Three‐body forces in ma2er and nuclei

Three‐body
forces
in
ma2er
and
nuclei
Thomas
Papenbrock
and
G.
Baardsen,
A.
Ekström,
C.
Forssen,
R.
J.
Furnstahl,
S.
Gandolfi,
G.
Hagen,
C.
Horowitz,
M.
Hjorth‐Jensen,
G.
R.
Jansen,
R.
Machleidt,
S.
More,
W.
Nazarewicz,
J.
Sarich,
K.
Wendt,
S.
M.
Wild
Three‐body
forces
from
ma2er
to
nuclei
ECT*
Trento,
May
5‐9,
2014
Research
partly
funded
by
the
US
Department
of
Energy
"Three‐body
force
status
and
needs"
Status:
Some
results
based
on
chiral
NN
interacUon
(N3LO,
E&M)
+
chiral
3NF
• 
Strong
regulator/scheme
dependence
of
chiral
3NF
• 
3NF
(a
NNLO
correcUon)
pivotal
for
basic
nuclear
structure
(magicity
of
48Ca,
drip
line
in
oxygen…)
• 
Poor
saturaUon
of
nuclear
maber
with
chiral
interacUons
Needs:
• 
Consistently
opUmized
(order
by
order)
chiral
interacUon
model
• 
Chiral
interacUon
model
with
acceptable
saturaUon
properUes
• 
ExploraUon
of
chiral
interacUon
model
at
NNLO
(NNLOopt
etc)
Coupled‐cluster,
Green’s
funcUon,
in‐medium
SRG
methods
…
Wave‐funcUon
based
methods
do
not
scale
well
with
increasing
mass
number
A.
Increase
in
compuUng
power
(Moore’s
law)
is
not
commensurate
with
the
scaling
of
this
problem.
• 
Number
of
single‐parUcle
states:
N
=
(const
×R
λ)3
for
cutoff
λ
and
radius
R
~
A1/3.
• 
Hamiltonian
has
dimension:
D
=
(N
choose
A)
≈
(const3
×
A
λ3
choose
A)
ExtrapolaUons
(address
const),
RG
(address
λ),
importance
truncaUons
(address
D),
alleviate
but
do
not
solve
the
scaling
problem.
• 
Reducing
λ
very
useful
(λ
≥
kF)
Induced
a‐body
forces
non‐negligible
at
cutoffs
Induced
3NF
(4NF)
non‐negligible
for
• 
Significant
components
in
wave
funcUon
have
size
κ
~
D‐1/2
Coupled-cluster method (in CCSD approximation)
[Recent review: Hagen, TP, Hjorth-Jensen, Dean, arXiv:1312.7872]
Ansatz:
  Scales
gently
(polynomial)
with
increasing
problem
size
o2u4
.
  TruncaUon
is
the
only
approximaUon.
  Size
extensive
(error
scales
with
A)

Most
efficient
for
doubly
magic
nuclei
CorrelaUons
are
exponen&ated
1p‐1h
and
2p‐2h
excitaUons.
Part
of
np‐nh
excitaUons
included!
Coupled
cluster
equaUons
AlternaDve
view:
CCSD
generates
similarity
transformed
Hamiltonian
with
no
1p‐1h
and
no
2p‐2h
excitaDons.
Frame
work
of
nuclear
coupled
cluster
Based
on
a
reference
state
(closed
subshell
in
j‐coupled
formulaUon)
Normal‐ordered
Hamiltonian
w.r.t.
reference
Decouples
reference
state
(and
its
2p‐2h
excitaUons)
from
remaining
Hilbert
space
CCSD:
effecUve
two‐body
Hamiltonian,
CCSDT:
effecUve
three‐body
Hamiltonian
Imaginary
Ume‐evoluUon
of
CCSD
technically
similar
to
SRG
evoluUon
toward
non‐HermiUan
Hamiltonian
Gamow
basis
for
descripUon
of
weakly
bound
and
unbound
nuclei
PracUcal
soluUon
of
center‐of‐mass
problem
for
intrinsic
Hamiltonian
Excited
states
(and
neighboring
nuclei)
from
equaUons
of
moUon
approach
Permits
user
to
employ
“bare”
interacUons
from
chiral
EFT
Gamow
shell
model
for
open
quantum
systems
Review:
Michel
et
al.,
J.
Phys.
G
36,
013101
(2009)
ComputaUonal
tool:
coupled‐cluster
method
[Coester
(1958);
Coester
&
Kümmel
(1960);
Kümmel,
Lührmann
&
Zabolitzky
(1978);
Bishop
(1991);
(…
many
others
…);
Hagen,
TP,
Dean,
&
Hjorth‐Jensen
arXiv:1312.7872]
6
Is
54Ca
a
magic
nucleus?
Ni
40Ca
in
our
bones
?
Ca
Magic
nuclei
determine
the
structure
of
enUre
regions
of
the
nuclear
chart
Ca:
Huck
et
al,
PRC
(1985);
Ti:
R.
Janssens
et
al,
PLB
(2002);
Cr:
Prisciandaro
et
al,
PRC
(2001)
Calcium isotopes from chiral interactions [Hagen,
Hjorth‐Jensen,
Jansen,
Machleidt,
TP,
Phys.
Rev.
Leb.
109,
032502
(2012).]
In‐medium
effecUve
3NF:
kF=0.95
fm‐1,
cE=0.735,
cD=‐0.2
[Holt,
Kaiser,
Weise
2009;
Hebeler
and
Schwenk
2010]
Sn(52Ca)
=
5.98MeV
(Gallant
et
al
PRL
2012)
Sn(54Ca)
=
3.84MeV
(Wienholtz
Nature
2013)
Shell structure of 52,54Ca
Hagen,
Hjorth‐Jensen,
Jansen,
Machleidt,
TP,
Phys.
Rev.
Leb.
109,
032502
(2012)
Indicators
of
shell
closure:
•  SeparaUon
energy
•  2+
excited
state
•  4+/2+
raUo
RIKEN
Experiment
★
suggest
that
54Ca
has
a
“so•”
(sub‐)shell
closure
similar
to
that
of
52Ca
Measurement at RIKEN
[Steppenbeck et al., J. Phys. G
2013; Nature 502, 207 (2013);]
confirms our prediction.
48Ca
2+
CC
3.58
Exp
3.83
52Ca
54Ca
4+
4+/2+
2+
4+
4+/2+
2+
4+
4+/2+
4.20
1.17
2.19
3.95
1.80
1.89
4.46
2.36
4.50
1.17
2.56
?
?
?
?
?
Spectra and shell evolution in Calcium isotopes RIKEN
1.  Our
predicUon
for
excited
5/2‐
and
½‐
states
in
53Ca
seen
at
RIKEN
2.  Inversion
of
9/2+
and
5/2+
states
in
neutron
rich
calcium
isotopes
3.  Harmonic
oscillator
gives
the
naïve
shell
model
order
ConUnuum
coupling
crucial
for
level
ordering
Chiral
interacUon
NNLOopt
[Ekström
et
al.,
Phys.
Rev.
Leb.
110,
192502
(2013)]
Kept
fixed
Adjusted
parameters
Weinberg;
van
Kolck;
Epelbaum,
Glöckle
&
Meiβner;
Entem
&
Machleidt;
Krebs;
…
OpUmizaUon
of
NN
potenUal
to
phase
shi•s;
χ2
from
data
Weights
for
contacts
scale
as
Q3;
for
pion‐nucleon
couplings
from
Nijmegen
analysis
Pion
nucleon
couplings
determined
from
fits
to
peripheral
d,
f,
g
parUal
waves
(NNLO
contacts
do
not
contribute
for
L≥2)
Phase
shi•s
of
NNLOopt
Oxygen
isotopes
(NN
only)
Drip
line
in
oxygen
from
NN
alone
(Oxygen
isotopes
not
included
in
opUmizaUon)
Calcium
isotopes
(NN
only)
Doubly
magic
48Ca
from
NN
alone
Nucleonic
maber
•  Use
plane‐wave
basis
with
generalized
boundary
condiUons
(Bloch
waves)
•  
Basis
respects
momentum
conservaUon
•  
CCD
(no
singles)
for
closed‐shell
references
•  
Three‐nucleon
forces
much
simpler
to
implement
than
in
shell‐model
basis
•  
Shell
effects
dominate
finite‐size
correcUons
Related
studies
with
chiral
interacUons:
Soma
&
Bozek
(2008);
Epelbaum,
Krebs,
Lee,
&
Meissner
(2009);
Hebeler
&
Schwenk
(2010);
Hebeler,
Bogner,
Furnstahl,
Nogga,
&
Schwenk
(2011);
Holt,
Kaiser,
&
Weise
(2012);
Carbone,
Polls,
&
Rios
(2013);
Gezerlis,
Tews,
Epelbaum,
Gandolfi,
Hebeler,
Nogga,
&
Schwenk
(2013);
…
MiUgate
shell
oscillaUons
“Twist
averaged”
boundary
condiUons:
average
over
all
possible
phases
of
Bloch
waves.
Exactly
removes
finite
size
effects
for
free
Fermi
gas.
[Gros,
Z.
Phys.
B
86,
359
(1992);
Lin,
Zong,
and
Ceperley,
Phys.
Rev.
E
64,
016702
(2001)]
Discrete
momenta
are
Nuclear
maber
(NN
only)
SCGF
=
Self‐consistent
Green’s
funcUon
at
T=5
MeV
[A.
Carbone,
A.
Polls,
and
A.
Rios,
Phys.
Rev.
C
88,
044302
(2013)].
Neutron
maber
is
perturbaUve
(NN
only)
Nuclear
maber
less
perturbaUve
(NN
only)
Overbinding
and
saturaUon
at
too
high
densiUes
Three
nucleon
force
Local
form
of
3NF
[NavraUl,
Few
Body
Syst.
41,
117
(2007)],
i.e.
cutoff
is
in
the
momentum
transfer
based
on
[Epelbaum
et
al.,
Phys.
Rev.
C
66,
064001
(2002)].
Fit
to
A=3,4
binding
energies
and
triton
half
life
(Gazit,
NavraUl,
&
Quaglioni)
cD=0.389,
cE=0.39
Inclusion
of
three‐nucleon
forces
3NFs
in
momentum
space
sUll
require
considerable
computer
Ume
but
less
human
Ume
Normal‐ordered
(w.r.t.
HF
vacuum)
Hamiltonian
Normal‐ordered
1‐body
interacUon
(contains
informaUon
about
NN
force
and
3NF)
Normal‐ordered
2‐body
interacUon
(contains
informaUon
of
3NF)
“Residual”
3NF
(enters
leading
triples
correcUon)
Nucleonic matter with NNLOopt and 3NFs
[Hagen et al., PRC 89 014319 (2013)]
•  3NFs
with
different
regulators
and
cutoffs
act
repulsively
in
neutron
maber
•  N2LOopt
+
3NFs
do
not
reproduce
saturaUon
density
and
binding
energy
of
nuclear
maber
•  Strong
dependence
on
regulator
and
cutoff
in
nuclear
maber
Where
actually
does
the
cutoff
cut
off?
For NNLOopt no combination of cD, cE simultaneously
satisfies light nuclei and nuclear matter 5%
error
bands
for
saturaUon
kf
,
E/N
and
binding
energy
of
3H

VariaUon
of
cD
and
cE
not
sufficient
to
simultaneously
bind
light
nuclei
and
nuclear
maber
Coupled-cluster effective interactions for the shell model
[G.
R.
Jansen,
J.
Engel,
G.
Hagen,
P.
NavraUl,
A.
Signoracci,
arXiv:1402.2563
(2014).]
•  Start
from
chiral
NN(N3LOEM)
+
3NF(N2LO)
interacUons
•  Solve
for
A+1
and
A+2
using
CC.
Project
A+1
and
A+2
CC
wave
funcUons
onto
the
s‐d
model
space
using
Lee‐Suzuki
similarity
transformaUon.
..
.
p Q
3/2
f7/2
P
d3/2
s1/2
d5/2
Q
p1/2
p3/2
s1/2
•  Diagonalize
the
effecUve
Hamiltonian
in
the
valence
space.
Comparison
between
coupled‐cluster
effecUve
interacUon
(CCEI)
and
“exact”
coupled‐cluster
calculaUon
with
inclusion
of
perturbaUve
triples
(Λ‐CCSD(T)).
Coupled-cluster effective interactions for the
shell model: Oxygen isotopes
Coupled-cluster effective interactions for the
shell model: Carbon isotopes
Convergence
in
finite
oscillator
spaces
CalculaUons
are
performed
in
finite
oscillator
spaces.
How
can
one
reliably
extrapolate
to
infinity?
What
is
the
equivalent
of
Lüscher’s
formula
for
the
harmonic
oscillator
basis
[Lüscher,
Comm.
Math.
Phys.
104,
177
(1986)]
?
Convergence
in
momentum
space
(UV)
and
in
posiUon
space
(IR)
needed
[Stetcu
et al.,
PLB
(2007);
Hagen
et al.,
PRC
(2010);
Jurgenson
et al.,
PRC
(2011);
Coon
et al.,
PRC
(2012)]
p
Nucleus
x
Phase
space
covered
by
oscillator
basis
with
(N,
b)
Nucleus
needs
to
“fit”
into
basis:
• 
Nuclear
radius
R
<
L
• 
cutoff
of
interacUon
Λ
<
ΛUV
Hard
wall
at
L2
is
in
the
spirit
of
EFT
ground
wave
funcUons
in
posiUon
space
(π/L2)2
is
the
lowest
eigenvalue
of
the
operator
p2.
Fourier
transforms
differ
only
at
large
momentum
The
difference
between
the
HO
basis
and
a
box
of
size
L2
can
not
be
resolved
at
low
momentum.
IR
correcUons
to
bound‐state
energies
Simple
view:
A
node
in
the
wave
funcUon
at
r=L2
requires
αE
=
–
exp(–2kEL2).
This
yields
a
(kineUc)
energy
correcUon
Model‐independent
approach
based
on
S‐matrix
theory
Final
results
[Furnstahl,
Hagen,
TP,
Phys.
Rev.
C
86,
031301
(2012);
More,
Ekström,
Furnstahl,
Hagen,
TP,
Phys.
Rev.
C
87,
044326
(2013);
Furnstahl,
More,
TP,
Phys.
Rev.
C
89,
044301
(2014)]
ANC2
Binding
momentum
only
observables
enter
Energy
extrapolaUon
explains
findings
by
[Coon
et
al,
Phys.
Rev.
C
86,
054002
(2012)]
CorrecUons
due
to
finite
Hilbert
spaces
• 
UV
pracUcally
converged
(because
λ
<
ΛUV)
• 
IR
convergence
is
slower
due
to
exponenUal
decay
of
wave
funcUon
• 
Dirichlet
boundary
condiUon
at
x=L
in
posiUon
space,
k∞
from
energy
fit
Summary
• 
Coupled
cluster
method
efficient
tool
for
many
nuclei
of
interest
• 
OpUmizaUon
of
chiral
interacUon
NNLOopt
• 
acceptable
χ2
≈
1
per
degree
of
freedom
for
lab
energies
<
125
MeV
• 
NN
interacUons
alone
reproduce
some
essenUal
features
in
isotopes
of
oxygen
and
calcium
• 
Nucleonic
maber
in
momentum
space
with
NNLOopt
• 
Finite
size
correcUons
under
control
• 
Neutron
maber
is
perturbaUve
–
nuclear
maber
somewhat
less
• 
VariaUon
of
cD
and
cE
not
sufficient
to
fit
light
nuclei
and
maber
• 
3NF
with
local
regulator
not
without
problems
(large
triples
correcUon)
• 
Infrared
extrapolaUons
based
on
well‐understood
box
size
L2