Realistic shell model and chiral three-body force Tokuro Fukui Yukawa Institute for Theoretical Physics, Kyoto University 22/October/2020
Biography 1 Apr. 2012-Mar. 2015 Ph. D. Apr. 2015-Sep. 2016 Postdoctoral Fellow Sep. 2016-Aug. 2018 Postdoctoral Fellow Sep. 2018-Oct. 2019 JSPS Overseas Research Fellow Nov. 2019-Present Research Assistant professor
Biography 1 Apr. 2012-Mar. 2015 Ph. D. March 2021 SPDR at Strangeness Nuclear Physics Lab. Apr. 2015-Sep. 2016 Postdoctoral Fellow Sep. 2016-Aug. 2018 Postdoctoral Fellow Sep. 2018-Oct. 2019 JSPS Overseas Research Fellow Nov. 2019-Present Research Assistant professor
Collaborators 2 INFN-Napoli A. Gargano A. Gargano L. Coraggio L. Coraggio N. Itaco N. Itaco L. De Angelis L. De Angelis G. De Gregorio G. De Gregorio Peking University F F . R. Xu . R. Xu Y. Z. Ma Y. Z. Ma South China Normal Univ.
Nuclear force | History 3 ‟ circle of history is closing” Machleidt & Entem, PR 503 , 1 (2011) Yukawa: Meson theory 1935 EFT Pion theories Chiral Symmetry 1990’s and beyond 1950’s 1-boson exchange QCD 1980’s 1960’s Diverse 2-pion exchange 1970’s
Nuclear force | State-of-the-art theories 4 Chiral effective field theory … Chiral symmetry ( ! N hierarchy) Many-body forces Realistic force ( χ 2 ≲ 1 ) Weinberg, PA 96 , 327 (1979) Machleidt & Entem, PR 503 , 1 (2011)
Nuclear force | State-of-the-art theories 5 Chiral effective field theory Nuclear force from Lattice QCD 100 600 1 S 0 3 S 1 500 OPEP 50 Future V C (r) [MeV] 400 300 0 200 -50 100 0.0 0.5 1.0 1.5 2.0 0 0.0 0.5 1.0 1.5 2.0 r [fm] … Ishii +, PRL 99 , 022001 (2007) Chiral symmetry ( ! N hierarchy) Many-body forces Realistic force ( χ 2 ≲ 1 ) Weinberg, PA 96 , 327 (1979) Machleidt & Entem, PR 503 , 1 (2011)
Motivation | Why chiral-EFT interaction? 6 Significance Microscopic origin (chiral symmetry and ! N hierarchy) Many-body forces on an equal footing χ 2 ≲ 1 Precise and hence realistic ( ) I expect to deepen and shed new light on the understanding of nuclear force and properties of nuclei. This work Relation between single-particle properties and chiral 3-nucleon force (3NF) elucidated by shell-model framework
Nuclear structure models | Realistic force 7 Realistic shell model (RSM) = Shell model with a realistic force Shell-model framework and model-space truncation Shell model ab initio Realistic force is applicable in a straightforward way Active nucleons Hartree-Fock method Difficult to find exact Kohn-Sham potential Configurations
8 RSM and 3NF 3NF contribution to RSM RSM Hamiltonian (in particular its monopole component) needs to be revised due to 3NF . Zuker, PRL 90 , 042502 (2003) cf .) Oxygen-drip line and 3NF Otsuka +, PRL 105 , 032501 (2010) The 3NF qualitatively accounts for the oxygen-drip line ( 24 O). Fujita-Miyazawa 3NF 4 Single-Particle Energy (MeV) (c) G-matrix NN + 3N ( ∆ ) forces Repulsive contribution 0 d3/2 -4 s 1/2 d5/2 -8 NN + 3N ( ∆ ) NN Fujita & Miyazawa, PTP 17 , 360 (1957) 8 1 4 1 6 20 Neutron Number ( N )
Motivation | RSM and 3NF 9 Shell evolution on pf -shell and RSM Holt +, PRC 90 , 024312 (2014) A crucial role played by 3NF (Chiral N 2 LO) for Ca isotopes. 0 ( a ) p f s h e l l 6 -30 5 Energy (MeV) Energy (MeV) -60 4 -90 3 2 -120 NN 1 NN+3N -150 0 NN NN NN+3N NN+3N Expt. GX KB3G 40 44 48 52 56 60 40 pf pfg 9/2 pf pfg 9/2 Mass Number A 3NF contributions need to be clarified in detail (monopole Hamiltonian)
Theoretical framework | Shell-model Hamiltonian 10 Realistic Hamiltonian (starting point) Single-particle Chiral 2NF Chiral 3NF energy at N 3 LO at N 2 LO + Coulomb
Theoretical framework | Shell-model Hamiltonian 10 Realistic Hamiltonian (starting point) 3-body matrix elements (3BMEs) Single-particle Chiral 2NF Chiral 3NF Our new formalism energy at N 3 LO at N 2 LO Parallelized code for HPC + Coulomb Fukui +, PRC 98 , 044305 (2018)
Theoretical framework | Shell-model Hamiltonian 10 Realistic Hamiltonian (starting point) 3-body matrix elements (3BMEs) Single-particle Chiral 2NF Chiral 3NF Our new formalism energy at N 3 LO at N 2 LO Parallelized code for HPC + Coulomb Fukui +, PRC 98 , 044305 (2018) Shell-model framework ab initio no-core shell model (NCSM) Diagonalization Eigenvalues Effective Hamiltonian Eigenvectors Many-body Diagonalization RSM perturbation theory Renormalization Navrátil +, PRL 84 , 5728 (2000) Optional Normal-order approximation Coraggio + , AP 327 , 2125 (2012)
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Chiral 3BMEs | Nonlocal regulator 12 High-momentum truncation by regulator with cutoff Λ Nonlocal regulator Epelbaum + , PRC 66 , 064001 (2002) 1.00 0.75 0.50 0.25 0.00 Necessary to retain 0 consistency of the contact term! Nonlocal 3BMEs with HO bases Pioneering work Navrátil, FBS 41 , 117 (2007) Only for the 1 " +contact and contact terms. Present work Fukui + , PRC 98 , 044305 (2018) New formalism for 2 " terms: Triple-fold multipole expansion
Chiral 3BMEs | 2 " terms 13 3BMEs of 2 " terms Fukui + , PRC 98 , 044305 (2018) 23 sums 26 3 nj symbols, etc. Triple-fold integration Computationally heavy! MPI + OpenMP parallelization MARCONI (CINECA, Italy) # of MEs Time Memory ~30 sec p -shell ~800 ~500 MB w/ 4 nodes, 48 threads ~10 min sd -shell ~20,000 ~3 GB w/ 60 nodes, 272 threads ~5 h pf-shell ~200,000 ~30 GB w/ 60 nodes, 272 threads
RSM calculations | Numerical details 14 Low-energy constants 2NF : N 3 LO Entem & Machleidt, PRC 68 , 041001(R) (2003) 3NF : N 2 LO Navrátil +, PRL 99 , 042501 (2007) Model space Many-body perturbation theory 0 p 1/2 0 p 3/2 Particle p -shell 2NF : Up to 3 rd -order folded-diagram expansion 3NF : Up to 1 st -order (normal-order approx.) Hole 0 s 1/2 Coraggio + AP 327 , 2125 (2012) Roth +, PRL 109 , 052501 (2012) 1 p 1/2 (1) Effective Hamiltonian involving H e ff 0 f 5/2 Particle 1 p 3/2 pf -shell Q -space effect . 0 f 7/2 (2) Theoretical single-particle energies Hole 0 s , 0 p , 0 d, 1 s and 2BMEs . No empirical inputs!
RSM calculations | First-order approximation 15 Many-body Hamiltonian and 3-body operator First-order (normal-order) approximation R. Roth +, PRL 109 , 052501 (2012) Normal-ordered 1-body term Normal-ordered 2-body term
RSM calculations | First-order approximation 15 Many-body Hamiltonian and 3-body operator Particle Excluded Excluded Hole First-order (normal-order) approximation R. Roth +, PRL 109 , 052501 (2012) Normal-ordered 1-body term Normal-ordered 2-body term
p -shell nuclei: Benchmark calculations
Benchmark calculations | p -shell nuclei 17 Comparison with NCSM Fukui + , PRC 98 , 044305 (2018) RSM and NCSM agree 2NF only with each other for low-lying states . Significant 3NF effect can be seen. 2NF + 3NF NCSM Navrátil +, PRL 99 , 042501 (2007).
Benchmark calculations | p -shell nuclei 18 Comparison with NCSM Fukui + , PRC 98 , 044305 (2018) Similar results for other p -shell nuclei RSM and NCSM agree → RSM: Simple and precise 2NF only with each other for low-lying states . 12 C 13 C 8 B 10 B 11 B Significant 3NF effect 8 Be can be seen. 6 Li 8 Li 4 He 2NF + 3NF NCSM Navrátil +, PRL 99 , 042501 (2007).
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