Ab initio framework for Nuclear Structure and Fundamental ... · For nuclear structure, typical...

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


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


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


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


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


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


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


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


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


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


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 — — — —


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


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)


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


Ab initio nuclear structure


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...


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


Green’s function Monte Carlo

Green’s function Monte Carlo

NN forcesdo not reproducebinding energiesand spectra:need 3N forces

Good agreementwith 3N forces


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


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.


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


Ab initio in general open-shell nuclei?


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)


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


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


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?


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


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


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'


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


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


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


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


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


Ab initio framework for Nuclear Structure and Fundamental ... · For nuclear structure, typical...

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