Three-nucleon forces and exotic nuclei Javier Menéndez Institut für Kernphysik (TU Darmstadt) and ExtreMe Matter Institute (EMMI) with Jason D. Holt (TU Darmstadt/EMMI), Achim Schwenk (EMMI/TU Darmstadt) and Johannes Simonis (TU Darmstadt/EMMI) NUSTAR Annual Meeting, GSI, 28 February 2013
Outline Theoretical Approach: NN+3N forces in Shell Model The nuclear interaction: need of 3N forces Shell Model interactions with microscopic chiral NN+3N forces Results for exotic nuclei Neutron rich O isotopes Neutron rich Ca isotopes Proton rich N=8 and N=20 isotopes 2 / 20
Outline Theoretical Approach: NN+3N forces in Shell Model The nuclear interaction: need of 3N forces Shell Model interactions with microscopic chiral NN+3N forces Results for exotic nuclei Neutron rich O isotopes Neutron rich Ca isotopes Proton rich N=8 and N=20 isotopes
The Nuclear interaction Ideally, QCD interaction (Latice QCD) ⇒ Hard problem: QCD non-perturbative at low energy Alternatively, bare NN potential (spirit of Shell Model) ⇒ Drawback: manipulate the interaction to solve the many-body nuclear problem Finally, use selected nuclei (Energy Density Functionals) ⇒ Drawback: loss of predictive power 3 / 20
The Nuclear interaction Ideally, QCD interaction (Latice QCD) ⇒ Hard problem: QCD non-perturbative at low energy Alternatively, bare NN potential (spirit of Shell Model) ⇒ Drawback: manipulate the interaction to solve the many-body nuclear problem Finally, use selected nuclei (Energy Density Functionals) ⇒ Drawback: loss of predictive power 3 / 20
Limitation of NN potentials Ab initio calculations (No Core Shell Model) with perfect NN potentials (AV18) fail to reproduce light nuclei spectra Navratil et al. PRL99 042501(2007) ⇒ Confirms experience of Shell Model (monopole adjustments) 4 / 20
Need of 3N forces Need 3N forces! Zuker PRL90 042502 (2003) 3N forces originate in the elimination of degrees of freedom (N-body forces appear in any effective theory) Bogner, Schwenk, Furnstahl PPNP65 94 (2010) But few NNN scattering data available! ⇒ Need a framework that, in a natural manner, describes 3N forces consistent with NN forces 5 / 20
Chiral EFT • Chiral EFT is a low energy approach to QCD valid for nuclear structure energies • Exploits approximate chiral symmetry of QCD: pions are special particles (pseudo-Goldstone bosons) • Nucleons interact via pion exchanges and contact interactions (physics non-resolved at nuclear structure energies) • Enables a systematic basis for strong interactions, expansion in powers of Q / Λ b Q ∼ m π , typical momentum scale Λ b ∼ 500 MeV, breakdown scale • Systematic expansion naturally includes NN, 3N, 4N... forces (at different orders) • Short-range couplings are fitted to experiment once 6 / 20
Chiral EFT NN+3N forces Systematic expansion: state-of-the-art chiral EFT forces • NN forces included up to N 3 LO • 3N forces included up to N 2 LO NN fitted to: • NN scattering data 3N fitted to: • 3 H Binding Energy • 4 He radius Weinberg, van Kolck, Kaplan, Savage, Wise, Epelbaum, Kaiser, Meißner... 7 / 20
Many Body Perturbation Theory Better convergence through V lowk transformation Single Particle Energies Two-Body Matrix Elements (SPEs) (TBMEs) Many-body Perturbation Theory up to third order to build an effective Shell Model interaction in a valence space Full diagonalizations using codes ANTOINE and NATHAN Caurier et al. RMP77 427(2005) and compare to experiment 8 / 20
3N Forces Treatment of 3N forces: normal-ordered 2B: 2 valence, 1 core particle ⇒ (effective) Two-body Matrix Elements (TBME) normal-ordered 1B: 1 valence, 2 core particles ⇒ (effective) Single particle energies (SPE) residual 3B: ⇒ Estimated to be suppressed by N valence / N core 9 / 20
Outline Theoretical Approach: NN+3N forces in Shell Model The nuclear interaction: need of 3N forces Shell Model interactions with microscopic chiral NN+3N forces Results for exotic nuclei Neutron rich O isotopes Neutron rich Ca isotopes Proton rich N=8 and N=20 isotopes
O isotopes: dripline anomaly O isotopes: ’anomaly’ in the dripline at 24 O, doubly magic nucleus stability line Z Si 2007 Al 2007 Mg � 2007 Na 2002 Ne 2002 F 1999 8 O 1970 stable isotopes N 1985 C 1986 unstable isotopes B 1984 unstable fluorine isotopes Be 1973 Li 1966 unstable oxygen isotopes 2 He 1961 neutron halo nuclei H 1934 2 8 20 28 N Theoretical calculations predict the dripline at 26 O or 28 O: a fit to this property is needed to correctly reproduce experiment (e.g. USD interactions, EDFs) 10 / 20
O isotopes: effective SPE’s Single-Particle Energy (MeV) 4 (a) Forces derived from NN theory (b) Phenomenological forces d3/2 d3/2 0 Evolution of d 3 / 2 orbit (and to a s 1/2 -4 less extent s 1 / 2 and d 5 / 2 orbits) s 1/2 d5/2 d5/2 incorrectly predicted by NN forces -8 G-matrix SDPF-M V low k USD-B Phenomenological interactions 8 14 16 20 8 14 16 20 Neutron Number ( N ) Neutron Number ( N ) include further repulsive Single-Particle Energy (MeV) 4 contributions 2 (c) G-matrix NN + 3N ( ∆) forces (d) V NN + 3N ( ∆ ,N LO ) forces low k 0 The effect of 3N forces is similar to d3/2 d3/2 phenomenological ’cures’ s -4 1/2 s 1/2 d5/2 d5/2 Otsuka et al. PRL105 032501 (2010) -8 2 NN + 3N (N LO) NN + 3N ( ∆) NN + 3N ( ∆) NN NN 8 14 16 20 8 14 16 20 Neutron Number ( N ) Neutron Number ( N ) 11 / 20
O isotopes: masses and spectra Chiral NN+3N forces give the correct picture for masses and spectra 0 (b) NN (c) NN + 3N -10 Otsuka et al. -20 Energy (MeV) PRL105 032501 (2010) -30 -40 Holt, JM, Schwenk -50 EPJA in press (2013) sd-shell (2nd) USDb -60 sd-shell sd-shell (3rd) sdf 7/2 p 3/2 -shell -70 sdf 7/2 p 3/2 -shell 9 -80 + 0 24 O 8 16 18 20 22 24 26 28 16 18 20 22 24 26 28 + 1 + 2 + + 4 Mass Number A Mass Number A 7 2 + 3 + 4 + + 0 3 + 2 Energy (MeV) 6 + 1 + 5 2 3N forces provide repulsion + 2 4 missing in NN-only forces 3 3N forces crucial also for reliable 2 1 description of spectra + + + 0 0 0 0 NN NN+3N Exp 12 / 20
4 2 Single-Particle Energy (MeV) (a) G-matrix NN + 3N ( forces (b) V NN + 3N ( ,N LO forces low k 0 d3/2 d3/2 s 1/2 -4 s 1/2 d5/2 d5/2 -8 2 NN + 3N (N LO) NN + 3N ( NN + 3N ( NN NN 8 14 16 20 8 14 16 20 Neutron Number ( N ) Neutron Number ( N ) Residual 3N Forces (c) 3-body interaction (d) 3-body interactions with one more neutron added to (c) In the most neutron-rich oxygen isotopes, 3N forces between 3 valence neutrons (remember, suppressed by N valence / N core ) 16 O core can give a relevant contribution + 2 R3B-LAND (this work) Residual 3N contributions are 1.5 MoNA/NSCL (2008, 2012) repulsive NN+3N + residual 3N NN+3N They are small compared to Energy (MeV) 1 + 3/2 normal-ordered 3N force, but + residual 3N increase with N 0.5 Very good agreement with + + 0 0 0 resonances in 25 O and 26 O Caesar, Simonis et al, arXiv:1209.0156 24 O 25 O 26 O 13 / 20
Ca isotopes: 2n separation energies Compare S 2 n theoretical calculations with experimental results S 2 n = − [ B ( N , Z ) − B ( N − 2 , Z )] New precision measurements change 18 previous slope from AME 2003 16 ∼ 2 MeV change in 52 Ca! 14 S 2n (MeV) 12 Very good agreement between TITAN 10 calculation and experimental trend AME2003 NN+3N (emp) 8 (Similar level as NN+3N (MBPT) phenomenological interactions) 6 4 Two sets of spe’s, empirical and 28 29 30 31 32 33 34 35 calculated, in pfg 9 / 2 valence space Gallant et al. PRL109 032506 (2012) 14 / 20
Nuclear Pairing Gaps Compare also to experimental three-point mass differences: = ( − 1 ) N ∆ ( 3 ) [ B ( N + 1 , Z ) + B ( N − 1 , Z ) − 2 B ( N , Z )] n 2 3 The experimental trend is very well reproduced by theory TITAN+ (3) (MeV) AME2003 2 Theoretical results systematically ∆ n 0.5 MeV higher than experiment 1 Prediction of sub-shell closure 0 candidates N = 32 (moderate closure) 28 29 30 31 32 33 34 35 Neutron Number N and N = 34 (no apparent closure) Gallant et al. PRL109 032506 (2012) 15 / 20
Shell closures in Ca isotopes 7 2 + 1 energies + Energy (MeV) 6 characterize shell closures 5 of Ca isotopes 4 3 Closure at N = 28 1 2 2 with 3N forces in ( pfg 9 / 2 ) 1 Holt et al. JPG39 085111(2012) NN 0 NN+3N [emp] NN+3N [MBPT] 42 44 46 48 50 52 54 56 58 60 62 64 66 68 Holt, JM, Schwenk, to be submitted Mass Number A 3N forces enhance closure at N = 32 (more moderate than N = 28) 3N forces reduce strong closure at N = 34 (no apparent closure) Predicted shell closure at N = 60, unaffected by 3N forces (but continuum missing in our calculations!) 16 / 20
Proton dripline at N = 8 Theory complements/improves 0 mass extrapolations and N=8 isomeric mass-multiplet formula (IMME) Ground-State Energy (MeV) -2 E ( A , T , T z ) = E ( A , T , − T z ) + 2 b ( A , T ) T z -4 NN forces oberbind -6 3N forces essential to describe masses and the predict the proton dripline -8 NN NN+3N Proton dripline not certain -10 NN+3N (sdf 7/2 p 3/2 ) predicted either in 20 Mg or 22 Si: AME2011 S 2 p = -0.12 (Theory) / +0.01 (IMME) -12 IMME Measurement needed! 16 17 18 19 20 21 22 23 24 Mass Number A Calculations in standard Holt, JM, Schwenk PRL110 022502 (2013) and extended spaces 17 / 20
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