Three-Nucleon Forces and the Structure of Exotic Nuclei Jason D. Holt

Three-Nucleon Forces and the Structure of Exotic Nuclei Jason D. Holt Based on T. Otsuka, T. Suzuki, JDH, A. Schwenk, Y. Akaishi, PRL (11) JDH, J. Menendez, A. Schwenk, arxiv:118.268 JDH, T. Otsuka, A. Schwenk, T. Suzuki JPG (212) JDH, J. Menendez, A. Schwenk, arxiv:127.159 JDH, J. Menendez, A. Schwenk, in prep. JDH and J. Engel, in prep. D. Lincoln, JDH et al., submited to PRL Supported by BMBF under 6DA747I (NuSTAR.DA)

Outline Goal: understand the role of 3N forces for structure of medium-mass exotic nuclei - What are the limits of nuclear existence? - How do magic numbers form and evolve? protons 28! 2! I. Microscopic approach: theoretical inputs consistently from 82! NN+3N II. Shell evolution in calcium a) Pairing gaps b) Spectra III. Proton-rich systems 5! IV. Neutrinoless double-beta decay 5! 82! 126! 8! 2! 2! 8! 2! 28! neutrons

Chiral Effective Field Theory: Nuclear Forces LO NLO Nucleons interact via pion exchanges and contact interactions Hierarchy: V NN > V 3N >... Consistent treatment of NN, 3N, electroweak operators Couplings fit to experiment once Evolve to low-momentum V low k (Improved convergence behavior) N 2 LO 3N constants fit to properties of light nuclei at low momentum N 3 LO Weinberg, van Kolck, Kaplan, Savage, Wise, Epelbaum, Kaiser, Meissner,

Solving the Nuclear Many-Body Problem Nuclei understood as many-body system starting from closed shell, add nucleons Interaction and energies of valence space orbitals from V low k Does not reproduce experimental data h,1f,2p g,1d,2s f,1p p - sd -valence space Active nucleons occupy valence space s Assume filled core

Solving the Nuclear Many-Body Problem Nuclei understood as many-body system starting from closed shell, add nucleons Interaction and energies of valence space orbitals from V low k Does not reproduce experimental data allow explicit breaking of core h,1f,2p g,1d,2s f,1p Strategy 1) Effective valence space interaction 3 rd order MBPT 2) HO basis of 13 major shells 3) Single-particle energies calculated self-consistently 4) Work in valence space as large as possible 5) Consistent treatment of chiral NN and 3N forces p - sd -valence space Active nucleons occupy valence space s Assume filled core

Convergence Properties NN matrix elements derived from: - Chiral N 3 LO (Machleidt, 5 MeV) using smooth-regulator V low k - Third order in MBPT - 13 major HO shells for intermediate state configurations -1.5 Energy (MeV) -11-11.5-12 18 O V low k (1st) V low k (2nd) V low k (3rd) -12.5 4 6 8 1 12 14 Major Shells Clear convergence with HO basis size Promising order-by-order behavior

Convergence Properties NN matrix elements derived from: - Chiral N 3 LO (Machleidt, 5 MeV) using smooth-regulator V low k - Third order in MBPT - 13 major HO shells for intermediate state configurations Energy (MeV) -11.6-11.8-12 -12.2-12.4-12.6 2 nd order 3 rd order V low k G-matrix 18 O 4 6 8 1 12 14 Major Shells G-matrix no sign of convergence 2 4 6 8 1 12 14 Major Shells

Benchmark against ab-initio Coupled Cluster at NN level SPEs: one-particle attached CC energies in 17 O and 41 Ca Energy [MeV] -1-2 -3-4 -5-6 -7-8 -9 Comparison to Coupled Cluster Oxygen CC MBPT Experiment (a) CC/MBPT -1 16 18 2 22 24 26 28 Mass Number A Energies relative to 16 O and 4 Ca Energy [MeV] -1-15 -2-25 -3 Small difference in many-body methods ~5% -5 Calcium CC MBPT CC/MBPT 4 44 48 52 56 6 Mass Number A

Limits of Nuclear Existence: Oxygen Anomaly Microscopic picture: NN-forces too attractive Fit to experiment

Limits of Nuclear Existence: Oxygen Anomaly Microscopic picture: NN-forces too attractive Fit to experiment -1 Incorrect prediction of oxygen dripline Extended-space more binding NN prediction Energy (MeV) -2-3 -4-5 -6-7 sd-shell sdf 7/2 p 3/2 shell USDb -8 16 18 2 22 24 26 28 Mass Number A

Single Particle Energies SPEs self-consistently from one-body diagrams + 3 rd order sd-shell: overbound, unreasonable spacing Orbit Exp USDb T+V NN (3 rd ) d 5/2-4.14-3.93-5.43 s 1/2-3.27-3.21-5.32 d 3/2.944 2.11 -.97 Typical approach: use empirical SPEs 3N forces eliminate need for adjusted parameters?

Chiral Effective Field Theory: 3N Forces Two new couplings at N 2 LO LO NLO c terms given from NN fits: constrained by NN, πn data c D c E fit to properties of light nuclei: Triton binding energy, 4 He radius N 2 LO N 3 LO

3N Forces for Valence-Shell Theories Normal-ordered 3N: contribution to valence nucleon interactions Effective one-body Effective two-body 16 O core 16 O core a V 3N,eff a' = 1 2 αβa V 3N αβa' αβ =core ab V 3N,eff a'b' = αab V 3N αa'b' α =core Combine with microscopic NN (Third Order): no empirical adjustments

Extended Valence Space SPEs 3N forces: additional repulsion comparable to phenomenology + 3 rd + 3N Orbit USDb T+V NN +V 3N SDPF-M T+V NN +V 3N d 5/2-3.93-3.78-3.95-3.46 s 1/2-3.21-2.42-3.16-2.2 d 3/2 2.11 1.45 1.65 1.92 f 7/2 3.1 3.71 p 3/2 3.1 7.72 Similar behavior in standard/extended spaces

Ground-State Energies of Oxygen Isotopes Valence-space interaction and SPEs from NN+3N Energy (MeV) -1-2 -3-4 -5-6 -7 sd-shell (2nd) sd-shell (3rd) sdf 7/2 p 3/2 -shell (b) NN -8 16 18 2 22 24 26 28 Mass Number A USDb sd-shell sdf 7/2 p 3/2 -shell (c) NN + 3N 16 18 2 22 24 26 28 Mass Number A JDH, Menendez, Schwenk, arxiv:118.268 Repulsive character improves agreement with experiment sd-shell results underbound; improved in extended space sdf 7/2 p 3/2

Impact on Spectra: 23 O Neutron-rich oxygen spectra with NN+3N 5/2 +, 3/2 + indicate position of d 5/2 and d 3/2 orbits Energy (MeV) 5 4 3 2 1 3/2 + 3/2 + 5/2 + 23 O 3/2 + 5/2 + (3/2 + ) S 3/2 + n 5/2 + (5/2 + ) sd-shell NN-only Wrong ground state 5/2 + too low 3/2 + bound NN+3N Improvements in extended valence space 1/2 + 1/2 + 1/2 + 1/2 + 1/2 + -1 NN sd 5/2 + NN sdf 7/2 p 3/2 NN+3N sd NN+3N sdf 7/2 p 3/2 Expt. JDH, Menendez, Schwenk, arxiv:118.268

Shell Formation/Evolution in Calcium Isotopes Goal: understand the role of 3N forces for structure of medium-mass exotic nuclei - What are the limits of nuclear existence? - How do magic numbers form and evolve? protons 28! 2! I. Microscopic approach: theoretical inputs consistently from 82! NN+3N II. Shell evolution in calcium a) Pairing gaps b) Spectra III. Proton-rich systems 5! IV. Neutrinoless double-beta decay 5! 82! 126! 8! 2! 2! 8! 2! 28! neutrons

Nuclear Pairing Pairing of even number of nucleons even/odd staggering Pairing gaps deduced from 3-point mass difference: Δ (3) n = ( 1)N 2 Allows comparison with experiment [ BE(N +1,Z) + BE(N 1,Z) 2BE(N,Z) ] Relative peak in pairing strength indicates shell closure

Pairing in EDF with 3N Forces In Energy Density Functional theory: 3N forces lower gaps systematically ~3% Lesinski, Hebeler, Duguet, Schwenk, JPG (212) What are the contributions from neglected many-body effects? (Core polarization)

Compare with Pairing in Calcium Isotopes: Ladders (3) Δ n calculated from microscopic NN+3N in calcium EDF HFB iterates ladders microscopically in pairing channel Compare with pp, hh ladders to 3 rd 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 (MeV) 2 1.5 n (3) 1.5 4 44 48 52 56 Mass Number A 4 44 48 52 56 Mass Number A 4 44 48 52 56 6 Mass Number A NN NN+3N AME23 TITAN AME23+TITAN

Pairing in Calcium Isotopes: Full 3 rd order Compare with EDF (3) Δ n calculated from microscopic NN+3N in calcium Full 3 rd -order MBPT Further increases gaps Correct odd/even staggering; more pronounced Good experimental reproduction with 3 rd -order NN+3N Can account for missing physics in EDF calculations JDH, Menendez, Schwenk, in prep. 3 (a) 3rd-order ladders (pf) (b) Full 3rd-order (pf) (c) Full 3rd (pfg 9/2 ) 2.5 (MeV) 2 1.5 n (3) 1.5 NN NN+3N (emp) NN+3N (MBPT) 4 44 48 52 56 Mass Number A 4 44 48 52 56 Mass Number A 4 44 48 52 56 6 64 68 Mass Number A

Pairing for Shell Evolution N=28 2 + Energy (MeV) 1 6 5 4 3 2 1 (d) pfg 9/2 -shell Peak in pairing gaps: complementary signature for shell closure Compare with 2 + energies for Ca NN NN+3N NN+3N (MBPT) N=28: strong peak, overprediction in both cases 44 48 52 56 6 64 68 3 (c) Full 3rd (pfg 9/2 ) 2.5 (MeV) n (3) 2 1.5 1.5 NN NN+3N (emp) NN+3N (MBPT) 4 44 48 52 56 6 64 68 Mass Number A

Pairing for Shell Evolution N=32 (MeV) n (3) 2 + Energy (MeV) 1 3 2.5 2 1.5 1.5 6 5 4 3 2 1 NN NN+3N NN+3N (MBPT) 44 48 52 56 6 64 68 (c) Full 3rd (pfg 9/2 ) (d) pfg 9/2 -shell NN NN+3N (emp) NN+3N (MBPT) 4 44 48 52 56 6 64 68 Mass Number A Peak in pairing gaps: complementary signature for shell closure Compare with 2 + energies for Ca N=32: moderate peak Close to experimental value with new TITAN data Experimental measurement of 53 Ca mass needed to reduce uncertainty

Evolution of Magic Numbers: N=34 N=34 magic number in calcium? Single-Particle Energy (MeV) -5-1 -15 (a) Phenomenological Forces f 5/2 p 1/2 p 3/2 f 7/2 KB3G GXPF1 4 44 48 52 56 6 Mass Number A 2 + Energy (MeV) 1 6 4 2 82! 82! 5 3 1 (a) Phenomenological Forces 5! Experiment GXPF1 KB3G 126! 42 44 46 48 5 52 54 56 58 Mass Number A Strong phenomenological disagreement for neutron-rich calcium

Pairing for Shell Evolution N=34 2 + Energy (MeV) 1 6 5 4 3 2 1 (d) pfg 9/2 -shell Peak in pairing gaps: complementary signature for shell closure Compare with 2 + energies for Ca NN NN+3N NN+3N (MBPT) N=34: suppressed with 3N forces 44 48 52 56 6 64 68 3 (c) Full 3rd (pfg 9/2 ) 2.5 (MeV) n (3) 2 1.5 1.5 NN NN+3N (emp) NN+3N (MBPT) 4 44 48 52 56 6 64 68 Mass Number A

Neutron-Rich Ca Spectra Near N=34 Neutron-rich calcium spectra with NN+3N Energy (MeV) 6 5 4 3 2 1 7/2-53 Ca? 7/2-7/2-7/2-1/2-9/2-7/2 - Energy (MeV) 8 7 6 5 4 3 2 1 3 + 4 + 2 + 3 + 2 + + 54 Ca 2 + 1 + 2 + + 3+ 4 + 3 + 2 + 4 + 3 + 2 + 4 + 2 + 3 + + 2 + 1/2-1/2-1/2 - NN+3N pfg 9/2 Expt. GXPF1 KB3G 1/2 - + + + + NN+3N pfg 9/2 Expt. GXPF1 KB3G JDH, Menendez, Schwenk, in prep. Dramatic difference in phenomenology NN+3N similar to KB3G no indication of N=34 magic number Consistent with predictions from Coupled-Cluster theory

6 49 Ca Impact on Spectra: 49 Ca Neutron-rich calcium spectra with NN+3N Energy (MeV) 5 4 3 2 1 NN pf 11/2-11/2-9/2 - - 9/2 9/2-7/2 - (1, 7/2 - ) 7/2 - (9/2 + ) 7/2-7/2-9/2-7/2-7/2-9/2-1/2-7/2-1/2-1/2-1/2-1/2-7/2-7/2-7/2-1/2-1/2-1/2-7/2 - NN pfg 9/2 NN+3N pf JDH, Menendez, Schwenk, in prep. NN+3N pfg 9/2 Expt. GXPF1 KB3G Spectrum typically too compressed NN+3N in pfg 9/2 correct 1/2 - Comparable to phenomenology (as for all lighter Ca isotopes)

Impact on Spectra: 51 Ca Neutron-rich calcium spectra with NN+3N 5 51 Ca 7/2-9/2 - Energy (MeV) 4 3 2 1 NN pf 11/2-11/2-1/2-1/2-1/2-7/2-7/2-1/2-7/2-1/2-1/2 - NN pfg 9/2 NN+3N pf 9/2-7/2-1/2-1/2 - NN+3N pfg 9/2 (?) (?) (?) (?) (?) ( ) (?) 7/2-1/2 - Expt. GXPF1 KB3G 1/2 7/2 - - 9/2-1/2 - JDH, Menendez, Schwenk, in prep. Possibility to assign spin/parity where unknown Gamma-ray spectroscopy needed

Pairing for Shell Evolution N=4 2 + Energy (MeV) 1 6 5 4 3 2 1 (d) pfg 9/2 -shell Peak in pairing gaps: complementary signature for shell closure Compare with 2 + energies for Ca NN NN+3N NN+3N (MBPT) N=4: robust signature of shell closure 44 48 52 56 6 64 68 3 (c) Full 3rd (pfg 9/2 ) 2.5 (MeV) n (3) 2 1.5 1.5 NN NN+3N (emp) NN+3N (MBPT) 4 44 48 52 56 6 64 68 Mass Number A

Proton-Rich Systems Move towards calculations of all sd- pf-shell nuclei Islands of inversion First step: proton-rich systems

Proton-Rich Systems Move towards calculations of all sd- pf-shell nuclei Islands of inversion First step: proton-rich systems Study N=8, 2 isotone chains I. Microscopic approach: theoretical inputs consistently from NN+3N II. Shell evolution in calcium a) Pairing gaps b) Spectra III. Proton-rich systems IV. Neutrinoless double-beta decay

Ground-State Energies of N=8 Isotones Ground-State Energy (MeV) -2-4 -6-8 -1 NN AME211 IMME N=8 Data limited use phenomenological isobaric multiplet mass equation (IMME) E(A,T,T z ) = E(A,T, T z ) + 2b(A,T)T z b =.768A 2 / 3.9133 NN-only: overbound -12 16 17 18 19 2 21 22 23 24 Mass Number A JDH, Menendez, Schwenk, arxiv:127.159

Ground-State Energies of N=8 Isotones Ground-State Energy (MeV) -2-4 -6-8 -1-12 NN NN+3N NN+3N (sdf 7/2 p 3/2 ) AME211 IMME N=8 16 17 18 19 2 21 22 23 24 Mass Number A Data limited use phenomenological isobaric multiplet mass equation (IMME) E(A,T,T z ) = E(A,T, T z ) + 2b(A,T)T z b =.768A 2 / 3.9133 NN-only: overbound NN+3N: improved agreement with experiment/imme JDH, Menendez, Schwenk, arxiv:127.159 Dripline uncertain: Unbound in AME211, NN+3N; bound in IMME

Excitation Energy (MeV) 6 5 4 3 2 1 18 Ne 3 + + 2 + 4 + 2 + + + 3 + 2 + + Spectra of N=8 Isotones 5/2 + 3/2 + Exp Exp Exp NN+3N: reasonable agreement with experiment New measurement: excited state in 2 Mg close to predicted 4 + -2 + doublet Predictions for proton-rich 21 Al, 22 Si spectra Closed sub-shell signature in 22 Si 2 + + 19 Na 2 Mg 1/2 + (1/2 + ) (3/2 + (5/2 + ) ) 4 + 2 + (4 + ) 2 + + 2 + + 21 Al 22 Si 4 + 5/2+ 7/2 + + 7/2 3/2 + 9/2 + 2 + + Unpublished; From I. Mukha 5/2 + 3/2 + 1/2 + 5/2 + 3 + + 2 + + JDH, Menendez, Schwenk, arxiv:127.159

Ground-State Energies of N=2 Isotones Ground-State Energy (MeV) -2-4 -6-8 -1 N=2 NN Exp and AME211 extrapolation IMME -12 4 41 42 43 44 45 46 47 48 Mass Number A NN-only: overbound beyond 45 Mn JDH, Menendez, Schwenk, arxiv:127.159

Ground-State Energies of N=2 Isotones Ground-State Energy (MeV) -2-4 -6 N=2-8 NN NN+3N NN+3N (pfg 9/2 ) -1 Exp and AME211 extrapolation IMME -12 4 41 42 43 44 45 46 47 48 Mass Number A NN-only: overbound beyond 45 Mn NN+3N: close to experiment/imme JDH, Menendez, Schwenk, arxiv:127.159 Dripline: Predicted to be 46 Fe in all calculations S 2p Expt. NN+3N (pf) NN+3N (pfg 9/2 ) -1.28(6) MeV -2.73 MeV -1.2 MeV Prediction for 48 Ni within 3keV of experiment

Spectra of N=2 Isotones Excitation Energy (MeV) 5 4 3 2 1 42 Ti 6 + 2 + 6 + 4 + 4 + (2 + ) + 2 + 43 V 44 Cr 45 Mn 46 Fe 47 Co 48 Ni 9/2-11/2-2 + 6 + 4 + 4 + 2 + 9/2-11/2-4 + 2 + 6 + 4 + 2 + 7/2-1/2-4 + 2 + + + + Exp 7/2 - + 7/2 - + 7/2 - NN+3N: reasonable agreement with measured 42 Ti Predictions for proton-rich spectra Mirror energy differences with Ca isotopes ~4keV Closed-shell signature in 48 Ni + JDH, Menendez, Schwenk, arxiv:127.159

Neutrinoless Double-Beta Decay Take first steps toward microscopic calculation First step: effective operator Study light νββ candidates I. Microscopic approach: theoretical inputs consistently from NN+3N II. Shell evolution in calcium a) Pairing gaps b) Spectra III. Proton-rich systems IV. Neutrinoless double-beta decay

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 Observed in nature with:

Nuclear Weak Processes: νββ-decay Nuclear weak processes: fundamental importance for particle physics Determines character of neutrino (Majorana/Dirac) Lepton number violation (T νββ 1/ 2 ) 1 2 2 = G ν (Q ββ,z) M ν m ββ

Nuclear Weak Processes: νββ-decay Nuclear weak processes: fundamental importance for particle physics Determines character of neutrino (Majorana/Dirac) Lepton number violation Neutrino mass scale (T νββ 1/ 2 ) 1 2 2 = G ν (Q ββ,z) M ν m ββ

Nuclear Weak Processes: νββ-decay Nuclear weak processes: fundamental importance for particle physics Two essential ingredients: Q-value (experiment) (T νββ 1/ 2 ) 1 2 2 = G ν (Q ββ,z) M ν m ββ Determines character of neutrino (Majorana/Dirac) Lepton number violation Neutrino mass scale New measurement of 82 Se at NSCL D. Lincoln, JDH et al., submitted to PRL

Nuclear Weak Processes: νββ-decay Nuclear weak processes: fundamental importance for particle physics Two essential ingredients: Q-value (experiment) Nuclear matrix element Determines character of neutrino (Majorana/Dirac) Lepton number violation Neutrino mass scale Nuclear structure required for NME (T νββ 1/ 2 ) 1 2 2 = G ν (Q ββ,z) M ν m ββ Need microscopic framework capable of accurate prediction

M ν M GT ν = f Nuclear Matrix Element = M GT ν g 2 V g M F 2 ν + Corrections ~ 3% A H(r ab ) σ a σ b τ + + a τ b i M F ν = f H(r ab )τ + + a τ b ab Shell model: arbitrary correlations in small single-particle space QRPA: simple correlations in large single-particle space ab i

Effective νββ-decay Operator Standard SM approach: phenomenological wavefunctions + bare operator Calculate effective νββ operator using formalism of effective interaction theory Diagrammatically similar: replace one interaction vertex with M ν M ν operator V low k Previous: G-matrix calculation in 2 nd -order MBPT non-convergent Low-momentum interactions: Improve convergence behavior? - Chiral N 3 LO (Machleidt, 5 MeV) using smooth-regulator V low k - 13 major HO shells for intermediate state configuration Engel and Hagen, PRC (29)

Calculate in MBPT: Effective νββ-decay Operator 2 nd order

Effective νββ-decay Operator Calculate in MBPT: 2 nd order 3 rd order

Intermediate-State Convergence First results in 82 Se (with phenomenological wavefunctions from A. Poves) 3.24 Nuclear Matrix Element 3.22 3.2 Gamow-Teller M ν GT 3.18 8 1 12 14 16 18 2 22 24 Nh JDH and Engel, in prep. Results well converged in terms of intermediate state excitations

Intermediate-State Convergence First results in 82 Se (with phenomenological wavefunctions from A. Poves) 3.24 3.4 Nuclear Matrix Element 3.22 3.2 Gamow-Teller M ν GT Nuclear Matrix Element 3.3 3.2 3.1 V low k (3rd) M ν GT G-matrix (2nd) 3.18 8 1 12 14 16 18 2 22 24 Nh Results well converged in terms of intermediate state excitations Comparison with G-matrix no sign of convergence Order-by-order convergence analysis in progress 3 6 8 1 12 14 16 18 2 22 24 Nh JDH and Engel, in prep.

Nuclear structure theory of medium-mass nuclei with 3N forces, extended spaces Robust repulsive 3N mechanism for T=1 neutron/proton-rich nuclei Oxygen isotopes Cures NN-only failings: dripline, shell evolution, spectra Calcium isotopes in pf- and pfg 9/2 -shells: Prediction of N=28 magic number in 48 Ca Shell evolution towards the dripline: modest N=34 closure, quenching of N=4 Pairing gaps reflect shell structure higher-order many-body processes essential Proton-rich N=8,2 isotones: similar improvements in g.s. energies/spectra First effective νββ operator with chiral NN interactions Clearly improvable upgrade path Conclusion

Acknowledgments Collaborators J. Menendez, A. Schwenk T. Otsuka J. Engel T. Suzuki (Nihon U.) G. Hagen, T. Papenbrock Computing support Travel support