ECT* workshop “Three-Nucleon Forces in Vacuum and in the medium”
Trento, Italy
July 11 (11-15), 2011
Problems and Ideas
at the Dawn of
Three-Body Force Effects in the Shell Model
Takaharu Otsuka
University of Tokyo / MSU
9時23分
Outline
1. Monopole problem in the shell model
2. Shell evolution in exotic nuclei
3. Solution by three-body force
Introduction to talks by
J. Holt, A. Schwenk and T. Suzuki
Spectra of Ca isotopes calculated by most
updated NN interaction microscopically obtained
By Y. Tsunoda and N. Tsunoda
N3LO
Vlow-k with L=2.0 fm-1
2nd and 3rd order Q-box
4hw and 6hw
s.p.e.
used
Present
GXPH1A
KB3G
GXPF1A
for comparison
KB3G
Two-body matrix elements (TBME) may be calculated
to a rather good accuracy
40Ca
core is not very stable yet -> 0+ energy lowered
48Ca
As N or Z is changed to a large extent in exotic nuclei,
the shell structure is changed (evolved) by
• Monopole component of the NN interaction
Averaged over possible orientations
Linearity: Shift
nj’ : # of particles in j’
<nj’ > can be ~ 10 in exotic nuclei
-> effect quite relevant to neutron-rich exotic nuclei
Strasbourg group made a major contribution in initiating systematic use of the
monopole interaction. (Poves and Zuker, Phys. Rep. 70, 235 (1981))
T=1 monopole
interactions
in the pf shell
GXPF1A
G-matrix
(H.-Jensen)
Tensor force
(p+r exchange)
Basic scale
~ 1/10 of T=0
What’s this ?
j = j’
j = j’
Repulsive
corrections
to G-matrix
T=1 monopole
interactions
in the sd shell
SDPF-M (~USD)
G-matrix
(H.-Jensen)
Tensor force
(p+r exchange)
Basic scale
~ 1/10 of T=0
Repulsive corrections
to G-matrix
j = j’
j = j’
T=0 monopole interaction
The correction is opposite !
T=0 monopole interactions in the pf shell
Tensor force
(p+r exchange)
GXPF1A
G-matrix
(H.-Jensen)
f-f
p-p
f-p
“Local pattern”  tensor force
T=0 monopole interactions in the pf shell
Tensor force
(p+r exchange)
GXPF1A
shell-model int.
G-matrix
(H.-Jensen)
Tensor
component
is
subtracted
Correction is
attractive
Outline
1. Monopole problem in the shell model
2. Shell evolution in exotic nuclei
3. Solution by three-body force
Treatment of tensor force by V
low k
and Q box (3rd order)
Monopole component
of tensor interactions
in pf shell
Bare (AV8’)
short-range correlation
by V low k
in-medium correction
with intermediate states
(> 10 hw, 3rd order)
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
almost equal ?
monopole component of
tensor force in free space
Shell evolution
due to proton-neutron tensor + central forces
Changes of single-particle properties due to these nuclear forces
T=1 NN interaction more relevant to ls splitting change
exotic nucleus
with neutron skin
stable nucleus
proton
neutron
r
dr/dr
ls
splitting smaller
From RIA Physics White Paper
Neutron single-particle energies at N=20 for Z=8~20
p3/2 low
energy (MeV)
Z
14
8
20
16
solid line : full VMU
(central + tensor)
dashed line : central only
Tensor force makes
changes more dramatic.
16
20
These single-particle
energies are “normal”
f7/2-p3/2 2~3 MeV
N=20 gap ~ 6 MeV
d5/2
s1/2
more exotic
d3/2
Z
PRL 104, 012501 (2010)
2+ level (MeV)
Increase of 2+ excitation
energy
Neutron number
Outline
1. Monopole problem in the shell model
2. Shell evolution in exotic nuclei
3. Solution by three-body force
Proton number 
Nuclear Chart
- Left Lower Part -
Why is the drip line of
Oxygen so near ?
Neutron number 
Single-Particle Energy for Oxygen isotopes
by microscopic eff. int.
G-matrix+ core-pol. : Kuo, Brown
V
low-k
: Bogner, Schwenk, Kuo
by phenomenological eff. int.
- G-matrix + fit SDPF-M
Utsuno, O., Mizusaki, Honma,
Phys. Rev. C 60, 054315 (1999)
USD-B
Brown and Richter,
Phys. Rev. C 74, 034315 (2006)
trend
What is the origin of
the repulsive modification of
T=1 monopole matrix elements ?
The same puzzle as in the pf shell
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 clue : Fujita-Miyazawa 3N mechanism
(D-hole excitation)
D particle
p
m=1232 MeV
S=3/2, I=3/2
D
p
Miyazawa, 2007
N
N
N
Renormalization of NN interaction
due to D excitation in the intermediate state
D
T=1
attraction
between NN
effectively
Modification to
bare NN interaction
(for NN scattering)
Pauli blocking effect on the renormalization of
single-particle energy
m
single
particle
states
m’
m
D
m
Renormalization of
single particle energy
due to
D-hole excitation
 more binding (attractive)
m’
m’
m
Another valence
particle in state m’
Pauli Forbidden
The effect is
suppressed
D
Inclusion of Pauli blocking
m
m’
m’
m
m’
D
m
Pauli forbidden
(from previous page)
D
m’
m
This Pauli effect is
included automatically
by the exchange term.
Most important message with Fujita-Miyazawa 3NF
m
m
m’
D
m
Renormalization
of single particle
energy
m’
+
Effective monopole
repulsive interaction
D
m’
m
Pauli blocking
m
same
Monopole part of
Fujita-Miyazawa m’
3-body force
m’
D
m
(i) D-hole excitation in a
conventional way
(ii) EFT with D
D-hole dominant
role in
determining
oxygen drip line
-> J.Holt, A. Schwenk,
T. Suzuki
(iii) EFT incl. contact
terms (N2LO)
O, Suzuki, Holt, O, Schwenk, Akaishi, PRL 105 (2010)
Ground-state energies of
oxygen isotopes
NN force + 3N-induced NN force
(Fujita-Miyazawa force)
Drip line
What was wrong with “microscopic theories” ?
N
N
N
N
D
N
N
Observed in NN scattering
(Effective) two-body interaction
N
present
picture
N
If the origin is
“forgotten”,
constant change of
single-particle energy
or
This is what happened
in “microscopic theories”,
leading to wrong drip line.
For neutron matter :
k
states below Fermi level
attractive
k
k
k
Brown and Green, Nucl.Phys. A137, 1 (1969
Fritsch, 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]
repulsive
For valence neutrons:
states outside the core
Attractive (single-particle energy
renormalization)
repulsive
(valence neutron
interaction)
Quick Summary More from J. Holt, A. Schwenk and T. Suzuki
Major monopole forces are due to
FM 3NF
+
V
+
m=1 fm
basic binding (T=0),
repulsive (T=1)
except for j=j’
variation of
shell structure
limit of existence,
shell structure at
far stability
Casablanca mechanism
Love = attractive force*
This love is reduced
by the presence of Rick
This love is reduced
by the presence of Victor
Rick
Victor
repulsion
*This
equation has
no proof.
END
The central force is modeled by a Gaussian function
V = V0 exp( -(r/m) 2)
(S,T dependences)
with V0 = -166 MeV, m=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 ?
Magic numbers
Mayer and
Jensen (1949)
Eigenvalues of
HO potential
126
5hw
82
4hw
50
3hw
28
20
2hw
8
1hw
2
Spin-orbit splitting
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
Brief history on our studies on 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
Tensor Interaction by pion exchange
VT = (t1t2) ( [s1s2](2) Y(2) (W) ) Z(r)
contributes
only to S=1 states
relative motion
p meson : primary source
s.
p
s.
Yukawa
r meson (~ p+p) : minor (~1/4) cancellation
Ref: Osterfeld, Rev. Mod. Phys. 64, 491 (92)
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 q with respect to the total spin S.
V ~ Y2,0 ~ 1 – 3 cos2q
q
S
q=0
attraction
q=p/2
repulsion
relative
coordinate
Monopole effects due to the tensor force
- An intuitive picture -
wave function of relative motion
spin of nucleon
large relative momentum
attractive
small relative momentum
repulsive
j> = l + ½, j< = l – ½
TO et al., Phys. Rev. Lett. 95, 232502 (2005)
wave function when two nucleons interact
- approx. by linear motion -
k2
k1
k = k1 – k2 , K = k1 + k2
k2
large relative
momentum k
small relative
momentum k
strong damping
wave function
of relative
coordinate
k1
loose damping
k2
k1
k1
k2
wave function
of relative
coordinate
TO. et al., Phys. Rev. Lett. 95, 232502 (2005)
j< = l – ½
General rule of
monopole interaction
of the tensor force
neutron
j> = l + ½
j’<
proton
j’>
Identity for tensor monopole interaction
( j’ j>)
(2j> +1) vm,T
( j’ j<)
+ (2j< +1) vm,T
vm,T : monopole strength for isospin T
= 0
The central force is modeled by a Gaussian function
V = V0 exp( -(r/m) 2)
(S,T dependences)
with V0 = -166 MeV, m=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 ?
T=0 monopole interactions in the pf shell
Tensor force
(p+r exchange)
GXPF1
G-matrix
(H.-Jensen)
Central (Gaussian)
- Reflecting
radial overlap f-f
p-p
f-p
Similarity to Chiral Perturbation of QCD
S. Weinberg, PLB 251, 288 (1990)
Central force:
strongly renormalized
In nuclei
finite
range
(Gaussian)
Tensor force is explicit
p+r
exchange
Monopole int. (MeV)
Central part changes as the cut-off L changes
Tensor (reminder)
T=1
T=0
j-j’
Measured spectroscopic factors
Ratio to naïve single-particle model
short-range
+
in-medium
corrections
Tensor force remains
almost unchanged !
Higher order effects
due to the tensor force
yield renormalization
of central forces.
from Dickhoff
Multipole component of tensor forces
- diagonal matrix elements -
Test by experiments
An example with
51Sb
isotopes with VMU interaction
Z =51 (= 50 + 1) isotopes
change driven
by neutrons in 1h11/2
g7/2
tensor force in VMU
(splitting increased by ~ 2 MeV)
h11/2 - h11/2 repulsive
h11/2 - g7/2
attractive
No mean field theory, (Skyrme,
Gogny, RMF) explained this before.
Consistent with recent experiment
- Position of p3/2
One of the Day 1 experiments at RIBF by Nakamura et al.
Proton single-particles levels of Ni isotopes
E (MeV)
Central Gaussian
+ Tensor
From
Grawe,
EPJA25,
357
Crossing here
is consistent
with exp. on
Cu isotopes
solid line:
full VMU effect
dotted line:
central only
g9/2 occupied
N
shaded area :
effect of
tensor force
Shell structure of a key nucleus
100Sn
solid line : full VMU
(central + tensor)
dashed line : central only
shaded area :
effect of tensor force
Zr
Sn
Exp. d5/2 and g7/2 should be close
Seweryniak et al.
Phys. Rev. Lett. 99, 022504 (2007)
Gryzywacz et al.
f7/2
neutron
d3/2
s1/2
d5/2
Si isotopes
SM calc. by Utsuno et al.
proton
exp.
Z=28 gap is
reduced also
Potential Energy Surface
42 Si
14 28
full
Tensor force removed
from cross-shell interaction
Strong oblate
Deformation ? Other calculations
show a variety of shapes.
42Si
Otsuka, Suzuki and Utsuno,
Nucl. Phys. A805, 127c (2008)
42Si:
B. Bastin, S. Grévy et al.,
PRL 99 (2007) 022503
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
Same interaction as the
one for 42Si
• pd5/2 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)
What happens, if the tensor
force is taken away ?
with full tensor force
s1/2
d3/2
d5/2
no tensor in the cross shell part
Summary
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.
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.
4. Fujita-Miyazawa 3N force can be the next subject for
the shell evolution.
Summary
1.
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, ７８Ni, 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
M. Honma
Y. Utsuno
N. Tsunoda
K. Tsukiyama
M. H.-Jensen
Nihon U.
Aizu
JAEA
Tokyo
Tokyo
Oslo
A. Schwenk Darmstadt
J. Holt
ORNL
K. Akaishi
RIKEN
END
Ca ground-state energy cont’d
SPE : GXPF1 f7:-8.62 f5: -1.38 p3: -5.68 p1: -4.14
Ca 2+ level systematics
2+ of 48Ca rises by 3N
becomes about right by using GXPF1A SPE
N=32, 34 higher 2+ levels
48Ca
M1 excitation
10
8-13MeV
GXPF1
spe : GXPF1
Spin quenching factor 0.8
Summary-2
Dominant monopole forces are due to
FM 3NF
+
V
+
m=1 fm
basic binding
variation of
shell structure
limit of
existence
古典力学での三体問題と三体力
2
2
酒井（英）氏より拝
2
P
P
P
GmE mM GmE mG GmM mG
H E  M  G 


2mE 2mM 2mG
rEM
rEG
rMG
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）
  
 V (rE , rM , rG )
（有効三体力）
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”
正真正銘の三体力は存在するか？
Ground-state energies of
oxygen isotopes
NN force + 3N-induced NN force
(Fujita-Miyazawa force)
Drip line
Collaborators
T. Suzuki
M. Honma
Y. Utsuno
N. Tsunoda
K. Tsukiyama
M. H.-Jensen
Nihon U.
Aizu
JAEA
Tokyo
Tokyo
Oslo
A. Schwenk TRIUMF/Darmstadt
J. Holt
ORNL
K. Akaishi
RIKEN