Towards Ab Initio Calculations of Double-Beta Decay Nuclear Matrix Elements

Jason D. Holt

R. Stroberg A. Schwenk S. Bogner H. Hergert

J. Engel M. Horoi How domagic numbers form evolve? and nuclear existence? What are limits the of proton/neutron-richWhat are properties the matter? of Worldwide joint nuclear science: frontiersExploring the of 2! 8!

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effort effort

82! 3N forces for essential exotic nuclei π 82! Advances inmany-bodymethods Self-Consistent Green’s Self-Consistent Function Many-BodyPerturbation Theory SRGIn-Medium Cluster Coupled π €

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How domagic numbers form evolve? and nuclear existence? What are limits the of proton/neutron-richWhat are properties the matter? of Worldwide joint nuclear science: frontiersExploring the of 2! 8!

2! protons Advances in 20! 8! 28! neutrons 20! experimental/ 28! Medium-Mass Exotic Nuclei Exotic Medium-Mass 50! Ab theoretical € Initio Nuclear Structure for

€

effort effort

82! 3N forces for essential exotic nuclei π 82! Advances inmany-bodymethods Self-Consistent Green’s Self-Consistent Function Many-BodyPerturbation Theory SRGIn-Medium Cluster Coupled π €

π

Improved 0νββ-Decay Calculations in Shell Model Standard SM approach: phenomenological wavefunctions + bare operator

Avenue towards ab initio shell-model calculations:

Consistent operators and wfs from chiral forces and currents

0⌫ 0⌫ 0⌫ MF 0⌫ M = MGT 2 + MT gA

M 0⌫ = f H(r ) ⌧ +⌧ + i GT h | ab a · b a b | i Xab

1) Wavefunctions currently phenomenological; calculate ab initio IM-SRG, CC, MBPT… Part II Improved 0νββ-Decay Calculations in Shell Model Standard SM approach: phenomenological wavefunctions + bare operator

Avenue towards ab initio shell-model calculations:

Consistent operators and wfs from chiral forces and currents

0⌫ 0⌫ 0⌫ MF 0⌫ M = MGT 2 + MT gA

M 0⌫ = f H(r ) ⌧ +⌧ + i GT h | abMeffa · b a b | i Xab

1) Wavefunctions currently phenomenological; calculate ab initio 2) Effective decay operator: correlations outside valence space MBPT… Part I Improved 0νββ-Decay Calculations in Shell Model Standard SM approach: phenomenological wavefunctions + bare operator

Avenue towards ab initio shell-model calculations:

Consistent operators and wfs from chiral forces and currents

0⌫ 0⌫ 0⌫ MF 0⌫ M = MGT 2 + MT gA

M 0⌫ = f H(r ) ⌧ +⌧ + i GT h | ab a · b a b | i Xab

1) Wavefunctions currently phenomenological; calculate ab initio 2) Effective decay operator: correlations outside valence space 3) Operator corrections: two-body currents

See talks of Menéndez, Gazit, Schwenk GT, E2, M1, EWoperators effective calculation Consistent of Towards 2! Calculation of nuclear Calculation of 8!

2! protons Calculations of Effective Operators (MBPT) 20! 8! 28! neutrons ab 20! initioelectroweak physics 0νββ 28! decay… • • • JDH Lincoln, Kwiatkowski PRC Engel, and JDH wavefunctions 50! , et Brunner al., PRL

✔

Impact on NMEon in Impact Keyphysics problems , et JDH al., PRC 82! 87 82! , 064315 (2013) , 064315 110 References , 012501 (2013) [LEBIT] (2013) , 012501 ? 50!

89

, 045502 (2014) [TITAN] (2014) , 045502 126! 48 Ca,

76 Ge,

82 Se

Effective 0νββ-Decay Operator c c c ccStandardd approach:c c V phenomenologicald c c d wavefunctions + bare operator d Vlow-k d d d Vlow-klow-k d d ˆ = CalculateˆQˆ = =consistent... effective+ 0νββ+...-decay... operator using MBPT Q + Q + + + a b a b a aaDiagrammaticallybbb a a similar:bb a a replacebb one interaction vertex with M0ν operator c d c d c cc Vlow-kddd c cc ddd c d c d c d + + Qˆ ++= +++ + ... Xˆ = M + Veff V Meff low k V a b a b a aa bbb a aa bbb a b a b a low k b c d c d c c c ddd c cc ddd c d c d c d € V low k ... € + + +++ ... +++ ++... + + + a a a aa b b a aa bb b b b b € a b a b a b c c d d c d c d + + + ... + + + ... a b a b a b a b Effective 0νββ-Decay Operator c c c ccStandardd approach:c c V phenomenologicald c c d wavefunctions + bare operator d Vlow-k d d d Vlow-klow-k d d ˆ = CalculateˆQˆ = =consistent... effective+ 0νββ+...-decay... operator using MBPT Q + Q + + + a b a b a aaDiagrammaticallybbb a a similar:bb a a replacebb one interaction vertex with M0ν operator c d c d c cc Vlow-kddd c cc ddd c d c d c d + + Qˆ ++= +++ + ... Xˆ = M + Veff V Meff low k V a b a b a aa bbb a aa bbb a b a b a low k b c d c d c c c ddd c cc ddd c d c d c d € V low k ... € + + +++ ... +++ ++... + + + a a a aa b b a aa bb b b b b € a b a b a b c d c d Previous: G-matrix calculation to c d c d + + + ... 1st-order MBPT non-convergent + + + ... Engela band Hagena, PRC (2009)b a b a b Low-momentum interactions: Improve convergence behavior? Effective 0νββ-Decay Operator Calculate in MBPT:

1st order (× 2) 2nd order (×3) 48 76 82 Calculations of M0ν in Ca, Ge, and Se Phenomenological wavefunctions from A. Poves, M. Horoi

48Ca Bare 0.77 76Ge Bare 3.12 82Se Bare 2.73 Effective 1.30 Effective 3.77 Effective 3.62

Lincoln, JDH et al., PRL (2013) Converged in 13 major oscillator shells JDH and Engel, PRC (2013) No order-by-order convergence in MBPT Kwiatkowski, et al., PRC (2014)

Overall ~25-30% increase for 76Ge, 82Se; 75% for 48Ca 48 76 82 Calculations of M0ν in Ca, Ge, and Se Phenomenological wavefunctions from A. Poves, M. Horoi

48Ca Bare 0.77 76Ge Bare 3.12 82Se Bare 2.73 Effective 1.30 Effective 3.77 Effective 3.62

Lincoln, JDH et al., PRL (2013) Converged in 13 major oscillator shells JDH and Engel, PRC (2013) No order-by-order convergence in MBPT Kwiatkowski, et al., PRC (2014)

Overall ~25-30% increase for 76Ge, 82Se; 75% for 48Ca

Promising first steps – up next…

- Consistent wavefunctions and operators in MBPT from chiral NN+3N

- Operator corrections: two-body currents from chiral EFT

- Nonperturbative methods (IM-SRG)

- Effects of induced 3-body operators 48 76 82 Calculations of M0ν in Ca, Ge, and Se Phenomenological wavefunctions from A. Poves, M. Horoi

48Ca Bare 0.77 76Ge Bare 3.12 82Se Bare 2.73 Effective 1.30 Effective 3.77 Effective 3.62

Lincoln, JDH et al., PRL (2013) Converged in 13 major oscillator shells JDH and Engel, PRC (2013) No order-by-order convergence in MBPT Kwiatkowski, et al., PRC (2014)

Overall ~25-30% increase for 76Ge, 82Se; 75% for 48Ca

Promising first steps – up next…

- Consistent wavefunctions and operators in MBPT from chiral NN+3N

- Operator corrections: two-body currents from chiral EFT

- Nonperturbative methods (IM-SRG)

- Effects of induced 3-body operators IM-SRG for Valence-Space Hamiltonians Tsukiyama, Bogner, Schwenk, PRC (2012)

In-Medium SRG continuous unitary trans. to decouple off-diagonal physics d od d H(s)=U(s)HU†(s) H (s)+H (s) H ( ) ⌘ ! 1 50 2v 1q1v 2q 3p1h 4p2h 2v 1q1v 2q 3p1h 4p2h 0g 9/2 H 0f5/2 eff 1p1/2 q 1p p 28 3/2 0f7/2 20 0d3/2 1s1/2 v 0d5/2

8 Inert 0p3/2 h core 0p1/2 H(s = 0) H( )

! 1 Separate p states into valence states (v) and those above valence space (q)

Define Hod to decouple valence space from excitations outside v

First nonperturbative construction of valence-space Hamitonians: Heff Benchmark in Oxygen with MBPT/CCEI Compare with Coupled-Cluster effective interactions from NN+3N forces

7 9 22 6 23 + 24 O O 1 O + + 1 8 4 + + 6 1 + 5 + 3/2 2 + + + 3/2 + 2 + 7 2 + (4 ) 2 1 4 + + (2 ) + 4 + (3/2 ) 5 + + 2 4 2 6 + 2 3 + + + 4 + 0 (0 ) 3/2 5 0 + + + 3 + + 0 + 3 5/2 5/2 (5/2 ) + 3 + 4 3 5/2 + 3 2 +

Energy (MeV) 2 + + 3 2 2 2 2 2 1 1 1 + + + + + + + + + + + + 0 0 0 0 0 0 1/2 1/2 1/2 1/2 0 0 0 0 0 MBPT CCEI IM-SRG Expt. MBPT CCEI IM-SRG Expt. MBPT CCEI IMSRG Expt. Bogner, Hergert, JDH, Schwenk et al., PRL (2014)

Hebeler, JDH, Menéndez, Schwenk, ARNPS (2015)

Many-Body Perturbation Theory in extended valence space (sdf7/2 p3/2)

IM-SRG/CCEI (sd shell) spectra agree within ~300 keV; improved quality Fully Open Shell: Neutron-Rich Fluorine Spectra Fluorine spectroscopy: MBPT and IM-SRG (sd shell) from NN+3N forces

+ + + 24 3/2+ 4 1 4 25 7/2 4 26 + + 3 + F + + 5 F 1/2+ F 2 (2 ) + 7/2 + 2 1+ + + + + 5/2 3.5 5 2 + + 2 (4 ,2 ) + 9/2 + 5/2 1+ 9/2 + + 4 + (5/2 ) 3 5/2+ 3 + 4 1/2 + + + 3 3 + + 3/2 (3/2 ) 3/2 + 3 3/2 + + + (3 ) + 9/2 + 4 0 + (3/2 ) + + + 0 3+ 4 + + 3 1 2.5 1 4 + + (4 ) (9/2 ) 2 3 + + + 1 3 2 + 2 2 1 + + 1 + 1 + 4 + 3/2 + + 3 1 2 1/2

Energy (MeV) 1.5 + (1/2 ) + + 2 1/2 1 1 + + 2+ 1 2+ + + + 4 + 5/2 4 2 + 2 2 0.5 2 + + 2 4 + + + + + + + + + + + + 0 1 1 1 1 0 3 3 3 3 0 1/2 5/2 (5/2 ) 5/2 CC IM-SRG Expt. USDB MBPT IM-SRG Expt. USDB MBPT IM-SRG Expt. USDB Bogner, Hergert, JDH, Schwenk, in prep. IM-SRG: competitive with phenomenology, good agreement with data

Preliminary results already for scalar operators: charge radii, E0 transitions

Upcoming: general operators M1, E2, GT, double-beta decay Stroberg et al. Effective Operators

Calculate unitary transformation directly

⌦(s) ⌦(s) 1 1 H(s)=e He = H + [⌦(s),H]+ [⌦(s), [⌦(s),H]] + 2 12 ··· Straightforward to generalize to arbitrary operators

⇤ ⌦(s) ⇤ ⌦(s) ⇤ 1 ⇤ 1 ⇤ (s)=e e = + ⌦(s), + ⌦(s), ⌦(s), + O O O 2 O 12 O ···

⇥ ⇤ ⇥ ⇥ ⇤⇤ First apply to scalar operators: charge radii, E0 transitions

Commutators induce important higher-order one-/two-body parts

Quantify importance of induced higher-body contributions! Mg Charge radii RMS Charge Radii in sd Shell Model Previous SM radii calculations rely on empirical input or as relative to core Absolute radii for entire sd shell calculated in shell model NN+3N

Stroberg, Bogner, Hergert, JDH, Schwenk , in prep Benchmarked against NCSMRadii in various for entire SM codessd-shell are accessible. ~10% too small – deficiencies expected to come from initial Hamiltonian Two-body part important 15-20%

Ragnar Stroberg (TRIUMF) E↵ective Operators w/ IMSRG May 22, 2015 19 / 23 spin-orbitpartners of Importance Heavier candidates accessible toSM Exotic decay mechanisms Two-body currents from chiral EFT Improvements inoperator Consistent 2! SRG-evolved operator 8! 2!

protons Horoi Ekstrom Menendez 20! 8! 28! , Brown neutrons , wavefunction Wendt 20! , Gazit , PRL (2013) 28! , et al., PRL (2014) , Schwenk Path to Full Calculation Shuster and operator and 50! , PRL (2011)

Horoi etal.

Independent calculations with uncertainties Move toheavier Strategy Benchmark SM 48 82! Ca 82! Understand origin of quenching inSM Understand origin of CCEI Nonperturbative

2νββ IM-SRG decay calculation 50! Jansen, Hagen Hagen Jansen,

Stroberg 48 76 Ca Ge withSM Ge , 0νββ SM 126! Bogner

wfs calculation against CC , Hergert /operators

, JDH, Schwenk

Fully Open Shell: Neutron-Rich Neon Spectra Neon spectroscopy: MBPT and IM-SRG (sd shell) from NN+3N forces

+ + 4 7/2 5 24 0+ 25 + + 5 26 + + 3 9/2 7/2 0 + 0 0 Ne + Ne + Ne+ + 4 + 3 + 3/2 9/2 2 0+ + 3 + + 3/2+ + 4 4 + 4 + 4 + + 5/2 4 + + + 5/2 2 + 4 2 2 2 3 + 2 2 3/2 + + + + 4 4 1/2 0 + + 2 7/2 3 3 + + + + 0 3/2 (3/2 ) 3/2 2 + 5/2 + + + 5/2 + + 5/2 (5/2 ) + + + 2 + 2 2 2 + 2 2 2 2 + 2 2 Energy (MeV) + 1 5/2 + 1 3/2 1

+ + + + + + + + + + + + 0 0 0 0 0 0 1/2 1/2 (1/2 ) 1/2 0 0 0 0 0 MBPT IM-SRG Expt. USDB MBPT IM-SRG Expt. USDB MBPT IM-SRG Expt. USDB Bogner, Hergert, JDH, Schwenk, in prep. IM-SRG: competitive with phenomenology, good agreement with data

Preliminary results already for scalar operators: charge radii, E0 transitions

Upcoming: general operators M1, E2, GT, double-beta decay R. Stroberg et al. 0νββ-Decay Operator Details of operator used in subsequent calculations

Closure approximation good to 8-10% Sen’kov, Horoi, Brown, PRC (2014)

1) Nuclear wavefunctions: currently phenomenological – calculate ab initio 2) Decay operator: correlations outside valence space; 2-body currents 82 76 Calculations of M0ν in Se and Ge

Phenomenological wavefunctions from A. Poves, M. Horoi (pf5/2 g9/2 space)

76Ge Bare matrix element 3.12 82Se Bare matrix element 2.73 Full 1st order 3.11 Full 1st order 2.79

Full 2nd order 3.77 Full 2nd order 3.62

Lincoln, JDH et al., PRL (2013) [LEBIT] Converged in 13 major oscillator shells JDH and Engel, PRC (2013) Little net effect at 1st order – no clear order-by-order convergence

Overall ~25-30% increase from bare value 48 Calculations of M0ν in Ca Phenomenological wavefunctions from GXPF1A (pf shell)

48Ca GT Fermi Tensor Sum Bare 0.675 0.130 −0.072 0.733

Final 1.211 0.160 −0.070 1.301 Kwiatkowski, JDH, Engel, Horoi et al., PRC (2014)

Converged in 13 major oscillator shells

Similar absolute increase as in Ge, Se – no clear order-by-order convergence

Overall ~75% increase from bare value 48 Calculations of M0ν in Ca Phenomenological wavefunctions from GXPF1A (pf shell)

48Ca GT Fermi Tensor Sum Bare 0.675 0.130 −0.072 0.733

Final 1.211 0.160 −0.070 1.301 Kwiatkowski, JDH, Engel, Horoi et al., PRC (2014)

Converged in 13 major oscillator shells

Similar absolute increase as in Ge, Se – no clear order-by-order convergence

Overall ~75% increase from bare value Promising first steps – up next…

- Consistent wavefunctions and operators in MBPT from chiral NN+3N

- Operator corrections: two-body currents from chiral EFT

- Nonperturbative methods (IM-SRG)

- Effects of induced 3-body operators 48 Calculations of M0ν in Ca Phenomenological wavefunctions from GXPF1A (pf shell)

48Ca GT Fermi Tensor Sum Bare 0.675 0.130 −0.072 0.733

Final 1.211 0.160 −0.070 1.301 Kwiatkowski, JDH, Engel, Horoi et al., PRC (2014)

Converged in 13 major oscillator shells

Similar absolute increase as in Ge, Se – no clear order-by-order convergence

Overall ~75% increase from bare value Promising first steps – up next…

- Consistent wavefunctions and operators in MBPT from chiral NN+3N

- Operator corrections: two-body currents from chiral EFT

- Nonperturbative method for wavefunction and operator (IM-SRG)

- Effects of induced 3-body operators? Nuclear Matrix Element

Improved calculations move in direction of other methods

Barea, Koltila, Iachello, PRL (2012)

Aim: first-principles framework capable of robust prediction Intermediate-State Convergence Convergence results in 82Se

82 Se 3.6

3.55

Matrix Element 3.5 0⌫ MGT 3.45 JDH and Engel, PRC (2013) 8 10 12 14 16 18 Nh

Results well converged in terms of intermediate state excitations

Improvement due to low-momentum interactions

Similar trend in 76Ge and 48Ca

Improvements from SRG-evolved operator? SRG Evolution of Operator Preliminary results with 0νββ-decay operator

Schuster, Engel, JDH, Navrátil, Quaglioni Minimal effects from SRG evolution

Negligible improvements in MBPT convergence? SRG Evolution of Operator Preliminary results with 0νββ-decay operator

1.2 48Ca

1.1

Matrix Element Bare SRG-evolved 1 8 10 12 14 16 18 Nhω Schuster, Engel, JDH, Navrátil, Quaglioni

Minimal improvement in intermediate-state convergence Strategy

1) Effective interaction: sum excitations outside valence space to 3rd order

2) Single-particle energies calculated self-consistently

3) Harmonic-oscillator basis of 13 major shells: converged

4) NN and 3N forces – fully to 3rd-order MBPT

5) Work in extended valence spaces

-18.5 -76 -77 -19 42 -78 48 Ca Ca -19.5 -79 1st order -80 -20 2nd order 3rd order -81 Ground-State Energy (MeV) -20.5 -82 2 4 6 8 10 12 14 16 18 2 4 6 8 10 12 14 16 18 _ _ Nh Nh Clear convergence with HO basis size

Promising order-by-order behavior Strategy

1) Effective interaction: sum excitations outside valence space to 3rd order

2) Single-particle energies calculated self-consistently

3) Harmonic-oscillator basis of 13 major shells: converged

4) NN and 3N forces – fully to 3rd-order MBPT

5) Work in extended valence spaces

-11.6 nd rd 2 order 3 order 18 -11.8 O -12

-12.2

Energy (MeV) V -12.4 low k G-matrix -12.6 4 6 8 10 12 14 2 4 6 8 10 12 14 Major Shells Major Shells

G-matrix – no signs of convergence (similar in pf-shell) Strategy

1) Effective interaction: sum excitations outside valence space to 3rd order

2) Single-particle energies calculated self-consistently

3) Harmonic-oscillator basis of 13 major shells: converged

4) NN and 3N forces – fully to 3rd-order MBPT

5) Work in extended valence spaces

-2 4 Neutron Proton -4 f f5/2 5/2 2 -6 p 0 p -8 1/2 1/2 f -2 f -10 7/2 7/2 p p 3/2 -4 3/2

Single-Particle Energy (MeV) -12

2 4 6 8 10 12 14 16 18 2 4 6 8 10 12 14 16 18 _ _ Nh Nh

Clear convergence with HO basis size Promising order-by-order behavior Nonperturbative In-Medium SRG Tsukiyama, Bogner, Schwenk, PRL (2011) In-Medium SRG continuous unitary trans. drives off-diagonal physics to zero

d od d H(s)=U(s)HU†(s) H (s)+H (s) H ( ) ⌘ ! 1 From uncorrelated Hartree-Fock ground state (e.g., 16O) define: Hod = p H h + pp H hh + +h.c. h | | i h | | i ···

i H j npnh H( ) core =0 h | | i h | 1 | i Drives all n-particle n-hole couplings to 0 – decouples core from excitations IM-SRG: Flow Equation Formulation

Define U(s) implicitly from particular choice of generator:

⌘(s) (dU(s)/ds) U †(s) ⌘ chosen for desired decoupling behavior – e.g., d od ⌘I (s)= H (s),H (s) Wegner (1994) ⇥ ⇤ Solve flow equation for Hamiltonian (coupled DEs for 0,1,2-body parts) dH(s) =[⌘(s),H(s)] ds

Hamiltonian and generator truncated at 2-body level: IM-SRG(2)

0-body flow drives uncorrelated ref. state to fully correlated ground state

Ab initio method for energies of closed-shell systems IM-SRG: Valence-Space Hamiltonians Tsukiyama, Bogner, Schwenk, PRC (2012) Open-shell systems

Separate p states into valence states (v) and those above valence space (q)

50 2v 1q1v 2q 3p1h 4p2h 2v 1q1v 2q 3p1h 4p2h 0g9/2 0f5/2 1p1/2 q 1p p 28 3/2 0f7/2 20 0d3/2 1s1/2 v 0d5/2

8 h 0p3/2 0p1/2 H(s = 0) H( ) ! 1 Redefine Hod to decouple valence space from excitations outside v Hod = p H h + pp H hh + v H q + pq H vv + pp H hv +h.c. h | | i h | | i h | | i h | | i h | | i IM-SRG: Valence-Space Hamiltonians Tsukiyama, Bogner, Schwenk, PRC (2012) Open-shell systems

Separate p states into valence states (v) and those above valence space (q)

50 2v 1q1v 2q 3p1h 4p2h 2v 1q1v 2q 3p1h 4p2h 0g 9/2 H 0f5/2 eff 1p1/2 q 1p p 28 3/2 0f7/2 20 0d3/2 1s1/2 v 0d 5/2 8 h 0p3/2 0p1/2 H(s = 0) H( ) ! 1

Core physics included consistently (absolute energies, radii…)

Inherently nonperturbative – no need for extended valence space

Non-degenerate valence-space orbitals IM-SRG Oxygen Ground-State Energies IM-SRG valence-space interaction and SPEs in sd shell

4 -130 d 3/2 0 -140 s 1/2 -150 -4 d -160 5/2 Energy (MeV) -8 -170 Exp. NN+3N-ind NN+3N-ind NN+3N-full

Single-Particle Energy (MeV) NN+3N-full -180 -12 16 18 20 22 24 26 28 16 18 20 22 24 26 28 Mass Number A Mass Number A Bogner et al., PRL (2014)

NN+3N-ind modest underbinding, dripline not reproduced

NN+3N-full modestly overbound, correct dripline trend

Weak ~ ! dependence Comparison with Large-Space Methods Large-space methods with same SRG-evolved NN+3N forces 4 -130 -120 obtained in large many-body spaces d 3/2 NN+3N-ind 0 -140 -130 s 1/2 -140 -150 -4 -150 d -160 5/2 -160 MR-IM-SRG Energy (MeV) Exp. Energy (MeV) -8 -170 -170 IT-NCSM NN+3N-ind NN+3N-ind SCGF NN+3N-full CC AME 2012 Single-Particle Energy (MeV) NN+3N-full -180 -180 -12 16 18 20 22 24 26 28 16 18 20 22 24 26 28 16 18 20 22 24 26 28 Mass Number A Mass Number A Bogner et al., PRL (2014) Mass Number A

Agreement between all methods with same input forces Comparison with Large-Space Methods Large-space methods with same SRG-evolved NN+3N forces 4 -130 -130 obtained in large many-body spaces d 3/2 NN+3N-full 0 -140 -140 s 1/2 -150 -150 -4 d -160 -160 MR-IM-SRG 5/2 IT-NCSM Energy (MeV) -8 -170 Exp. Energy (MeV) -170 SCGF NN+3N-ind NN+3N-ind Lattice EFT NN+3N-full CC AME 2012 Single-Particle Energy (MeV) NN+3N-full -180 -180 -12 16 18 20 22 24 26 28 16 18 20 22 24 26 28 16 18 20 22 24 26 28 Mass Number A Mass Number A Bogner et al., PRL (2014) Mass Number A Hebeler, JDH, Menéndez, Schwenk, ARNPS (2015) Agreement between all methods with same input forces

Clear improvement with NN+3N-full

Validates valence-space results week ending PRL 111, 062501 (2013) PHYSICAL REVIEW LETTERS 9 AUGUST 2013

(Fig. 1) is over bound at 130:8 1 MeV but in close Figures 3 and 4 collect our results for the oxygen, nitro- agreement with the 130À:5 1 MeVð Þ obtained from gen and fluorine isotopes calculated with ! 24 MeV À ð Þ 1 ¼ IM-SRG [11], giving further confirmation of the accuracy and !SRG 2:0 fmÀ . The top panel of Fig.@ 3 shows the achieved by different many-body methods. Note that the predicted¼ evolution of neutron single particle spectrum energies of 15O and 23O can be obtained in two different (addition and separation energies) of oxygen isotopes in ways, from either neutron addition or removal on neigh- the sd shell. Induced 3NFs reproduce the overall trend but boring subshell closures. Results in Fig. 2 differ by at most predict a bound d3=2 when the shell is filled. Adding pre- 400 keV, again within the estimated uncertainty of our existing 3NFs—the full Hamiltonian—raises this orbit many-body truncation scheme. The c.m. correction in above the continuum also for the highest masses.week ending This Eq.PRL (111,10) is062501 crucial (2013) to obtain this agreement.PHYSICAL For ! REVIEWgives a LETTERS first principle confirmation of the repulsive9 AUGUST effects 2013 1 15¼ 24 MeV and !SRG 2:0 fmÀ , the discrepancy in@ O of the two-pion exchange Fujita-Miyazawa interaction ((Fig.23O)1 is) is 1.65 over MeV bound¼ (1.03 at MeV)130:8 when1 MeV neglectingbut in close the Figures 3 and 4 collect our results for the oxygen, nitro- À ð Þ discussed in Ref. [3]. The consequences of this trend are agreementchanges in kinetic with the energy130 of:5 the1 c.m.MeV butobtained it reduces from to demonstratedgen and fluorine by isotopes the calculated calculated ground with state! 24 energies MeV À ð Þ 1 ¼ IM-SRGonly 190 [ keV11], giving (20 keV) further when confirmation this is accounted of the accuracyfor. This shownand !SRG in the2:0 bottom fmÀ . panel The top and panel in Fig. of Fig.4@: the3 shows induced the achievedgives us confidence by different that many-body a proper separation methods. of Note the center that the of predicted¼ evolution of neutron single particle spectrum 15 23 Hamiltonian systematically under binds the whole isotopic energiesmass motion of isO beingand reached.O can be obtained in two different (additionchain and and erroneously separation places energies) the drip of line oxygen at 28O isotopesdue to the in ways,Figure from2 also either gives neutron a first addition remarkable or removal demonstration on neigh- of thelacksd ofshell. repulsion Induced in the 3NFsd reproduceorbit. The the contribution overall trend from but boring subshell closures. Results in Fig. 2 differ by at most 3=2 the predictive power of chiral 2N 3N interactions: predictfull 3NFs a bound increased3= with2 when the the mass shell number is filled. up to Adding24O, when pre- 400 keV, again within the estimatedþ uncertainty of our accounting for the precision of our many-body approach theexisting unbound 3NFs—thed orbit full starts Hamiltonian—raises being filled. Other this bound orbit many-bodyand dependence truncation on ! scheme.found in The Ref. c.m. [28], correction we expect an in 3=2 SRG abovequasihole the states continuum are lowered also for resulting the highest in additional masses. overall This accuracyEq. (10) of is at crucial least 5% to obtainon binding this energies. agreement. All For calculated! gives a first principle confirmation of the repulsive effects 1 15¼ binding. As a result, the inclusion of NNLO 3NFs consis- values24 MeV agreeand with!SRG the2 experiment:0 fmÀ , the within discrepancy this limit. in@ NoteO of the two-pion exchange Fujita-Miyazawa interaction 23 ¼ tently brings calculations close to the experiment and (thatO) the is interactions 1.65 MeV employed (1.03 MeV) were when only constrainedneglecting the by discussed in Ref. [3]. The consequences24 of this trend are 3 4 reproduces the observed dripline at O [41–43]. Our cal- 2changesN and inH and kineticHe energydata. of the c.m. but it reduces to demonstrated by25 the calculated ground state energies only 190 keV (20 keV) when this is accounted for. This culations predict O to be particle unbound by 1.54 MeV, shownlarger than in the the bottom experimental panel and value in of Fig. 7704: keV the [ induced44] but gives us confidence that a proper separation of the center of Hamiltonian systematically under binds the whole isotopic mass motion is being reached. within the estimated errors. The ground state28 resonance for 6 28chain and erroneously places the drip line at O due to the 2N 3N ind O is suggested to be unbound by 5.2 MeV with respect to Figure 2 also gives a first remarkable demonstration of lack of repulsion in the d orbit. The contribution from 4 1d3 2 2N 3N full 24O. However, this estimate3=2 is likely to be affected by the the predictive power of chiral 2N 3N interactions: full 3NFs increase with the mass number up to 24O, when accounting2 for the precision of our many-bodyþ approach presence of the continuum which is important for this thenucleus unbound but neglectedd3=2 orbit in thestarts present being work. filled. Other bound and dependence0 on !SRG found in Ref. [28], we expect an

MeV quasiholeThe same states mechanism are lowered affects resulting neighboring in additional isotopic overall accuracy of at least 5% on2 bindings energies. All calculated 1 2 1 2 chains.binding. This As a is result, demonstrated the inclusion in Fig. of NNLO4 for the 3NFs semimagic consis- A valuesi agree with the experiment within this limit. Note 4 odd-eventently brings isotopes calculations of nitrogen close and to fluorine. the experiment Induced 3NF and Oxygen Driplinethat the interactions Mechanism employed were only constrained by 24 3 4 forcesreproduces consistently the observed under dripline bind these at O isotopes[41–43]. and Our even cal- 2N and 6H and He data.1d5 2 25 culationspredict a predict27N closeO into energybe particle to 23 unboundN. This by is 1.54 fully MeV, cor- Self-consistent Green’s Function with same8 SRG-evolved NN+3N forces largerrected thanby full the 3NFs experimental that strongly value bind of23 770N with keV respect [44] but to 4 60 within27 the estimated errors. The ground state resonance for 6 N, in accordance with the experimentally observed drip -130 24 MeV 28O is suggested to be unbound by 5.2 MeV with respect to 1 2N 3N ind line. The repulsive effects of filling the d3=2 is also d 80 SRG 2.0 fm 24 3/2 4 1d3 2 2N 3N full O. However, this estimate is likely to be affected by the 0 -140 s 2 presence of the13 continuum15 which21 is23 important27 for this 1/2 100 N N N N N -150 nucleus but neglected in the present work. 0 80 24 MeV MeV -4 MeV 120 2N 3N ind The same mechanism affects1 neighboring isotopic .

s 2s 1 1 2 2.0 fm . 2 SRG d g -160 2N 3N full chains. This is demonstrated in Fig. 4 for the semimagic A i 100 5/2 E 140 Exp

4 Energy (MeV) odd-even isotopes of nitrogen and fluorine. Induced 3NF -8 -170 Exp. 120 6 forces consistently under bind these isotopes and even

1d NN+3N-ind MeV NN+3N-ind 160 5 2 27 23 .

NN+3N-full predicts a N close in energy to N. This is fully cor- . 140

Single-Particle Energy (MeV) NN+3N-full

-180 g 8 23 -12 180 rectedE by full 3NFs that strongly bind N with2N 3N respectfull to 16 18 20 22 24 26 28 60 14O 1616O 18 2022O 2224O 24 28O26 28 27N, in160 accordance with the experimentally2N observed3N ind drip Exp Mass Number A Bogner et al., PRL (2014) Cipollone,24 MeV Barbieri,Mass NumberNavrátil, APRL (2013) line. The repulsive effects of filling the d is also 2.0 fm 1 180 3=2 FIG. 3 (color80 online). Top.SRG Evolution of single particle ener- gies for neutron addition and removal around sub-shell closures 15 17 23 25 29 Robust mechanism driving dripline behavior100 F 13NF 15N F 21NF 23N F 27N of oxygen isotopes. Bottom. Binding energies obtained from the 80 24 MeV KoltunMeV SR and the poles of propagator (1), compared to experi- FIG. 4 (color online). Binding energies of odd-even nitrogen d 120 2N 3N ind 1 3N repulsion raises 3/2, lessens decrease. across shell s 2.0 fm ment. (bars) [44,46,47]. All points are corrected for the kinetic and fluorine isotopes calculatedSRG for induced (red squares) g 2N 3N full 100 energyE of140 the c.o.m. motion. For all lines, redExp squares (blue dots) and full (green dots) interactions. Experimental data are from Similar to first MBPT NN+3N calculations in oxygen refer to induced (full) 3NFs. [45–47].120

160 MeV . s

. 140 g 180 062501-4 E 2N 3N full 14O 16O 22O 24O 28O 160 2N 3N ind Exp 180 FIG. 3 (color online). Top. Evolution of single particle ener- gies for neutron addition and removal around sub-shell closures 15F 17F 23F 25F 29F of oxygen isotopes. Bottom. Binding energies obtained from the Koltun SR and the poles of propagator (1), compared to experi- FIG. 4 (color online). Binding energies of odd-even nitrogen ment (bars) [44,46,47]. All points are corrected for the kinetic and fluorine isotopes calculated for induced (red squares) energy of the c.o.m. motion. For all lines, red squares (blue dots) and full (green dots) interactions. Experimental data are from refer to induced (full) 3NFs. [45–47].

062501-4 IM-SRG Oxygen Spectra Oxygen spectra: extended-space MBPT and sd-shell IM-SRG

+ 8 4

+ 22 + 7 2 + (4 ) 4 + O (2 ) + 2 6 + + 3 4 + 2 + + 0 (0 ) 5 + + + 3 0 + 3 0+ 4 3 + 2 + 2 + 3 2 Energy (MeV) + 2 2 JDH, Menéndez, Schwenk, EPJ (2013) 1 Bogner et al., PRL (2014)

+ + + + 0 0 0 0 0 MBPT IM-SRG IM-SRG Expt. NN+3N-ind NN+3N-full

Clear improvement with NN+3N-full

IM-SRG: comparable with phenomenology IM-SRG Oxygen Spectra Oxygen spectra: extended-space MBPT and IM-SRG

+ 5 3/2 23 O + 4 (3/2 )

+ + 3/2 3/2 3 + + + 5/2 5/2 (5/2 )

+ 5/2 2 Energy (MeV)

1 JDH, Menéndez, Schwenk, EPJ (2013) Bogner et al., PRL (2014) + + + + 0 1/2 1/2 1/2 1/2 MBPT IM-SRG IM-SRG Expt. NN+3N-ind NN+3N-full

Clear improvement with NN+3N-full

Continuum neglected: expect to lower d3/2 IM-SRG Oxygen Spectra Oxygen spectra: IM-SRG predictions beyond the dripline

7 24 25 26 + O O O 6 1 + + 2 + 1 5/2

5 + 2 + 1/2 4

3

Energy (MeV) 2 + 2 1 Bogner et al., PRL (2014)

+ + + + 0 0 0 3/2 0 Exp. 24 + O closed shell (too high 2 )

Continuum neglected: expect to lower spectrum

Experimental Connection: 26O Spectrum Oxygen spectra: IM-SRG predictions beyond the dripline

7 24 25 26 + O O O 6 1 + + 2 + 1 5/2

5 + 2 + 1/2 4

3

Energy (MeV) 2 + 2 1 Bogner et al., PRL (2014)

+ + + + Kondo et al., PRELIMINARY 0 0 0 3/2 0 Exp.

New measurement at RIKEN: excited states in 26O

Existence of excited state 1.3MeV IM-SRG prediction: one natural-parity state below 7MeV at 1.22MeV Comparison with MBPT/CCEI Oxygen Spectra Oxygen spectra: Effective interactions from Coupled-Cluster theory

7 9 22 6 23 + 24 O O 1 O + + 1 8 4 + + 6 1 + 5 + 3/2 2 + + + 3/2 + 2 + 7 2 + (4 ) 2 1 4 + + (2 ) + 4 + (3/2 ) 5 + + 2 4 2 6 + 2 3 + + + 4 + 0 (0 ) 3/2 5 0 + + + 3 + + 0 + 3 5/2 5/2 (5/2 ) + 3 + 4 3 5/2 + 3 2 +

Energy (MeV) 2 + + 3 2 2 2 2 2 1 1 1 + + + + + + + + + + + + 0 0 0 0 0 0 1/2 1/2 1/2 1/2 0 0 0 0 0 MBPT CCEI IM-SRG Expt. MBPT CCEI IM-SRG Expt. MBPT CCEI IMSRG Expt.

Hebeler, JDH, Menéndez, Schwenk, ARNPS (2015)

MBPT in extended valence space

IM-SRG/CCEI spectra agree within ~300 keV Experimental Connection: 24F Spectrum 24F spectrum: IM-SRG (sd shell), full CC, USDB

+ + 4 1 4 24 + F + + 2 2 (2 ) + + + + 3.5 5 (4 ,2 ) 2+ 1+ + 4 3 + 3 + 3 + 3 + + (3 ) 4 0 + + + 0 4 + + 2.5 1 (4 ) 4

+ 2 + 2 1 + + 1 1 + 1 Energy (MeV) 1.5 1 + + + 2 2 0.5 2 + 2 Ekström et al., PRL (2014) Cáceres et al., arXiv:1501.01166 + + + + 0 3 3 3 3 Hebeler, JDH, Menéndez, Schwenk, ARNPS (2015) CC IM-SRG Expt. USDB

New measurements from GANIL

IM-SRG: comparable with phenomenology, good agreement with new data Fully Open Shell: Neutron-Rich Fluorine Spectra Fluorine spectra: extended-space MBPT and IM-SRG (sd shell)

+ + 3/2 3/2 + + 23 25 7/2 4 26 3 + + + F 7/2 (3/2 ) + 5 F 1/2+ F 4 3/2 + 7/2 + 5/2 1+ + + + 2 9/2 9/2 9/2 + + 9/2 + 5/2 + (5/2 ) 5/2+ + + 4 + + + 5/2 5/2 1/2+ 3/2 (3/2 ) 3 3/2 3/2+ 3 + + 9/2 + + + 7/2 (3/2 ) 3+ 7/2 + 3 1 + 7/2 + + 5/2 + (9/2 ) 2 1/2 3 + + 1 3 + 2 2 1/2 + + 4 3/2 + + 3 2 1/2

Energy (MeV) + (1/2 ) + + 3/2 + 2 + 1/2 1 1 1/2 + + 2+ 2 1 + 4 + + 5/2 4 2 + 4 + + + + + + + + + + + + 0 5/2 5/2 5/2 5/2 0 1/2 5/2 (5/2 ) 5/2 0 1 1 1 1 MBPT IM-SRG Expt. USDB MBPT IM-SRG Expt. USDB MBPT IM-SRG Expt. USDB Bogner, Hergert, JDH, Schwenk, in prep.

MBPT: clear deficiencies

IM-SRG: competitive with phenomenology, good agreement with data Fully Open Shell: Neutron-Rich Neon Spectra Neon spectra: extended-space MBPT and IM-SRG (sd shell)

+ + 4 7/2 5 24 0+ 25 + + 5 26 + + 3 9/2 7/2 0 + 0 0 Ne + Ne + Ne+ + 4 + 3 + 3/2 9/2 2 0+ + 3 + + 3/2+ + 4 4 + 4 + 4 + + 5/2 4 + + + 5/2 2 + 4 2 2 2 3 + 2 2 3/2 + + + + 4 4 1/2 0 + + 2 7/2 3 3 + + + + 0 3/2 (3/2 ) 3/2 2 + 5/2 + + + 5/2 + + 5/2 (5/2 ) + + + 2 + 2 2 2 + 2 2 2 2 + 2 2 Energy (MeV) + 1 5/2 + 1 3/2 1

+ + + + + + + + + + + + 0 0 0 0 0 0 1/2 1/2 (1/2 ) 1/2 0 0 0 0 0 MBPT IM-SRG Expt. USDB MBPT IM-SRG Expt. USDB MBPT IM-SRG Expt. USDB Bogner, Hergert, JDH, Schwenk, in prep.

MBPT: clear deficiencies

IM-SRG: competitive with phenomenology, good agreement with data Alternative Approach: Magnus Expansion Morris, Parzuchowski, Bogner, in prep. Magnus expansion: explicitly construct unitary transformation

U(s)=exp⌦(s)

With flow equation:

d⌦(s) 1 1 = ⌘(s)+ [⌦(s), ⌘(s)] + [⌦(s), [⌦(s), ⌘(s)]] + ... ds 2 12

Leads to commutator expression for evolved Hamiltonian

⌦(s) ⌦(s) 1 1 H(s)=e He = H + [⌦(s),H]+ [⌦(s), [⌦(s),H]] + 2 12 ···

Nested commutator series – in practice truncate numerically

All calculations truncated at normal-ordered two-body level GT, E2, M1, EWoperators effective calculation Consistent of Towards 2! Prospect for Applications to Double-Betaforto Decay Prospect Applications Calculation of nuclear Calculation of 8!

2! protons 20! 8! 28! neutrons ab 20! initioelectroweak physics 0νββ 28! decay… wavefunctions 50! €

2-body currents2-body essential EW operators inmany-bodymethods? JDH and Engel; Engel; and JDH Klos

✔ €

, Menéndez, , Menéndez, π 82! ? π 50! Stroberg Gazit € , Schwenk

et al. (IM-SRG);et al. (CC) Jansen 126! ; Ekström π , Wendt

Improved 0νββ-Decay Calculations in Shell Model

Avenues towards ab initio shell model calculations

Consistent operators and wavefunctions

M 0ν M 0ν = M 0ν − F + M 0ν GT g2 T !A ! M 0ν = f H(r )σ ⋅σ τ +τ + i GT ∑ ab Maeff b a b ab 0ν + + M F = f ∑H(rab )τ aτ b i ab Meff

1) Nuclear wavefunctions: currently phenomenological – calculate ab initio 2) Effective decay operator: correlations outside valence space E0 Transitions and Radii

Seldom calculated in nuclear shell model In single HO shell:

2 1 2 f ⇢E0 i ij where ⇢E0 = eiri | h | | i | / e2R i X Must resort to phenomenological gymnastics

IM-SRG: straightforward to calculate effective valence-space operator:

⌦(s) ⌦(s) 1 ⇢ (s)=e ⇢ e = ⇢ + [⌦(s), ⇢ ]+ E0 E0 E0 2 E0 ··· Commutators induce important higher-order and two-body parts

Quantify importance of induced higher-body contributions! How domagic numbers form evolve? and nuclear existence? What are limits the of proton/neutron-richWhat are properties the matter? of Worldwide joint nuclear science: frontiersExploring the of 2! 8! The Nuclear Landscape TowardtheExtremes The NuclearLandscape

2! protons 20! 8! 28! neutrons 20! DripLinesand MagicNumbers: experimental/theoretical 28! 50! effort effort 82! 82! 50! 126!

How domagic numbers form evolve? and nuclear existence? What are limits the of proton/neutron-richWhat are properties the matter? of Worldwide joint nuclear science: frontiersExploring the of 2! 8!

2! protons Advances in 20! 8! 28! neutrons 20! experimental/theoretical 28! Medium-Mass Exotic Nuclei Exotic Medium-Mass 50! ab € initio Nuclear Structure for

€ effort effort

82! 3N forces for essential exotic nuclei π 82! Advances inmany-bodymethods Self-Consistent Green’s Self-Consistent Function Many-BodyPerturbation Theory SRGIn-Medium Cluster Coupled (Hagen, (Hagen, ( (JDH, (JDH, Bogner ( π Barbieri 50! Hjorth , Papenbrock Hergert € , Soma,

-Jensen, -Jensen, , JDH, 126! Duguet , Dean,Roth) Schwenk π Schwenk

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How domagic numbers form evolve? and nuclear existence? What are limits the of proton/neutron-richWhat are properties the matter? of Worldwide joint nuclear science: frontiersExploring the of 2! 8!

2! protons Advances in 20! 8! 28! neutrons 20! experimental/ 28! Medium-Mass Exotic Nuclei Exotic Medium-Mass 50! ab theoretical € initio Nuclear Structure for

€

effort effort

82! 3N forces for essential exotic nuclei π 82! Advances inmany-bodymethods Self-Consistent Green’s Self-Consistent Function Many-BodyPerturbation Theory SRGIn-Medium Cluster Coupled (Hagen, (Hagen, ( (JDH, (JDH, Bogner ( π Barbieri Hjorth , Papenbrock Hergert € , Duguet

-Jensen, -Jensen, , JDH, , Soma) , Roth) Schwenk π Schwenk

)

, Stroberg )

How domagic numbers form evolve? and nuclear existence? What are limits the of proton/neutron-richWhat are properties the matter? of Worldwide joint nuclear science: frontiersExploring the of 2! 8!

2! protons Advances in 20! 8! 28! neutrons 20! experimental/ 28! Medium-Mass Exotic Nuclei Exotic Medium-Mass 50! ab theoretical € initio Nuclear Structure for

€

effort effort

82! 3N forces for essential exotic nuclei π 82! Advances inmany-bodymethods Self-Consistent Green’s Self-Consistent Function Many-BodyPerturbation Theory SRGIn-Medium Cluster Coupled (Hagen, (Hagen, ( (JDH, (JDH, Bogner ( π Barbieri Hjorth , Papenbrock Hergert € , Soma,

-Jensen, -Jensen, , JDH, Duguet , Dean,Roth) Schwenk π Schwenk

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operators Effective for GT, E2 M1, symmetries? Fundamental 2! ab 8! initio calculation of initiocalculation of

2! protons 20! 8! 28! neutrons 20! Neutrinoless 28! Prospect for Applications to 0νββ 50! decay €

Double-Beta Decay €

82! π 82! JDH and Engel; Engel; and JDH 2-body currents2-body essential Advances inmany-body methods? Menendez, Menendez, π 50! € Gazit

Ekstrom , 126! Schwenk π etal.; ; Ekstrom Stroberg

etal. etal.

Nuclear Weak Processes: ββ-Decay Nuclear weak processes: fundamental importance for particle physics

Rare cases when single β-decay is energetically forbidden Can undergo ββ-decay

Second-order weak process

Observed in 15 nuclei with:

2νββ −1 19 (T1/2 ) ≈ 10 y Fundamental Symmetries: 0νββ-Decay Assuming exchange of light neutrinos

Two essential ingredients: Q-value (experiment) Nuclear matrix element

2 2 " % 0νββ −1 0ν 0ν mββ 3 (T ) = G (Q , Z) M $ ' m = Ueimi 1/2 ββ $ ' ββ ∑i=1 # me & Character of neutrino (Majorana/Dirac) Lepton number violation Neutrino mass scale Nuclear Matrix Element Recent calculations of NME: pronounced differences for lighter candidates

Barea, Koltila, Iachello, PRL (2012)

Shell model: exact correlations in truncated single-particle space

QRPA, EDF, IBM: schematic correlations in large single-particle space Aim: first-principles framework capable of robust prediction Improved 0νββ-Decay Calculations in Shell Model

Avenues towards ab initio shell model calculations

Consistent operators and wavefunctions

0ν 0ν 0ν M F 0ν M = MGT − 2 + MT gA 0ν ! ! + + MGT = f ∑H(rab )σ a ⋅σ bτ aτ b i ab 0ν + + M F = f ∑H(rab )τ aτ b i ab

1) Nuclear wavefunctions: currently phenomenological – calculate ab initio Effective 0νββ-Decay Operator Calculate in MBPT:

1st order (× 2) Improved 0νββ-Decay Calculations in Shell Model

Avenues towards ab initio shell model calculations

Consistent operators and wavefunctions

M 0ν M 0ν = M 0ν − F + M 0ν GT g2 T !A ! M 0ν = f H(r )σ ⋅σ τ +τ + i GT ∑ ab Maeff b a b ab 0ν + + M F = f ∑H(rab )τ aτ b i ab Meff

1) Nuclear wavefunctions: currently phenomenological – calculate ab initio 2) Effective decay operator: correlations outside valence space Improved 0νββ-Decay Calculations in Shell Model

Avenues towards ab initio shell model calculations

Consistent operators and wavefunctions

0ν 0ν 0ν M F 0ν M = MGT − 2 + MT gA 0ν ! ! + + MGT = f ∑H(rab )σ a ⋅σ bτ aτ b i ab 0ν + + M F = f ∑H(rab )τ aτ b i ab

1) Nuclear wavefunctions: currently phenomenological – calculate ab initio 2) Effective decay operator: correlations outside valence space 3) Improvements in operator: two-body currents The Nuclear Many-Body Problem Nuclei understood as many-body system starting from closed shell, add nucleons

Calculate valence-space Hamiltonian inputs from nuclear forces

Interaction matrix elements

Single-particle energies (SPEs)

112 c c c 0h, 1f, 2p d Vlow-k d d ˆ = + + ... 70 Q 0g,1d,2s a b a b a b 40 0f,1p c d c d 20 0d3/2 1s1/2 “sd”-valence space + + 0d 5/2 V 8 a b a b 0p Active nucleons occupy c d c d 2 €valence space 0s Inert core: does not reproduce+ experiment+ + ... a b a b The Nuclear Many-Body Problem Nuclei understood as many-body system starting from closed shell, add nucleons

Calculate valence-space Hamiltonian inputs from nuclear forces

Interaction matrix elements

Single-particle energies (SPEs)

112 c d c Vlow-kc d dc c Vlow-kd d c d 0h, 1f, 2p Solution: allow breaking of core ˆ = ˆ + = + ...+ + ... 70 Q Q 0g,1d,2s Effective Hamiltoniana b for valencea a nucleonsb ba a b b a b 40 0f,1p c d c c d d c d 20 0d 3/2 + + + + 1s1/2 “sd”-valence space Veff 0d 5/2 V 8 a b a a b b a b 0p Active nucleons occupy c d c cd d c d 2 €valence space 0s + + + + ...+ + ...

a b a a bb a b c d c Vlow-k d c d Many-BodyQˆ = Perturbation+ +Theory... First calculate energya dependentb a effectiveb interactiona b c c c d c Vlow-k d c cc d dd c cVcVVlow-klow-kd dd c cc d dd ⌢ 1 low-k ˆ ... Q( ω) = PVP + PVQ Q QVP+= + QˆQˆQˆ ==+= +++ +++...... ω − QHQ a a a b a b a aa !b bb a aa b bb a aa b bb Effects of excitations outside valencec space ωc = PH0P c d c d c cc d dd c cc d dd ⌢ V € Approximation: sum Q ( ω ) to+ finite order+ +++ ... +++

€ a b b a b a aa b bb a aa b bb € € c d c d c cc c cc € d dd d dd + + +++ ... +++ +++......

a b a b a aa b bb a aa b bb

c d c Vlow-k d c d Many-BodyQˆ = Perturbation+ +Theory... First calculate energya dependentb a effectiveb interactiona b c c c d c Vlow-k d c cc d dd c cVcVVlow-klow-kd dd c cc d dd ⌢ 1 low-k ˆ = ... Q( ω) = PVP + PVQ Q QVP+ + QˆVQˆQˆ ==+= +++ +++...... ω − QHQ eff a a a b a b a aa !b bb a aa b bb a aa b bb Effects of excitations outside valencec space ωc = PH0P c d c d c cc d dd c cc d dd ⌢ V € Approximation: sum Q ( ω ) to+ finite order+ +++ ... +++

€ a b b a b a aa b bb a aa b bb € € c d c d c cc c cc Nonperturbative€ folded diagrams d dd d dd to remove energy dependence + + +++ ... +++ +++...... a b a b ∞ m ⌢ a aa b bb a aa b bb m (n) ⌢ 1 d Q (n -1) V = Q + V eff ∑ m! dω m { eff } m =1

Eigenvalues converge to exact values of A-body Hamiltonian with

largest model-space overlap

€

Perturbative Valence-Space Strategy

1) Effective interaction: sum excitations outside valence space to 3rd order

2) Single-particle energies calculated self consistently

3) Harmonic-oscillator basis of 13-15 major shells: converged c c c c d c Vlow-k d c c d dd c Vlow-kVlow-kd d c d d rd 4) NNˆ and 3N forces from chiral EFT –... to 3 -order MBPT Qˆ = + Qˆ Q = =+ ... + + + +... 5) Explore extended valence spaces a b a b a aa b bb a a b b a a b b

c cc cc c d c d c Vlow-kd dd c d dd + + VQˆ + += + ++ + ... eff V a b a b a aa b bb a aa b bb c c cc c dd c cc dd d € d d d + + + ++ ... + ++ + +...... a b a b a aa b b b a aa b bb

c d c d + + + ...

a b a b Perturbative Valence-Space Strategy

1) Effective interaction: sum excitations outside valence space to 3rd order

2) Single-particle energies calculated self consistently

3) Harmonic-oscillator basis of 13-15 major shells: converged

4) NN and 3N forces from chiral EFT – to 3rd-order MBPT

5) Explore extended valence spaces

-18.5 -76 -77 -19 42 -78 48 Ca Ca -19.5 -79 1st order -80 -20 2nd order 3rd order -81 Ground-State Energy (MeV) -20.5 -82 2 4 6 8 10 12 14 16 18 2 4 6 8 10 12 14 16 18 _ _ Nh Nh Clear convergence with HO basis size

Promising order-by-order behavior – compare with ab initio methods Chiral Effective Field Theory: Nuclear Forces Nucleons interact via pion exchanges and contact interactions

LO Consistent treatment of NN, 3N,… 2-body currents

NLO 3N couplings fit to properties of light nuclei at low momentum

N2LO Improve convergence of many-body methods:

V low k or VSRG

N3LO Weinberg, van Kolck, Kaplan, Savage, Wise, € Epelbaum, Kaiser, Meissner,… Perturbative Valence-Space Strategy

1) Effective interaction: sum excitations outside valence space to 3rd order

2) Single-particle energies calculated self consistently

3) Harmonic-oscillator basis of 13-15 major shells: converged

4) NN and 3N forces from chiral EFT – to 3rd-order MBPT

5) Explore extended valence spaces

NN matrix elements 3 500MeV V ( ) - Chiral N LO (Machleidt, Λ NN = ); smooth-regulator low k Λ 3N force contributions

- Chiral N2LO € € -1 cD, cE fit to properties of light nuclei with Vlow k ( Λ = Λ 3N = 2.0 fm )

- Included to 5 major HO shells

€ Perturbative Valence-Space Strategy

1) Effective interaction: sum excitations outside valence space to 3rd order

2) Single-particle energies calculated self consistently

3) Harmonic-oscillator basis of 13-15 major shells: converged

4) NN and 3N forces from chiral EFT – to 3rd-order MBPT

5) Explore extended valence spaces

Philosophy: diagonalize in largest possible valence space (where orbits relevant)

50 50 0g 0g 9/2 9/2 Treats higher orbits nonperturbatively 0f 0f 1p5/2 1p5/2 1/2 1/2 When important for exotic nuclei? 1p 1p 28 3/2 28 3/2 0f 0f 7/2 7/2 Best option for double-beta decay? 20 20 0d3/2 0d3/2 1s1/2 1s1/2 0d5/2 0d5/2 16 8 O 8 0p3/2 0p3/2 0p1/2 0p1/2 How domagic numbers form evolve? and nuclear existence? What are limits the of proton/neutron-richWhat are properties the matter? of Worldwide jointexperimental/theoretical effort nuclear science: frontiersExploring the of 2! 8!

2! protons 20! Nuclear Nuclear 8! 28! neutrons 20! 28! • • • • • JDH, JDH, JDH, Wienholtz Gallant Wavefunctions 50! Menendez Otsuka Menendez

et al., PRL

et al., Nature al., Nature , Schwenk , , Schwenk Simonis 109 Spectra,- rates transition evolution Shell - through Keyphysics problems: 82! , 032506 (2012) (2012) , 032506 , JPGSuzuki, 82! 486 References , , JPG Schwenk , 346 (2013) ,(2013) 346 : CalciumIsotopes 50! 40 , 075105 (2013) (2013) , 075105 , PRC

39 , 085111 (2012) 90 126! , 024312 (2014) (2014) , 024312 52

Ca , 54 Ca

Calcium Isotopes: Magic Numbers 50 0g9/2 (a) Phenomenological Forces (b) NN-only Theory (c) NN + 3N (d) NN + 3N (pfg shell) 0f 9/2 5/2 g 1p 0 f5/2 f 9/2 1/2 f5/2 5/2 f5/2 1p p 28 3/2 1/2 p p p 0f 1/2 1/2 1/2 7/2 -5 p p 40 p3/2 3/2 3/2 20 p3/2 Ca 0d3/2 1s f f7/2 1/2 -10 f 7/2 f 0d 7/2 7/2 5/2 KB3G G V Single-Particle Energy (MeV) GXPF1 low k -15 8 40 44 48 52 56 60 40 44 48 52 56 60 0p3/2 Mass Number A Mass Number A 0p1/2 GXPF1: Honma, Otsuka, Brown, Mizusaki (2004) KB3G: Poves, Sanchez-Solano, Caurier, Nowacki (2001) Phenomenological Forces 6 Large gap at 48Ca 5 (a) Phenomenological Forces (b) NN-only Theory 4 Discrepancy at N=34

3 Microscopic NN Theory 2 ? 48 Energy (MeV) 1 + 1 Small gap at Ca 2 G V 0 low k Experiment G [SPE_KB3G] N=28: first standard magic GXPF1 Vlow k [SPE_KB3G] KB3G Vlow k [fp+g9/2] number not reproduced 42 44 46 48 50 52 54 56 58 42 44 46 48 50 52 54 56 58 in microscopic NN theories Calcium Ground State Energies and Dripline Signatures of shell evolution from ground-state energies? 0 NN -30 NN+3N (emp) NN+3N (MBPT) -60

-90

Energy (MeV) -120

-150 Holt, Otsuka, Schwenk, Suzuki, JPG (2012) 40 44 48 52 56 60 64 68 Mass Number A No clear dripline; flat behavior past 54Ca – Halos beyond 60Ca?

S 2 n = − [ BE ( N , Z ) − BE ( N − 2 , Z ) ] sharp decrease indicates shell closure

€ MR-TOF Measurements 54 Experimental Connection:Implementation Mass of multi-reflection of timeCa-of-flight mass separator (MR-TOF MS) opened a wide range of possibilities New precision mass measurement of 53,54Ca at ISOLTRAPSupport Penning-trap: multi-reflectionmass spectrometry on fast time ToF scales MR-TOF plus detector as a stand-alone system " #& #H$ !H' ## !H$

86F-;G3896:; $H'

#$ #= $H$ !* E$H' E!H$ 8@56A;3E3< !( #= E!H' D< E#H$ !& %$ %! %# %% %& 52Ca >6?@/A=3=?,B6/3! Versatile tool allows for:

3896:; ISOLTRAP Higher contamination Measurement yield – 106:1

#= !# lower production yield – 10/s52 !$ Sharp decrease past Ca lower half-lives – tens of ms 52

Unambiguous High repetition rateclosed-shell – 20Hz Ca * +F-6/.,6=@ statistical uncertainty 7 J>P%>389QCM; RR38S176=36@312H; ? & TQ%U MBPT NN+3N # UVCW!O #* #" %$ %! %# %% %& %' %( %) %* Excellent agreement with new data >6?@/A=3=?,B6/3! 48,52 Reproduces closed-shell Ca ! Wienholtz et al., Nature (2013) 54 #& #H$ Weak closed sell signature past Ca !H' ## !H$ $H' #$ 86F-;G3896:; #= $H$ !* E$H' E!H$ 8@56A;3E3<

!( #= E!H' D< E#H$ !& %$ %! %# %% %& >6?@/A=3=?,B6/3! 3896:; <

#= !# !$ +F-6/.,6=@ * J+YW$ ( X>+YW! SWQ#! & 6?@/A=3=?,B6/3! # !I#* * !I%# J

(

'

&

% +,-./.012345622371-3896:; < #

!" #$ #! ## #% #& #' #( #) #* #" %$ %! C/A@A=3=?,B6/3"# Neutron-Rich Ca Spectra Near N=34 Neutron-rich calcium spectra with NN+3N

6 53 54 + - 6 2+ 7/2- 1+ Ca 5/2 Ca 2 - + 7/2- 3 5 3/2 + - 4+ 5/2- 5 0 7/2 + 4+ - 3 3/2 - 4 - 1/2- 9/2- + 3/2- + + 4 1/2- 3/2 4 2 - 3 + 3/2 - (3 ) 2+ - 9/2- 3 5/2- 7/2- 7/2 - 5/2 + 3 5/2 4 + + 3 2 4 - - (3/2 ) - + + 3/2 - 3/2 2+ 2 3/2 3 + 2 (2 ) Energy (MeV) - Energy (MeV) (5/2 ) 2 + + 2 3+ + 0+ - 0 2 5/2 1 - 5/2 1

- - - - + + + + 0 1/2 (1/2 ) 1/2 1/2 0 0 0 0 0 NN+3N Expt. GXPF1A KB3G NN+3N Expt. GXPF1A KB3G pfg9/2 pfg9/2 JDH, Menendez, Schwenk, JPG (2013) JDH, Menendez, Simonis, Schwenk, PRC (2014) Phenomenology: inconsistent predictions Hagen et al., PRL (2013)

NN+3N: signature of new N=34 magic number (also predicted in CC theory)

Agrees with new measurements from RIKEN

Steppenbeck et al., Nature (2013) Transition Rates Neutron-rich calcium B(E2) rates 100

] 10 4 fm 2

B(E2) [e 1 KB3G GXPF1A NN + 3N (MBPT) Raman et al. (2001)JDH, Menendez, Schwenk, in prep. Montanari et al. (2012)

0.1 46 46 46 47 48 49 50 +Ca + +Ca + +Ca + -Ca - +Ca + -Ca - +Ca + 2 →0 4 →2 6 →4 3/2 →7/2 2 →0 7/2 →3/2 2 →0 JDH, Menendez, Simonis, Schwenk, PRC (2014) Reasonable agreement with experiment – comparable to phenomenology

Uses effective charges – need calculated effective operators!

N=28 Magic Number: M1 Transition Strength B ( M 1 : 0 + → 1 + ) concentration indicates a single particle (spin-flip) transition gs Not reproduced in phenomenology von Neumann-Coesel, et al. (1998)

NN-only: highly fragmented strength, well below experiment

€ 6 Experiment 5 NN NN + 3N (emp) ]

2 4

N NN + 3N (MBPT)

µ KB3G 3 GXPF1A

B(M1) [ 2

1

0 6 7 8 9 10 11 12 Excitation Energy (MeV) JDH, Menendez, Simonis, Schwenk, PRC (2014) pfg9/2-shell: 3N gives additional concentration Peak close to experimental energy

Supports N=28 magic number operators Effective for GT, E2 M1, symmetries Fundamental 2! ab 8!

2! protons initio calculation of initiocalculation of 20! 8! 28! neutrons 20! Calculations of Effective Operators 28! • • • JDH Lincoln, Kwiatkowski PRC Engel, and JDH 0νββ 50! decay , et Brunner al., PRL - NMEon in Impact - Keyphysics problems: Quenching of Quenching , et JDH al., PRC 82! 87 82! , 064315 (2013) , 064315 110 References , 012501 (2013) [LEBIT] (2013) , 012501 50!

g 89

A

in , 045502 (2014) [TITAN] (2014) , 045502 126! 48 sd Ca, - and and -

76 Ge,

pf -shell -shell 82

Se 0νββ-Decay Operator Details of operator used in subsequent calculations

Closure approximation good to 8-10% Sen’kov, Horoi, Brown, PRC (2014)

1) Nuclear wavefunctions: currently phenomenological – calculate ab initio 2) Decay operator: correlations outside valence space; 2-body currents Effective 0νββ-Decay Operator c c c ccStandardd approach:c c V phenomenologicald c c d wavefunctions + bare operator d Vlow-k d d d Vlow-klow-k d d ˆ = CalculateˆQˆ = =effective... 0νββ+ -decay operator+...... using MBPT Q + Q + + + a b a b a aaDiagrammaticallybbb a a similar:bb a a replacebb one interaction vertex with M 0 ν operator c d c d c cc Vlow-kddd c cc ddd c d c d c d + + Qˆ ++= +++ + ... ˆ = M + Veff MXeff V low k € V low k a b a b a aa bbb a aa bbb a b a b a b c d c d c c c ddd c cc ddd c d c d c d € V low k ... € + + +++ ... +++ ++... + + + a a a aa b b a aa bb b b b b € a b a b a b c c d d c d c d + + + ... + + + ... a b a b a b a b Effective 0νββ-Decay Operator c c c ccStandardd approach:c c V phenomenologicald c c d wavefunctions + bare operator d Vlow-k d d d Vlow-klow-k d d ˆ = CalculateˆQˆ = =effective... 0νββ+ -decay operator+...... using MBPT Q + Q + + + a b a b a aaDiagrammaticallybbb a a similar:bb a a replacebb one interaction vertex with M 0 ν operator c d c d c cc Vlow-kddd c cc ddd c d c d c d + + Qˆ ++= +++ + ... ˆ = M + Veff MXeff V low k € V low k a b a b a aa bbb a aa bbb a b a b a b c d c d c c c ddd c cc ddd c d c d c d € V low k ... € + + +++ ... +++ ++... + + + a a a aa b b a aa bb b b b b € a b a b a b c d c d Previous: G-matrix calculation to c d c d + + + ... 1st-order MBPT non-convergent + + + ... Engela band Hagena, PRC (2009)b a b a b Low-momentum interactions: Improve convergence behavior Effective 0νββ-Decay Operator Calculate in MBPT:

1st order (× 2) Effective 0νββ-Decay Operator Calculate in MBPT:

1st order (× 2) 2nd order (×3) Effective 0νββ-Decay Operator Calculate in MBPT:

Final result requires expansion of wavefunction norms and folded diagrams: products of Q-box and X-box derivatives

$ ⌢ 2 ⌢ ⌢ 2 ' 1 dQ( ε) 1 d Q( ε) ⌢ 3$ dQ( ε)' M eff = &1 + + Q( ε) + + ) 0ν & 2 & ) ") % 2 dε 2 dε 8% dε ( ( , / ⌢ ⌢ ∂X(ε,ε ) ∂X(ε f ,ε) ⌢ ×. X (ε) + Q( ε) i + Q( ε) +"1 . 1 ∂εi ε =ε ∂ε f - i ε f =ε 0 ⌢ ⌢ ⌢ 2 $ 1 dQ( ) 1 d 2Q( ) ⌢ 3$ dQ( )' ' & ε ε ε ) × 1+ + 2 Q( ε) + & ) +" & 2 dε 2 dε 8% dε ( ) % (

P. Ellis (1976)

€Q-box – sum of interaction diagrams X-box – sum of 0νββ diagrams Intermediate-State Convergence Convergence results in 82Se

82 Se 3.6

3.55

Matrix Element 3.5 M0ν

3.45 JDH and Engel, PRC (2013) 8 10 12 14 16 18 € Nh

Results well converged in terms of intermediate state excitations

Improvement due to low-momentum interactions

Similar trend in 76Ge and 48Ca

Improvements from SRG-evolved operator? SRG Evolution of Operator Promising preliminary results with schematic of 0νββ operator

Schuster et al Full 0νββ operator in progress

Forthcoming: test in effective-operator calculations 82 76 Calculations of M0ν in Se and Ge

Phenomenological wavefunctions from A. Poves, M. Horoi (pf5/2 g9/2 space)

76Ge Bare matrix element 3.12 82Se Bare matrix element 2.73 Full 1st order 3.11 Full 1st order 2.79

Full 2nd order 3.77 Full 2nd order 3.62

Lincoln, JDH et al., PRL (2013) [LEBIT] Converged in 13 major oscillator shells JDH and Engel, PRC (2013) Little net effect at 1st order

Overall ~25-30% increase from bare value 48 Calculations of M0ν in Ca Phenomenological wavefunctions from GXPF1A (standard pf-shell)

48Ca GT Fermi Tensor Sum Bare 0.675 0.130 −0.072 0.733

Final 1.211 0.160 −0.070 1.301 Kwiatkowski, Brunner, JDH et al., PRC (2014)

Converged in 13 major oscillator shells

Little net effect at 1st order – larger effect from 2nd order than in Ge, Se

Overall ~75% increase from bare value Nuclear Matrix Element

Improved calculations move in direction of other methods

Barea, Koltila, Iachello, PRL (2012)

Aim: first-principles framework capable of robust prediction Effective 0νββ-Decay Operator Calculate in MBPT:

Final result requires expansion of wavefunction norms and folded diagrams: products of Q-box and X-box derivatives

$ ⌢ 2 ⌢ ⌢ 2 ' 1 dQ( ε) 1 d Q( ε) ⌢ 3$ dQ( ε)' M eff = &1 + + Q( ε) + + ) 0ν & 2 & ) ") % 2 dε 2 dε 8% dε ( ( , / ⌢ ⌢ ∂X(ε,ε ) ∂X(ε f ,ε) ⌢ ×. X (ε) + Q( ε) i + Q( ε) +"1 . 1 ∂εi ε =ε ∂ε f - i ε f =ε 0 ⌢ ⌢ ⌢ 2 $ 1 dQ( ) 1 d 2Q( ) ⌢ 3$ dQ( )' ' & ε ε ε ) × 1+ + 2 Q( ε) + & ) +" & 2 dε 2 dε 8% dε ( ) % (

P. Ellis (1976)

€Q-box – sum of interaction diagrams X-box – sum of 0νββ diagrams Quenching of GT Strength

Investigate many-body effects on quenching of gA

Set of two-body diagrams contributing to renormalization of gA in 2νββ decay: product of 2 single-beta-decay

M2ν

€

82 Quenching of 38% in Se (gA ~ 1.0)

Quenching of GT Strength

Investigate many-body effects on quenching of gA

Directly look at one-body effective GT

Use low-momentum interactions and 3N forces

Quenching of GT Strength

Investigate many-body effects on quenching of gA

Calculate GT transitions in sd- and pf-shells

Net quenching between 0.8-0.9

Consistent with findings from 2νββ decay

Larger quenching when spin-orbit partners absent from valence space

Converged using low-momentum interactions – little effect from 3N

Conclusion/Outlook • Nuclear structure theory of medium-mass nuclei with 3N forces, extended spaces • Robust repulsive 3N mechanism for T=1 neutron/proton-rich nuclei

• Calcium isotopes in pf- and pfg9/2-shells: • Prediction of N=28 magic number in 48Ca • Shell evolution towards the dripline: small N=34 closure • Pairing gaps reflect shell structure – higher-order many-body processes essential • NN+3N predictions confirmed in new TITAN and ISOLTRAP experiments

• New directions for fundamental symmetries • Effective operators for 0vBB decay • 30% increase for Ge, Se; 75% increase for 48Ca • Effective one-body GT: quenching 0.8-0.9

• Many improvements needed

Map Exotic cores inversion in of Islands near Continuum effects Moving beyond stability Revisit cross-shell theory Towards full Ab Benchmark in48Ca 2! initiovalence-shell Hamiltonians 8!

2! protons driplines 20! 8! 28! sd neutrons

20! in sd - and - and sd 28! pf region?

pf sd/pf Path to Full Calculation -shells -shells driplines 50! regions

82! 82! WIMP-nucleus scattering electroweak Effective operators symmetries Fundamental Non-empirical calculation of calculationNon-empirical of 50! 126!

0νββ decay Evaluating Center-of-Mass Contamination Nonperturbative Lee-Suzuki (LS) transformation from extended space

Transform for two-body systems H ψn = E n ψn (e.g., 18O, 42Ca) Q Extended-space spectrum free of CM contamination € PH LS P P Project into standard space € eff φn = ε n φn onto eigenenergies from ε ⊂ E { n } { n } extended space calculation

P

€ HCM = 0 €

sd sdf7 / 2 p3 / 2 € € P Q

LS Use H eff as new two-body Hamiltonian in sd-shell valence-space calculations € € € €

€ 82 76 Calculations of M0ν in Se and Ge Calculation of 82Se (phenomenological wavefunctions from A. Poves, M. Horoi)

76Ge Bare matrix element 3.12 82Se Bare matrix element 2.73 + pp, hh ladders (1st) 5.44 + pp, hh ladders (1st) 4.86 + 3p-1h 2.20 + 3p-1h 2.40 + 2nd-order 4.14 + 2nd-order 3.92 Large effects cancel at 1st order JDH and Engel, PRC (2013)

Conclusion/Outlook • Nuclear structure theory of medium-mass nuclei with 3N forces, extended spaces

• Non-empirical valence-space methods • First calculations based on NN+3N forces • Extended valence spaces needed • Cures NN-only failings: dripline, shell evolution, spectra • Residual 3N forces improve predictions beyond dripline • New directions • Promising first results for F/Ne ground states to • Non-perturbative IM-SRG – excellent binding energies, spectra in sd shell only!

• Large-space ab-initio methods • Similar improvements with NN+3N as in valence-space methods • Agreement between methods encouraging for future – benchmarking valuable!

Nuclear Matrix Element 2 GT gV F M0ν = M0ν − 2 M0ν +! gA GT ! ! + + F + + M0ν = f ∑H(rab )σ a ⋅ σ bτ aτ b i M0ν = f ∑H(rab )τaτb i ab ab Shell model: arbitrary correlations in truncated single-particle space € QRPA: simple correlations in large single-particle space € € Nuclear Matrix Element

2 GT gV F M0ν = M0ν − 2 M0ν +! gA GT ! ! + + F + + M0ν = f ∑H(rab )σ a ⋅ σ bτ aτ b i M0ν = f ∑H(rab )τaτb i ab ab Shell model: arbitrary correlations in truncated single-particle space € QRPA: simple correlations in large single-particle space € € Pronounced differences for lighter candidates

Explore shell model improvements

Barea, Koltila, Iachello, PRL (2012) Backup Oxygen week ending PRL 105, 032501 (2010) PHYSICAL REVIEW LETTERS 16 JULY 2010

for the G matrix formalism, where a standard pion-N-Á coupling [22] was used and all 3N diagrams of the same order as Fig. 3(d) are included. We observe that the repul- sive FM 3N contributions become significant with increas- ing N and the resulting SPE structure is similar to that of phenomenological forces, where the d3=2 orbital remains high. Next, we calculate the SPEs from chiral low- momentum interactions Vlow k, including the changes due to the leading (N2LO) 3N forces in chiral EFT [23], see Figs. 3(f)–3(h). We consider also the SPEs where 3N-force contributions are only due to Á excitations [24]. The lead- ing chiral 3N forces include the long-range two-pion- exchange part, Fig. 3(f), which takes into account the excitation to a Á and other resonances, plus shorter-range 3N interactions, Figs. 3(g) and 3(h), that have been con- strained in few-nucleon systems [25]. The resulting SPEs in Fig. 2(d) demonstrate that the long-range contributions FIG. 3 (color online). Processes involving 3N contributions. due to Á excitations dominate the changes in the SPE The external lines are valence neutrons. The dashed and thick evolution and the effects of shorter-range 3N interactions lines denote pions and Á excitations, respectively. Nucleon-hole are smaller. We point out that 3N forces play a key role for lines are indicated by downward arrows. The leading chiral 3N the magic number N 14 between d5=2 and s1=2 [26], and forces include the long-range two-pion-exchange parts, diagram that they enlarge the N¼ 16 gap between s and d [5]. (f), which take into account the excitation to a Á and other ¼ 1=2 3=2 resonances, plus shorter-range one-pion exchange, diagram (g), The contributions from Figs. 3(f)–3(h) (plus all ex- and 3N contact interactions, diagram (h). change terms) to the monopole components take into ac- count the normal-ordered two-body parts of 3N forces, where one of the nucleons is summed over all nucleons Drip stateLines have been exchangedand andMagic this leads to the Numbers: exchange of in the core. This is also motivated by recent coupled-cluster the final (or initial) orbital labels j, m and j0, m0. Because calculations [27], where residual 3N forces between three 3N Forcesthis process in reflects Medium-Mass a cancellation of the lowering of the Nucleivalence states were found to be small. In addition, the SPE, the contribution from Fig. 3(d) has to be repulsive for effects of 3N forces among three valence neutrons should Exploring the frontierstwo neutrons.of nuclear Finally, we science: can rewrite Fig. 3(d) as the FM be generally weaker due to the Pauli principle. 3N force of Fig. 3(e), where the middle nucleon is summed Finally, we take into account many-body correlations by Worldwide joint experimental/theoreticalover core nucleons. The effort importance of the cancellation diagonalization in the valence space. The resulting ground- between Figs. 3(a) and 3(e) was recognized for nuclear state energies of the oxygen isotopes are presented in What are the properties ofmatter proton/neutron-rich in Ref. [21]. matter? Fig. 4. Figure 4(a) (based on phenomenological forces) The process in Fig. 3(d) corresponds to a two-valence- implies that many-body correlations do not change our What are the limits of nuclearneutron existence? monopole interaction, 82 schematically! illustrated in picture developed from the SPEs: The energy decreases

Fig. 4(d). The resulting SPE evolution is shown in Fig. 2(c) to N 16, but the d neutrons added out to N 20 How do magic numbers form and evolve? Heaviest oxygen¼ isotope3=2 ¼ 0 126! (a) Energies calculated (b) Energies calculated (c) Energies calculated (d) Schematic picture of two- from phenomenological from G-matrix NN from Vlow k NN valence-neutron interaction 2 forces + 3N (∆) forces + 3N (∆,N LO) forces induced from 3N force −20 50! −40 Energy (MeV) Energy Exp. Exp. Exp. 16 2 O core SDPF-M NN + 3N (∆) NN + 3N (N LO) USD-B 82! NN NN + 3N (∆) NN −60

protons 814 16 20 814 16 20 8162014 28! Neutron Number (N) Neutron Number (N) Neutron Number (N) 20! FIG.50 4! (color online). Ground-stateOtsuka energies, Suzuki, of oxygen JDH, isotopes Schwenk measured from, Akaishi,16O, including PRL (2010) experimental values of the bound 16– 24 O. Energies obtained from (a) phenomenological forces SDPF-M [13] and USD-B [14], (b) a G matrix and including FM 3N forces 2 due to Á excitations, and (c) from low-momentum interactions Vlow k and including chiral EFT 3N interactions at N LO as well as only 8! 28! due to Á excitations [25]. The changes due to 3N forces based on Á excitations are highlighted by the shaded areas. (d) Schematic illustration of a two-valence-neutronπ interactionπ generated by 3N forcesπ with a nucleon in the 16O core. 2! 20! 032501-3 2! 8! neutrons € € € How domagic numbers form evolve? and nuclear existence? What are limits the of proton/neutron-richWhat are properties the matter? of Worldwide jointexperimental/theoretical effort nuclear science: frontiersExploring the of 2! 8!

2! protons 20! 8! 28! neutrons 20! 3N Forces in Medium-Mass Nuclei Medium-Mass in Forces 3N DripLinesand MagicNumbers: 28! 50! €

€

82! π 82!

+ 2 1 Energy (MeV) New magic numbers incalcium 0 2 4 6 3 5 1 π 50! JDH, Menendez, JDH, Menendez, JDH, 44 25 06 68 64 60 56 52 48 44 € Otsuka NN+3N (MBPT) NN+3N NN

Mass NumberA 126! , Schwenk Schwenk π , Suzuki, JPG (2012) , JPG Suzuki, (2012) , JPG (2013)

Pairing for Shell Evolution N=28

5 Peak in pairing gaps: complementary pfg9/2 shell 4 signature for shell closure

3 Compare with 2+ energies for Ca 2 Energy (MeV) + 2 1 Agreement with CC throughout chain 0 NN NN+3N (emp) Hagen et al. PRL (2012) NN+3N (MBPT)

44 48 52 56 60 64 68 N=28 strong peak Mass Number A 3 3 (a) 3rd-order ladders (pf) pfg9/2 shell 2.5 2.5 2 2 1.5 1.5 (MeV) (MeV) n n (3) (3)

1 1 NN 0.5 0.5 NN+3N (emp) NN+3N (MBPT) 0 0 40 44 48 52 56 40 44 48 52 56 60 64 68 Mass Number A Mass Number A JDH, Menendez, Schwenk, JPG (2013) Pairing for Shell Evolution N=32

5 Peak in pairing gaps: complementary pfg9/2 shell 4 signature for shell closure

3 Compare with 2+ energies for Ca 2 Energy (MeV) + 2 1 Agreement with CC throughout chain 0 NN NN+3N (emp) Hagen et al. PRL (2012) NN+3N (MBPT)

44 48 52 56 60 64 68 N=28 strong peak Mass Number A 3 (a) 3rd-order ladders (pf) 3 pfg shell 9/2 N=32 moderate peak

2.5 2.5 Close to data with new TITAN value 2 2 53 1.5 1.5 Experimental measurement of Ca (MeV) (MeV) n n (3) (3) mass needed to reduce uncertainty

1 1 NN 0.5 0.5 NN+3N (emp) NN+3N (MBPT) 0 0 40 44 48 52 56 40 44 48 52 56 60 64 68 Mass Number A Mass Number A JDH, Menendez, Schwenk, JPG (2013) How domagic numbers form evolve? and nuclear existence? What are limits the of proton/neutron-richWhat are properties the matter? of Worldwide jointexperimental/theoretical effort nuclear science: frontiersExploring the of 2! 8!

2! protons 20! 8! • • • • 28! JDH, Bogner Caesar Otsuka neutrons 20! Menendez , JDH, Suzuki, ,

et 28! Hergert al. (R3B), , JDH, , Schwenk 50! Simonis OxygenIsotopes Schwenk Schwenk , EPJA Physics- beyond the neutron dripline of new nucleiclosed-shell Properties - Location- of dripline Keyphysics problems: , JDH, , et al., PRL, Akaishi 49 References 82! Menendez , (2013) 39 82! PRL

50! 113 105

, Schwenk

, 142501 (2014) (2014) , 142501 , 032501 (2010) , 032501

126! PRC PRC

88

, 034313 (2013) (2013) , 034313

22,24 O week ending PRL 105, 032501 (2010) PHYSICAL REVIEW LETTERS 16 JULY 2010

In Fig. 2, we show the single-particle energies (SPEs) of ergies and spectra. This shows a striking difference com- the neutron d5=2, s1=2 and d3=2 orbitals at subshell closures pared to Fig. 2(a): As neutrons occupy the d5=2 orbital, with N 8, 14, 16, and 20. The evolution of the SPEs is due to N evolving from 8 to 14, the d3=2 orbital remains almost at interactions¼ as neutrons are added. For the SPEs based on the same energy and is not well bound out to N 20. The ¼ NN forces in Fig. 2(a), the d3=2 orbital decreases rapidly as dominant differences between Figs. 2(a) and 2(b) can be neutrons occupy the d5=2 orbital, and remains well bound traced to the two-body monopole components, which de- from N 14 on. This leads to bound oxygen isotopes out termine the average interaction between two orbitals. The to N 20¼ and puts the neutron drip line incorrectly at 28O. monopole components of a general two-body interaction V This¼ result appears to depend only weakly on the renor- are given by an angular average over all possible orienta- malization method or the NN interaction used. We dem- tions of the two nucleons in orbitals lj and l0j0 [15],

onstrate this by showing SPEs calculated in the G matrix mono V jmj0m0 V jmj0m0 = 1; (1) formalism [10], which sums particle-particle ladders, and j;j0 ¼ h j j i m;m0 m;m0 based on low-momentum interactions Vlow k [11] obtained X X from chiral NN interactions at next-to-next-to-next-to- where the sum over magnetic quantum numbers m and m 3 0 leading order (N LO)[12] using the renormalization can be restricted by antisymmetry (see [16,17] for details). group. Both calculations include core polarization effects The SPE of the orbital j is effectively shifted by Vmono perturbatively [including diagram Fig. 3(d) with the Á j;j0 multiplied by the occupation number of the orbital j . This replaced by a nucleon and all other second-order diagrams] 0 leads to the change in the SPE and determines shell struc- and start from empirical SPEs [13] in 17O. The empirical ture and the location of the drip line [16–19]. SPEs contain effects from the core and its excitations, The comparison of Figs. 2(a) and 2(b) suggests that the including effects due to 3N forces. monopole interaction between the d and d orbitals We next show in Fig. 2(b) the SPEs obtained from the 3=2 5=2 obtained from NN theories is too attractive, and that the phenomenological forces SDPF-M [13] and USD-B [14] oxygen anomaly can be solved by additional repulsive Limits of Nuclearthat Existence: have been fit to reproduceOxygen experimental Anomaly binding en- contributions to the two-neutron monopole components, which approximately cancel the average NN attraction on 4 (a) Forces derived from NN theory (b) Phenomenological forces the d3=2 orbital. With extensive studies based on NN d3/2 forces, it is unlikely that such a distinct property would d3/2 Microscopic picture: 0 have been missed, and it has been argued that 3N forces Fit to experiment may be important for the monopole components [20]. s 1/2 NN-forces too attractive -4 Next, we show that 3N forces among two valence neu- s 16 d5/2 1/2 d5/2 trons and one nucleon in the O core give rise to repulsive monopole interactions between the valence neutrons. -8 G-matrix SDPF-M

Single-Particle Energy (MeV) While the contributions of the FM 3N force to other Vlow k USD-B quantities can be different, the shell-model configurations 8164 1 20 8164 1 20 Neutron Number (N) Neutron Number (N) composed of valence neutrons probe the long-range parts of 3N forces. The repulsive nature of this 3N mechanism 4 2 (c) G-matrix NN + 3N (∆) forces (d) Vlow k NN + 3N (∆,N LO) forces can be understood based on the Pauli exclusion principle. Figure 3(a) depicts the leading contribution to NN forces 0 due to the excitation of a Á, induced by the exchange of d d3/2 3/2 pions with another nucleon. Because this is a second-order -4 s 1/2 perturbation, its contribution to the energy and to the two- s 1/2 d5/2 d5/2 neutron monopole components has to be attractive. This is part of the attractive d3=2 d5=2 monopole component -8 NN + 3N (N 2 LO) À Single-Particle Energy (MeV) NN + 3N (∆) obtained from NN forces. NN NN + 3N (∆) NN In nuclei, the process of Fig. 3(a) leads to a change of the 8164 1 20 814 16 20 SPE of the j, m orbital due to the excitation of a core Neutron Number (N) Neutron Number (N) nucleon to a Á, as illustrated in Fig. 3(b) where the initial valence neutron is virtually excited to another j0, m0 orbital. FIG. 2 (color online). Single-particle energies of the neutron As discussed, this lowers the energy of the j, m orbital d , s and d orbitals measured from the energy of 16O as a 5=2 1=2 3=2 and thus increases its binding. However, in nuclei this function of neutron number N. (a) SPEs calculated from a G process is forbidden by the Pauli exclusion principle, if matrix and from low-momentum interactions Vlow k. (b) SPEs obtained from the phenomenological forces SDPF-M [13] and another neutron occupies the same orbital j0, m0, as shown USD-B [14]. (c),(d) SPEs including contributions from 3N in Fig. 3(c). The corresponding contribution must then be forces due to Á excitations and chiral EFT 3N interactions at subtracted from the SPE change due to Fig. 3(b). This is N2LO [25]. The changes due to 3N forces based on Á excitations taken into account by the inclusion of the exchange dia- are highlighted by the shaded areas. gram, Fig. 3(d), where the neutrons in the intermediate 032501-2 week ending PRL 105, 032501 (2010) PHYSICAL REVIEW LETTERS 16 JULY 2010

In Fig. 2, we show the single-particle energies (SPEs) of ergies and spectra. This shows a striking difference com- the neutron d5=2, s1=2 and d3=2 orbitals at subshell closures pared to Fig. 2(a): As neutrons occupy the d5=2 orbital, with N 8, 14, 16, and 20. The evolution of the SPEs is due to N evolving from 8 to 14, the d3=2 orbital remains almost at interactions¼ as neutrons are added. For the SPEs based on the same energy and is not well bound out to N 20. The ¼ NN forces in Fig. 2(a), the d3=2 orbital decreases rapidly as dominant differences between Figs. 2(a) and 2(b) can be neutrons occupy the d5=2 orbital, and remains well bound traced to the two-body monopole components, which de- from N 14 on. This leads to bound oxygen isotopes out termine the average interaction between two orbitals. The to N 20¼ and puts the neutron drip line incorrectly at 28O. monopole components of a general two-body interaction V This¼ result appears to depend only weakly on the renor- are given by an angular average over all possible orienta- malization method or the NN interaction used. We dem- tions of the two nucleons in orbitals lj and l0j0 [15],

onstrate this by showing SPEs calculated in the G matrix mono V jmj0m0 V jmj0m0 = 1; (1) formalism [10], which sums particle-particle ladders, and j;j0 ¼ h j j i m;m0 m;m0 based on low-momentum interactions Vlow k [11] obtained X X from chiral NN interactions at next-to-next-to-next-to- where the sum over magnetic quantum numbers m and m 3 0 leading order (N LO)[12] using the renormalization can be restricted by antisymmetry (see [16,17] for details). group. Both calculations include core polarization effects The SPE of the orbital j is effectively shifted by Vmono perturbatively [including diagram Fig. 3(d) with the Á j;j0 multiplied by the occupation number of the orbital j . This replaced by a nucleon and all other second-order diagrams] 0 leads to the change in the SPE and determines shell struc- and start from empirical SPEs [13] in 17O. The empirical ture and the location of the drip line [16–19]. SPEs contain effects from the core and its excitations, The comparison of Figs. 2(a) and 2(b) suggests that the including effects due to 3N forces. monopole interaction between the d and d orbitals We next show in Fig. 2(b) the SPEs obtained from the 3=2 5=2 obtained from NN theories is too attractive, and that the phenomenological forces SDPF-M [13] and USD-B [14] oxygen anomaly can be solved by additional repulsive Limits of Nuclearthat Existence: have been fit to reproduceOxygen experimental Anomaly binding en- contributions to the two-neutron monopole components, which approximately cancel the average NN attraction on 4 (a) Forces derived from NN theory (b) Phenomenological forces the d3=2 orbital. With extensive studies based on NN d3/2 forces, it is unlikely that such a distinct property would d3/2 Microscopic picture: 0 have been missed, and it has been argued that 3N forces Fit to experiment may be important for the monopole components [20]. s 1/2 NN-forces too attractive -4 Next, we show that 3N forces among two valence neu- s 16 d5/2 1/2 d5/2 trons and one nucleon in the O core give rise to repulsive monopole interactions between the valence neutrons. -8 G-matrix SDPF-M

Single-Particle Energy (MeV) While the contributions of the FM 3N force to other Vlow k USD-B quantities can be different, the shell-model configurations 8164 1 20 8164 1 20 Neutron Number (N) Neutron Number (N) composed of valence neutrons probe the long-range parts of 3N forces. The repulsive nature of this 3N mechanism 4 2 can be understood based on the Pauli exclusion principle. (c) G-matrix NN + 3N (∆) forces 0 (d) Vlow k NN + 3N (∆,N LO) forces Figure 3(a) depicts the leading contribution to NN forces (c) NN + 3N 0 -10 due to the excitation of a Á, induced by the exchange of d d3/2 3/2 -20 pions with another nucleon. Because this is a second-order -30 s -4 s 1/2 perturbation, its contribution to the energy and to the two- NNd prediction1/2 5/2 -40 d5/2 neutron monopole components has to be attractive. This is part of the attractive d d monopole component -50 2 3=2 5=2 -8 NN + 3N (N LO) À Energy (MeV) Single-Particle Energy (MeV) NN + 3N (∆) sd-shell obtained from NN forces. USDb NN -60 NN + 3N (∆) NNsdf7/2p3/2 shell In nuclei, the process of Fig. 3(a)sd-shellleads to a change of the -70 sdf p shell 8164 1 20 8USDb 14 16 20 SPE of the j, m orbital due to7/2 the3/2 excitation of a core Incorrect prediction of oxygen dripline Neutron Number (N)-80 Neutron Number (N) 16 18 20 22 24 26 28 nucleon to a Á, as illustrated16 18 in20 Fig.223(b)24 where26 28 the initial Extended-space – more binding Mass Number A valence neutron is virtually excitedMass toNumber another A j0, m0 orbital. FIG. 2 (color online). Single-particle energies of the neutron As discussed, this lowers the energy of the j, m orbital d , s and d orbitals measured from the energy of 16O as a 5=2 1=2 3=2 and thus increases its binding. However, in nuclei this function of neutron number N. (a) SPEs calculated from a G process is forbidden by the Pauli exclusion principle, if matrix and from low-momentum interactions Vlow k. (b) SPEs obtained from the phenomenological forces SDPF-M [13] and another neutron occupies the same orbital j0, m0, as shown USD-B [14]. (c),(d) SPEs including contributions from 3N in Fig. 3(c). The corresponding contribution must then be forces due to Á excitations and chiral EFT 3N interactions at subtracted from the SPE change due to Fig. 3(b). This is N2LO [25]. The changes due to 3N forces based on Á excitations taken into account by the inclusion of the exchange dia- are highlighted by the shaded areas. gram, Fig. 3(d), where the neutrons in the intermediate 032501-2 Two-Neutron Separation Energies: Mass of 52Ca Compare with AME2003 data

18 NN+3N Predictions AME2003 16 Reproduce 48Ca shell closure

50 14 Predictions too bound past Ca (MeV)

2n 12 S

10 NN+3N (MBPT) NN+3N (emp) 8 3 28 29 30 31 32

TITAN+ 2 AME2003 (MeV) n (3)

1

0 28 29 30 31 32 Neutron Number N 4 42 2 (a) G-matrix NN + 3N (∆) forces (b) Vlow k NN + 3N (∆,N LO(a)) forces G-matrix NN + 3N (∆) forces (b) Vlow k NN + 3N (∆,N LO) forces

0 0 d3/2 d3/2 d3/2 d3/2

-4 s-41/2 s 1/2 s 1/2 s 1/2 d5/2 d5/2 d5/2 d5/2

-8 2 -8 2 NN + 3N (∆) NN + 3N (N LO) NN + 3N (∆) NN + 3N (N LO) ∆) ∆) Single-Particle Energy (MeV) NN NN + 3N ( Single-Particle Energy (MeV) NN NN + 3N ( 3N Forces for Valence-ShellNN Theories NN 814 16 20 814 16 8 20 14 16 20 814 16 20 Normal-orderedNeutron Number (3NN) : contributionNeutron to Number valence-space (N) Neutron NumberHamiltonian (N) Neutron Number (N)

Effective(c) 3-body interaction one-body (d) 3-body interactions withEffective one(c) 3-body interaction two-body (d) 3-body interactions with one more neutron added to (c) more neutron added to (c)

16O core 16O core

a a b a' a' b'

1 a V a' = αβa V αβa' ab V a'b' = αab V αa'b' 3N,eff 2 ∑ 3N 3N,eff ∑ 3N αβ =core α =core

Combine with NN (Third Order): no empirical adjustments € € week ending week ending PRL 105, 032501 (2010) PHYSICAL REVIEWPRL 105, 032501 LETTERS (2010) PHYSICAL16 REVIEW JULY 2010 LETTERS 16 JULY 2010

for the G matrix formalism, where a standard pion-N-forÁ the G matrix formalism, where a standard pion-N-Á week ending week ending PHYSICAL REVIEWPHYSICALcoupling LETTERS [22] was REVIEW used and all 3N LETTERSdiagrams of the samecoupling [22] was used and all 3N diagrams of the same PRL 105, 032501 (2010) PRL 105, 032501 (2010) order as Fig. 3(d) are included. We observe that16 the JULY repul-order 2010 as Fig. 3(d) are16 included. JULY 2010 We observe that the repul- In Fig. 2, we show the single-particleIn Fig. 2, we energies show the (SPEs) single-particle of ergiessive energies FM and3N spectra. (SPEs)contributions of This shows becomeergies a and significant striking spectra. difference with This increas- shows com-sive FM a striking3N contributions difference become com- significant with increas- ing N and the resulting SPE structure is similar to thating of N and the resulting SPE structure is similar to that of the neutron d , s and thed neutronorbitalsd at subshell, s and closuresd orbitalspared at subshell to Fig. 2(a) closures: As neutronspared occupy to Fig. the2(a)d: Asorbital, neutrons with occupy the d orbital, with 5=2 1=2 3=2 5=2 1=2 3=2 phenomenological forces, where the d orbital5=2 remainsphenomenological5=2 forces, where the d orbital remains N 8, 14, 16, and 20. TheN evolution8, 14, 16, of the and SPEs 20. The is due evolution to N ofevolving the SPEs from is due 8 to to 14, theN evolvingd orbital from3=2 remains 8 to 14, almost the d at orbital remains almost at 3=2 high. Next, we calculate the3=2 SPEs from chiral low-high.3=2 Next, we calculate the SPEs from chiral low- interactions¼ as neutrons areinteractions added.¼ For as the neutrons SPEs based are added. on Forthe the same SPEs energy based and on is notthe well same bound energy out and to N is not20 well. The bound out to N 20. The momentum interactions V , including the changes duemomentum interactions V , including the changes due NN forces in Fig. 2(a), theNNd forcesorbital in decreases Fig. 2(a), rapidly the d asorbital decreases rapidly as low k ¼ ¼ low k 3=2 3=2 dominant differences2 betweendominant Figs. differences2(a) and 2(b) betweencan be Figs. 2(a) and2 2(b) can be neutrons occupy the d neutronsorbital, and occupy remains the d wellorbital, bound andtracedto remains the to leading the well two-body (boundN LO) 3 monopoleNtracedforces to in components, the chiral two-body EFT [23 which monopole], seeto de- the leading components, (N LO) which3N forces de- in chiral EFT [23], see 5=2 5=2 Figs. 3(f)–3(h). We consider also the SPEs where 3N-forceFigs. 3(f)–3(h). We consider also the SPEs where 3N-force from N 14 on. This leadsfrom toN bound14 oxygenon. This isotopes leads to out boundtermine oxygen the isotopes average out interactiontermine between the average two orbitals. interaction The between two orbitals. The contributions are only due to Á excitations [24]. The lead-contributions are only due to Á excitations [24]. The lead- to N 20¼ and puts the neutronto N drip20¼ lineand incorrectly puts the neutron at 28O drip. linemonopole incorrectly components at 28O. of amonopole general two-body components interaction of a generalV two-body interaction V ing chiral 3N forces include the long-range two-pion-ing chiral 3N forces include the long-range two-pion- This¼ result appears to dependThis¼ result only weakly appears on to the depend renor- only weaklyare given on by the an renor- angular averageare given over by all an possible angular averageorienta- over all possible orienta- exchange part, Fig. 3(f), which takes into account theexchange part, Fig. 3(f), which takes into account the malization method or themalizationNN interaction method used. or the WeNN dem-interactiontions of used. the twoWe dem- nucleonstions in orbitals of thelj twoand nucleonsl0j0 [15], in orbitals lj and l0j0 [15], excitation to a Á and other resonances, plus shorter-rangeexcitation to a Á and other resonances, plus shorter-range onstrate this by showing SPEsonstrate calculated this by showing in the G SPEsmatrix calculated in themonoG matrix mono 3N interactions,V Figs. 3(g)jmjand0m0 V3(h)jmj,V that0m0 have= been1; jmj con-03mN(1)0 interactions,V jmj0m0 = Figs.1; 3(g) and(1)3(h), that have been con- formalism [10], which sumsformalism particle-particle [10], which ladders, sums and particle-particle ladders,j;j0 and j;j0 strained in few-nucleon¼ h systemsj [25j ]. Thei resulting¼ h SPEsstrainedj j in few-nucleoni systems [25]. The resulting SPEs based on low-momentumbased interactions on low-momentumV [11] obtained interactions V [11] obtainedm;mX0 m;mXm;mX0 0 m;mX0 low k inlow Fig.k 2(d) demonstrate that the long-range contributionsin Fig. 2(d) demonstrate that the long-range contributions from chiral NN interactionsfrom at chiral next-to-next-to-next-to-NN interactions at next-to-next-to-next-to- FIG. 3 (color online). Processes involving 3N contributions.FIG. 3wheredue (color to the online).Á sumexcitations over Processes magnetic dominate involvingwhere quantum the the3N changessumcontributions. numbers over in magnetic them and SPEduem quantum0 to Á excitations numbers dominatem and m0 the changes in the SPE leading order (N3LO)[leading12] using order the (N renormalization3LO)[12] using the renormalization The external lines are valence neutrons. The dashed and thickThe externalcanevolution be lines restricted and are valencethe by effects antisymmetry neutrons. ofcan shorter-range be The restricted (see dashed [163 andN, by17interactions thick] antisymmetry for details).evolution (see and [16 the,17 effects] for details). of shorter-range 3N interactions group. Both calculations includegroup. Both core calculations polarization include effects core polarization effects mono mono lines denote pions and Á excitations, respectively. Nucleon-holelines denoteTheare smaller. SPE pions of and We theÁ pointexcitations, orbital outj thatThe respectively.is3 effectivelyN SPEforces of Nucleon-hole play the shifted orbitala key role byj Vis forare effectively smaller. We shifted point out by thatV 3N forces play a key role for perturbatively [includingperturbatively diagram Fig. [including3(d) with the diagramÁ Fig. 3(d) with the Á j;j0 j;j0 lines are indicated by downward arrows. The leading chiral 3linesN aremultipliedthe indicated magic numberby by thedownward occupationN arrows.14multipliedbetween number The leadingd by ofand the the chirals occupation orbital3[N26],j and.the This number magic number of the orbitalN 14j .between This d and s [26], and replaced by a nucleon andreplaced all other bysecond-order a nucleon and diagrams] all other second-order diagrams] 5=2 1=2 0 0 5=2 1=2 forces include the long-range two-pion-exchange parts, diagramforcesleads includethat they to the the enlarge long-range change the inN two-pion-exchange¼ the16leads SPEgap and to between the determines parts, changes diagramand in shelld the SPE[ struc-5that]. and they determines enlarge the N¼ shell16 struc-gap between s and d [5]. and start(f), which from empirical take into account SPEsand start [ the13] excitationfrom in 17 empiricalO. The to a empiricalÁ SPEsand other [(f),13] which in 17O take. The into empirical account the excitation to a Á and1=2 other3=2 1=2 3=2 tureThe and contributions the location of from¼ theture Figs. drip and line3(f) the– [163(h) location–19(plus]. of all the ex- dripThe line contributions [16–19]. from¼ Figs. 3(f)–3(h) (plus all ex- SPEsresonances, contain effects plus shorter-range fromSPEs the one-pion contain core and exchange, effects its excitations, from diagram the (g),resonances, core and plus its shorter-range excitations, one-pion exchange, diagram (g), changeThe comparison terms) to the of monopole Figs. 2(a)The componentsand comparison2(b) suggests take of Figs. into that ac-change2(a) theand terms)2(b) tosuggests the monopole that the components take into ac- includingand 3 effectsN contact due interactions, to 3Nincludingforces. diagram effects (h). due to 3N forces.and 3N contact interactions, diagram (h). monopolecount the interaction normal-ordered betweenmonopole two-body the d interactionpartsand of 3dN betweenforces,orbitalscount the thed normal-orderedand d orbitals two-body parts of 3N forces, We next show in Fig. 2(b)Wethe next SPEs show obtained in Fig. from2(b) thethe SPEs obtained from the 3=2 5=2 3=2 5=2 obtainedwhere one from of theNN nucleonstheoriesobtained is is summed too fromattractive, overNN alltheories and nucleons thatwhere is the too one attractive, of the nucleons and that is the summed over all nucleons phenomenological forcesphenomenological SDPF-M [13] and forces USD-B SDPF-M [14]Oxygen [13] and USD-B Anomaly [14] state have been exchanged and this leads to the exchangestate of haveoxygenin the been core. anomaly exchanged This is canalso and motivated be thisoxygen solved leads by to byanomaly the recent additionalexchange coupled-cluster can of be repulsive solvedin the core. by additionalThis is also motivated repulsive by recent coupled-cluster that have been fit to reproducethat have experimental been fit to binding reproduce en- experimental binding en- the final (or initial) orbital labels j, m and j0, m0. Becausethe finalcontributionscalculations (or initial) [ orbital27 to], the where labels two-neutron residualcontributionsj, m and3N monopolej0forces, m0 to. Because between the components, two-neutron threecalculations monopole [27], where components, residual 3N forces between three this process reflects a cancellationFirst calculations of the lowering of thethis processwhichvalence approximatelyreflects states a were cancellation found cancelwhich to of the be the approximately average small. lowering InNN addition, ofattraction the cancel thevalence the on average statesNN wereattraction found to on be small. In addition, the 4 (a) Forces derived from NN theory 4 (a) Forces derived from NN theory SPE, the contribution fromusing Fig.(b)3(d) NN+3NPhenomenologicalhas to be repulsive forces forSPE,(b) Phenomenologicalthe effects contributiond3=2 oforbital.3 forcesN forces from With Fig. among3(d) extensivethe threehasd3= to valence2 be studiesorbital. repulsive neutrons based With for should extensive oneffectsNN of studies3N forces based among on threeNN valence neutrons should two neutrons. Finally, we can rewrite Fig. 3(d)das3/2 the FMtwo neutrons.forces,be generally it Finally, isd unlikely weaker3/2 we can due that rewrite toforces, such the Fig.Pauli a it distinct3(d) is principle. unlikelyas the property FM that would suchbe generally a distinct weaker property due to would the Pauli principle. d3/2 d3/2 0 3N force of Fig. 3(e), where0 the middle nucleon is summed3N forcehaveFinally, of been Fig. 3(e) wemissed,, take where into and the account it middlehave has been many-body been nucleon argued missed, is summed correlations that and3N it hasforces byFinally, been argued we take that into3 accountN forces many-body correlations by over core nucleons. The importance of the cancellationovermay corediagonalization benucleons. important The in the for importance valence themay monopole space. be of important the The components cancellation resulting for the ground- [20 monopole].diagonalization components in the valence [20]. space. The resulting ground- s 1/2 s 1/2 -4 between Figs. 3(a) and 3(e)-4 was recognized for nuclearbetweenstateNext, Figs. energies we3(a) showand of that3(e) the oxygen3wasN forces recognizedNext, isotopes among we show for are two nuclear presented that valence3N forcesstate in neu- energies among two of the valence oxygen neu- isotopes are presented in s s 16 16 matterd5/2 in Ref. [211/2]. d5/2 d5/2 1/2 mattertronsFig.Probe ind Ref.5/2 and4. Figure[limits21 one]. nucleon 4(a)of nuclear(based in thetrons on existenceO phenomenological andcore one give nucleonwith rise 3N to in forces) repulsive forces theFig.O 4core. Figure give4(a) rise(based to repulsive on phenomenological forces) The process in Fig. 3(d) corresponds to a two-valence-Themonopoleimplies processF dripline that in interactions Fig. many-body3(d) corresponds between correlationsmonopole to the a two-valence- dointeractions valence not change neutrons. between ourimplies the that valence many-body neutrons. correlations do not change our -8 neutronG-matrix monopole interaction,-8 schematicallyG-matrixSDPF-M illustratedneutron in SDPF-Mpicture monopole developed interaction, from schematically the SPEs: The illustrated energy decreases in picture developed from the SPEs: The energy decreases Single-Particle Energy (MeV) Single-Particle Energy (MeV) 3N repulsion amplifiedWhile with the contributions N: crucial Whilefor of theneutron-rich the FM contributions3N force nuclei to of other the FM 3N force to other Vlow k V USD-Blow k ? USD-B Fig. 4(d). The resulting SPE evolution is shown in Fig. 2(c)Fig. 4(d)quantitiesto.N The resulting16 can, but be SPE the different,d evolution3=2 neutronsquantities the is shell-model shown added can in be outFig. different, configurations to2(c)N 20to theN shell-model16, but the configurationsd3=2 neutrons added out to N 20 8164 1 20 88161644 11 20 20 248¼164 1 20 ¼ ¼ ¼ Neutron Number (N) NeutronNeutrond3/2 Number Numberunbound ( N(N) ) at O composedwithNeutron 3N Number offorces ( valenceN) neutronscomposed probe of the valence long-range neutrons parts probe the long-range parts of 3N forces. The repulsiveof 3 natureN forces. of this The3N repulsivemechanism nature of this 3N mechanism 4 0 4 0 2 2 (d) Schematic picture of two- (d) Schematic picture of two- (c) G-matrix NN + 3N(a) ( ∆Energies) forces calculated(c)(d) G-matrix Vlow k NN NN + +3N (b)3N (∆ (Energies,N∆) LOforces) forces calculated(d) V lowcan k NN be + 3N understood(a) (c)(∆ ,N EnergiesEnergies LO) forces calculatedcalculated based oncan Isotopes the be Pauli understood(b) Energies exclusionunbound calculated based principle. on the Pauli(c) Energies exclusion calculated principle. valence-neutron interaction valence-neutron interaction from phenomenological from G-matrix NN Figure 3(a)from depicts phenomenological from Vlow the k NN leading Figure contribution3(a) fromdepicts G-matrix to theNN NN leading forces contribution from V tolow NNk NN forces 2 24 2 0 0 forces + 3N (∆) forces due to the + excitation 3N ( ∆ ,N LO ) forcesforces of adueÁ,beyond induced to the induced excitation by +O from 3N the ( 3N∆) exchangeforces force of a Á, induced of + 3N by (∆ the,N LO exchange) forces of induced from 3N force −d20 d d3/2 −20d3/2 3/2 3/2 pions with another nucleon.pions Because with another this is a nucleon. second-order Because this is a second-order -4 -4 s 1/2 perturbation,s 1/2 its contributionperturbation, to the energy its contribution and to the two- to the energy and to the two- s 1/2 s 1/2 d5/2 dd5/25/2 d5/2neutron monopole componentsneutron has monopole to be attractive. components This is has to be attractive. This is −40 −40 Energy (MeV) Energy Exp. Exp. (MeV) Energy Exp.Exp. Exp. Exp. part of the attractive d partFirstd of16 the microscopicmonopole attractive componentd d monopole component16 -8 -8 2 2 2 3=2 5O= 2core 3=2 5=2 2 O core SDPF-M NN + 3N (NNN LO)+ 3N (∆) NN + 3N NN(NSDPF-M +LO) 3N (N LO) À NN + 3N (∆) À NN + 3N (N LO) Single-Particle Energy (MeV) NN + 3N (∆) Single-Particle Energy (MeV) NN + 3N (∆) obtained from NN forces.obtained from NN forces. NN USD-B NN NN + 3N (NN∆) NN + 3N NN(∆USD-B) + 3N (∆) explanationNN of oxygen NN + 3N (∆) NN NNIn nuclei,NN the process of Fig.In nuclei,3(a) leads the to process a change of Fig. of the3(a)NNleads to a change of the −60 −60 88164 1 14 16 20 88 20 816144 11614 16 20 20 8 20SPE816208 of14 the1614j, m 16 20orbital 20SPE due8anomaly to of the the excitationj14, m 16orbital of 20 duea81620 core to the excitation14 of a core NeutronNeutron Number Number(N) (N) Neutron NeutronNeutron Number Number (Number N(N)) (N) nucleon NeutronNeutron Number to a Number Á(N,) as illustrated (N) nucleon inNeutron Fig. to3(b) a ÁNumber,where as illustrated (N) the initial inNeutron Fig. 3(b) Numberwhere (N) the initial valence neutron is virtuallyvalence excited neutron to another is virtuallyj , m orbital. excited to another j , m orbital. Otsuka, Suzuki, JDH, Schwenk, Akaishi, PRL (2010) 16 0 0 16 0 0 FIG. 2FIG. (color 4 (color online). online). Single-particle Ground-stateFIG. 2 (color energies energies online). of of oxygen the Single-particle neutron isotopesFIG. measured 4 energiesAs (color discussed, online). from of theO, neutron this Ground-state including lowers experimentalAs energies the discussed, energy of values oxygen of this of the isotopes the lowersj bound, m measuredorbital 16– the energy from ofO, including the j, m experimentalorbital values of the bound 16– d , s and d orbitals measuredd , s fromand d the energyorbitals of measured16O as a from the energy of 16O as a 5=2 241=2 O. Energies3=2 obtained from5=2 (a)1=2 phenomenological3=2 forces SDPF-M24 O.and Energies [13] and thus obtained USD-B increases [14 from], (b) (a) its a phenomenologicalG binding.andmatrix thus and However, including increases forces FM in SDPF-M its3 nucleiN binding.forces [13 this] and However, USD-B [14], in (b) nuclei a G matrix this and including FM 3N forces functiondue of to Á neutronexcitations, number andfunction (c)N. from (a) SPEs low-momentumof neutron calculated number interactions fromN. a (a)GdueV SPEs to Áand calculatedexcitations, including from andchiral (c) a EFT fromG 3N low-momentuminteractions at interactionsN2LO as wellV as onlyand including chiral EFT 3N interactions at N2LO as well as only low kprocess is forbidden by theprocess Pauli is exclusion forbidden principle, bylow thek Pauli if exclusion principle, if matrixdue and to fromÁ excitations low-momentum [25matrix]. The interactions changes and from due low-momentum toVlow3Nk.forces (b) SPEs baseddue interactions on toÁÁexcitationsexcitationsVlow k. are [ (b)25 highlighted]. SPEs The changes by the due shaded to 3N areas.forces (d) based Schematic on Á excitations are highlighted by the shaded areas. (d) Schematic obtainedillustration from the of phenomenological a two-valence-neutronobtained from forces interaction the SDPF-M phenomenological generated [13] and byillustration3N forcesforcesanother SDPF-Mof with a neutron two-valence-neutron a nucleon [13 occupies] and in the 16O theanother interactioncore. same neutron orbital generated occupiesj0, m by0,3 asN forces the shown same with orbital a nucleonj0, inm the0, as16 shownO core. USD-B [14]. (c),(d) SPEsUSD-B including [14 contributions]. (c),(d) SPEs from including3N contributionsin Fig. 3(c). The from corresponding3N in Fig. contribution3(c). The corresponding must then be contribution must then be 032501-3 032501-3 forces due to Á excitationsforces and chiral due to EFTÁ excitations3N interactions and chiral at EFTsubtracted3N interactions from the at SPEsubtracted change due from to Fig. the3(b) SPE. This change is due to Fig. 3(b). This is N2LO [25]. The changes dueN to2LO3N [forces25]. The based changes on Á dueexcitations to 3N forcestaken based into on Á accountexcitations by thetaken inclusion into account of the exchange by the inclusion dia- of the exchange dia- are highlighted by the shadedare areas. highlighted by the shaded areas. gram, Fig. 3(d), where thegram, neutrons Fig. 3(d) in the, where intermediate the neutrons in the intermediate 032501-2 032501-2 Ground-State Energies of Oxygen Isotopes Valence-space interaction and SPEs from NN+3N

0 -10 (b) NN (c) NN + 3N -20 -30 -40 -50

Energy (MeV) sd-shell (2nd) USDb -60 sd-shell (3rd) sd-shell -70 sdf p -shell sdf7/2p3/2-shell 7/2 3/2 -80 16 18 20 22 24 26 28 16 18 20 22 24 26 28 Mass Number A Mass Number A JDH, Menendez, Schwenk, EPJA (2013)

Repulsive character improves agreement with experiment

sd-shell results underbound; improved in extended space

Impact on Spectra: 23O Neutron-rich oxygen spectra with NN+3N

5/2+, 3/2+ energies reflect 22,24O shell closures 5 23 sd-shell NN only O + + Wrong ground state 4 3/2 (3/2 ) + + 5/2 too low S 3/2 n + + 3 5/2 + 3/2 bound (5/2 )

+ 3/2 NN+3N 2 + 3/2 Clear improvement in

+ Energy (MeV) 1 5/2 extended valence space + 5/2 + + + + + 0 1/2 1/2 1/2 1/2 1/2

+ 5/2 -1 NN NN NN+3N NN+3N Expt. sd sdf7/2p3/2 sd sdf7/2p3/2 JDH, Menendez, Schwenk, EPJA (2013) Experimental Connection: Beyond the Dripline Hoffman, Kanungo, Lunderberg… PRLs (2008+) Valence-space Hamiltonian from NN + 3N + residual 3N

+ 2 R3B-LAND (this work) 1.5 MoNA/NSCL (2008, 2012) NN+3N + residual 3N NN+3N 16O core 1 + 3/2 + residual 3N MBPT Exp. 0.5 S (25O)

Energy (MeV) n + 742 keV 770 keV + 0 0 0 MBPT Exp. S (26O) Caesar, Simonis et al., PRC (2013) 2n 40 keV <120 keV 24 25 26 O O O Repulsion more pronounced for neutron-rich systems: 400 keV at 26O

Improved agreement with new data beyond 24O dripline

Future: include coupling to continuum In-Medium SRG: Basics

In-Medium SRG applies continuous unitary transformation to drive off- diagonal physics to zero Tsukiyama, Bogner, Schwenk, PRL (2011)

H(s) = U(s)HU† (s) ≡ H d (s) + H od (s) →H d (∞)

Decouples reference state from excitations npnh H(∞) Φc = 0

€

€

In-Medium SRG: Basics

In-Medium SRG applies continuous unitary transformation to drive off- diagonal physics to zero Tsukiyama, Bogner, Schwenk, PRL (2011)

H(s) = U(s)HU† (s) ≡ H d (s) + H od (s) →H d (∞)

Where U is defined by the generator:

η ( s ) ≡ ( d U ( s ) d s ) U † ( s ) chosen for desired decoupling behavior € Taking

d d od η(s) = [H (s),H(s)] = [H (s),H (s)] € Drives Hod to 0 (Wegner,1994)

€ Closed-shell reference state: drives all n-particle n-hole couplings to 0 dH(s) = [η(s),H(s)] npnh H(∞) Φc = 0 ds

€ € IM-SRG for Valence-Space Hamiltonians

In-Medium SRG applies continuous unitary transformation to drive off- diagonal physics to zero Tsukiyama, Bogner, Schwenk, PRC (2012)

Open shell systems: split particle states into valence states, v, and those above valence space, q Redefine “off-diagonal” to exclude valence particles

2v 1q1v 2q 3p1h 4p2h 2v 1q1v 2q 3p1h 4p2h

H(s = 0) →H(∞)

€ IM-SRG for Valence-Space Hamiltonians

In-Medium SRG applies continuous unitary transformation to drive off- diagonal physics to zero Tsukiyama, Bogner, Schwenk, PRC (2012)

Open shell systems: split particle states into valence states, v, and those above valence space, q Redefine “off-diagonal” to exclude valence particles

2v 1q1v 2q 3p1h 4p2h 2v 1q1v 2q 3p1h 4p2h sd

H(s = 0) →H(∞)

Defines new effective valence-space Hamiltonian Heff States outside valence space are decoupled €

€ Nonperturbative Valence-Space Strategy

1) Effective interaction: nonperturbative from IM-SRG

2) Single-particle energies: nonperturbative from IM-SRG

3) Hartree-Fock basis of e max = 2 n + l = 14 converged

4) NN and 3N forces from chiral EFT

5) Explore extended valence spaces – in progress €

NN matrix elements 3 - Chiral N LO (Machleidt, Λ NN = 500 MeV ); free-space SRG evolution

λ =1.88 − 2.24fm−1 - Cutoff variation SRG - Vary ! ω = 20 − 24MeV - Consistently include€ 3N forces induced by SRG evolution (NN+3N-ind) Initial 3N€ force contributions € 2 - Chiral N LO Λ 3N = 400 MeV (NN+3N-full)

- Included with cut: e1 + e2 + e3 ≤ E 3 max =14

€ € IM-SRG Oxygen Ground-State Energies Valence-space interaction and SPEs from IM-SRG in sd shell

4 -120 (a) (b) -140 d3/2 0 -160 s1/2 -180 -4 d -200 5/2 Exp. Energy (MeV) -8 -220 NN NN+3N-ind NN+3N-ind NN+3N-full Single-Particle Energy (MeV) NN+3N-full -240 -12 16 18 20 22 24 26 28 16 18 20 22 24 26 28 Mass Number A Mass Number A Bogner et al., PRL (2014)

NN+3N-induced reproduce exp well, not dripline NN+3N-full modestly overbound – good behavior past dripline

Good dripline properties Very weak ! ω dependence

€ week ending PRL 111, 062501 (2013) PHYSICAL REVIEW LETTERS 9 AUGUST 2013

(Fig. 1) is over bound at 130:8 1 MeV but in close Figures 3 and 4 collect our results for the oxygen, nitro- agreement with the 130À:5 1 MeVð Þ obtained from gen and fluorine isotopes calculated with ! 24 MeV À ð Þ 1 ¼ IM-SRG [11], giving further confirmation of the accuracy and !SRG 2:0 fmÀ . The top panel of Fig.@ 3 shows the achieved by different many-body methods. Note that the predicted¼ evolution of neutron single particle spectrum energies of 15O and 23O can be obtained in two different (addition and separation energies) of oxygen isotopes in ways, from either neutron addition or removal on neigh- the sd shell. Induced 3NFs reproduce the overall trend but boring subshell closures. Results in Fig. 2 differ by at most predict a bound d3=2 when the shell is filled. Adding pre- 400 keV, again within the estimated uncertainty of our existing 3NFs—the full Hamiltonian—raises this orbit many-body truncation scheme. The c.m. correction in above the continuum also for the highest masses. This Eq. (10) is crucial to obtain this agreement. For ! gives a first principle confirmation of the repulsive effects 1 15¼ 24 MeV and !SRG 2:0 fmÀ , the discrepancy in@ O of the two-pion exchange Fujita-Miyazawa interaction 23 ¼ ( O) is 1.65 MeV (1.03 MeV) when neglecting the discussed in Ref. [3]. The consequences of this trend are changes in kinetic energy of the c.m. but it reduces to demonstrated by the calculated ground state energies only 190 keV (20 keV) when this is accounted for. This shown in the bottom panel and in Fig. 4: the induced gives us confidence that a proper separation of the center of Hamiltonian systematically under binds the whole isotopic mass motion is being reached. chain and erroneously places the drip line at 28O due to the Figure 2 also gives a first remarkable demonstration of lack of repulsion in the d3=2 orbit. The contribution from the predictive power of chiral 2N 3N interactions: 24 þ full 3NFs increase with the mass number up to O, when accounting for the precision of our many-body approach the unbound d orbit starts being filled. Other bound and dependence on ! found in Ref. [28], we expect an 3=2 SRG quasihole states are lowered resulting in additional overall accuracy of at least 5% on binding energies. All calculated binding. As a result, the inclusion of NNLO 3NFs consis- values agree with the experiment within this limit. Note tently brings calculations close to the experiment and that the interactions employed were only constrained by reproduces the observed dripline at 24O [41–43]. Our cal- 2N and 3H and 4He data. culations predict 25O to be particle unbound by 1.54 MeV, larger than the experimental value of 770 keV [44] but within the estimated errors. The ground state resonance for 6 28 2N 3N ind O is suggested to be unbound by 5.2 MeV with respect to 4 1d3 2 2N 3N full 24O. However, this estimate is likely to be affected by the 2 presence of the continuum which is important for this nucleus but neglected in the present work. 0

MeV The same mechanism affects neighboring isotopic 2s IM-SRG Oxygen1 Ground-State2 1 2 Energies chains. This is demonstrated in Fig. 4 for the semimagic A i 4 odd-even isotopes of nitrogen and fluorine. Induced 3NF forces consistently under bind these isotopes and even Valence-space interaction and SPEs from6 IM-SRG1 din5 2 sd shell predict a 27N close in energy to 23N. This is fully cor- 8 rected by full 3NFs that strongly bind 23N with respect to 4 -120 60 27N, in accordance with the experimentally observed drip 24 MeV (a) (b) 1 line. The repulsive effects of filling the d3=2 is also -140 80 SRG 2.0 fm d3/2 0 -160 100 13N 15N 21N 23N 27N s1/2 80 24 MeV

MeV -180 120 2N 3N ind 1 -4 . s 2.0 fm . SRG g 2N 3N full 100 d -200 E Exp 5/2 Exp. 140 Energy (MeV) 120 -8 -220 NN NN+3N-ind 160 MeV NN+3N-ind . s

. 140 NN+3N-full g

Single-Particle Energy (MeV) NN+3N-full -240 E 180 2N 3N full -12 14O 16O 22O 24O 28O 160 2N 3N ind 16 18 20 22 24 26 28 16 18 20 22 24 26 28 Exp Mass Number A Mass Number A 180 BognerFIG. et al., 3 (colorPRL (2014) online). Top.Cipollone, Evolution Barbieri, of single Navratil particle, PRL ener- (2013)

gies for neutron addition and removal around sub-shell closures 15F 17F 23F 25F 29F of oxygen isotopes. Bottom. Binding energies obtained from the NN+3N-induced reproduce expKoltun well, SR and not the dripline poles of propagator (1), compared to experi- FIG. 4 (color online). Binding energies of odd-even nitrogen NN+3N-full modestly overboundment (bars) – good [44,46 behavior,47]. All points past are dripline corrected for the kinetic and fluorine isotopes calculated for induced (red squares) energy of the c.o.m. motion. For all lines, red squares (blue dots) and full (green dots) interactions. Experimental data are from Good dripline properties refer to induced (full) 3NFs. [45–47].

Very weak ! ω dependence 062501-4

€ week ending PRL 111, 062501 (2013) PHYSICAL REVIEW LETTERS 9 AUGUST 2013

(Fig. 1) is over bound at 130:8 1 MeV but in close Figures 3 and 4 collect our results for the oxygen, nitro- agreement with the 130À:5 1 MeVð Þ obtained from gen and fluorine isotopes calculated with ! 24 MeV À ð Þ 1 ¼ IM-SRG [11], giving further confirmation of the accuracy and !SRG 2:0 fmÀ . The top panel of Fig.@ 3 shows the achieved by different many-body methods. Noteweek that the endingpredicted¼ evolution of neutron single particle spectrum PRL 110, 242501 (2013) PHYSICAL REVIEWenergies LETTERS of 15O and 23O can be obtained in two14 different JUNE 2013(addition and separation energies) of oxygen isotopes in ways, from either neutron addition or removal on neigh- the sd shell. Induced 3NFs reproduce the overall trend but (a) (b) truncation of the many-body expansion, while the effect of boring-80 subshell closures. Results in Fig. 2 differ by at most predict a bound d3=2 when the shell is filled. Adding pre- 14O the NO2B approximation is found to be independent of !SRG. 400 keV, again within the estimated uncertainty of our existing 3NFs—the full Hamiltonian—raises this orbit many-body truncation scheme. The c.m. correction14 inO For Ã3N 350 MeV=c we do not expect significant -100 NN+3N-ind. above the continuum also for the highest masses. This ¼ Eq. (10) is crucial to obtain this agreement.NN+3N-full For ! gives a first principle confirmation of the repulsive effects induced 4N interactions [27]. As !SRG is reduced, we 1 15¼ 24 MeV and !SRG 2:0 fm16À , the discrepancy in@ O of the two-pion exchange Fujita-Miyazawa interaction capture additional repulsive 3N strength in matrix elements 23-120 ¼ O

([MeV] O) is 1.65 MeV (1.03 MeV) when neglecting the 18O 16O discussed in Ref. [3]. The consequences of this trend are with e e e E . We also speed up the conver- E 1 2 3 3 max changes in kinetic energy of20O the c.m. but it reduces to demonstrated by the calculated ground state energies þ þ  -140 18O gence of the many-body expansion and reduce the error due only 190 keV (20 keV) when22O this is accounted for. This 20 shown in the bottom panel and in Fig. 4: the induced gives us confidence that a proper24O separation of the center ofO to the MR-IM-SRG(2) truncation, but for the resolution 26 22 Hamiltonian systematically under binds the whole isotopic -160 O O scales considered here, this effect is already saturated. In mass motion is being reached. 26O chain and erroneously places the drip line at 28O due to the Figure 2 also gives a first remarkable demonstration24 ofO total, we find a slight artificial increase of the ground-state lack of repulsion in the d3=2 orbit. The contribution from the predictive2468 10 power12 14 16of18 chiral2462N 83N 10 12interactions: 14 16 18 N N full 3NFs increase with the mass number up to 24O, when energies as we lower !SRG [13]. max þ max accounting for the precision of our many-body approach the unbound d orbit starts being filled. Other bound For our standard choice Ã3N 400 MeV=c, effects and dependence on ! found in Ref. [28], we expect an 3=2 ¼ FIG. 3 (color online). IT-NCSMSRG ground-state energies of thequasihole states are lowered resulting in additional overall from omitted 4N interactions, the E cut, and the accuracy of at least 5% on binding energies. All calculated 3 max even oxygen isotopes for the NN 3N-induced (a) and NN binding. As a result, the inclusion of NNLO 3NFs consis- many-body truncation cancel, and the !SRG dependence values agree with the experimentþ within this1 limit. Note þ 3N-full Hamiltonians (b) at !SRG 1:88 fmÀ . Solid lines in-tently brings calculations close to the experiment and of the energies in Fig. 2 is extremely weak [13]. The that the interactions employed were¼ only constrained by 24 dicate the energy3 4 extrapolation based on Nmax 8–12 data;reproduces the observed dripline at O [41–43]. Our cal- 2N and H and He data. ¼ 25 omission of 4N interactions becomes the dominant source dotted lines guide the eye for smaller Nmax. Uncertainties dueculations predict O to be particle unbound by 1.54 MeV, of uncertainty as we increase Ã3N to 450 MeV=c, resulting to the importance truncation are smaller than the symbols used tolarger than the experimental value of 770 keV [44] but in an enhanced !SRG dependence of the ground-state ener- represent the data. All energies are obtained at optimal . within the estimated errors. The ground state resonance for 6 28 gies of the heavier oxygen isotopes. This is consistent with 2N 3N ind @ O is suggested to be unbound by 5.2 MeV with respect to 4 1d3 2 2N 3N full 24O. However, this estimate is likely to be affected by the the even stronger !SRG dependence for Ã3N 500 MeV=c ¼ demonstration2 of the predictive power of current chiralpresence of the continuum which is important for this observed in Refs. [23,26,27]. NN 3N Hamiltonians, at least for ground-state energies.nucleus but neglected in the present work. To assess the quality of our MR-IM-SRG(2) ground- 0 þ MeV The same mechanism affects neighboring isotopic For further confirmation, we2s perform CC calculations with state energies, we compare them to results from the 1 2 1 2 chains. This is demonstrated in Fig. 4 for the semimagic A Comparison withsingles Large-Space andi doubles (CCSD), Methods as well as perturbative triples IT-NCSM, which yields the exact NCSM results within [Ã-CCSD(T)]4 [15,22,34,35] for oxygen isotopes with sub-odd-even isotopes of nitrogen and fluorine. Induced 3NF forces consistently under bind these isotopes and even quantified uncertainties from the importance truncation shell closures.6 Using the same1d5 2 Hamiltonians in the NO2B [26,32]. In the IT-NCSMLarge-space calculations, methods we with use thesame full SRG-evolved NN+3N forces predict a 27N close in energy to 23N. This is fully cor- approximation,8 the MR-IM-SRG energies are bracketed 23 3N interaction without the NO2B approximation, and the rected by full 3NFs that strongly bind N with respect to 4 -120 by the CC60 results, and similar to the Ã-CCSD(T) values,27N, in accordance with the experimentally observed drip E3 max cut is naturally compatible with the IT-NCSM 24 MeV (a) (b) consistent with the closed-shell results1 discussed in Ref. [13].line. The repulsive effects of filling the d3=2 is also model-space truncation-140 [13]. In Fig. 3 we show the 80 SRG 2.0 fm d3/2 0 convergence of the oxygen ground-state energies for the 13 15 21 23 27 -160 100 N N N N N s NN 3N-induced and NN 3N-full Hamiltonians as a -50 1/2 80 24 MeV þ þ MeV 120 2N 3N ind 1 . function of N , along-180 with exponential fits which ex- s 2.0 fm max . -75 SRG -4 g 2N 3N full 100

26 E trapolate Nmax [26,32,33]. With the exception of O, 140 Exp d !1 -200 -100 5/2 all isotopes converge well, and theExp. uncertainties of the 120 MeV Energy (MeV) 160 MeV

-8 threshold and model spaces truncationsNN of the IT-NCSM -125 . s

-220 . 140 26 g NN+3N-indresults are typically about 1 MeV.NN+3N-ind For O, the rate of E -150180 2N 3N full NN+3N-full 14O 16O 22O 24O 28O 160 Single-Particle Energy (MeV) NN+3N-full -240 2N 3N ind convergence is significantly worse, which is expected due (a) NN+3N-induced -12 Cipollone, Barbieri, Navratil, PRL (2013) Exp to the resonance nature of this ground state. -175 180 16 18 20 22 24 26 28 16 18 20 22 24 26 28 FIG. 3 (color online). Top. Evolution of single particle ener- The neutron-rich oxygen isotopes are the heaviest nuclei -50 MR-IM-SRG(2) Mass Number A Mass Number A gies for neutron addition and removal around sub-shell closures 15F 17F 23F 25F 29F studied so far in the IT-NCSM with full 3N interactions. For of oxygen isotopes. Bottom. Binding energiesIT-NCSM obtained from the 26 -75 CCSD O, the computation of the complete Nmax sequence shown Koltun SR and the poles of propagator (1), compared to experi- FIG. 4 (color online). Binding energies of odd-even nitrogen ment (bars) [44,46,47]. All points are corrected-CCSD(T) for the kinetic and fluorine isotopes calculated for induced (red squares) in Fig. 3 requiresClear about 200 improvement 000 CPU hours. with In contrast, full NN+3N a -100 energy of the c.o.m. motion. For all lines, red squares (blue dots) and full (green dots) interactions. Experimental data are from corresponding sequence of single-particle basis sizes in the MeV refer to-125 induced (full) 3NFs. [45–47]. MR-IM-SRG requiresConfirms only about valence-space 3000 CPU hoursresults on a

-150 comparable system. Overall, the method scales polynomially 062501-4 with O N6 to largerRemarkable basis sizes N agreement, which makes with it ideally same forces (b) NN+3N-full ð Þ -175 suited for the description of medium- and heavy-mass nuclei. 12 14 16 18 20 22 24 26 In Fig. 4, we compare the MR-IM-SRG(2) and A IT-NCSM ground-state energies of the oxygen isotopes, for Hergert et al., PRL (2013) the NN 3N-induced and NN 3N-full Hamiltonians FIG. 4 (color online). Oxygen ground-state energies for the þ 1 þ NN 3N-induced (a) and NN 3N-full (b) Hamiltonian with with !SRG 1:88 fmÀ to experiment. For the latter, the þ þ ¼ Ã3N 400 MeV=c. MR-IM-SRG(2), CCSD, and Ã-CCSD(T) overall agreement between the two very different many-body results¼ are obtained at optimal , using 15 major oscillator approaches and experiment is striking: Except for slightly shells and E3 max 14. The IT-NCSM@ energies are extrapolated larger deviations in 12O and 26O, we reproduce experimental to infinite model¼ space. Experimental values are indicated by binding energies within 2–3 MeV. This is a remarkable black bars [28,36].

242501-4 week ending PRL 111, 062501 (2013) PHYSICAL REVIEW LETTERS 9 AUGUST 2013

(Fig. 1) is over bound at 130:8 1 MeV but in close Figures 3 and 4 collect our results for the oxygen, nitro- agreement with the 130À:5 1 MeVð Þ obtained from gen and fluorine isotopes calculated with ! 24 MeV À ð Þ 1 ¼ IM-SRG [11], giving further confirmation of the accuracy and !SRG 2:0 fmÀ . The top panel of Fig.@ 3 shows the achieved by different many-body methods. Note that the predicted¼ evolution of neutron single particle spectrum energies of 15O and 23O can be obtained in two different (addition and separation energies) of oxygen isotopes in ways, from either neutron addition or removal on neigh- the sd shell. Induced 3NFs reproduce the overall trend but boring subshell closures. Results in Fig. 2 differ by at most predict a bound d3=2 when the shell is filled. Adding pre- 400 keV, again within the estimated uncertainty of our existing 3NFs—the full Hamiltonian—raises this orbit many-body truncation scheme. The c.m. correction in above the continuum also for the highest masses.week ending This Eq.PRL (111,10) is062501 crucial (2013) to obtain this agreement.PHYSICAL For ! REVIEWgives a LETTERS first principle confirmation of the repulsive9 AUGUST effects 2013 1 15¼ 24 MeV and !SRG 2:0 fmÀ , the discrepancy in@ O of the two-pion exchange Fujita-Miyazawa interaction ((Fig.23O)1 is) is 1.65 over MeV bound¼ (1.03 at MeV)130:8 when1 MeV neglectingbut in close the Figures 3 and 4 collect our results for the oxygen, nitro- À ð Þ discussed in Ref. [3]. The consequences of this trend are agreementchanges in kinetic with the energy130 of:5 the1 c.m.MeV butobtained it reduces from to demonstratedgen and fluorine by isotopes the calculated calculated ground with state! 24 energies MeV À ð Þ 1 ¼ IM-SRGonly 190 [ keV11], giving (20 keV) further when confirmation this is accounted of the accuracyfor. This shownand !SRG in the2:0 bottom fmÀ . panel The top and panel in Fig. of Fig.4@: the3 shows induced the achievedgives us confidence by different that many-body a proper separation methods. of Note the center that the of predicted¼ evolution of neutron single particle spectrum 15 23 Hamiltonian systematically under binds the whole isotopic energiesmass motion of isO beingand reached.O can be obtained in two different (additionchain and and erroneously separation places energies) the drip of line oxygen at 28O isotopesdue to the in ways,Figure from2 also either gives neutron a first addition remarkable or removal demonstration on neigh- of thelacksd ofshell. repulsion Induced in the 3NFsd reproduceorbit. The the contribution overall trend from but boring subshell closures. Results in Fig. 2 differ by at most 3=2 the predictive power of chiral 2N 3N interactions: predictfull 3NFs a bound increased3= with2 when the the mass shell number is filled. up to Adding24O, when pre- 400 keV, again within the estimatedþ uncertainty of our accounting for the precision of our many-body approach theexisting unbound 3NFs—thed orbit full starts Hamiltonian—raises being filled. Other this bound orbit many-bodyand dependence truncation on ! scheme.found in The Ref. c.m. [28], correction we expect an in 3=2 SRG abovequasihole the states continuum are lowered also for resulting the highest in additional masses. overall This accuracyEq. (10) of is at crucial least 5% to obtainon binding this energies. agreement. All For calculated! gives a first principle confirmation of the repulsive effects 1 15¼ binding. As a result, the inclusion of NNLO 3NFs consis- values24 MeV agreeand with!SRG the2 experiment:0 fmÀ , the within discrepancy this limit. in@ NoteO of the two-pion exchange Fujita-Miyazawa interaction 23 ¼ tently brings calculations close to the experiment and (thatO) the is interactions 1.65 MeV employed (1.03 MeV) were when only constrainedneglecting the by discussed in Ref. [3]. The consequences24 of this trend are 3 4 reproduces the observed dripline at O [41–43]. Our cal- 2changesN and inH and kineticHe energydata. of the c.m. but it reduces to demonstrated by25 the calculated ground state energies only 190 keV (20 keV) when this is accounted for. This culations predict O to be particle unbound by 1.54 MeV, shownlarger than in the the bottom experimental panel and value in of Fig. 7704: keV the [ induced44] but gives us confidence that a proper separation of the center of Hamiltonian systematically under binds the whole isotopic mass motion is being reached. within the estimated errors. The ground state28 resonance for 6 28chain and erroneously places the drip line at O due to the 2N 3N ind O is suggested to be unbound by 5.2 MeV with respect to Figure 2 also gives a first remarkable demonstration of lack of repulsion in the d orbit. The contribution from 4 1d3 2 2N 3N full 24O. However, this estimate3=2 is likely to be affected by the the predictive power of chiral 2N 3N interactions: full 3NFs increase with the mass number up to 24O, when accounting2 for the precision of our many-bodyþ approach presence of the continuum which is important for this thenucleus unbound but neglectedd3=2 orbit in thestarts present being work. filled. Other bound and dependence0 on !SRG found in Ref. [28], we expect an

MeV quasiholeThe same states mechanism are lowered affects resulting neighboring in additional isotopic overall accuracy of at least 5% on2 bindings energies. All calculated 1 2 1 2 chains.binding. This As a is result, demonstrated the inclusion in Fig. of NNLO4 for the 3NFs semimagic consis- A valuesi agree with the experiment within this limit. Note tently brings calculations close to the experiment and Driplinethat the interactions4 Mechanism employed were only constrained by odd-even isotopes of nitrogen and fluorine. Induced 3NF reproduces the observed dripline at 24O [41–43]. Our cal- 2N and 36H and 4He data.1d forces consistently under bind these isotopes and even 5 2 culations predict27 25O to be particle23 unbound by 1.54 MeV, Compare to large-space methods with same SRG-evolved NN+3N forces predict a N close in energy to N. This is fully cor- 8 larger than the experimental value of23 770 keV [44] but 4 0 rected by full 3NFs that strongly bind N with respect to 60 within27N, in the accordance estimated with errors. the The experimentally ground state observed resonance drip for -10 6 28 24 MeV 2N 3N ind O is suggested to be unbound by 5.2 MeV with respect to d 1 line. The repulsive effects of filling the d3=2 is also 3/2 -20 80 SRG 2.0 fm 24 0 4 1d3 2 2N 3N full O. However, this estimate is likely to be affected by the -30 presence of the continuum which is important for this s 1002 13N 15N 21N 23N 27N 1/2 -40 nucleus but neglected in the present work. 0 80 24 MeV MeV -4 MeV -50 120 2N 3N ind The same mechanism affects1 neighboring isotopic .

s 2s 1 1 2 2.0 fm . 2 SRG d -60g 2N 3N full chains. This is demonstrated in Fig. 4 for the semimagic A i 100 5/2 E Exp. Exp

Energy (MeV) 140 -70 4 NN odd-even isotopes of nitrogen and fluorine. Induced 3NF -8 120 -80 6 NN+3N-ind forces consistently under bind these isotopes and even NN+3N-ind 1d5 2 MeV 160 27 23 NN+3N-full . predicts a N close in energy to N. This is fully cor- -90 . 140 Single-Particle Energy (MeV) NN+3N-full 8 g 23 E -12 -100180 rected by full 3NFs that strongly bind N with2N 3N respectfull to 16 18 20 22 24 26 28 1660 1814O 2016O 22 24 2622O 2824O 28O 27N, in160 accordance with the experimentally2N observed3N ind drip Exp Mass Number A Bogner et al., PRL (2014)Mass Number24 MeVCipollone, A Barbieri, Navratil, PRL line.(2013) The repulsive effects of filling the d is also 2.0 fm 1 180 3=2 FIG. 3 (color80 online). Top.SRG Evolution of single particle ener- gies for neutron addition and removal around sub-shell closures 15 17 23 25 29 100 F 13NF 15N F 21NF 23N F 27N Robust mechanism driving driplineof oxygen behavior isotopes. Bottom. Binding energies obtained from the 80 24 MeV KoltunMeV SR and the poles of propagator (1), compared to experi- FIG. 4 (color online). Binding energies of odd-even nitrogen 120 2N 3N ind 1 .

s 2.0 fm ment. (bars) [44,46,47]. All points are corrected for the kinetic and fluorine isotopes calculatedSRG for induced (red squares) 3N repulsion raises d3/2, lessens decreaseg across shell 2N 3N full 100

E energy of140 the c.o.m. motion. For all lines, redExp squares (blue dots) and full (green dots) interactions. Experimental data are from Similar to initial MBPT NN+3Nrefer calculations to induced (full) in 3NFs. oxygen [45–47].120

160 MeV . s

. 140 g 180 062501-4 E 2N 3N full 14O 16O 22O 24O 28O 160 2N 3N ind Exp 180 FIG. 3 (color online). Top. Evolution of single particle ener- gies for neutron addition and removal around sub-shell closures 15F 17F 23F 25F 29F of oxygen isotopes. Bottom. Binding energies obtained from the Koltun SR and the poles of propagator (1), compared to experi- FIG. 4 (color online). Binding energies of odd-even nitrogen ment (bars) [44,46,47]. All points are corrected for the kinetic and fluorine isotopes calculated for induced (red squares) energy of the c.o.m. motion. For all lines, red squares (blue dots) and full (green dots) interactions. Experimental data are from refer to induced (full) 3NFs. [45–47].

062501-4 IM-SRG Oxygen Spectra Oxygen spectra: extended-space MBPT and IM-SRG

+ 8 4

+ 22 + 7 2 + (4 ) 4 + O (2 ) + 2 6 + + 3 4 + 2 + + 0 (0 ) 5 + + + 3 0 + 3 0+ 4 3 + 2 + 2 + 3 2 Energy (MeV) + 2 2 JDH, Menendez, Schwenk, EPJ (2013) 1 Bogner et al., PRL (2014)

+ + + + 0 0 0 0 0 MBPT IM-SRG IM-SRG Expt. NN+3N-ind NN+3N-full

Clear improvement with NN+3N-full

IM-SRG: comparable with phenomenology IM-SRG Oxygen Spectra Oxygen spectra: extended-space MBPT and IM-SRG

+ 5 3/2 23 O + 4 (3/2 )

+ + 3/2 3/2 3 + + + 5/2 5/2 (5/2 )

+ 5/2 2 Energy (MeV)

1 JDH, Menendez, Schwenk, EPJ (2013) Bogner et al., PRL (2014) + + + + 0 1/2 1/2 1/2 1/2 MBPT IM-SRG IM-SRG Expt. NN+3N-ind NN+3N-full Similar CC valence-space results: G. Jansen et al., arXiv:1402.2563

Clear improvement with NN+3N-full

Continuum neglected: expect to lower d3/2 IM-SRG Oxygen Spectra Oxygen spectra: IM-SRG predictions beyond the dripline

7 24 25 26 + O O O 6 1 + + 2 + 1 5/2

5 + 2 + 1/2 4

3

Energy (MeV) 2 + 2 1 Bogner et al., PRL (2014)

+ + + + 0 0 0 3/2 0 Exp. 24 + O closed shell (too high 2 )

Continuum neglected: expect to lower spectrum

Only one excited state in 26O below 6.5MeV Experimental Connection: 26O Spectrum Oxygen spectra: IM-SRG predictions beyond the dripline

7 24 25 26 + O O O 6 1 + + 2 + 1 5/2

5 + 2 + 1/2 4

3

Energy (MeV) 2 + 2 Kondo et al., PRELIMINARY 1 Bogner et al., PRL (2014)

+ + + + 0 0 0 3/2 0 Exp.

New measurement at RIKEN on excited states in 26O

Existence of excited state “just over 1.0 MeV” (uncertainty not finalized) Towards Full sd-Shell with MBPT: Fluorine Next challenge: valence protons + neutrons

Neutron-rich fluorine and neon

Fluorine: at least 6 additional neutrons

sd shell filled at 29F/30Ne

Need extended-space orbits Towards Full sd-Shell with MBPT: Fluorine Next challenge: valence protons + neutrons

Neutron-rich fluorine and neon 0 0

F -20 Ne -20 -40 -40 -60

-60 -80

Energy (MeV) -100 -80 NN NN NN+3N -120 NN+3N SDPF-M SDPF-M -100 -140 16 20 24 28 32 36 20 24 28 32 36 Mass Number A Mass Number A JDH, Menendez, Simonis, Schwenk, in prep. NN only: severe overbinding

NN+3N: good experimental agreement through 29F

Sharp increase in ground-state energies beyond 29F: incorrect dripline Towards Full sd-Shell with MBPT: Neon Next challenge: valence protons + neutrons

Neutron-rich fluorine and neon 0 00 0

F -20 FNe -20 Ne -20 -20 -40 -40 -40 -40 -60 -60

-60 -80-60 -80

Energy (MeV) Energy (MeV) -100 -100 -80 NN -80 NNNN NN NN+3N -120 NN+3NNN+3N -120 NN+3N SDPF-M SDPF-MSDPF-M SDPF-M -100 -100 -140 -140 16 20 24 28 32 36 16 2020 24 24 28 28 32 32 3636 20 24 28 32 36 Mass Number A MassMass NumberNumber AA Mass Number A JDH, Menendez, Simonis, Schwenk, in prep. Similar behavior in Neon isotopes

Revisit cross-shell valence space theory – non-degenerate valence spaces

Tsunoda, Hjorth-Jensen, Otsuka IM-SRG energies overbound in F/Ne Experimental Connection: 24F Spectrum Fluorine spectra: extended-space MBPT and (sd-shell) IM-SRG

+ 4 4 24 + + (2 ) 2 + + F + (4 ,2 ) 3.5 2+ 1 + 4 + 3 + 3 + 3 + (3 ) + 0 0 + + 4 + 2.5 4 (4 )

+ 2 1 + 1 + + 1 1 1.5 + 0 Energy (MeV) 1

+ + + 2 2 0.5 2+ + 3 2 + + + + 0 1 3 3 3 Caceres et al., in prep. MBPT USDb IM-SRG Expt. NN+3N-full

New measurements from GANIL

IM-SRG: comparable with phenomenology in good agreement with new data 25Ne Spectrum Neon spectra: extended-space MBPT and (sd-shell) IM-SRG

+ 4 7/2 25 + + 7/2 9/2 Ne + + 9/2 3/2 + 3/2+ 5/2 + 5/2 3 + 3/2 + 1/2 + 7/2

+ + + 3/2 3/2 (3/2 ) 2 + 5/2 + + + 5/2 5/2 (5/2 )

Energy (MeV) + 1 5/2 + 3/2

+ + + + 1/2 1/2 1/2 (1/2 ) 0 Bogner, Hergert, JDH, Schwenk, in prep MBPT USDb IM-SRG Expt. NN+3N-full

Limited experimental data

IM-SRG: comparable with phenomenology, good agreement with data Limits of Nuclear Existence: Oxygen Anomaly Where is the nuclear dripline? Limits defined as last isotope with positive neutron separation energy - Nucleons “drip” out of nucleus Neutron dripline experimentally established to Z=8 (Oxygen)

Regular dripline trend… except anomalous oxygen Adding one proton binds 6 additional neutrons Single Particle Energies SPEs self-consistently from one-body diagrams

+ 3rd order

sd-shell: overbound, unreasonable spacing

Orbit “Exp” USDb rd T+VNN (3 )

d5/2 -4.14 -3.93 -5.43

s1/2 -3.27 -3.21 -5.32

d3/2 0.944 2.11 -0.97

Typical approach: use empirical SPEs

3N forces eliminate need for adjusted parameters? One-Body 3N: Single Particle Energies NN-only microscopic SPEs yield poor results – rely on empirical adjustments

sd-shell: SPEs much too bound, unreasonable splitting 3N forces: additional repulsion – comparable to phenomenology

Orbit USDb T+VNN+V3N

d5/2 -3.93 -3.78

s1/2 -3.21 -2.42

d3/2 2.11 1.45 c dc c Vlow-kd dc Vlow-kc d dc d Qˆ = Qˆ = + + + ... + ... a ba a b ba a b ba b c dc c d dc d + + +Single-Particle+ Energies with NN+3N Forces a b3Na forcesa b: additionalba repulsionb improves SPEs c cc a c a a d d d d ...... rd + + +[KE + + + + + 3 + 3N] = E a a a a a b b a b a b a ∈ valence space

Orbit a USDb SDPF-M € T+VNN T+VNN+V3N T+VNN+V3N € d5/2 -3.93€ -5.43€ -3.78 -3.95 -3.46 s 1/2 -3.21 -5.32 -2.42 -3.16 -2.20

d3/2 2.11 -0.97 1.45 1.65 1.92

f7/2 3.10 3.71

p 3.10 7.72 3/2 JDH, Menendez, Schwenk, EPJA (2013) Similar contributions in standard/extended valence spaces

Comparable with phenomenology Non-Observability of Shell Gaps

SPE evolution with 3N forces in sd and sdf7/2p3/2 spaces: 8 2 p 2 (a) sd-shell 3/2 (b) sdf7/2p3/2 (c) sd-shell d3/2 6 d3/2 0 f 0 4 7/2 s s -2 1/2 2 -2 1/2 d3/2 -4 0 -4 d 5/2 s d -2 1/2 5/2

Energy (MeV) -6 -6 -4 d5/2 -8 -8 USDb NN+3N -6 NN+3N (LS-trans) -10 -8 -10 16 18 20 22 24 26 28 16 18 20 22 24 26 28 16 18 20 22 24 26 28 Mass Number A Mass Number A Mass Number A 4 22 Duguet, Hergert, JDH, Soma, in prep. NN+3NO extended space: No3 gap at 22O, yet enhanced closed-shell features (high 2+, etc.)

Lee-Suzuki2 transformation into sd-shell 22

Energy (MeV) Usual shell gap emerges at O 1 + Similar evolution to USDb2 Hamiltonian d5/2 - s1/2 gap 0 sd sdf7/2p3/2 sd USDb LS-trans Two-body 3N: Monopoles in sd-shell

1 Dominant effect from 0.5 T=1 one-Δ – as expected 0 from cutoff variation -0.5 3N forces produce clear -1 repulsive shift in monopoles V -1.5 low k +3N (Δ)

V(ab;T) [MeV] -2 2 + 3N (N LO) -2.5 USDa -3 USDb -3.5 d5d5 d5d3 d5s1 d3d3 d3s1 s1s1 First calculations to show missing monopole strength due to neglected 3N

Future: Improved treatment of high-lying orbits Treat as holes in 40Ca core Continuum effects Impact on Spectra: 21O Neutron-rich oxygen spectra with NN+3N

Spectrum sensitive to s1/2 shell closure

4 S 21 NN-only n O + 7/2 Low-lying states + (7/2+) 3 + (5/2 ) too compressed + + 5/2 7/2 7/2 + + + 7/2 7/2 -5/2 too wide

+ (3/2 ) 2 Microscopic NN+3N + 5/2 +

+ 3/2 Improvement in sd + 5/2 5/2 + Energy (MeV) (1/2 ) Extended orbits essential 1 + + 1/2 3/2 Improved spacing in all

+ + 3/2 + levels 3/2 1/2 + + + + + 1/2+ 1/2+ 5/2 5/2 5/2 0 5/2 5/2 NN NN NN+3N NN+3N Expt. sd sdf7/2p3/2 sd sdf7/2p3/2 JDH, Menendez, Schwenk, EPJ (2013)

3N improvements largely due to higher calculated s1/2 orbital

Need proper treatment of non-degenerate spaces Impact on Spectra: 22O Neutron-rich oxygen spectra with NN+3N 22O: N=14 new magic number – not reproduced with NN

+ 8 4 22 NN-only O + + 7 + (4 ) 2 + 2 too low S (2 ) n Spectrum too compressed 6 + + 3 4+ + + 3 4 3 + + 4 2 + + 5 + 2 (0 ) 2+ + + + 3 0 3 0

+ Microscopic NN+3N 4 + 0 2 + + 0 2 Extended space essential 3 + + 2 Energy (MeV) 2 + Reproduces N=14 magic 2 2 number

1

+ + + + + 0 0 0 0 0 0 NN NN NN+3N NN+3N Expt. sd sdf7/2p3/2 sd sdf7/2p3/2 JDH, Menendez, Schwenk, EPJ (2013)

Contributions from 3N and extended valence orbitals important Impact on Spectra: 23O Neutron-rich oxygen spectra with NN+3N

5/2+, 3/2+ energies reflect 22,24O shell closures 5 23 sd-shell NN only O + + Wrong ground state 4 3/2 (3/2 ) + + 5/2 too low S 3/2 n + + 3 5/2 + 3/2 bound (5/2 )

+ 3/2 NN+3N 2 + 3/2 Clear improvement in

+ Energy (MeV) 1 5/2 extended valence space + 5/2 + + + + + 0 1/2 1/2 1/2 1/2 1/2

+ 5/2 -1 NN NN NN+3N NN+3N Expt. sd sdf7/2p3/2 sd sdf7/2p3/2 JDH, Menendez, Schwenk, EPJA (2013) Coupled Cluster Benchmark Benchmark against ab-initio Coupled-Cluster Theory

SPEs: one-particle attached CC energies in 17O and 41Ca

Oxygen Calcium 0 0 0 -10 NN only (b) NNCC/MBPTNN only (c) NN + 3N -10 -50 -20 -20 -30 -100 -40 -30 -150 -50 -40 -60 -50 -200 Energy [MeV] Energy (MeV) -70 CC sd-shell (2nd)CC USDb -60 -250 -80 MBPT sd-shell (3rd)MBPT sd-shell -70 sdf p -shell sdf7/2p3/2-shell -90 -300 7/2 3/2 -100 -80 16 18 20 22 24 26 28 16 18 2040 2244 244826 5228 1656 1860 20 22 24 26 28 Mass Number A Mass NumberMass A Number A Mass Number A

Energies relative to 16O and 40Ca

Small difference in many-body methods ~5%

Explore further benchmarks: NCSM, IM-SRG… IM-SRG vs. MBPT: Monopoles

Monopoles: angular average of interaction

Determines interaction of orbit a with b; evolution of orbital energies 1 (2J +1) V JT 0.5 T=1 T ∑J abab Vab = 0 (2J +1) ∑J -0.5 -1 -1.5 MBPT [VSRG]

V(ab;T) [MeV] -2 € IMSRG -2.5 IMSRG (HF Int) USDb -3 -3.5 d5d5 d5d3 d5s1 d3d3 d3s1 s1s1 Similar trends in MBPT/IM-SRG – attractive shift in IM-SRG

Promising trends for IM-SRG in HF basis using intrinsic Hamiltonian Perturbative vs. Nonperturbative SPEs 3N forces: additional repulsion improves SPEs

Orbit USDb MBPT MBPT IM-SRG IM-SRG IM-SRG

NN NN+3N NN NN+3N-ind NN+3N-full

d5/2 -3.93 -5.43 -3.78 -7.90 -3.77 -4.62 s1/2 -3.21 -5.32 -2.42 -6.87 -2.46 -2.96

d3/2 2.11 -0.97 1.45 1.41 2.33 3.17 JDH, Menendez, Schwenk, EPJA (2013) Bogner et al., PRL (2014)

Similar contributions in standard/extended valence spaces

Comparable with phenomenology Towards Full sd-Shell: Fluorine Next challenge: valence protons + neutrons

Explore physics of neutron-rich fluorine and neon isotopes

Prediction of unbound 28,30F

Agrees well with phenomenology

SDPF-U NN+3N AME 2011 S (28F) n -451 keV -458 keV -419 keV JDH, Menendez, Schwenk, in prep. Non-Degenerate Valence Space Theory Strong non-Hermiticity (or divergence) for non-degenerate valence space:

1 1

ε d −ε b −ε p −ε h εa −εc −ε p −εh

€ Idea: replace starting energy with arbitrary parameter€ ω = PH0P → E 1 In both cases: avoids divergence and non-Hermiticity E − (εc +ε p +εb −εh ) € Final results should not depend on E – true when folded diagrams are calculated

Required€ for use of Hartree-Fock basis

Test calculations show more attractive cross-shell monopoles

Lower f7/2 p3/2 orbitals for neutron-rich F, Ne

Backup Calcium Evolution of Shell Structure

SPE evolution with 3N forces in pf and pfg9/2 spaces:

(a) Phenomenological Forces (b) NN-only Theory (c) NN + 3N (d) NN + 3N (pfg9/2 shell) 0 f5/2 f f5/2 5/2 g9/2 p f 1/2 p p 5/2 1/2 1/2 p -5 1/2 p p 3/2 3/2 p p3/2 3/2

f f7/2 -10 f7/2 7/2 f7/2 G G + 3N() KB3G 2 2 V V + 3N(N LO) V + 3N(N LO) [MBPT] Single-Particle Energy (MeV) GXPF1 low k low k low k -15 40 44 48 52 56 60 40 44 48 52 56 60 40 44 48 52 56 60 40 44 48 52 56 60 Mass Number A Mass Number A Mass Number A Mass Number A NN+3N pf-shell: JDH, Otsuka, Schwenk, Suzuki JPG (2012) Trend across: improved binding energies Increased gap at 48Ca: enhanced closed-shell features

Include g9/2 orbit, calculated SPEs Different behavior of ESPEs (not observable, model dependent)

+ Small gap can give large 2 energy: due to many-body correlations Duguet, Hagen, PRC (2012) Duguet, Hergert, JDH, Soma, in prep. Two-body 3N: Monopoles in pf-shell

0.4 Dominant effect from T=1 one-Δ – as expected 0.2 from cutoff variation 0 3N forces produce clear -0.2 repulsive shift in monopoles -0.4 Similar to sd-shell V(ab;T) [MeV] -0.6 GXPF1 KB3G V -0.8 low k +3N(Δ) 2 -1 +3N(N LO) f7f7 f7p3 f7f5 f7p1 p3p3 f5p3 p3p1 f5f5 f5p1 p1p1

First calculations to show missing monopole strength due to neglected 3N " #& #H$ !H' ## !H$

86F-;G3896:; $H'

#$ #= $H$ !* E$H' E!H$ 8@56A;3E3< !( #= E!H' D< E#H$ !& %$ %! %# %% %& >6?@/A=3=?,B6/3! 3896:;

#= !# !$

* +F-6/.,6=@ J>P%>389QCM; MR-TOF Measurements & RR38S176=36@312H; 54 ExperimentalTQ%U Connection:Implementation Mass of multi-reflection of timeCa-of-flight mass separator (MR-TOF MS) # UVCW!O opened a wide range of possibilities 53,54 New #*precision #" %$ mass %! %# measurement %% %& %' %( of %) %* Ca at ISOLTRAPSupport Penning-trap: multi-reflectionmass spectrometry on fast time ToF scales >6?@/A=3=?,B6/3! MR-TOF plus detector as a stand-alone system ! #& #H$ !H' ## !H$ $H' #$ 86F-;G3896:; #= $H$ !* E$H' E!H$ 8@56A;3E3<

!( #= E!H' D< E#H$ !& %$ %! %# %% %& 52Ca >6?@/A=3=?,B6/3! Versatile tool allows for:

3896:; < ISOLTRAP Higher contamination Measurement yield – 106:1

#= !# lower production yield – 10/s52 Sharp decrease past Ca !$ lower half-lives – tens of ms +F-6/.,6=@ 52

* J+YW$ R. N. Wolf et al., NIM A 686, 82 (2012) 21 m / m 310 ( X>+YW! Test predictions of various models

SWQ#! & 6?@/A=3=?,B6/3! # Closed-shell features not reproduced !I#* Wienholtz et al., Nature (2013) 48,52 * !I%# for Ca J

(

'

&

% +,-./.012345622371-3896:; < #

!" #$ #! ## #% #& #' #( #) #* #" %$ %! C/A@A=3=?,B6/3"# 0 (a) Phenomenological Forces (b) NN-only Theory -30

-60

-90

-120 Experiment G Energy (MeV) Calcium Ground State Energies and Dripline V Extrapolation low k Ground-150 stateGXPF1 energies using NN+3N G [SPE_KB3G] KB3G V [SPE_KB3G] 46 low k NN-only-180 : overbinds beyond ~ Ca 40 44 48 52 56 60 40 44 48 52 56 60 0

(c) NN + 3N (d) NN + 3N (pfg9/2 shell) -30

-60

-90 V (NN) -120 low k Energy (MeV) G + 3N() Vlow k (NN) V + 3N() 2 -150 low k + 3N(N LO) 2 2 Vlow k + 3N(N LO) + 3N(N LO) [MBPT] -180 40 44 48 52 56 60 40 44 48 52 56 60 64 68 Mass Number A Mass Number A Holt, Otsuka, Schwenk, Suzuki, JPG (2012) pf-shell: 3N forces correct binding energies; good experimental agreement

pfg -shell: calculate to 70Ca; modest overbinding near 52Ca 9/2 Heaviest calcium isotope ~ 58-60Ca; flat behavior past 54Ca Pairing in Calcium Isotopes: Ladders (3) Compare with Δ n calculated from microscopic NN+3N in calcium

HFB iterates ladders microscopically in pairing channel EDF Compare with pp, hh ladders to 3rd order € Improved agreement with experiment

Convergence in order-by-order ladders

Suppression from 3N forces as in EDF

Incorrect odd/even staggering

JDH, Menendez, Schwenk, in prep

3 (a) 1st order (b) + 2nd-order ladders (c) + 3rd-order ladders 2.5 2

(MeV) 1.5 (3) n

1 NN 0.5 NN+3N AME2003 0 TITAN AME2003+TITAN 40 44 48 52 56 40 44 48 52 56 40 44 48 52 56 60 Mass Number A Mass Number A Mass Number A N=28 Magic Number in Calcium: Spectrum

+ + 0 0 + 7 + + 2 48 0 3 + + + 3 Ca 2 2+ + 2 4 + + + 4 2 6 5 + + 0 + + + 2 0 5 + 0+ + + 5 5 5 3+ (+) 4 3- + + + 4 3 0 + + 3+ + 3+ 2 2 4+ 4 2 + + 0 + + 3 4 + + 4 4 2 + + 0 + 2 + 4 0 4 + 2 + 2 + 3 + 3+ + 2 4 4 + 3 + 2 0+ + 2 0 + Energy (MeV) 2 2 + 2 + + 0 0 1

+ + + + + + + 0 0 0 0 0 0 0 0 JDH, Menendez, Schwenk, in prep. NN NN NN+3N NN+3N Expt. GXPF1 KB3G pf pfg9/2 pf pfg9/2 Generally spectrum too compressed

pfg -shell: With NN+3N spectrum comparable to phenomenology 9/2 Both 3N and extended space essential Impact on Spectra: 49Ca Neutron-rich calcium spectra with NN+3N 6 49 Ca - - 5 3/2 11/2 - - 11/2- - 9/2 9/2 9/2- - - - 7/2 3/2 3/2- - 3/2- - 7/2 4 3/2 5/2 - 5/2 - - (1,3/2 ) - 5/2 5/2 - 7/2- 5/2 5/2 - - + 5/2 3/2 (9/2 ) - - - - 7/2 3/2 - 3 9/2- - 7/2 5/2 7/2 - 5/2- 7/2- 9/2 5/2 - - - - 7/2 5/2 1/2- 3/2 - - 3/2 5/2 1/2 - 2 - - Energy (MeV) - 1/2 - 3/2 1/2 1/2 5/2- - 7/2- 7/2 5/2 - - - 1/2 7/2 1 3/2- - 1/2 1/2 - 7/2 JDH, Menendez, Schwenk, in prep.

------0 3/2 3/2 3/2 3/2 3/2 3/2 3/2 NN NN NN+3N NN+3N Expt. GXPF1 KB3G pf pfg9/2 pf pfg9/2 Spectrum typically too compressed

NN+3N in pfg correct 1/2- 9/2 Comparable to phenomenology (as for all lighter Ca isotopes) Impact on Spectra: 51Ca Neutron-rich calcium spectra with NN+3N

- 5 7/2- 51 9/2 Ca (?) - (?) - 4 3/2 3/2- (?) - 1/2- 5/2 7/2- - 9/2- 9/2 (?) - 5/2 - 7/2 7/2- 1/2- (?) - 3 - - 3/2 5/2- 11/2 3/2 - 5/2- 1/2 - - 1/2 - 5/2 - 11/2 - 5/2 (5/2 ) - - 3/2 - 5/2 - 5/2- 3/2 - 5/2 1/2 - 3/2 2 - 5/2 - 3/2 - (?) - 3/2 - 1/2 1/2

Energy (MeV) 3/2 - - 5/2 5/2 - - 1/2 5/2- 1 3/2 - 7/2 - - 7/2- - 1/2 1/2 7/2 ------3/2- 3/2 3/2 3/2 3/2 3/2 3/2 0 1/2 NN NN NN+3N NN+3N Expt. GXPF1 KB3G pf pfg pf pfg 9/2 9/2 JDH, Menendez, Simonis, Schwenk, in prep.

Possibility to assign spin/parity where unknown

Gamma-ray spectroscopy needed N=28 Magic Number: M1 Transition Strength B ( M 1 : 0 + → 1 + ) concentration indicates a single particle (spin-flip) transition gs Not reproduced in phenomenology von Neumann-Coesel, et al. (1998)

NN-only: highly fragmented strength, well below experiment

€ 5 (a) pf-shell Experiment

] 4

2 GXPF1 N G µ 3 NN

2 B(M1) [

1

0 6 7 8 9 10 11 12 Excitation Energy (MeV) N=28 Magic Number: M1 Transition Strength B ( M 1 : 0 + → 1 + ) concentration indicates a single particle (spin-flip) transition gs Not reproduced in phenomenology von Neumann-Coesel, et al. (1998)

NN-only: highly fragmented strength, well below experiment

€ 5 (a) pf-shell pf-shell: Experiment

] 4 3N concentrates strength

2 GXPF1 N G µ 3 G + 3N() Peaks below experiment NN 2 NN + 3N B(M1) [

1 JDH, Otsuka, Schwenk, Suzuki, JPG (2012) 0 6 7 8 9 10 11 12 Excitation Energy (MeV) N=28 Magic Number: M1 Transition Strength B ( M 1 : 0 + → 1 + ) concentration indicates a single particle (spin-flip) transition gs Not reproduced in phenomenology von Neumann-Coesel, et al. (1998)

NN-only: highly fragmented strength, well below experiment

€ 5 (a) pf-shell pf-shell: Experiment

] 4 3N concentrates strength

2 GXPF1 N G µ 3 G + 3N() Peaks below experiment NN NN + 3N 2 B(M1) [

1 JDH, Otsuka, Schwenk, Suzuki, JPG (2012) 0 5 (b) pfg9/2-shell pfg9/2-shell: NN

] 4 NN + 3N (emp) 2 3N gives additional concentration N NN + 3N (MBPT) µ 3 Peak close to experimental energy

2 B(M1) [ 1 Supports N=28 magic number 0 6 7 8 9 10 11 12 Excitation Energy (MeV) Effective Operators

Investigate many-body effects on effective charges and quenching of gA

Use low-momentum interactions and 3N forces

Evaluating Center-of-Mass Contamination

LS Apply new H eff to calculate spectra in neutron-rich oxygen 5

23 + 4 O (3/2 ) + 3/2 + € 3/2 + + 3 5/2 3/2 + (5/2 ) + 3/2 + 3/2 2 + 5/2 Energy (MeV) 1 + + 5/2 H = 0 5/2 CM + + + + + + 1/2 1/2 0 1/2 1/2 1/2 1/2 + 5/2 NN+3N NN+3N€ NN+3N Exp sd sd p sd(LS) f7/2 3/2

Clear improvement from standard sd-shell – not due to center of mass

Missing physics involving N > 2 neutrons in extended space

€ Evaluating Center-of-Mass Contamination

LS Apply new H eff to calculate spectra in neutron-rich oxygen

8 + 22 4 O + + 4 7 2 + + 4 2 € 6 + + + 2 + 4 + 3 3 + 4 3 + + + + 0 2 + 5 2 3 + 2 + + + 4+ 3 0 3 0 4 + + + 0 2 0 + + 0 2 + 3 2 + + 2 Energy (MeV) 2 + 2 2

HCM = 0 1 + + + + + + 0 0 0 0 0 0 0 NN+3N NN+3N NN+3N Exp sd sd p sd(LS) f7/2 3/2 € Improvements from standard sd-shell – not due to center of mass

Work in progress: involving N > 2 neutrons in extended space

€ IM-SRG Oxygen Ground-State Energies Valence-space interaction and SPEs from NN+3N in sd-shell

4 0 -10

d3/2 -20 0 -30 s 1/2 -40 -4 -50 -60 d5/2 Exp.

Energy (MeV) -70 -8 NN NN+3N-ind -80 NN+3N-ind -90 NN+3N-full Single-Particle Energy (MeV) NN+3N-full -12 -100 16 18 20 22 24 26 28 16 18 20 22 24 26 28 Mass Number A Mass Number A Hergert, JDH, Bogner, Schwenk, in prep.

Repulsive 3N gives unbound d3/2

Good dripline properties 2! 8!

2! protons 20! 8! 28! neutrons 20! 28! New Directions andNew Directions Outlook 50! 82! 82! 50! 126!

Heavier semi-magic chains: Nickel Tin and 2! 8!

2! protons 20! 8! 28! neutrons 20! 28! New Directions andNew Directions Outlook 50! 82!

82! 50! 126!

Revisit cross-shell theory Next step: Fluorine Neon and Towards full Nuclei with Heavier semi-magic chains: Nickel Tin and 2! 8!

2! protons 20! 8! 28! sd neutrons p/n

20! sd valence degrees of freedom valence degrees of - and - and 28! pf New Directions andNew Directions Outlook

pf -shells -shells 50! 82!

82! 50! 126!

Exotic cores inversion in of Islands near Continuum effects Moving beyond stability Revisit cross-shell theory First step: Fluorine Neon and Towards full Nuclei with Heavier semi-magic chains: Nickel Tin and 2! 8!

2! protons 20! 8! 28! sd neutrons p/n

20! sd valence degrees of freedom valence degrees of - and - and 28! pf New Directions andNew Directions Outlook

pf sd/pf -shells -shells driplines 50! regions

82!

82! 50! 126!

NN+3N interactions for community Exotic cores inversion in of Islands near Continuum effects Moving beyond stability Revisit cross-shell theory First step: Fluorine Neon and Towards full Nuclei with Heavier semi-magic chains: Nickel Tin and 2! 8!

2! protons 20! 8! 28! sd neutrons p/n

20! sd valence degrees of freedom valence degrees of - and - and 28! pf New Directions andNew Directions Outlook

pf sd/pf -shells -shells driplines 50! regions

82!

82! 50! 126!

NN+3N interactions for community Exotic cores inversion in of Islands near Continuum effects Moving beyond stability Revisit cross-shell theory First step: Fluorine Neon and Towards full Nuclei with Heavier semi-magic chains: Nickel Tin and 2! 8!

2! protons 20! 8! 28! sd neutrons p/n

20! sd valence degrees of freedom valence degrees of - and - and 28! pf New Directions andNew Directions Outlook

pf sd/pf -shells -shells driplines 50! regions • • • JDH Lincoln, Kwiatkowski PRC Engel, and JDH

82!

, 82! et Brunner WIMP-nucleus scattering electroweak Effective operators symmetries Fundamental al., PRL Non-empirical calculation of calculationNon-empirical of 50! , et JDH al., [TITAN]PRC 87 References , 064315 (2013) , 064315 110 , 012501 (2013) [LEBIT] (2013) , 012501

126!

0νββ decay

Fundamental Symmetries: 0νββ-Decay What is the nature and mass of the neutrino?

Two essential ingredients: Q-value (experiment) Nuclear matrix element

2 0νββ −1 2 (T1/ 2 ) = G0ν (Qββ ,Z) M0ν mββ

g2 M = f M GT − v M F +! i 0ν 0ν g2 0ν € A

€ 48 76 82 Calculations of M0ν in Ca, Ge, and Se Phenomenological wavefunctions from A. Poves, M. Horoi

48Ca Bare 0.77 76Ge Bare 3.12 82Se Bare 2.73 Effective 1.30 Effective 3.77 Effective 3.62

Lincoln, JDH et al. [LEBIT], PRL (2013) Converged in 13 major oscillator shells JDH and Engel, PRC (2013)

Kwiatkowski, Brunner, JDH et al. [TITAN], PRC (2014)

Overall ~25-30% increase for 76Ge, 82Se; 75% for 48Ca

Promising first steps – up next…

- Consistent wavefunctions and operators from NN+3N

- Improve operator: 2-body currents from chiral EFT

Menendez, Gazit, Schwenk, PRL (2011) - Non-perturbative methods (IM-SRG)

- Effects of induced 3-body operators Shukla, Engel, Navratil, PRC (2011) NN+3N interactions for community Exotic cores inversion in of Islands near Continuum effects Moving beyond stability Revisit cross-shell theory First step: Fluorine Neon and Towards full Nuclei with Heavier semi-magic chains: Nickel Tin and 2! 8!

2! protons 20! 8! 28! sd neutrons p/n

20! sd valence degrees of freedom valence degrees of - and - and 28! pf New Directions andNew Directions Outlook

pf sd/pf -shells -shells driplines 50! regions

82!

82! WIMP-nucleus scattering electroweak Effective operators symmetries Fundamental Ab Non-empirical calculation of calculationNon-empirical of -initio optical model potential modelpotential -initio optical 50! with 126!

J . W. Holt

0νββ decay NN+3N interactions for community Exotic cores inversion in of Islands near Continuum effects Moving beyond stability Revisit cross-shell theory First step: Fluorine Neon and Towards full Nuclei with Heavier semi-magic chains: Nickel Tin and 2! 8!

2! protons 20! 8! 28! sd neutrons p/n

20! sd valence degrees of freedom valence degrees of - and - and 28! pf New Directions andNew Directions Outlook

pf sd/pf -shells -shells driplines 50! regions

82!

82! WIMP-nucleus scattering electroweak Effective operators symmetries Fundamental Beyond for MBPT valence-shell interactions Ab Nonperturbative Non-empirical calculation of calculationNon-empirical of -initio optical model potential modelpotential -initio optical with 50! S. Bogner 126! , H. In-MediumSRG Hergert

, A. 0νββ Schwenk decay

NN+3N interactions for community Exotic cores inversion in of Islands near Continuum effects Moving beyond stability Revisit cross-shell theory First step: Fluorine Neon and Towards full Nuclei with Heavier semi-magic chains: Nickel Tin and 2! 8!

2! protons 20! 8! 28! sd neutrons p/n

20! sd valence degrees of freedom valence degrees of - and - and 28! pf New Directions andNew Directions Outlook

pf sd/pf -shells -shells driplines 50! regions

82!

82! WIMP-nucleus scattering electroweak Effective operators symmetries Fundamental Error analysis Constraints ondensity Neutron drops for clusterCoupled theory Beyond for MBPT valence-shell interactions Ab Nonperturbative Non-empirical calculation of calculationNon-empirical of -initio optical model potential modelpotential -initio optical 50! with G. Hagen wrt 126! In-MediumSRG exact diagonalization ab , T. -initio benchmarking functionals Papenbrock

0νββ decay

NN+3N interactions for community Exotic cores inversion in of Islands near Continuum effects Moving beyond stability Revisit cross-shell theory First step: Fluorine Neon and Towards full Nuclei with Heavier semi-magic chains: Nickel Tin and 2! 8!

2! protons 20! 8! 28! sd neutrons p/n

20! sd valence degrees of freedom valence degrees of - and - and 28! pf New Directions andNew Directions Outlook

pf sd/pf -shells -shells driplines 50! regions

82!

82! WIMP-nucleus scattering electroweak Effective operators symmetries Fundamental Guidance forGuidance Map Error analysis Constraints ondensity Neutron drops for clusterCoupled theory Beyond for MBPT valence-shell interactions Ab Nonperturbative Non-empirical calculation of calculationNon-empirical of -initio optical model potential modelpotential -initio optical driplines 50! in r with -process region wrt sd 126! L. In-MediumSRG - and - and exact diagonalization ab Caballero -initio benchmarking functionals pf

-shell regions-shell , A. 0νββ Arcones decay

NN+3N interactions for community Exotic cores inversion in of Islands near Continuum effects Moving beyond stability Revisit cross-shell theory First step: Fluorine Neon and Towards full Nuclei with Heavier semi-magic chains: Nickel Tin and 2! 8!

2! protons 20! 8! 28! sd neutrons p/n

20! sd valence degrees of freedom valence degrees of - and - and 28! pf New Directions andNew Directions Outlook

pf sd/pf -shells -shells driplines 50! regions

82!

82! WIMP-nucleus scattering electroweak Effective operators symmetries Fundamental Guidance forGuidance Map Error analysis Constraints ondensity Neutron drops for clusterCoupled theory Beyond for MBPT valence-shell interactions Ab Nonperturbative Non-empirical calculation of calculationNon-empirical of -initio optical model potential modelpotential -initio optical driplines 50! in r -process region wrt sd 126! In-MediumSRG - and - and exact diagonalization ab -initio benchmarking functionals pf

-shell regions-shell 0νββ decay

Proton Dripline StudyN=8, 20 step: systems First proton-rich Towardscalculations of all 2! 8!

2! protons 20! 8! 28! neutrons 20! 28! • • isotone JDH, Gallant Menendez , et al., PRL Proton-RichSystems chains chains 50! sd - and and - , Schwenk 113 pf Reference - - Two-protonemission Proton- Keyphysics problems: -shell nuclei: -shell of inversion islands , (2014) ,[TITAN] (2014) Spectra, featuresclosed-shell 82!

82! , PRL

driplines 110 50! , 022052 (2013) (2013) , 022052

126!

Ground-State Energies of N=8 Isotones

0 Data limited – use phenomenological N=8 -2 isobaric multiplet mass equation (IMME)

-4 E(A,T,T ) = E(A,T,−T ) + 2b(A,T)T z z z 2 / 3 -6 b = 0.7068A − 0.9133

-8 NN-only: overbound NN € AME2011 -10 IMME € Ground-State Energy (MeV) -12 16 17 18 19 20 21 22 23 24 Mass Number A JDH, Menendez, Schwenk, PRL (2013) Ground-State Energies of N=8 Isotones

0 Data limited – use phenomenological N=8 -2 isobaric multiplet mass equation (IMME)

-4 E(A,T,T ) = E(A,T,−T ) + 2b(A,T)T z z z 2 / 3 -6 b = 0.7068A − 0.9133

-8 NN-only: overbound NN € NN+3N -10 NN+3N: improved agreement with NN+3N (sdf7/2p3/2) €

Ground-State Energy (MeV) experiment/IMME AME2011 -12 IMME 16 17 18 19 20 21 22 23 24 Mass Number A JDH, Menendez, Schwenk, PRL (2013)

22 Dripline unclear: Si unbound in AME, NN+3N; bound in IMME

Ground-State Energies of N=8 Isotones

0 Data limited – use phenomenological N=8 -2 isobaric multiplet mass equation (IMME)

-4 E(A,T,T ) = E(A,T,−T ) + 2b(A,T)T z z z 2 / 3 -6 b = 0.7068A − 0.9133

-8 NN-only: overbound NN € NN+3N -10 NN+3N: improved agreement with NN+3N (sdf7/2p3/2) €

Ground-State Energy (MeV) experiment/IMME AME2011 -12 IMME 16 17 18 19 20 21 22 23 24 Mass Number A JDH, Menendez, Schwenk, PRL (2013)

22 Dripline unclear: Si unbound in AME, NN+3N; bound in IMME

22 Si possible two-proton emitter IMME NN+3N (sd) NN+3N (sdf p 7/2 3/2 ) S2p Mass measurement needed! 0.01 MeV -1.63 MeV -0.12 MeV N=8 and N=20 Proton SPEs Interaction and self-consistent SPEs from NN+3N sdf p - and pfg -spaces 7/2 3/2 9/2

NN-only: empirical SPEs in sd- and pf-shells only

Orbit Empirical Orbit Empirical T+VNN+V3N T+VNN+V3N

d5/2 -0.60 -0.41 f7/2 -1.07 -0.86

s1/2 -0.10 0.95 p3/2 -0.63 1.40

d3/2 4.40 4.57 p1/2 2.38 3.94

f7/2 - 9.73 f5/2 5.00 5.36

p3/2 - 12.64 g9/2 - 6.40 Spectra of N=8 Isotones

6 18 19 20 21 22 Ne+ Na Mg Al Si 3 + Unpublished; 2 + From I. Mukha 5 + + 0 0 0 + + + + 3 2 3 4+ + 2 (4 ) 4 + 2 + + + 0 + 0 + 5/2+ 4 4 7/2 + 7/2+ + 3 + 3/2 2 9/2 + 5/2 + + 2 2 + 2 + 2 + 1/2 2 + 3/2

+

Excitation Energy (MeV) 1 (1/2 ) + 1/2 + + + + + + + + 5/2+ (3/2+) 0 0 0 3/2 (5/2 ) 0 0 5/2 0 Exp Exp Exp

JDH, Menendez, Schwenk, PRL (2012) NN+3N: reasonable agreement with experiment

New measurement: excited state in 20Mg close to predicted 4+-2+ doublet

Predictions for proton-rich 21Al, 22Si spectra

22 Closed sub-shell signature in Si Ground-State Energies of N=20 Isotones

0 NN-only: overbound beyond 45Mn

N=20 -2

-4

-6

-8 NN Exp and AME2011 extrapolation

Ground-State Energy (MeV) -10 IMME -12 40 41 42 43 44 45 46 47 48 Mass Number A JDH, Menendez, Schwenk, PRL (2013) Ground-State Energies of N=20 Isotones

0 NN-only: overbound beyond 45Mn N=20 -2 NN+3N: close to experiment/IMME

-4

-6

-8 NN NN+3N

NN+3N (pfg9/2)

Ground-State Energy (MeV) -10 Exp and AME2011 extrapolation IMME -12 40 41 42 43 44 45 46 47 48 Mass Number A JDH, Menendez, Schwenk, PRL (2013)

Dripline: Predicted to be 46Fe in all calculations

Expt. NN+3N (pf) 48 NN+3N (pfg9/2 ) Prediction for Ni within S2p -1.28(6) MeV -2.73 MeV -1.02 MeV 300keV of experiment Dossat et al (2005); Pomorski et al (2012) Spectra of N=20 Isotones

42 43 44 45 46 47 48 5 Ti V Cr Mn Fe Co Ni + 4 + 4+ 2 + + 0 4 + 2+ 2+ 6 6 - + + + 6 + 4 4 5/2 6 3 + - 4 + 7/2 4 + - + 3/2 (2 ) - 4 9/2 - - - + 3/2 - 9/2 1/2 2 0 11/2 - + 11/2- 2 + + 3/2 + 2 2 - 2 3/2 - 3/2 - 1 3/2 Excitation Energy (MeV)

- - 5/2 + + 5/2- + - + - + 0 0 0 7/2 0 7/2 0 7/2 0 Exp JDH, Menendez, Schwenk, PRL (2013) 42 NN+3N: reasonable agreement with measured Ti

Predictions for proton-rich spectra

Mirror energy differences with Ca isotopes ~400keV

48 Closed-shell signature in Ni Intermediate-State Convergence Results in 82Se (with phenomenological wavefunctions from A. Poves)

82 Se 3.6

3.55

Matrix Element 3.5 M0ν

3.45 JDH and Engel, PRC (2013) 8 10 12 14 16 18 € Nh

Results well converged in terms of intermediate state excitations

Improvement due to low-momentum interactions

Similar trend in 76Ge and 48Ca Quenching of GT Strength

Investigate many-body effects on quenching of gA

Set of two-body diagrams contributing to renormalization of gA in 2νββ decay: product of 2 single-beta-decay

M2ν

€

82 Quenching of 38% in Se (gA ~ 1.0)

Quenching of GT Strength

Investigate many-body effects on quenching of gA

Calculate GT transitions in sd- and pf-shells

Net quenching between 0.8-0.9

Consistent with findings from 2νββ decay

Larger quenching when spin-orbit partners absent from valence space

Converged using low-momentum interactions – little effect from 3N

Backup Details Renormalization of Nuclear Interactions

Textbook nuclear potentials in r-space - Hard core, long-range one-pion exchange Also treat in momentum space - Strong high-momentum repulsion, coupling of low to high momenta Renormalization of Nuclear Interactions

Evolve momentum resolution scale of chiral interactions from initial Λ χ Remove coupling to high momenta, low-energy physics unchanged Bogner, Kuo, Schwenk, Furnstahl €

AV18

Universal at low-momentum

N3LO

Vlow k(Λ): lower cutoffs advantageous for nuclear structure calculations Chiral Effective Field Theory: 3N Forces Two new couplings at N2LO

LO

NLO c terms given from NN fits: constrained by NN, πN data

cD cE fit to properties of light nuclei: Triton binding energy, 4He radius

N2LO

N3LO Strategy

1) Effective interaction: sum excitations outside valence space to 3rd order

2) Single-particle energies calculated self-consistently

3) Harmonic-oscillator basis of 13 major shells: converged

4) NN and 3N forces – fully to 3rd-order MBPT

5) Work in extended valence spaces

-18.5 -76 -77 -19 42 -78 48 Ca Ca -19.5 -79 1st order -80 -20 2nd order 3rd order -81 Ground-State Energy (MeV) -20.5 -82 2 4 6 8 10 12 14 16 18 2 4 6 8 10 12 14 16 18 _ _ Nh Nh Clear convergence with HO basis size

Promising order-by-order behavior Strategy

1) Effective interaction: sum excitations outside valence space to 3rd order

2) Single-particle energies calculated self-consistently

3) Harmonic-oscillator basis of 13 major shells: converged

4) NN and 3N forces – fully to 3rd-order MBPT

5) Work in extended valence spaces

-11.6 nd rd 2 order 3 order 18 -11.8 O -12

-12.2

Energy (MeV) V -12.4 low k G-matrix -12.6 4 6 8 10 12 14 2 4 6 8 10 12 14 Major Shells Major Shells

G-matrix – no signs of convergence (similar in pf-shell) Strategy

1) Effective interaction: sum excitations outside valence space to 3rd order

2) Single-particle energies calculated self-consistently

3) Harmonic-oscillator basis of 13 major shells: converged

4) NN and 3N forces – fully to 3rd-order MBPT

5) Work in extended valence spaces

-2 4 Neutron Proton -4 f f5/2 5/2 2 -6 p 0 p -8 1/2 1/2 f -2 f -10 7/2 7/2 p p 3/2 -4 3/2

Single-Particle Energy (MeV) -12

2 4 6 8 10 12 14 16 18 2 4 6 8 10 12 14 16 18 _ _ Nh Nh

Clear convergence with HO basis size Promising order-by-order behavior Convergence Properties NN matrix elements derived from: - Chiral N3LO (Machleidt, 500 MeV) using smooth-regulator V low k - Third order in MBPT

- 13 major HO shells for intermediate state configurations

-10.5 18 18 -11 O O

V (1st) -11.5 low k Vlow k (2nd)

Vlow k (3rd)

Energy (MeV) -12

-12.5

4 6 8 10 12 14 Major Shells Clear convergence with HO basis size Promising order-by-order behavior Convergence Properties NN matrix elements derived from: - Chiral N3LO (Machleidt, 500 MeV) using smooth-regulator V low k - Third order in MBPT

- 13 major HO shells for intermediate state configurations

-11.6 nd rd 2 order 3 order 18 -11.8 O -12

-12.2

Energy (MeV) V -12.4 low k G-matrix -12.6 4 6 8 10 12 14 2 4 6 8 10 12 14 Major Shells Major Shells G-matrix – no sign of convergence 3N Forces for Valence-Shell Theories Effects of residual 3N between 3 valence nucleons? Normal-ordered 3N: microscopic contributions to inputs for CI Hamiltonian Effects of residual 3N between 3 valence nucleons? Coupled-Cluster theory with 3N: benchmark of 4He

0- 1- and 2-body of 3NF dominate Residual 3N can be neglected Work on 16O in progress

Hagen, Papenbrock et al. (2007)

Approximated residual 3N by summing over valence nucleon – Nucleus-dependent: effect small, not negligible by 24O Backup Intro From QCD to Nuclear Interactions How do we determine interactions between nucleons?

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Only degrees of freedom relevant to energy scale

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Λchiral ~ 700MeV

Appropriate separation of scales

Typical momenta in nucleus Q~ mπ

Increased Resolution Effective theory: only nucleons and pions Monopole Part of Valence-Space Interactions Microscopic MBPT – effective interaction in chosen model space

Works near closed shells: deteriorates beyond this Deficiencies improved adjusting particular two-body matrix elements (2J +1) V JT Monopoles: T ∑J abab Vab = Angular average of interaction (2J +1) ∑J Determines interaction of orbit a with b: evolution of orbital energies 1 Microscopic low-momentum interactions 0.5 T=1€ 0 Phenomenological USD interactions -0.5 -1 Clear shifts in low-lying orbitals: V -1.5 low k - T=1 repulsive shift USDa

V(ab;T) [MeV] -2 USDb -2.5 Origin of shifts: Neglected 3N forces -3 -- Zuker (2003) -3.5 d5d5 d5d3 d5s1 d3d3 d3s1 s1s1 How domagic numbers form evolve? and nuclear existence? What are limits the of proton/neutron-richWhat are properties the matter? of Worldwide joint nuclear science: frontiersExploring the of 2! 8! The Nuclear Landscape TowardtheExtremes The NuclearLandscape

2! protons 20! 8! 28! neutrons 20! DripLinesand MagicNumbers: experimental 28! 50! /theoretical Theory: points tokey measurements Provide for tests nuclear models, forces Different techniques, goals similarscience effort effort 82! 82! 50! known nuclei known 126!

Perturbative Valence-Space Strategy

1) Effective interaction: sum excitations outside valence space to 3rd order

2) Single-particle energies calculated self consistently

3) Harmonic-oscillator basis of 13-15 major shells: converged

4) NN and 3N forces from chiral EFT – to 3rd-order MBPT

5) Explore extended valence spaces

-2 4 Neutron Proton -4 f f5/2 5/2 2 -6 p 0 p -8 1/2 1/2 f -2 f -10 7/2 7/2 p p 3/2 -4 3/2

Single-Particle Energy (MeV) -12

2 4 6 8 10 12 14 16 18 2 4 6 8 10 12 14 16 18 _ _ Nh Nh Clear convergence with HO basis size Promising order-by-order behavior How domagic numbers form evolve? and nuclear existence? What are limits the of proton/neutron-richWhat are properties the matter? of Worldwide jointexperimental/theoretical effort nuclear science: frontiersExploring the of 2! 8! The Nuclear Landscape TowardtheExtremes The NuclearLandscape

2! protons 20! 8! 28! neutrons 20! DripLinesand MagicNumbers: 28! 50! 82! 82! 50! Novel quantumphenomena Halo 126! 11 Li Skin 80 Ni