Realistic shell model and chiral three-body force Tokuro Fukui - - PowerPoint PPT Presentation

Tokuro Fukui

22/October/2020

Realistic shell model 
 and 
 chiral three-body force

Yukawa Institute for Theoretical Physics,
 Kyoto University

• 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.


• 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

March 2021


SPDR at Strangeness Nuclear Physics Lab.

Collaborators

INFN-Napoli

F . R. Xu F . R. Xu

• Y. Z. Ma
• Y. Z. Ma

Peking University

2

• A. Gargano
• A. Gargano
• N. Itaco
• N. Itaco
• L. Coraggio
• L. Coraggio
• L. De Angelis
• L. De Angelis
• G. De Gregorio
• G. De Gregorio

South China Normal Univ.

Nuclear force | History 3

1950’s

Pion theories

1960’s

1-boson exchange

1970’s

Diverse 2-pion 
 exchange

1980’s

QCD

1990’s and beyond

EFT
 Chiral Symmetry Yukawa: 
 Meson theory

1935

Machleidt & Entem, PR 503, 1 (2011)

‟circle of history is closing”

Nuclear force | State-of-the-art theories 4

Chiral effective field theory

Weinberg, PA 96, 327 (1979) Machleidt & Entem, PR 503, 1 (2011)

…

Chiral symmetry (!N hierarchy)
 Many-body forces Realistic force ( )

χ2 ≲ 1

100 200 300 400 500 600 0.0 0.5 1.0 1.5 2.0 VC(r) [MeV] r [fm]

• 50

50 100 0.0 0.5 1.0 1.5 2.0

1S0 3S1

OPEP

Ishii +, PRL 99, 022001 (2007)

Nuclear force 
 from Lattice QCD

5

Future

Nuclear force | State-of-the-art theories

Chiral effective field theory

Weinberg, PA 96, 327 (1979) Machleidt & Entem, PR 503, 1 (2011)

…

Chiral symmetry (!N hierarchy)
 Many-body forces Realistic force ( )

χ2 ≲ 1

6

Significance

Motivation | Why chiral-EFT interaction?

Microscopic origin (chiral symmetry and !N hierarchy) 
 Many-body forces on an equal footing Precise and hence realistic (

)

χ2 ≲ 1

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

Realistic shell model (RSM)

= Shell model with a realistic force

Shell-model framework and model-space truncation

ab initio Configurations Active nucleons Shell model Realistic force is applicable
 in a straightforward way Hartree-Fock method Difficult to find exact Kohn-Sham potential

7

RSM and 3NF

Otsuka +, PRL 105, 032501 (2010)

The 3NF qualitatively accounts for the oxygen-drip line (24O).

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 Fujita-Miyazawa 3NF

Repulsive contribution

Fujita & Miyazawa, PTP 17, 360 (1957)

8 20 4 1 6 1 Neutron Number (N)

s 1/2

(c) G-matrix NN + 3N (∆) forces

d3/2 d5/2

NN NN + 3N (∆)

Single-Particle Energy (MeV)

4

• 4
• 8

8

Shell evolution on pf-shell and RSM

Holt +, PRC 90, 024312 (2014)

A crucial role played by 3NF (Chiral N2LO) for Ca isotopes.

Motivation | RSM and 3NF

3NF contributions need to be clarified in detail 


(monopole Hamiltonian)

40 44 48 52 56 60

Mass Number A

• 150
• 120
• 90
• 60
• 30

Energy (MeV)

NN NN+3N

40 l l e h s f p

) a (

NN pf NN pfg9/2 NN+3N pf NN+3N pfg9/2

• Expt. GX

1 2 3 4 5 6

Energy (MeV)

KB3G

9

Theoretical framework | Shell-model Hamiltonian

Realistic Hamiltonian (starting point)

Chiral 3NF
 at N2LO Single-particle energy Chiral 2NF at N3LO
 + Coulomb

10

Theoretical framework | Shell-model Hamiltonian

Realistic Hamiltonian (starting point)

Chiral 3NF
 at N2LO Single-particle energy Chiral 2NF at N3LO
 + Coulomb

10

Our new formalism Parallelized code for HPC 3-body matrix elements (3BMEs)

Fukui +, PRC 98, 044305 (2018)

Theoretical framework | Shell-model Hamiltonian

Realistic Hamiltonian (starting point)

Chiral 3NF
 at N2LO Single-particle energy Chiral 2NF at N3LO
 + Coulomb

Shell-model framework

Many-body 
 perturbation theory Diagonalization Effective Hamiltonian Eigenvalues
 Eigenvectors Diagonalization ab initio no-core shell model (NCSM) RSM Renormalization Normal-order approximation Optional Our new formalism Parallelized code for HPC 3-body matrix elements (3BMEs)

Fukui +, PRC 98, 044305 (2018) Navrátil +, PRL 84, 5728 (2000) Coraggio +, AP 327, 2125 (2012)

10

Chiral 3BMEs | Harmonic-oscillator (HO) bases

How to compute 3BMEs

Talmi transformation Talmi, HPA 25, 185 (1952)

Nogga +, PRC 73 , 064002 (2006)

Center-of-mass separation Diagonalization of 
 antisymmetrizer

Navrátil +, PRC 61, 044001 (2000)

Antisymmetrization

Weinberg, PLB 295, 114 (1992) van Kolck, PRC 49, 2932 (1994)

Chiral N2LO 3NF 2π (

) c1, c3, c4

1π + contact (

) cD

Contact (

) cE

hV3Ni =

3BMEs

=

A

Dh⇥ ⇤ i

JT

• V3N
• h⇥

⇤ i

JT

E

A

11

Pioneering work

Only for the 1"+contact and contact terms.


Present work

New formalism for 2" terms:
 Triple-fold multipole expansion

Chiral 3BMEs | Nonlocal regulator

High-momentum truncation by regulator with cutoff Λ

Navrátil, FBS 41, 117 (2007)

Nonlocal 3BMEs with HO bases Nonlocal regulator

Epelbaum +, PRC 66, 064001 (2002) Fukui +, PRC 98, 044305 (2018) 0.00 0.25 0.50 0.75 1.00 Necessary to retain
 consistency


• f the contact term!

12

3BMEs of 2" terms

MARCONI (CINECA, Italy)

Fukui +, PRC 98, 044305 (2018)

Chiral 3BMEs | 2" terms

Computationally heavy!

MPI + OpenMP parallelization

23 sums 26 3nj symbols, etc. Triple-fold integration # of MEs Time Memory p-shell ~800 ~30 sec
 w/ 4 nodes, 48 threads ~500 MB sd-shell ~20,000 ~10 min w/ 60 nodes, 272 threads ~3 GB pf-shell ~200,000 ~5 h w/ 60 nodes, 272 threads ~30 GB

13

RSM calculations | Numerical details

Low-energy constants

2NF: N3LO 3NF: N2LO

Entem & Machleidt, PRC 68, 041001(R) (2003) Navrátil +, PRL 99, 042501 (2007)

Model space

0p1/2 0p3/2 0s1/2 Particle Hole 1p1/2 0f5/2 0f7/2 Particle Hole 1p3/2 0s, 0p, 0d, 1s

Many-body perturbation theory

2NF: Up to 3rd-order folded-diagram expansion
 3NF: Up to 1st-order (normal-order approx.)

Coraggio + AP 327, 2125 (2012) Roth +, PRL 109 , 052501 (2012)

(1) Effective Hamiltonian involving 
 Q-space effect. (2) Theoretical single-particle energies
 and 2BMEs.

Heff

p-shell pf-shell No empirical inputs!

14

RSM calculations | First-order approximation

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

15

RSM calculations | First-order approximation

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 Particle Hole Excluded Excluded

15

p-shell nuclei:

Benchmark calculations

Benchmark calculations | p-shell nuclei

Comparison with NCSM

Navrátil +, PRL 99, 042501 (2007).

NCSM RSM and NCSM agree
 with each other 
 for low-lying states.
 Significant 3NF effect
 can be seen. 2NF only 2NF + 3NF

Fukui +, PRC 98, 044305 (2018)

17

Navrátil +, PRL 99, 042501 (2007).

NCSM RSM and NCSM agree
 with each other 
 for low-lying states.
 Significant 3NF effect
 can be seen.

Benchmark calculations | p-shell nuclei

Comparison with NCSM

2NF only 2NF + 3NF

4He 6Li 8Li 8Be 8B 10B 11B 12C 13C

Similar results for other p-shell nuclei → RSM: Simple and precise

Fukui +, PRC 98, 044305 (2018)

18

Benchmark calculations | p-shell nuclei

Ground-state energies

At maximum 
 ~ 1 MeV discrepancy Higher-order 3NF contribution may be 
 necessary for single-particle energies. 2NF + 3NF

6 8 10 12

A

• 70
• 60
• 50
• 40
• 30
• 20
• 10

N=Z isotopes Expt NCSM RSM

2NF only

6 8 10 12

A

• 70
• 60
• 50
• 40
• 30
• 20
• 10

g.s . energy (MeV) N=Z isotopes Expt NCSM RSM

We focus on the relative energies!

Fukui +, PRC 98, 044305 (2018)

19

Individual contributions | 6Li ground-state energy

• 12
• 10
• 8
• 6
• 4
• 2

2 4

2NF 2NF +3NF Exp

2! dominant ( in particular) Large coherent effects

c4

20

Monopole interaction

Core Monopole interaction = J-averaged 2-body interaction Effective single-particle energy (ESPE) Monopole-Hamiltonian ME = Evolved single-particle energy Spherical mean field

Tool to study system in an effective way

21

Monopole property | p-shell nuclei

Effective single-particle energies

0p1/2 0p3/2

2NF only decreases with A, and 
 the two orbits become inverted.

0p1/2 0p3/2

2NF + 3NF is almost constant. →

2NF+3NF 2NF only

4 8 12 A

• 3
• 2
• 1

1 2 3 4 5 6 ESPE (MeV) Proton Neutro n

8 Better closure properties

Fukui +, PRC 98, 044305 (2018)

22

Short summary 1

Benchmark test for p-shell Our RSM calculations with 3NF 
 are satisfactorily comparable to 
 the ab initio results (relative energies).

0p1/2 0p3/2

Spin-orbit splitting stabilized by 3NF

2NF+3NF 2NF only

4 8 12 A

• 3
• 2
• 1

1 2 3 4 5 6 ESPE (MeV) Proton Neutro n

8 Probably due to the tensor force

(under investigation).

23

pf-shell nuclei:

Shell evolution and drip line

Shell evolution

http://griffincollaboration.github.io/griffin-website/physics.html

Shell structure is related to 
 the robustness of the energy gap.
 
 “Observed” from
 two-neutron separation energies
 and first excitation energies

25

pf-shell nuclei | Monopole properties

Even 2NF reasonably accounts for experimental behavior of 
 2n-separation energies S2n 
 and closure properties.

Ca isotopes

22 24 26 28 30 N 1 2 3 4 2

+ excitation energy (MeV)

22 24 26 28 30 32 34 N 5 10 15 20 S2n (MeV)

Exp 2NF 2NF + 3NF

B(E2) 
 [e2fm4]

Ma +, PRC 100, 034324 (2019)

26

Even 2NF reasonably accounts for experimental behavior of 
 2n-separation energies S2n 
 and closure properties. The 3NF monopole component 
 is small except for 48Ca.

pf-shell nuclei | Monopole properties

22 24 26 28 30 N 1 2 3 4 2

+ excitation energy (MeV)

Ca isotopes

22 24 26 28 30 32 34 N 5 10 15 20 S2n (MeV)

Exp 2NF 2NF + 3NF 2NF + 3NF monopole

B(E2) 
 [e2fm4]

Ma +, PRC 100, 034324 (2019)

26

2NF fails and 3NF plays an important role to explain experimental data.

pf-shell nuclei | Monopole properties

Ni isotopes

22 24 26 28 30 N 20 25 30 35 S2n (MeV) 20 22 24 26 28 30 N 1 2 3 4 2

+ excitation energy (MeV)

Exp 2NF 2NF + 3NF

B(E2) 
 [e2fm4]

Ma +, PRC 100, 034324 (2019)

27

pf-shell nuclei | Monopole properties

Ni isotopes

2NF fails and 3NF plays an important role to explain experimental data. The 3NF monopole component is essentially improves the calculations.

22 24 26 28 30 N 20 25 30 35 S2n (MeV) 20 22 24 26 28 30 N 1 2 3 4 2

+ excitation energy (MeV)

Exp 2NF 2NF + 3NF 2NF + 3NF monopole

B(E2) 
 [e2fm4]

Why opposite from Ca?

Ma +, PRC 100, 034324 (2019)

27

20 22 24 26 28 N 2.5 5 7.5 10 12.5

Neutron ESPE for Ca 2NF only 2NF + 3NF

f7/2 f5/2 p3/2 p1/2

ESPEs relevant for Ca isotopes

< 1 MeV Very small difference from 2NF to 2NF + 3NF .

pf-shell nuclei | Monopole properties 28

ESPEs relevant for Ni isotopes

Drastic evolution of ESPEs 
 induced by the 3NF . 0f7/2-1p3/2 gap is enlarged. Proton 0f5/2 ESPE is enhanced.

pf-shell nuclei | Monopole properties

20 22 24 26 28 N

• 5
• 2,5

2,5 5 7,5

f7/2 f5/2 p3/2 p1/2

2,5 5 7,5 10 12,5

f7/2 p3/2

Neutron ESPE for Ni

f5/2 p1/2

Proton ESPE for Ni 2NF only 2NF + 3NF neutron-proton interaction 
 induced by the 2" term in Ni 
 is more relevant than that in Ca.

Naively…

The term has the operator

c4

which vanishes for identical particles.

29

Experimental and theoretical efforts

pf-shell nuclei | Beyond 1-major shell

Extended model space

1p1/2 0f5/2 0f7/2 Particle Hole 1p3/2 0s, 0p, 0d, 1s 0g9/2

Where is the Ca-drip line?

Tarasov +, PRL 121, 022501 (2018) Neufcourt +, PRL 122, 062502 (2019)

30

Towards drip line of Ca isotopes

pf-shell nuclei | Beyond 1-major shell

22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 N 4 8 12 16 20

2n Heff 4 orbitals Heff 5 orbitals Expt

12 S2n (MeV)

Exp fp + g9/2 fp only

H H Expt H

22 24 26 28 30 32 34 36 38 40 42 44 46 48 N 1 2 3 4 21

+ excitation energy (MeV)

Expt Heff 5 orbitals Heff 4 orbitals

Exp fp + g9/2 fp only

H H Expt H

g9/2 g9/2

Bound 60Ca: 
 Consistent with experiment

Coraggio+, arXiv:2006.15196 (PRC accepted) Tarasov +, PRL 121, 022501 (2018)

Bound 70Ca:
 Consistent with other predictions

Density functional theories 
 
 Bayesian analysis

Kortelainen +, PRC 85, 024304 (2012) Goriely +, PRC 88, 024308 (2013) Wang +, PLB 734, 215 (2014) Neufcourt +, PRL 122, 062502 (2019)

31

Short summary 2

Neutron-drip line of Ca isotopes Monopole properties & spin-orbit splitting stabilized by 3NF The 3NF-induced monopole 
 Hamiltonian is essential to explain 
 the measured shell evolution. Probably due to the tensor force


(under investigation).

20 22 24 26 28 30 N 1 2 3 4 2

+ excitation energy (MeV)

1p1/2 0f5/2 0f7/2 1p3/2

28

1p1/2 0f5/2 0f7/2 Particle Hole 1p3/2 0s, 0p, 0d, 1s 0g9/2

70Ca: Possibly bound state,


consistent with other predictions.

32

sd-shell nuclei:

Neutron-drip line and beyond

Beyond drip line | Oxygen isotopes

Interplay between continua and 3NF

1

Otsuka +, PRL, 105, 032501 (2010)

OK as to measured drip line 24O, but quantitatively insufficient Bound 28O, 
 conflicting with experiment

Fossez +, PRC 96, 024308 (2017)

Naive 
 model Continuum 3NF

?

Repulsive Attractive

Contradiction

Realistic Gamow shell model

= Shell model with continuum + chiral EFT

Neutron Number (N) 8 20 16 14

d3/2 d5/2 s 1/2

NN + 3N (N LO) NN NN + 3N (∆)

low k

(d) V NN + 3N (∆,N LO) forces

2 2

Single-Particle Energy (MeV)

• 4
• 8

Continuum states are missing. Present work

34

Ma +, PLB 802, 135257 (2020)

Beyond drip line | Oxygen isotopes

Interplay between continua and 3NF

is repulsive. Quantitatively reasonable.

?

Realistic Gamow-shell model

1

Otsuka +, PRL, 105, 032501 (2010)

OK as to measured drip line 24O, but quantitatively insufficient Bound 28O, 
 conflicting with experiment

Fossez +, PRC 96, 024308 (2017)

Naive 
 model Continuum 3NF

?

Repulsive Attractive

Contradiction

35

Perspectives

Extension of theoretical framework Study of neutrinoless double β decay

RSM calculations with chiral 3NF 
 for 76Ge, 82Se, 130Te, 136Xe, etc. Beyond 1st-order contribution … 13 diagrams for 2nd-order one-body term

36

Current interest | Cluster physics with chiral EFT

→ Range of local Gaussian: 2 fm Nucleon-wave packets

Summary Phenomenological 
 from chiral N3LO 2NFs

̂ Veff

Relative strengths; unchanged

Brink model for 8Be

Central interactions only

Consistent with Volkov result

Energy minimum at ~3 fm 3 fm LO NLO + Short-range contacts + NLO Long-range 2"

5 10 15 20 25 1 2 3 4 5 6 7 8 9 10 Eg.s (MeV) α α Full LOct LOct+NLO 3 4 5 6 7 α-α distance (fm) 1 2 3 4 5 6 7 8 9 10 α-α distance (fm)

Energy from threshold

37