Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model

Problems and Ideasat the Dawn of

Three-Body Force Effects in the Shell Model

Takaharu OtsukaUniversity of Tokyo / MSU

午前 6:12

ECT* workshop “Three-Nucleon Forces in Vacuum and in the medium”Trento, Italy

July 11 (11-15), 2011

Outline

1. Monopole problem in the shell model

2. Shell evolution in exotic nuclei

3. Solution by three-body force

Introduction to talks byJ. Holt, A. Schwenk and T. Suzuki

Spectra of Ca isotopes calculated by mostupdated NN interaction microscopically obtained

By Y. Tsunoda and N. Tsunoda

N3LO

Vlow-k with =2.0 fm-1

2nd and 3rd order Q-box

4hw and 6hw

s.p.e. used

PresentGXPH1A

KB3G

GXPF1AKB3G

for comparison

Two-body matrix elements (TBME) may be calculatedto a rather good accuracy

40Ca core is not very stable yet -> 0+ energy lowered

48Ca

• Monopole component of the NN interaction

　　 Averaged over possible orientations 　

As N or Z is changed to a large extent in exotic nuclei, the shell structure is changed (evolved) by

Strasbourg group made a major contribution in initiating systematic use of the monopole interaction. (Poves and Zuker, Phys. Rep. 70, 235 (1981))

<nj’ > can be ~ 10 in exotic nuclei -> effect quite relevant to neutron-rich exotic nuclei

nj’ : # of particles in j’Linearity: Shift

T=1 monopole interactions in the pf shell

j = j’ j = j’

Tensor force (+exchange)

G-matrix (H.-Jensen)

GXPF1A

What’s this ?

Basic scale ~ 1/10 of T=0

Repulsive corrections to G-matrix

T=1 monopole interactions in the sd shell

SDPF-M (~USD) G-matrix

(H.-Jensen)


j = j’ j = j’

Basic scale ~ 1/10 of T=0

Repulsive corrections to G-matrix

T=0 monopole interaction

The correction is opposite !


GXPF1A



f-f p-p f-p

“Local pattern” tensor force


GXPF1Ashell-model int.



Tensor component

is subtracted

Correction isattractive

Outline




Treatment of tensor force by V low k and Q box (3rd order)

Monopole component of tensor interactions in pf shell

short-range correlationby V low k

in-medium correctionwith intermediate states(> 10 hw, 3rd order)

Bare (AV8’)

only for comparison

Systematic description of monopole properties of exotic nuclei can be obtained by an extremely simple interaction as

Parameters are fixed for all nuclei

monopole component of

tensor force in nuclear

medium monopole component of

tensor force in free space

almost equal ?

Shell evolutiondue to proton-neutron tensor + central forces

Changes of single-particle properties due to these nuclear forces

ls splitting smaller

exotic nucleus with neutron skin

proton

neutron

d/dr

stable nucleus

T=1 NN interaction more relevant to ls splitting change

From RIA Physics White Paper

Neutron single-particle energies at N=20 for Z=8~20

Z

8 14Z

16 20

These single-particleenergies are “normal”

f7/2-p3/2 2~3 MeV N=20 gap ~ 6 MeV

energ

y (

MeV

)

d5/2 s1/2 d3/2

16

20

solid line : full VMU (central + tensor)

dashed line : central only

p3/2 low

Tensor force makeschanges more dramatic.

PRL 104, 012501 (2010)more exotic

Increase of 2+ excitationenergy

Neutron number

2+

level (M

eV

)

Outline




Pro

ton

nu

mb

er

Neutron number

Nuclear Chart- Left Lower Part -

Why is the drip line of

Oxygen so near ?

Single-Particle Energy for Oxygen isotopes

Utsuno, O., Mizusaki, Honma, Phys. Rev. C 60, 054315 (1999)

G-matrix+ core-pol. : Kuo, Brown

SDPF-M

- G-matrix + fit -

V low-k : Bogner, Schwenk, Kuo

trend

trend

by microscopic eff. int. by phenomenological eff. int.

USD-B Brown and Richter, Phys. Rev. C 74, 034315 (2006)

What is the origin of

the repulsive modification of

T=1 monopole matrix elements ?

A solution within bare 2-body interaction is very unlikely

(considering efforts made so far) Zuker, Phys. Rev. Lett. 90, 042502 (2003)

3-body interaction

The same puzzle as in the pf shell

The clue : Fujita-Miyazawa 3N mechanism-hole excitation)

particle m=1232 MeV S=3/2, I=3/2

N N N

Miyazawa, 2007

Renormalization of NN interaction due to excitation in the intermediate state

T=1attraction

between NNeffectively

Modification tobare NN interaction(for NN scattering)

Pauli blocking effect on the renormalization of single-particle energy

Pauli blocking effect on the renormalization of single-particle energy

Pauli ForbiddenThe effect is

suppressed

m

m

m’ m’

Renormalization of single particle energy

due to -hole excitation more binding

(attractive)

m

m

m’

singleparticle states

Another valenceparticle in state m’

Pauli forbidden(from previous page) This Pauli effect is

included automaticallyby the exchange term.

m

mm’

m’

m

m’ m’

m

Inclusion of Pauli blocking

Most important message with Fujita-Miyazawa 3NFMost important message with Fujita-Miyazawa 3NF

+

Renormalizationof single particle

energy

m

m

m’

same

Monopole part ofFujita-Miyazawa3-body force

Pauli blockingm

mm’

m’

Effective monopole repulsive interaction

(i) -hole excitation in a conventional way

-hole dominantrole in determiningoxygen drip line

(ii) EFT with

(iii) EFT incl. contact terms (N2LO)

-> J.Holt, A. Schwenk, T. Suzuki

Ground-state energies of oxygen isotopes

Drip line

NN force + 3N-induced NN force (Fujita-Miyazawa force)

O, Suzuki, Holt, O, Schwenk, Akaishi, PRL 105 (2010)

(Effective) two-body interaction

constant change ofsingle-particle energy

If the origin is“forgotten”,

This is what happenedin “microscopic theories”,leading to wrong drip line.

N N

NN

N

NN

N

or

What was wrong with “microscopic theories” ?

Observed in NN scattering

presentpicture

For neutron matter : states below Fermi level

attractive

repulsivek

k

k

k

Brown and Green, Nucl.Phys. A137, 1 (1969Fritsch, Kaiser and Weise, Nucl. Phys. A750, 259 (2005); Tolos, Friman and Schwenk, Nucl.Phys. A806}, 105 (2008); Hebeler and Schwenk, arXiv:0911.0483 [nucl-th]

For valence neutrons: states outside the core

Attractive (single-particle energy renormalization)

repulsive

(valence neutron interaction)

Major monopole forces are due to

+V

=1 fm

Quick Summary More from J. Holt, A. Schwenk and T. Suzuki

+

FM 3NF

basic binding (T=0),repulsive (T=1) except for j=j’

variation ofshell structure

limit of existence,shell structure at far stability

Casablanca mechanism

RickVictor

This love is reduced by the presence of Rick

This love is reduced by the presence of Victor

repulsion

Love = attractive force*

*This equation hasno proof.

E N D

The central force is modeled by a Gaussian function

V = V0 exp( -(r/) 2) (S,T dependences)

with V0 = -166 MeV, =1.0 fm,

(S,T) factor (0,0) (1,0) (0,1) (1,1) -------------------------------------------------- relative strength 1 1 0.6 -0.8

Can we explain the difference between f-f/p-p and f-p ?

Spin-orbit splitting

Eigenvalues of HO potential

Magic numbers

Mayer and Jensen (1949)126

8

20

28

50

82

2

5h

4h

3h

2h

1h

density saturation + short-range NN interaction + spin-orbit splitting

Mayer-Jensen’s magic number

with rather constant gaps (except for gradual A dependence)

robust feature -> nuclear forces not included in the above can change it

-> tensor force

Magic numbers may change due to spin-isospin nuclear forces

Tensor force produces unique and sizable effect

Tensor and central forces -> Weinberg-type model

Brief history on our studies on tensor force

VT = (2) Y(2 Z(r)

contributes only to S=1

states

relative motion

meson (~ +) : minor (~1/4) cancellation

meson : primary source

Ref: Osterfeld, Rev. Mod. Phys. 64, 491 (92)

Tensor Interaction by pion exchange

Yukawa

How does the tensor force work ?

Spin of each nucleon is parallel, because the total spin must be S=1

The potential has the following dependence on the angle with respect to the total spin S.

V ~ Y2,0 ~ 1 – 3

cos2

attraction=0

repulsion=/2

S

relativecoordinate

Monopole effects due to the tensor force

- An intuitive picture -

wave function of relative motion

large relative momentum small relative momentum

attractive repulsive

spin of nucleon

TO et al., Phys. Rev. Lett. 95, 232502 (2005)j> = l + ½, j< = l – ½

wave function when two nucleons interact

k1k2

k1k2

- approx. by linear motion -

k = k1 – k2 , K = k1 + k2

large relativemomentum k

strong damping

wave functionof relative coordinate

k1k2

wave functionof relative coordinate

small relativemomentum k

loose damping

k1k2

j> = l + ½

j< = l – ½

j’<

proton

neutron

j’>

General rule of monopole interaction

of the tensor force

Identity for tensor monopole interaction

(2j> +1) vm,T + (2j< +1) vm,T = 0( j’ j>) ( j’ j<)

vm,T : monopole strength for isospin T

TO. et al., Phys. Rev. Lett. 95, 232502 (2005)

The central force is modeled by a Gaussian function

V = V0 exp( -(r/) 2) (S,T dependences)

with V0 = -166 MeV, =1.0 fm,

(S,T) factor (0,0) (1,0) (0,1) (1,1) -------------------------------------------------- relative strength 1 1 0.6 -0.8

Can we explain the difference between f-f/p-p and f-p ?

GXPF1


Central (Gaussian)- Reflecting

radial overlap -



f-f p-p f-p

S. Weinberg, PLB 251, 288 (1990)

Tensor force is explicit

Central force:strongly renormalized

finite range(Gaussian)

In nuclei

+ exchange

Similarity to Chiral Perturbation of QCD

Central part changes as the cut-off changes

T=0 T=1

j-j’

Mon

opol

e in

t. (M

eV)

Tensor (reminder)

from Dickhoff

Rat

io to

naï

ve s

ingl

e-pa

rtic

le m

odel

Measured spectroscopic factors

short-range +in-mediumcorrections

Tensor force remains almost unchanged !

Higher order effectsdue to the tensor forceyield renormalization of central forces.

Multipole component of tensor forces- diagonal matrix elements -

Test by experiments

Z =51 (= 50 + 1) isotopes

No mean field theory, (Skyrme, Gogny, RMF) explained this before.

An example with 51Sb isotopes with VMU interaction

tensor force in VMU (splitting increased by ~ 2 MeV)

change driven by neutrons in 1h11/2

g7/2

Consistent with recent experiment

One of the Day 1 experiments at RIBF by Nakamura et al.

- Position of p3/2

FromGrawe, EPJA25,357

g9/2 occupied

E (

MeV

)

N

Central Gaussian+ Tensor

solid line: full VMU effect

dotted line: central only

Crossing hereis consistentwith exp. onCu isotopes

Proton single-particles levels of Ni isotopes

shaded area : effect of tensor force

solid line : full VMU (central + tensor)

dashed line : central only

shaded area : effect of tensor force

Shell structure of a key nucleus 100Sn

Exp. d5/2 and g7/2 should be closeSeweryniak et al. Phys. Rev. Lett. 99, 022504 (2007) Gryzywacz et al.

Zr Sn

Potential Energy Surface

s1/2

Z=28 gap is reduced alsoproton

neutronf7/2d3/2

d5/2

full

Tensor force removedfrom cross-shell interaction

Strong oblateDeformation ?

Otsuka, Suzuki and Utsuno,Nucl. Phys. A805, 127c (2008)

exp.

4214Si28

42Si

Other calculationsshow a variety of shapes. 42Si: B. Bastin, S. Grévy et al.,

PRL 99 (2007) 022503

Si isotopesSM calc. by Utsuno et al.

Spectroscopic factors obtained by (e,e’p)

on 48Ca and the tensor force

Collaboration with Utsuno and Suzuki

Spectroscopic factor for 1p removal from 48Ca

• d5/2 deep hole state

– More fragmentation

• Distribution of strength

– quenching factor 0.7 is needed (as usual).

– Agreement between

experiment and theory for both position and strength (e,e’p): Kramer et al., NP A679, 267 (2001)

Same interaction as theone for 42Si

What happens, if the tensor force is taken away ?

no tensor in the cross shell part

s1/2

d3/2

d5/2

with full tensor force

Summary

2. The tensor force remain ~unchanged by the treatments of short-range correlations and in-medium correction. This feature is very unique.

3. (e, e’p) data on 48Ca suggests the importance of the tensor force, which is consistent with exotic feature of 42Si.

Direct reactions with RI beam should play important roles in exploring structure of exotic nuclei driven by nuclear forces.

1. Changes of shell structure and magic numbers in exotic nuclei are a good probe to see effects of nuclear forces. Such changes are largely due to tensor force, as have been described by VMU. Transfer reactions have made important contributions.

4. Fujita-Miyazawa 3N force can be the next subject for the shell evolution.

Summary1. Monopole interactions : effects magnified in neutron-rich nuclei

2. Tensor force combined with central force : a unified description particularly for proton-neutron monopole correlation. -> N=20 Island of inversion, 42Si, ７８ Ni, 100Sn, Sb, 132Sn, Z=64,…

Tensor force in nuclear medium is very similar to the bare one.This central force may be a challenge for microscopic theories.

3. Fujita-Miyazawa 3-body force produces repulsive effective interaction between valence neutrons in general.

The spacings between neutron single-particle levels can become wider as N increases, and new magic numbers may arise.Examples are shown for O and Ca isotopes with visible effects.

<--> shell quenching 4. Structure change on top of the shell evolution -> diagonalization with super computer

Collaborators

T. Suzuki Nihon U.M. Honma AizuY. Utsuno JAEAN. Tsunoda TokyoK. Tsukiyama Tokyo M. H.-Jensen Oslo

A. Schwenk 　 DarmstadtJ. Holt ORNLK. Akaishi RIKEN

E N D

SPE : GXPF1 f7:-8.62 f5: -1.38 p3: -5.68 p1: -4.14

Ca ground-state energy cont’d

Ca 2+ level systematics

2+ of 48Ca rises by 3N becomes about right by using GXPF1A SPE

N=32, 34 higher 2+ levels

Spin quenching factor 0.8

8-13MeV

GXPF1

spe : GXPF1

48Ca M1 excitation 10

Dominant monopole forces are due to

+V

=1 fm

Summary-2

+

FM 3NF

basic binding variation ofshell structure

limit of existence

古典力学での三体問題と三体力

（有効三体力）

MG

GM

EG

GE

EM

ME

G

G

M

M

E

E

r

mGm

r

mGm

r

mGm

m

P

m

P

m

PH

222

222

),,( GME rrrV

ここでの三体力は、二体力＋超多体問題を回避するための“有効”相互作用 .

正真正銘の三体力は存在するか？

GPS の位置をこの方程式を数値的に解いても正確には求まらない。 GPS の役割を果たさない！

それは地球が変形するから（もちろん相対論効果もあるが）

酒井（英）氏より拝借

Ground-state energies of oxygen isotopes

Drip line

NN force + 3N-induced NN force (Fujita-Miyazawa force)

Collaborators

T. Suzuki Nihon U.M. Honma AizuY. Utsuno JAEAN. Tsunoda TokyoK. Tsukiyama Tokyo M. H.-Jensen Oslo

A. Schwenk 　 TRIUMF/DarmstadtJ. Holt ORNLK. Akaishi RIKEN

Documents

Problems and Ideas at the Dawn of Three-Body Force Effects in the Shell Model