S tructure and reactions at the dripline with coupled-cluster theory.

1. Structure and reactions at the dripline with coupled-cluster theory. Gaute Hagen (ORNL) Collaborators: Thomas Papenbrock (UT/ORNL) MortenHjorth-Jensen (UiO/CMA) Gustav Jansen (UiO/CMA) RuprechtMachleidt (UI) Nicolas Michel (UT/ORNL) NN2012 San Antonio, May 31, 2012

2. Roadmap for Theory of Nuclei Main goal : To arrive at at comprehensive description of all nuclei and low-energy reactions from the basic interactions between the constituent nucleons

3. Energy scales and relevant degrees of freedom Chiral effective field theory Energy Fig.: Bertsch, Dean, Nazarewicz, SciDAC review (2007)

4. Nuclear forces from chiral effective field theory [Weinberg; van Kolck; Epelbaumet al.; Entem & Machleidt; …] [Epelbaum, Hammer, Meissner RMP 81, 1773 (2009)] CD CE Low energy constants from fit of NN data, A=3,4 nuclei, or light nuclei.

5. Effective three-nucleon forces By integrating over the third leg in infinite nuclear matter we can derive density dependent corrections to the nucleon-nucleon interaction. J. Holt . Phys.Rev.C81, 024002, (2010) CD CE

6. Coupled-cluster method (in CCSD approximation) • Scales gently (polynomial) with increasing problem size o2u4 . • Truncation is the only approximation. • Size extensive (error scales with A) •  Most efficient for doubly magic nuclei Ansatz: Correlations are exponentiated 1p-1h and 2p-2h excitations. Part of np-nh excitations included! Coupled cluster equations Alternative view: CCSD generates similarity transformed Hamiltonian with no 1p-1h and no 2p-2h excitations.

7. Toward medium-mass nucleiChiral N3LO (500 MeV) by Entem & Machleidt, NN only Binding energy per nucleon Benchmarking different methods: Our CC results for 16O agree with IT-NCSM (R. Roth et al PRL 107, 072501 (2011)) and UMOA (Fujii et al., Phys. Rev. Lett. 103, 182501 (2009) Hagen,Papenbrock, Dean, Hjorth-Jensen, Phys. Rev. Lett. 101, 092502 (2008)

8. Open Quantum Systems The Berggren completeness treats bound, resonant and scattering states on equal footing. Has been successfully applied in the shell model in the complex energy plane to light nuclei. For a review see N. Michel et al J. Phys. G 36, 013101 (2009).

9. Physics of neutron rich nuclei Physics of neutron rich nuclei Many-body Correlations: Coupled-Cluster theory Interactions: NN + 3NFs from Chiral EFT Open channels: Gamow Shell Model

10. Is 28O a bound nucleus? • Experimental situation • “Last” stable oxygen isotope 24O • 25O unstable (Hoffman et al 2008) • 26,28O not seen in experiments • 31F exists (adding on proton shifts drip line by 6 neutrons!?) 28O? Shell model (sd shell) with monopole corrections based on three-nucleon force predicts 2nd O as last stable isotope of oxygen. [Otsuka, Suzuki, Holt, Schwenk, Akaishi, PRL (2010), arXiv:0908.2607]

11. Oxygen isotopes from chiral interactions Chiral three-nucleon force at order N2LO. kf=1.05fm-1, CD = 0.2, CE = 0.71 (fitted to to the binding energy of 16O and 22O). • NN only underbinds and wrongly places driplinebeyond 28O • Inclusion of effective 3NF places dripline at25O. • Overall the odd-even staggering in the neutron rich oxygen is well reproduced. • We find 26O to unbound with respect to 24O by ~100keV, this agrees with recent experiments on 26O byE. Lunderberg et al., Phys. Rev. Lett. 108 (2012) 142503 • Wefind 28O to be unboundwith a resonancewidthof ~2MeV CD CE G. Hagen, M. Hjorth-Jensen, G. R. Jansen, R. Machleidt, T. Papenbrock, accepted for Publication in PRL, arXiv:1202.2839 (2012)

12. Oxygen isotopes from chiral interactions The effects of three-nucleon forces decompress the spectra and brings it in good agreement with experiment. We predict the newly observed resonance at ~7.5MeV in 24O to be a super position of several states with spin and parity 4+,3+,2+ Matter and charge radii for 21-24O Computed from intrinsic densities and Compared to experiment. Excited states in 24O computed with EOM-CCSD and compared to experiment

13. Evolution of shell structure in neutron rich Calcium isotopes • How do shell closures and magic numbers evolve towards the dripline? • Is the naïve shell model picture valid at the neutron dripline?

14. Calcium isotopes from chiral interactions Main Features: Total binding energies agree wellwith experimental masses. Masses for 40-52Ca are converged in 19 major shells. 60Ca weakly bound/unbound 60Ca is bound with respect to 1n emission. 61-62Ca are located right at threshold. kF=0.95fm-1, cD=-0.2, cE=0.735 Nmax= 18, hw = 26MeV G. Hagen, G. Jansen, M. Hjorth-Jensen, R. Machleid, T. Papenbrock, arXiv:1204.3612 (2012)

15. 2+ systematics in Calcium isotopes Main Features: Very nice agreement between theory and experiment. Our calculations for 2+ and 4+ in 54Ca do not point to a new magic shell closure. G. Hagen, G. Jansen, M. Hjorth-Jensen, R. Machleid, T. Papenbrock, arXiv:1204.3612 (2012) CC Exp

16. Spectra and shell evolution in Calcium isotopes Due to continuum coupling,we find an inversion of the 9/2+ and 5/2+ resonant states in 53,55,61Ca We find the ground state of 61Ca to be ½+ located right at threshold. A harmonic oscillator basis gives the naïve shell model ordering of states. Continuum coupling is crucial. New penning trap measurement of masses of 51,52Ca A. T. Gallant et al arXiv:1204.1987v1 (2012)

17. Towards nuclear reactions with coupled-cluster theory One-nucleon overlap functions Elastic scattering, capture and transfer reactions of a nucleon on/to a target nucleus with mass A is determined by the one-nucleon overlap function Microscopic definition of Spectroscopic Factors SF is the norm of the overlap function and quantifies the degree of correlations SFs are not observables and depend on the resolution scale Asymptotic properties of the one-nucleon overlap functions The overlap functions satisfy a one-body Schrodinger like equation, and outside the range of the interaction the overlap function is proportional to a single-particle wave function Dfd Bound states Scattering states

18. Quenching of spectroscopic factors for proton removal in neutron rich oxygen isotopes Spectroscopic factor is a useful tool to study correlations towards the dripline. SF for proton removal in neutron rich 24O show strong “quenching” pointing to large deviations from a mean-field like picture. G. Hagen et al Phys. Rev. Lett. 107, 032501 (2011). Strong asymmetry dependence on the SF for proton and neutron removal in neutron rich oxygen isotopes. SF~1 for neutron removal while protons are strongly correlated SF ~0.6-0.7 in 22,24,28O

19. Threshold effects and spectroscopic factors Near the scattering threshold for one-neutron decay the spectroscopic factors are Significantly influenced by the presence of The continuum. The standard shell model approximation to spectroscopic factors completely fails in this region. N. Michel et al Phys. Rev. C 75, 031301 (2007) N. Michel et al Nucl. Phys. A 794, 29 (2007) Top and middle: Bottom :

20. Elastic proton/neutron scattering on 40Ca The one-nucleon overlap function: Beyond the range of the nuclear interaction the overlap functions take the form: G. Hagen and N. Michel, in preparation (2012). Preliminary

21. Elastic proton/neutron scattering on 40Ca Differential cross section for elastic proton scattering on 40Ca. Good agreement between theory and Experiment for low-energy scattering. Preliminary G. Hagen and N. Michel In Preparation (2012).

22. Densities and radii from coupled-cluster theory We solve for the right and left ground state of the similarity transformed Hamiltonian The density matrix is computed within coupled-cluster method as: The coupled-cluster wave function factorizes to a good approximation into an intrinsic and center of mass part, where the center of mass part is a Gaussian with a fixed oscillator frequency independent of single-particle basis GH, T. Papenbrock and D. Dean et al, Phys. Rev. Lett. 103, 062503 (2009) We can obtain the intrinsic density by a deconvolution of the laboratory density B. G. Giraud, Phys. Rev. C 77, 014311 (2008) Intrinsic density Lab. density Center of mass part

23. Densities and radii from coupled-cluster theory Relative energies in 21-24O depend weakly on the resolution scale We clearly see shell structure appearing in the matter densities for 21-24O Matter and charge radii depend on the resolution scale, however relative difference which is relevant for isotope shift measurements does not

24. 23O interaction cross section (scattering off 12C target @ GSI) Experiment Ozawa et al. (2001) Kanungo et al (2011) E ≈ 900A MeV Experimental radii extracted from matter distribution within Glauber model. Main result of new measurement: 23O follows systematics; interaction cross section consistent with separation energies. R. Kanungoet al Phys. Rev. C 84, 061304 (2011)

25. Resolving the anomaly in the cross section of 23O The anamoly of 23O New measurements (R. Kanungo) of the 23O cross section and coupled cluster calculations show that 23O is not consistent with a one-neutron halo picture

26. Summary Interactions from Chiral EFT probed in nuclei CC calculations for oxygen and calcium with effects of 3NF and continuum give significant improvement in binding energy and spectra. Predict spin and parity of newly observed resonance in 24O. Predict weak sub-shell closure in 54Ca. Level ordering in the gds shell in neutron calcium is reversed compared to naïve shell model. Quenching of spectroscopic factors near neutron dripline show role of continuum induced correlations for protons Densities and cross sections from coupled-cluster theory help resolve long-standing anomoly of 23O

27. Ab initio description of proton-halo state in 17F • Separation energy of halo state is only 105 keV • Continuum has to be treated properly • Focus is on single-particle states • Previous study: shell model in the continuum with16O core • [K. Bennaceur, N. Michel, F. Nowacki, J. Okolowicz, M. Ploszajczak, Phys. Lett. B 488, 75 (2000)]

28. Bound states and resonances in 17F • Single-particle basis consists of bound, resonance and scattering states • Gamow basis for s1/2 d5/2 and d3/2 single-particle states • Harmonic oscillator states for other partial waves • Computation of single-particle states via “Equation-of-motion CCSD” • Excitation operator acting on closed-shell reference • Here: superposition of one-particle and 2p-1h excitations • Gamow basis weakly dependent on oscillator frequency • d5/2 not bound; spin-orbit splitting too small • s1/2 proton halo state close to experiment [G. Hagen, TP, M. Hjorth-Jensen, Phys. Rev. Lett. 104, 182501 (2010)]

29. Variation of cutoff probes omitted short-range forces 17F • Proton-halo state (s1/2) only weakly sensitive to variation of cutoff • Spin-orbit splitting increases with decreasing cutoff [G. Hagen, TP, M. Hjorth-Jensen, Phys. Rev. Lett. 104, 182501 (2010)]