Shortrange and tensor correlations in light nuclei studied

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Short-range and tensor correlations in light nuclei studied with antisymmetrized molecular dynamics (TOAMD) Takayuki

Short-range and tensor correlations in light nuclei studied with antisymmetrized molecular dynamics (TOAMD) Takayuki MYO Mengjiao LYU Masahiro ISAKA Hiroshi TOKI Hisashi HORIUCHI Kiyomi IKEDA Tadahiro SUHARA Taiichi YAMADA QNP 2018. 11, Tsukuba (RCNP) (Hosei) (RCNP) (RIKEN) (Matsue) (Kanto Gakuin)

Outline • F

Outline • F

Deuteron properties & tensor force Energy S D -2. 24 Me. V S 11.

Deuteron properties & tensor force Energy S D -2. 24 Me. V S 11. 31 D 8. 57 Kinetic 19. 88 Central -4. 46 SD -18. 93 Tensor -16. 64 DD 2. 29 r [fm] LS -1. 02 P(L=2) S D Radius 5. 77% 1. 96 fm S 2. 00 D 1. 22 d-wave is “spatially compact” (high-momentum)

Prog. Theor. Exp. Phys. 2015, 073 D 02 (38 pages) DOI: 10. 1093/ptep/ptv 087

Prog. Theor. Exp. Phys. 2015, 073 D 02 (38 pages) DOI: 10. 1093/ptep/ptv 087 Tensor-optimized antisymmetrized molecular dynamics in nuclear physics (TOAMD) Takayuki Myo, Hiroshi Toki, Kiyomi Ikeda, Hisashi Horiuchi, and Tadahiro Suhara 4

General formulation of TOAMD tensor • short-range tensor short-range Power series expansion, but, all

General formulation of TOAMD tensor • short-range tensor short-range Power series expansion, but, all F are independent nucleon w. f. Gaussian wave packet spin-isospin dependent Eigenvalue problem

Matrix elements of correlated operator Correlated Hamiltonian Correlated Norm Classify the connections of F,

Matrix elements of correlated operator Correlated Hamiltonian Correlated Norm Classify the connections of F, H into many-body operators using cluster expansion method F† F 3 -body (2 -body)2 4 -body bra V ket i j k l (2 -body)3 relative single particle

Results PLB 769 (2017) 213 PRC 95 (2017) 044314 PRC 96 (2017) 034309 PTEP

Results PLB 769 (2017) 213 PRC 95 (2017) 044314 PRC 96 (2017) 034309 PTEP (2017) 073 D 01 PTEP (2017) 111 D 01 • • Single F AMD Double F F F 2 p-2 h 3 p-3 h F F F 4 p-4 h

 AV 8’ F F S D SS SD DD DS F are independent

AV 8’ F F S D SS SD DD DS F are independent AMD 3 H Correlation functions

 AV 8’ F F S D SS SD DS DD PLB 769 (2017)

AV 8’ F F S D SS SD DS DD PLB 769 (2017) 213 3 H Kinetic/2 F are independent • Reproduce the Hamiltonian components of 3 H Central Tensor Correlation functions

 AV 8’ F F S D SS SD DD DS F are independent

AV 8’ F F S D SS SD DD DS F are independent AMD 4 He Correlation functions

 Amplitude FS 3 E intermediate FS 1 E • r 2×FD 3 E

Amplitude FS 3 E intermediate FS 1 E • r 2×FD 3 E long

 3 H Energy 3 -body 4 He 3 -body 4 -body Energy 2

3 H Energy 3 -body 4 He 3 -body 4 -body Energy 2 -body • Many-body terms play a decisive role for energy saturation • NO energy saturation within 2 -body term 12 PTEP (2017) 073 D 01

High-Momentum Antisymmetrized Molecular Dynamics (HM-AMD) TM et al. Lyu et al. TM Lyu et

High-Momentum Antisymmetrized Molecular Dynamics (HM-AMD) TM et al. Lyu et al. TM Lyu et al. Zhao et al. PTEP (2017) 111 D 01 Tensor correlation vs. TOSM PTEP (2018) 011 D 01 HM-AMD + TOAMD with AV 8’ PTEP (2018) 031 D 01 Short-range correlation with AV 4’ ar. Xiv: 1807. 11213 Submitted to RPC ar. Xiv: 1811. 00271 Submitted to RPC 13

TOAMD for p-shell nuclei • 14

TOAMD for p-shell nuclei • 14

High-momentum AMD (HM-AMD) • cf. TOAMD PTEP (2017) 111 D 01, (2018) 011 D

High-momentum AMD (HM-AMD) • cf. TOAMD PTEP (2017) 111 D 01, (2018) 011 D 01

Lyu’s work: HM-AMD + TOAMD (HM-TOAMD) (RCNP, Osaka) • F

Lyu’s work: HM-AMD + TOAMD (HM-TOAMD) (RCNP, Osaka) • F

Energy, radius and Hamiltonian components • 4 He with AV 8’ bare interaction •

Energy, radius and Hamiltonian components • 4 He with AV 8’ bare interaction • Superpose the basis states with different D’s for pairs. 4 He Add bases successively Lyu et al. PTEP 2018 (2018) 011 D 01, PRC submitted

Energy, radius in 4 He with HM-TOAMD • 4 He with AV 8’ bare

Energy, radius in 4 He with HM-TOAMD • 4 He with AV 8’ bare interaction • Superpose the basis states with different D’s for pairs. HM-TOAMD-F 2 GFMC Energy -24. 74 Kinetic 95. 17 97. 06 102. 3 Central -52. 33 -53. 12 -55. 05 Tensor -63. 80 -64. 84 -68. 05 LS -3. 77 -3. 83 -4. 75 Radius 1. 51 1. 50 -25. 93 [Me. V] 1. 49 [fm] Lyu et al. PTEP 2018 (2018) 011 D 01, PRC • p-shell nuclei with HM-TOAMD is in progress.

Summary • Tensor-Optimized AMD (TOAMD) – Successive variational method for nuclei to treat VNN

Summary • Tensor-Optimized AMD (TOAMD) – Successive variational method for nuclei to treat VNN directly. – Correlation functions : FD (tensor) , FS (short-range). – F are independently optimized, better than Jastrow method. PLB 769 (2017) 213 PRC 96 (2017) 034309 • High-momentum AMD (HM-AMD) PTEP (2017) 111 D 01 – High-momentum pairs using imaginary centroids in Gaussian wave packets – Comparison with shell model : “One high-momentum pair” = 2 p-2 h correlation – Hybrid : HM-TOAMD using VNN PTEP (2018) 011 D 01

Backup 20

Backup 20

 AV 8’ F F S 4 He D SS SD Kinetic/2 DD DS

AV 8’ F F S 4 He D SS SD Kinetic/2 DD DS F are independent Central Tensor Correlation functions

TOAMD & Jastrow method • R. Jastrow, Phys. Rev. 98, 1479 (1955) f f

TOAMD & Jastrow method • R. Jastrow, Phys. Rev. 98, 1479 (1955) f f

TOAMD & VMC with Jastrow VNN : AV 6 for central & tensor forces

TOAMD & VMC with Jastrow VNN : AV 6 for central & tensor forces Energy (Me. V) 3 H Few-body omit LS, L 2, (LS)2 from AV 14 TOAMD (power series) gives better energy than VMC (Jastrow) from variational point of view Independent optimization of all F 4 He PRC 96 (2017) 034309 Correlation functions JPS magazine (2017) Dec. 867

 AV 6 S D SS SD DS DD same Few-body F Kinetic/2 F

AV 6 S D SS SD DS DD same Few-body F Kinetic/2 F Central 3 H Tensor Correlation functions Free F

Diagrams of cluster expansion - VNN bra V 2 -body F F V V

Diagrams of cluster expansion - VNN bra V 2 -body F F V V ket 3 -body F 5 -body 6 -body 4 -body

Diagrams of cluster expansion, Kinetic energy 1 -body 2 -body bra T ket F

Diagrams of cluster expansion, Kinetic energy 1 -body 2 -body bra T ket F F T T F uncorrelated kinetic energy 5 -body 3 -body 4 -body

Convergence of HM-AMD • VNN : Central : Volkov No. 2 with M=0. 6

Convergence of HM-AMD • VNN : Central : Volkov No. 2 with M=0. 6 LS & Tensor : G 3 RS bare force 4 He HM-AMD with 1 pair (0 s)4 Add bases successively Tensor-optimized shell model (TOSM), complete 2 p-2 h. “Criterion”

Energy surface with the shift Dz • VNN : Central Volkov No. 2 with

Energy surface with the shift Dz • VNN : Central Volkov No. 2 with M=0. 6, Coulomb, LS & Tensor : G 3 RS bare force (0 s)4 TM, K. Kato, K. Ikeda 2 bases : (0 s)4 + pn pair with Dz 4 He Superpose by increasing Dz

Hamiltonian Component in HM-AMD 4 He 29

Hamiltonian Component in HM-AMD 4 He 29

HM-AMD & TOSM for 4 He • Tensor-Optimized Shell Model (Me. V) HM-AMD TOSM

HM-AMD & TOSM for 4 He • Tensor-Optimized Shell Model (Me. V) HM-AMD TOSM Energy − 64. 7 − 65. 2 Kinetic 83. 6 84. 8 Central − 83. 6 − 84. 1 Tensor − 66. 3 − 67. 7 LS 0. 8 0. 9 Coulomb 0. 9 Myo, Sugimoto, Kato, Toki, Ikeda Prog. Theor. Phys. 117 (2007) 257 NO truncation of particle states in TOSM using Gaussian expansion with high-L more than 10ℏ HM-AMD = TOSM (1 pair) (full 2 p-2 h) Next, bare VNN (Lyu, Isaka)

Tensor-optimized Shell Model for 4 He NO truncation of particle states using Gaussian expansion

Tensor-optimized Shell Model for 4 He NO truncation of particle states using Gaussian expansion Myo, Sugimoto, Kato, Toki, Ikeda Prog. Theor. Phys. 117 (2007) 257 • 9 Gaussian bases for each orbit with L • “Criterion” to examine 2 p-2 h effect (0 s)4 (single particle orbit)

 AV 8’ F F Single SS SD DS 3 H Single +SS +SD+DS

AV 8’ F F Single SS SD DS 3 H Single +SS +SD+DS +DD DD F are independent • Getting close to the GFMC energy. • n-dependence is small due to the flexibility of F.

 Kinetic 3 H F Energy 4 He LS Central Tensor • 33

Kinetic 3 H F Energy 4 He LS Central Tensor • 33

 uncorrelated kinetic Kinetic Full T+ 2 -body 3 -body same trend in central

uncorrelated kinetic Kinetic Full T+ 2 -body 3 -body same trend in central & tensor 2 -body > 3 -body not negligible Tensor Central 34

Many-body Hamiltonian terms in 4 He PTEP (2017) 073 D 01 Kinetic Full 1+2

Many-body Hamiltonian terms in 4 He PTEP (2017) 073 D 01 Kinetic Full 1+2 body 3 -body 4 -body Central Higher-body term tends to give smaller scale, but, not ignored. Tensor 35

 AV 8’ S D SS SD DS DD PTEP (2017) 073 D 01

AV 8’ S D SS SD DS DD PTEP (2017) 073 D 01 cf. Jastrow ansatz 3 H DE=1. 4 Me. V Correlation functions F are independent as a full variation

 AV 8’ S D SS SD DS DD same Kinetic/2 3 H F

AV 8’ S D SS SD DS DD same Kinetic/2 3 H F F Few-body Correlation functions Central Tensor Free F

 AV 8’ S D SS SD DS DD PTEP (2017) 073 D 01

AV 8’ S D SS SD DS DD PTEP (2017) 073 D 01 cf. Jastrow ansatz 4 He DE=2. 3 Me. V Correlation functions F are independent as a full variation

 AV 8’ S D SS SD 4 He DS DD Kinetic/2 same F

AV 8’ S D SS SD 4 He DS DD Kinetic/2 same F F Central Tensor Correlation functions Free F Independent optimization of all F