ab initio framework for nuclear structure and fundamental ... · for nuclear structure, typical...
TRANSCRIPT
Ab initio frameworkfor Nuclear Structure
and Fundamental Symmetries
Javier Menéndez
JSPS Fellow, The University of Tokyo
14th CNS International Summer School26th − 29th August 2015
Outline
1 Why ab initio calculations?
2 Nuclear forces
3 Nuclear structure calculations
4 Neutrinoless ββ decay
5 Dark Matter scattering off nuclei
Javier Menéndez (JSPS / U. Tokyo) Ab initio Nuclear Structure CNS Summer School, 2015 2 / 38
Similarity renormalization groupThe SRG-evolved interactions decouple low and high momenta
As expected,evolved interactionsquite different at high scalesconverge at lower scales
SRG evolution eliminatesinformation from high energiesfrom the potential
Javier Menéndez (JSPS / U. Tokyo) Ab initio Nuclear Structure CNS Summer School, 2015 3 / 38
Three-nucleon forces
3N forces known for a long timeFujita and Miyazawa (1957), Towner (1987)...
3N forces originate in the elimination of degrees of freedom(Many-body forces appear in any effective theory)
The ∆ isobar, with M∆ = 1232 MeVrelatively low excitation of the nucleon, MN = 939 MeV
3N forces difficult to constrain directly, limited NNN scattering data available
Need 3N forces consistent framework with NN forces
Javier Menéndez (JSPS / U. Tokyo) Ab initio Nuclear Structure CNS Summer School, 2015 4 / 38
Theory for nuclear forces
Difficult to find NN potential with consistentNNN forces and connected to QCD...
Use concept of separation of scales!
The energy scale relevantdetermines the degrees of freedom
For nuclear structure,typical energies of interestpoint to nucleons and pions(pions are particularly light mesons!)
Effective theory with nucleons and pionsas degrees of freedom,with connection to QCD
Javier Menéndez (JSPS / U. Tokyo) Ab initio Nuclear Structure CNS Summer School, 2015 5 / 38
Effective theories
Effective theory:approximation of the full theoryvalid at relevant scales
Expansion in terms of small parameter:typical scale / breakdown scale
In an effective theorythe physics resolvedat relevant energies is explicit
Terms at different orders given bysymmetries of the full theory
Unresolved physicsencoded in Low Energy Couplings
Javier Menéndez (JSPS / U. Tokyo) Ab initio Nuclear Structure CNS Summer School, 2015 6 / 38
Chiral effective field theory (EFT)
Chiral EFT is a low energy approach to QCD
Exploits approximate chiral symmetry of QCD:pions are special particles (pseudo-Goldstone bosons)
Nucleons interact via pion exchanges and contact terms(physics non-resolved at low energies)
Systematic basis for consistent nuclear forces and hadronic currents,expansion in powers of Q/ΛbQ ∼ mπ, typical momentum scaleΛb ∼ 500 MeV, breakdown scale
Systematic expansion naturally includes, at different ordersNN, 3N, 4N... forces and 1b, 2b, 3b... currents (interactions)
Short-range couplings are fitted to experiment once:NN scattering, π − N scattering, 3H, 4He
Javier Menéndez (JSPS / U. Tokyo) Ab initio Nuclear Structure CNS Summer School, 2015 7 / 38
Chiral Effective Field Theory
Chiral EFT: low energy approach to QCD, nuclear structure energies
Approximate chiral symmetry: pion exchanges, contact interactions
Systematic expansion: nuclear forces and electroweak currents
2N LO
N LO3
NLO
LO
3N force 4N force2N force
N
N
e ν
N
N
e
N
π
N ν e ν
N
NN
N
Weinberg, van Kolck, Kaplan, Savage, Weise, Epelbaum, Meißner...
Park, Gazit, Klos
Short-range couplingsfitted to experiment once
Javier Menéndez (JSPS / U. Tokyo) Ab initio Nuclear Structure CNS Summer School, 2015 8 / 38
How does chiral EFT work?
The chiral EFT Lagrangian is an expansion, in different ordersof pion-pion, pion-nucleon and nucleon-nucleon parts
LχEFT =L(0) + L(1) + L(2) + · · ·=L(0)
ππ + L(0)πN + L(0)
NN + L(1)ππ + L(1)
πN + L(1)NN + · · ·
For example:
L(0)ππ =
f 2π
4Tr[∂µU∂µU† + m2
π
(U + U†
)], U = exp
[iπ · τ
fπ
]L(0)πN = N
(iγµDµ +
gA
2γµγ5u(π)µ −M
)N
· · ·
Evaluate these expressions to lowest orders in pion fieldsobtain Feynman diagrams for each vertex λi
The chiral order of a (simple) diagram is ν = 2N + 2L +∑
i λi
with N nucleons, and L loops
Javier Menéndez (JSPS / U. Tokyo) Ab initio Nuclear Structure CNS Summer School, 2015 9 / 38
Examples of chiral EFT diagrams
Feynman diagrams are read off from the Lagrangian:
qµa
(a)
kµab
qµ
(b) (c)
qµa
LO − gA2Fπγ5/qτ
a 14F2πεabc/qτc —
NLO —2iF2π
�c4ε
abc τc
2kµqνσ
µν − c3kµqµδab − 2c1m2
πδab
� d1Fπτaσ1 · q+ (1↔ 2)
+ d2Fπ(τ1×τ2)aq · (σ1×σ2)
for the lowest order pion-nucleon diagrams from L(0)ππ + L(0)
πN + L(1)NN
Javier Menéndez (JSPS / U. Tokyo) Ab initio Nuclear Structure CNS Summer School, 2015 10 / 38
Chiral EFT NN forces to N3LO
Leading order
Next−to−next−to−next−to−leading order
Next−to−leading order
Next−to−next−to−leading order
Interaction parts(pion-exchange, spin-orbit) andLow Energy Couplingsshow up in chiral expansion
V (0)NN =
g2A
4f 2π
σ1 · qσ1 · qq2 + m2
π
τ1 · τ2
+ CS + CTσ1 · σ2
V (1)NN =0
V (2)NN =C1q2 + C2k2+
(C3q2 + C4k2)σ1 · σ2
+ C512
(σ1 + σ2) · q × k
+ C6σ1 · qσ1 · q + C7σ1 · kσ1 · k+ · · ·
V (3)NN = · · · c1 · · · c3 · · · c4 · · ·
V (4)NN = · · ·
15 new Low Energy Couplings
Javier Menéndez (JSPS / U. Tokyo) Ab initio Nuclear Structure CNS Summer School, 2015 11 / 38
Chiral EFT 3N forces to N3LO
Next−to−next−to−leading order
Next−to−next−to−next−to−leading order
3N forces consistent(same Low Energy Couplings)with NN forces
V (3)NN = · · · c1 · · · c3 · · · c4 · · ·
+ CDgA
8f 2π
σ3q3
q23 + m2
π
τ1 · τ3σ1q3
+ CEτ2 · τ3
2 new Low Energy Couplings
V (4)NN = · · ·
No new Low Energy Couplings!
N3LO 3N forcesmuch more involved thanleading N2LO 3N forces
Javier Menéndez (JSPS / U. Tokyo) Ab initio Nuclear Structure CNS Summer School, 2015 12 / 38
Chiral EFT currentsIn addition, from the Chiral EFT Lagrangian we obtain the currentson how nucleons (and pions) couple to different probesof scalar, vector, axial... character
This is consistent with nuclear forces (same couplings)!
a
(a)
akµ
b
(b)
akµ
b
(c)
a
kµc
(d)
qµb
LO axial i gAγµγ5
τa
2− i
Fπεabcγµ τ
c
2Fπkµδab —
LO vector iγµ τa
2−i gA
Fπεabcγµγ5
τc
2— −εabckµ
NLO axial —2
Fπ
�−c4ε
abc τc
2kνσ
µν + c3kµδab�
— —
NLO vector — — — —
Javier Menéndez (JSPS / U. Tokyo) Ab initio Nuclear Structure CNS Summer School, 2015 13 / 38
Chiral EFT interactions
NN chiral EFT interactions at order N3LO good agreement to datasimilar to standard (Bonn, Argonne) NN potentials
0
20
40
60
PhaseShift[deg]
1S0
0
50
100
150 3S1
-40
-30
-20
-10
0
PhaseShift[deg]
1P1
0
10
20
3P0
0 50 100 150 200 250
Lab. Energy [MeV]
-30
-20
-10
0
PhaseShift[deg]
3P1
0 50 100 150 200 250
Lab. Energy [MeV]
0
10
20
30
3P2
0
5
10
PhaseShift[deg]
1D
2
-30
-20
-10
0
3D
1
0
10
20
30
PhaseShift[deg]
3D
2
0
5
10 3D
3
0 50 100 150 200 250
Lab. Energy [MeV]
0
2
4
6
PhaseShift[deg]
ε1
0 50 100 150 200 250
Lab. Energy [MeV]
-3
-2
-1
0
ε2
Additionally, resolution variation indicates missing physics at this order
Javier Menéndez (JSPS / U. Tokyo) Ab initio Nuclear Structure CNS Summer School, 2015 14 / 38
Chiral EFT interactions: order-by-order
NN chiral EFT interactions at order N3LO good agreement to datasimilar to standard (Bonn, Argonne) NN potentials
0 50 100 150 200 250
Elab
[MeV]
-20
0
20
40
60PhaseShift[deg]
1S0
Agreement improves at higher orders: NLO (brown), N2LO (blue), N3LO (red)
Javier Menéndez (JSPS / U. Tokyo) Ab initio Nuclear Structure CNS Summer School, 2015 15 / 38
Summary: nuclear forces
Ab initio calculations connect many-body nuclear structureto the underlying theory QCD
Separation of scalesset the relevant degrees of freedomin nuclei (neutrons, protons)and for nuclear forces (pions, contact terms)
Chiral EFT provides a systematic organization ofnuclear forcesat nuclear structure energies (p ∼ mπ)connected to the symmetries of QCD
Chiral EFT provides also gives frameworkfor nuclear currents (interactions)consistent with nuclear forces
Renormalization Group approaches (SRG, RG)simplify the many-body calculations,variation of resolution evaluate missing physics
Javier Menéndez (JSPS / U. Tokyo) Ab initio Nuclear Structure CNS Summer School, 2015 16 / 38
Ab initio nuclear structure
Javier Menéndez (JSPS / U. Tokyo) Ab initio Nuclear Structure CNS Summer School, 2015 17 / 38
The (no core) Shell Model
The (no core) Shell Model
Many-body wave functionlinear combination ofSlater Determinantsfrom single particle states in the basis(3D harmonic oscillator)
|i〉 = |ni li jimji mti 〉|φα〉 = a+
i1a+j2...a
+kA |0〉
|Ψ〉 =∑
α
cα |φα〉
H |Ψ〉 = E |Ψ〉
Dim ∼(
(p + 1)(p + 2)νN
)((p + 1)(p + 2)π
Z
) Dimensions increasecombinatorially...
Javier Menéndez (JSPS / U. Tokyo) Ab initio Nuclear Structure CNS Summer School, 2015 18 / 38
No Core Shell Model
Ab initio many-body calculationsfeasible in light nuclei
No Core Shell ModelGreen’s Function Monte Carlo
NN forcesdo not reproducebinding energiesand spectra:need 3N forces
Good agreementwith 3N forces
Javier Menéndez (JSPS / U. Tokyo) Ab initio Nuclear Structure CNS Summer School, 2015 19 / 38
Green’s function Monte Carlo
Green’s function Monte Carlo
NN forcesdo not reproducebinding energiesand spectra:need 3N forces
Good agreementwith 3N forces
Javier Menéndez (JSPS / U. Tokyo) Ab initio Nuclear Structure CNS Summer School, 2015 20 / 38
Ab initio calculations for oxygen
Ab initio calculations by different approaches,treating explicitly all nucleons as degrees of freedom
No-core shell model(Importance-truncated)
In-medium SRG
Self-consistentGreen’s function
Coupled-cluster
16 18 20 22 24 26 28
Mass Number A
-180
-170
-160
-150
-140
-130
Ener
gy (
MeV
)
MR-IM-SRG
IT-NCSM
SCGF
Lattice EFT
CC
obtained in large many-body spaces
AME 2012
Benchmark with the same initial HamiltonianJavier Menéndez (JSPS / U. Tokyo) Ab initio Nuclear Structure CNS Summer School, 2015 21 / 38
Coupled Cluster, In-Medium SRGThe Coupled Cluster method is based on a reference stateand acting particle-hole excitation operatorsimpose no particle-hole excitations in the reference state
|Ψ〉 = e−(T1+T2+T3··· ) |Φ〉
with T1 =∑α,α
tαα{
a†α, aα},T2 =
∑αβ,αβ
tαβαβ{
a†αa†β, aαaβ
}, · · ·
solve⟨Φαα∣∣ e∑
Ti He−∑
Ti |Φ〉 = 0 ,⟨
Φαβαβ
∣∣∣ e∑Ti He−
∑Ti |Φ〉 = 0
The In-medium similarityrenormalization group methoduses a similarity (unitary)transformationto decouple reference statefrom particle-hole excitations
〈i|H(0) |j〉 〈i|H(∞) |j〉
0p0h 1p1h 2p2h 3p3h 0p0h 1p1h 2p2h 3p3h0p0h
1p1h
2p2h
3p3h
0p0h
1p1h
2p2h
3p3h
Javier Menéndez (JSPS / U. Tokyo) Ab initio Nuclear Structure CNS Summer School, 2015 22 / 38
Ab initio calculations of semimagic nuclei
Coupled-cluster calculations performed in semimagic nuclei up to 132Sn
4
-10
-9
-8
-7
-6NN+3N-induced
� N3LO� N2LOopt
(a)
exp
-0.5
0.5 (b)
-10
-9
-8
-7
.
E/A
[MeV
]
NN+3N-full
� Λ3N = 400 MeV/c
� Λ3N = 350 MeV/c
(c)
exp
16O24O
36Ca40Ca
48Ca52Ca
54Ca48Ni
56Ni60Ni
62Ni66Ni
68Ni78Ni
88Sr90Zr
100Sn106Sn
108Sn114Sn
116Sn118Sn
120Sn132Sn
-0.5
0.5 (d)
FIG. 5: (Color online) Ground-state energies from CR-CC(2,3) for (a) the NN 3N-induced Hamiltonian starting from the N3LO and N2LO-optimized NN interaction and (c) the NN 3N-full Hamiltonian with 3N 400 MeV c and 3N 350 MeV c. The boxes represent thespread of the results from 0 04 fm4 to 0 08 fm4, and the tip points into the direction of smaller values of . Also shown are thecontributions of the CR-CC(2,3) triples correction to the (b) NN 3N-induced and (d) NN 3N-full results. All results employ 24 MeVand 3N interactions with E3max 18 in NO2B approximation and full inclusion of the 3N interaction in CCSD up to E3max 12. Experimentalbinding energies [32] are shown as black bars.
ies have shown that for both cuto s, the induced 4N inter-action are small up into the sd-shell [6, 9]. For heavier nuclei,Fig. 5(c) reveals that the -dependence of the ground-stateenergies remains small for 3N 400 MeV c up to the heav-iest nuclei. Thus, the attractive induced 4N contributions thatoriginate from the initial NN interaction are canceled by ad-ditional repulsive 4N contributions originating from the ini-tial chiral 3N interaction. By reducing the initial 3N cutoto 3N 350 MeV c, the repulsive 4N component resultingfor the initial 3N interaction is weakened [9] and the attrac-tive induced 4N from the initial NN prevails, leading to anincreased -dependence indicating an attractive net 4N con-tribution. All of these e ects are larger than the truncation un-certainties of the calculations, such as the cluster truncation,as is evident by the comparatively small triples contributionsshown in Fig. 5(b) and (d).Taking advantage of the cancellation of induced 4N terms
for the NN 3N-full Hamiltonian with 3N 400 MeV c wecompare the energies to experiment. Throughout the di erentisotopic chains starting from Ca, the experimental pattern ofthe binding energies is reproduced up to a constant shift ofthe order of 1 MeV per nucleon. The stability and qualitativeagreement of the these results over an unprecedented massrange is remarkable, given the fact that the Hamiltonian wasdetermined in the few-body sector alone.When considering the quantitative deviations, one has to
consider consistent chiral 3N interaction at N3LO, and theinitial 4N interaction. In particular for heavier nuclei, the
contribution of the leading-order 4N interaction might be siz-able. Another important future aspect is the study of otherobservables, such as charge radii. In the present calcula-tions the charge radii of the HF reference states are sys-tematically smaller than experiment and the discrepancy in-creases with mass. For 16O, 40Ca, 88Sr, and 120Sn the cal-culated charge radii are 0 3 fm, 0 5 fm, 0 7 fm, and 1 0 fmtoo small [32]. These deviations are larger than the ex-pected e ects of beyond-HF correlations and consistent SRG-evolutions of the radii. This discrepancy will remain a chal-lenge for future studies of medium-mass and heavy nucleiwith chiral Hamiltonians.
Conclusions. In this Letter we have presented the firstaccurate ab initio calculations for heavy nuclei using SRG-evolved chiral interactions. We have identified and eliminateda number of technical hurdles, e.g., regarding the SRG modelspace, that have inhibited state-of-the-art medium-mass ap-proaches to address heavy nuclei. As a result, many-bodycalculations up to 132Sn are now possible with controlled un-certainties on the order of 2%. The qualitative agreement ofground-state energies for nuclei ranging from 16O to 132Snobtained in a single theoretical framework demonstrates thepotential of ab initio approaches based on chiral Hamiltoni-ans. This is a first direct validation of chiral Hamiltonians inthe regime of heavy nuclei using ab initio techniques. Futurestudies will have to involve consistent chiral Hamiltonians atN3LO considering initial and SRG-induced 4N interactionsand provide an exploration of other observables.
Javier Menéndez (JSPS / U. Tokyo) Ab initio Nuclear Structure CNS Summer School, 2015 23 / 38
Ab initio calculations of medium-mass isotopic chains
Calculations with Self-Consistent Green’s Function and In-medium SRGcalculate medium-mass isotopic chains (even-even nuclei)
Ca
18 20 22 24 26 28 30 320
10
20
30
40
50
60
N
S2n [
MeV
]
Ar
K
Ca
Sc
Ti
-10
0
10
20
30
40
.
S2n[M
eV]
ANi
(a) NN+3N-induced
λSRG=1.88 ( )/2.24 fm−1( )
E3max = 14
52 56 60 64 68 72 76 80 84 88
A
-10
0
10
20
30
40
.S2n[M
eV]
(b) NN+3N-full
Λ3N [MeV/c ]
/ 350
❍/● 400λSRG=1.88/2.24 fm
−1
E3max = 14
Start from chiral interaction, treat all nucleons as degrees of freedom,approximate (non-perturbative) solution to the many-body problem
Javier Menéndez (JSPS / U. Tokyo) Ab initio Nuclear Structure CNS Summer School, 2015 24 / 38
Ab initio in general open-shell nuclei?
Javier Menéndez (JSPS / U. Tokyo) Ab initio Nuclear Structure CNS Summer School, 2015 25 / 38
Shell ModelThe Shell Model is the method of choice for shell model nuclei:energies, deformation, electromagnetic and beta transition rates...
0
02
4
6
8
10
0
2
4
6
8
10
12
14
16
3
5
7
9
11
13
2
4
12
14
16
0
0
2
4
6
8
10
0
2
4
6
8
10
12
14
16
3
5
7
9
11
13
2
4
12
14
16
812
1014
1062
1090
263
502
210
179
184
188
190
192
181
1110
1124
1134
1094176
18
1.7
116
21
13521
49
3227179
10
1.0
579
813
874
906
844
292
397
346
227
161
75
112
49
214
211
175
110
80
133
546
557
429
427 2.7
12
7.8
43
1.8
0.1
6.8
19
583.0
49
202891
16
0.2
284
Exp. Th.
40Ca
Excita
tio
n E
ne
rgy (
Me
V)
0
2
4
6
8
10
12
14
16
18
20
22
24
26
0
2
4
6
Ex (
MeV
)
(a) Yrast levels
0
2
4
6
Ex (
MeV
)
(b) Yrare levels
0
1000
30 40 50
B(E
2;
0+ 1 →
2+ 1)
(e2fm
4)
N
(c) B(E2; 0+1 → 2
+1)
Javier Menéndez (JSPS / U. Tokyo) Ab initio Nuclear Structure CNS Summer School, 2015 26 / 38
Shell Model (with core)
The Shell Model solves the many-bodyproblem by direct diagonalization in arelatively small configuration space
The total space is separated into
Outer orbits:orbits that are always empty
Valence space: the space inwhich we explicitly solve theproblem
Inner core:orbits that are always filled
Diagonalization in valence space: H |Ψ〉 = E |Ψ〉 → Heff |Ψ〉eff = E |Ψ〉eff
where Heff includes the effect of inner core and outer orbits
Javier Menéndez (JSPS / U. Tokyo) Ab initio Nuclear Structure CNS Summer School, 2015 27 / 38
Effective Shell Model interactionsContributions to the effective interaction from orbitals outside the valencespace can be obtained within many-body perturbation theory (to third order)
H |Ψ〉 = E |Ψ〉 → Heff |Ψ〉eff = E |Ψ〉eff
Heff = εeff + Veff
a
c
b
d
Q =
a
c
b
d
+
a
c
b
d
a b
dc
+
a
c
b
d
+
V
+ + . . .
Veff
b
d
=
a
c
b
dVlow-k
+
a
c
+ . . .
b
dlow-k
εeff
a a
a
a
a a x
V
Vlow-k
Javier Menéndez (JSPS / U. Tokyo) Ab initio Nuclear Structure CNS Summer School, 2015 28 / 38
Convergence in many-body perturbation theory
How good convergence in MBPT approach?
εeff Veff
2 4 6 8 10 12 14 16 18
Nh_
ω
-12
-10
-8
-6
-4
-2
Sin
gle
-Par
ticl
e E
ner
gy (
MeV
)
2 4 6 8 10 12 14 16 18
Nh_
ω
-4
-2
0
2
4Neutron Proton
p3/2
f7/2
p1/2
f5/2
f5/2
p1/2
f7/2
p3/2
2 4 6 8 10 12 14 16 18
Nh_
ω
-20.5
-20
-19.5
-19
-18.5
Gro
und-S
tate
Ener
gy (
MeV
)
2 4 6 8 10 12 14 16 18
Nh_
ω
-82
-81
-80
-79
-78
-77
-76
1st order
2nd order
3rd order
42Ca
48Ca
Intermediate-state excitations seem to be under control
Associated uncertainty difficult to quantify,3rd order reasonable but 4th order very expensive:
Is it possible to obtain Shell Model interactions non-perturbatively?
Javier Menéndez (JSPS / U. Tokyo) Ab initio Nuclear Structure CNS Summer School, 2015 29 / 38
Effective Shell Model interactions
Coupled Cluster:Solve coupled-cluster equations forcore (reference state |Φ〉), A + 1 and A + 2 systemsProject the coupled-cluster solution into valence space(Okubo-Lee-Suzuki transformation)
In-medium similarityrenormalization groupdecouplecore from excitationsdecouple A particles invalence space from rest
Drives all n-particle n-hole couplings to 0 – decouples core from excitations
〈i|H|j〉
!" # $!"$# %!"%# &!"&#
&!"&#
%!"%#
$!"$#
!" #
⟨ ∣∣ !
∣∣"⟩
����� ����� ����� �����
�����
�����
�����
�����
〈npnh|H(∞)|Φcore〉 = 0
In addition to Heff , these non-perturbative methods provide the core energy
Javier Menéndez (JSPS / U. Tokyo) Ab initio Nuclear Structure CNS Summer School, 2015 30 / 38
Non-perturbative Shell Model interactions
Valence-shell interactions derived non-perturbativecomparable to MBPT and phenomenological interactions,but so far limited to oxygen isotopes
MBPT CCEI IM-SRG Expt.
0
1
2
3
4
5
6
7
8
9
Ener
gy (
MeV
)
0+
2+
2+ 2
+
0+
0+
(2+)
2+
2+
(0+)0
+
0+
4+
4+ (4
+)
4+
2+
0+
2+
0+
3+
3+ 3
+
3+
0+
22O
MBPT CCEI IM-SRG Expt.
0
1
2
3
4
5
6
1/2+
5/2+
5/2+
1/2+
3/2+
3/2+
1/2+
5/2+
3/2+
(5/2+)
1/2+
(3/2+)
23O
MBPT CCEI IMSRG Expt.
0
1
2
3
4
5
6
7
0+
2+
0+
2+
1+
0+
1+
1+
2+
0+
2+
1+ 24
O
Javier Menéndez (JSPS / U. Tokyo) Ab initio Nuclear Structure CNS Summer School, 2015 31 / 38
3N Forces: normal ordering
Shell Model codes usually do not diagonalize 3N forcesUse normal-ordering approximation:
normal-ordered 2B: 2 valence, 1 core particle⇒ Two-body Matrix Elements
N
NN
π
N N
N
π
N
N
N
N
N
N
π
N
N
N
N N
N
〈a′b′|V eff3N |ab〉 =
∑c 〈a′b′c|V3N |abc〉
normal-ordered 1B: 1 valence, 2 core particles⇒ Single particle energies
N
NN
π
N N
N
π
N
N
N
N
N
N
π
N
N
N
N N
N
〈a| εeff3N |a〉 =
∑c1 c2〈ac1c2|V3N |ac1c2〉
O core
'b'
Javier Menéndez (JSPS / U. Tokyo) Ab initio Nuclear Structure CNS Summer School, 2015 32 / 38
Residual 3N Forces
In the most neutron-rich isotopes,3N forces between 3 valence neutrons(suppressed by Nvalence/Ncore)
Evaluated perturbatively: 〈Ψ|V 3N |Ψ〉
d3/2
s 1/2 d5/2
(a) G-matrix NN + 3N (Δ) forces
d3/2
s 1/2d5/2
4
-4
0
-8
Single-ParticleEnergy(M
eV)
Neutron Number (N)8 201614
Neutron Number (N)8 201614
NN + 3N (N LO)
NNNN + 3N (Δ)NN
NN + 3N (Δ)
low k(b) V NN + 3N (Δ,N LO) forces2
2
(d) 3-body interactions with onemore neutron added to (c)
(c) 3-body interaction
O core16
11 12 13 14 15 16 17 18 19 20
Neutron Number N
0.2
0.4
0.6
0.8
∆E3N
,res
(M
eV)
O
8 9 10 11 12 13 14 15 16 17 18 19 20
Neutron Number N-60
-50
-40
-30
-20
-10
0
Ene
rgy
(MeV
)
NNNN+3NUSDbAME 2003
Residual 3N repulsive and small compared to overall 3N forces
Javier Menéndez (JSPS / U. Tokyo) Ab initio Nuclear Structure CNS Summer School, 2015 33 / 38
Oxygen dripline anomaly and 3N forces
O isotopes: ’anomaly’ in the dripline at 24O, doubly magic nucleusChiral NN+3N forces provided repulsion needed to describe O dripline
2 8 20 28
Z
N
2
8
stab
ility
line
O 1970
F 1999
N 1985
C 1986
B 1984
Be 1973
Li 1966
H 1934
He 1961
Ne 2002
Na 2002
Mg 2007
Al 2007
Si 2007
unstable oxygen isotopes
unstable fluorine isotopes
stable isotopes
unstable isotopes
neutron halo nuclei
Based on many-body perturbation theoryOtsuka et al. PRL105 032501 (2010)
8 201614Neutron Number (N) Neutron Number (
s 1/2
(c) G-matrix NN + 3N (∆) forces
d3/2
d5/2
NN
NN + 3N (∆)Sin
gle
-Par
ticl
e E
ner
gy (
MeV
)
4
-4
0
-8
Sin
gle
-Par
ticl
e E
ner
gy (
MeV
)
8 201614
d3/2
d5/2s 1/2
(a) Forces derived from NN theory
V
G-matrix
Neutron Number (N)
low k
4
-4
0
-8
Javier Menéndez (JSPS / U. Tokyo) Ab initio Nuclear Structure CNS Summer School, 2015 34 / 38
Oxygen dripline in ab initio calculationsOxygen dripline including chiral NN+3N forces correctly reproducedconfirmed in ab initio calculations by different approaches,treating explicitly all nucleons as degrees of freedom
2s1 2
1d5 2
1d3 2
8
6
4
2
0
2
4
6
iA1
MeV
2N 3N full
2N 3N ind
60
16 18 20 22 24 26 28
Mass Number A
-12
-8
-4
0
4
Sin
gle
-Par
ticl
e E
ner
gy (
MeV
)
NN+3N-ind
NN+3N-full
d5/2
d3/2
s1/2
(a)
16 18 20 22 24 26 28
Mass Number A
-180
-170
-160
-150
-140
-130
Ener
gy (
MeV
)
MR-IM-SRG
IT-NCSM
SCGF
Lattice EFT
CC
obtained in large many-body spaces
AME 2012
Javier Menéndez (JSPS / U. Tokyo) Ab initio Nuclear Structure CNS Summer School, 2015 35 / 38
Shell closures in calcium isotopesAb initio calculations able to describe shell closure at N = 28
28 29 30 31 32 33 34 35 36
Neutron Number N
0
2
4
6
8
10
12
14
16
18
20
22
S2
n (
MeV
)
MBPT
CC
SCGF
MR-IM-SRG
42 44 46 48 50 52 54 56
Mass Number A
0
1
2
3
4
5
2+ E
ner
gy (
MeV
)
MBPT
CC
Prediction of shell closures at 52Ca, 54Ca inagreement to recent experiments
51,52Ca / 53,54Ca masses [TRIUMF/ISOLDE]54Ca 2+
1 state excitation energy [RIBF]Javier Menéndez (JSPS / U. Tokyo) Ab initio Nuclear Structure CNS Summer School, 2015 36 / 38
Full sd-shell calculation
-100
10203040
S 2n (M
eV)
-10010203040
01020304050
10 12 14 16 18 20 10 12 14 16 18 2001020304050
10 12 14 16 18 201020304050
10 12 14 16 18 20 10 12 14 16 18 20
Neutron Number N
1020304050
2nd order3rd orderAME 2012
8O 9F 11Na
13Al
10Ne
14Si
12Mg
15P
19K18Ar
16S 17Cl
20Ca
Full sd-shell calculationwith chiral NN+3N forces
Fit only to two, three andfour-body systems
Uncertainly band includingdifferent SRG resolution scalesand low-energy couplings
Second and third-orderMBPT results
-20-10
0102030
S 2p (M
eV)
-20-100102030
-100
10203040
10 12 14 16 18 20 10 12 14 16 18 20-10010203040
10 12 14 16 18 20
010203040
10 12 14 16 18 20 10 12 14 16 18 20
Proton Number Z
010203040
2nd order3rd orderAME 2012
N=8 N=9 N=11
N=13
N=10
N=14
N=12
N=15
N=19N=18
N=16 N=17
N=20
Javier Menéndez (JSPS / U. Tokyo) Ab initio Nuclear Structure CNS Summer School, 2015 37 / 38
SummaryAb initio many-body calculations with chiral NN+3N forcesextending over the nuclear chart
Calculations with all nucleons activerestricted to light nuclei or the vicinity of semimagic nuclei(No core shell model, Coupled Cluster, In-medium SRG...)
Combination to the Shell Model:Microscopic foundation of Shell Model successeffective interactions based on perturbativeand non-perturbative approaches
Long standing puzzles like oxygen driplineor N = 28 shell-closure clarifiedbased on NN+3N forces
Medium-mass nuclei within reachof ab initio calculations
28 29 30 31 32 33 34 35 36
Neutron Number N
0
2
4
6
8
10
12
14
16
18
20
22
S2
n (
MeV
)
MBPT
CC
SCGF
MR-IM-SRG
2 8 20 28
Z
N
2
8
stab
ility
line
O 1970
F 1999
N 1985
C 1986
B 1984
Be 1973
Li 1966
H 1934
He 1961
Ne 2002
Na 2002
Mg 2007
Al 2007
Si 2007
unstable oxygen isotopes
unstable fluorine isotopes
stable isotopes
unstable isotopes
neutron halo nuclei
Javier Menéndez (JSPS / U. Tokyo) Ab initio Nuclear Structure CNS Summer School, 2015 38 / 38