1. Strong tensor correlation in light nuclei with tensor-optimized antisymmetrized molecular dynamics (TOAMD) International symposium on “High-resolution Spectroscopy and Tensor interactions” (HST15), Osaka, 2015.11 Takayuki MYO

2. 2 Outline Tensor Optimized Shell Model (TOSM) Importance of 2p2h excitation involving high- momentum component for tensor correlation Applications to He, Li, Be isotopes Tensor Optimized AMD (TOAMD) Toward clustering with tensor correlation Formulation given in PTEP 2015, 073D02

3. 3 Pion exchange interaction & V tensor  interaction Yukawa interaction involve large momentum Tensor operator -V tensor produces the high momentum component.

4. S D Energy -2.24 MeV Kinetic19.88 Central -4.46 Tensor-16.64 LS -1.02 P( L=2 ) 5.77% Radius 1.96 fm V central V tensor AV8’ Deuteron properties & tensor force R m (s)=2.00 fm R m (d)=1.22 fm d-wave is “spatially compact” (high momentum) r

5. c.m. excitation is excluded by Lawson’s method (0p0h+1p1h+2p2h) particle state Gaussian expansion for each orbit Gaussian basis function Hiyama, Kino, Kamimura PPNP51(2003)223 Shell model type configuration with mass number A Tensor-optimized shell model (TOSM) TM, Sugimoto, Kato, Toki, Ikeda PTP117(2007)257

6. particle hole 6 Tensor force matrix elements 6 Centrifugal potential (1GeV@0.5fm) pushes away D-wave.  +  (Bonn) AV8’ V T = V residual V T ≠ V residual M SD (r) = r 2 ·  S (r,b S ) · V T ·  D (r,b D ) Integrand of Tensor ME b D ~ b S b D ~ b S  0.5 1 st order 0p0h-2p2h 3.8 MeV 15.0 MeV high momentum

7. He, Li, Be isotopes in TOSM 7 TM, A. Umeya, H. Toki, K. Ikeda PRC84 (2011) 034315 TM, A. Umeya, H. Toki, K. Ikeda PRC86 (2012) 024318 TM, A. Umeya, K. Horii, H. Toki, K. Ikeda PTEP (2014) 033D01 TM, A. Umeya, H. Toki, K. Ikeda PTEP (2015) 073D02

8. T VTVT V LS VCVC E (exact) Kamada et al. PRC64 (Jacobi) Gaussian expansion at most with 9 bases variational calculation TM, H. Toki, K. Ikeda PTP121(2009)511 4 He in TOSM + central UCOM good convergence High- k components bring large kinetic energy

9. Selectivity of the tensor coupling in 4 He 9  L = 2  S =  2 Selectivity of S 12 s1s1 s2s2 s3s3 s4s4 1 + (pn) s1s1 s2s2 s3s3 s4s4 L=2 s1s1 s2s2 s3s3 s4s4 VTVT VTVT l 1 = l 2 = l 3 = l 4 =0 l 1 = l 2 =0 l 3 =1 l 4 =1 l 3 =0 l 4 =2

10. 5-8 He with TOSM+UCOM Excitation energies in MeV Excitation energy spectra are reproduced well TM, A. Umeya, H. Toki, K. Ikeda PRC84 (2011) 034315

11. 5-9 Li with TOSM+UCOM Excitation energies in MeV Excitation energy spectra are reproduced well TM, A. Umeya, H. Toki, K. Ikeda PRC86(2012) 024318

12. 12 Radius of He & Li isotopes I. Tanihata et al., PLB289(‘92)261 G. D. Alkhazov et al., PRL78(‘97)2313 O. A. Kiselev et al., EPJA 25, Suppl. 1(‘05)215. P. Mueller et al., PRL99(2007)252501 Expt HaloSkin A. Dobrovolsky, NPA 766(2006)1 TOSM with AV8’

13. 8 Be spectrum 13  -  structure Argonne Group –Green’s function Monte Carlo C. Pieper, R. B. Wiringa, Annu.Rev.Nucl.Part.Sci.51 (2001) TUNL  ~ few MeV  < 1MeV

14. 8 Be in TOSM  AV8’  V T  1.1, V LS  1.4 –simulate 4 He benchmark (Kamada et al., PRC64) correct level order (T=0,1) tensor contribution : T=0 > T=1  0p0h+2p2h with high- k –2  needs  4p4h. –spatial asymptotic form of 2  14   clustering TOSM Expt. (TUNL)  TOAMD

15. Tensor-Optimized Antisymmetrized Molecular Dynamics (TOAMD) 15 TM, H. Toki, K. Ikeda, H. Horiuchi, T. Suhara, PTEP 2015, 073D02 Toward the clustering with tensor correlation explicitly

16. (TOAMD) tensor correlation short-range correlation 2p2h 0p0h Gaussian expansion

17. Formulation of TOAMD 17 2p2h 4p4h 0p0h

18. Matrix elements of multi-body operator VTVT FDFD FDFD 2-body VTVT FDFD FDFD 4-body Matrix elements particle coordinate centroid range

19. Matrix elements with Fourier trans. Fourier transformation of the interaction V & F D, F S. –Y. Goto and H. Horiuchi, Prog. Theor. Phys., 62 (1979) 662 –Gaussian expansion of V, F D, F S for relative motion –Multi-body operators are represented in the separable form for particle coordinates. –Three-body interaction can be treated in the same manner. 19 relative single particle tensor-type overlap Quadratic + Linear terms of k

20. Matrix elements with Fourier trans. Example : F D  V T VTVT FDFD 2-body (2-body)  (2-body) = (2-body)+(3-body)+(4-body) k -integral spatial part (tensor) 2 spin-isospin partantisymmetrization a1a1 a2a2 B ij =  i  j  overlap

21. Matrix elements with Fourier trans. Example : F D  V T 21 VTVT FDFD 3-body (2-body)  (2-body) = (2-body)+(3-body)+(4-body) k -integral spatial part (tensor) 2 spin-isospin partantisymmetrization a1a1 a2a2 B ij =  i  j  overlap

22. TOSM vs. TOAMD 22 TOSMTOAMD Correlation 1p1h, 2p2h (single particle) Relative motion in F D, F S CM excitationLawson method Nothing with single Hole states harmonic oscillator basis Can optimize in each basis Short-range repulsion central-UCOM Correlation function, F S

23. Results 3-body 2-body VTVT FDFD FDFD FDFD FDFD VTVT others... FDFD FDFD VTVT FSFS FSFS T FSFS FSFS T

24. 3 H energy surface 24 Good convergence for Gaussian numbers N G. Obtain the saturation property with good radius. Gaussian expansion compactwide Radius=1.76 fm

25. Three-body term in energy of 3 H 25 No saturation point with one- & two-body terms. Three-body term has a saturation behavior. compactwide 3-body term 1& 2-body terms Full Radius=1.76 fm

26. Energy components E AMD E TOAMD TVCVC VTVT V LS PDPD 3H 3H16.9-5.037.9-17.4-24.1-1.36.4 4 He41.4-13.361.5-34.9-38.3-1.67.3 8 Be110.3-27.6134.7-91.0-69.1-2.29.4 26 in MeV preliminary in progress

27. Correlation functions F D, F S in 3 H 27 2  r 2 F D 3 E F S 1 E F S 3 E same trend in 3 H, 4 He, 8 Be preliminary

28. 28 Summary Tensor-Optimized Shell Model (TOSM) using V bare. –Strong tensor correlation from 0p0h-2p2h involving high- k components. –He, Li, Be isotopes : Energy spectra, Radius of n -rich nuclei. –For 8 Be, two aspects : grand state region & highly excited states, which indicates of more configurations, such as  states. Tensor-Optimized AMD (TOAMD). –Two-kinds of correlation functions F D (tensor) & F S (short-range) –Three-body term contributes to the energy saturation. –Ranges of F D, F S are not short. –Spatially compact behavior of F D produces high- k component.