave function 80~90% Shell model High momentum Component Momentum Distribution Shell model Tensor state Matrix element

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)

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

Effect of delta in deuteron Effect of delta in deuteron Deuteron 1 + AV14 AV28 L・SL・S L2L (L ・ S) P NN [ 3 S 1 ] % P NN [ 3 D 1 ] 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 ] AV14AV28 Kinetic +Mass Central Energy Tensor (NN)

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, (2003) A. Picklesimer etal. Phys. Rev. C46 (1992)

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] Kinetic 10.51(NN=9.23) 9.51 (NN=9.30) Central 0.90(NN=0.70) (NN=-0.12) Tensor L・SL・S L2L2 (L ・ S) P NN [ 1 S 0 ] % P ΔΔ [ 1 D 0 ] P ΔΔ [ 5 D 0 ] P NΔ [ 5 D 0 ] Odd channels

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

Delta with and without 7 G 1 Delta with and without 7 G 1 Deuteron 1 + FullNo G-wave L・SL・S L2L (L ・ S) P NN [ 3 S 1 ] % P NN [ 3 D 1 ] P ΔΔ [ 3 S 1 ] P ΔΔ [ 3 D 1 ] P ΔΔ [ 7 D 1 ] P ΔΔ [ 7 G 1 ] FullNo G-wave Kinetic Central Energy Tensor (NN) (NN) 7.05 Horii et al.

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)

4 He with UCOM