1. Resonance and continuum in atomic nuclei
Furong Xu
I. Model
Core Gamow Shell Model (CGSM) with realistic nuclear forces
(resonance + continuum)
II. Calculations
Neutron-rich oxygen isotopes
Excited-state spectra
Stockholm June 7, 2017

2. γ-ray spectra Spectra of resonance states
Energies and resonance widths against particle emissions
188Pb: prolate and oblate bands
J. Pakarinen et al., PRC 72, (R) (2005)
N. Michel, W. Nazarewicz, J. Okolowicz, M. Ploszjczak, Nucl. Phys. A 752, 335c (2005)

3. P Q
Core
A time-dependent many-body problem

4. Gamow Shell Model
T. Berggren, Nucl. Phys. A109 (1968) 265 Single-particle basis in complex-k plane describe bound, resonance and scattering on equal footing.
The radial wave function
Outgoing solution at large distance
Orthogonality and Completeness
Discretized

5. Woods-Saxon potential, CD-Bonn, 16O core
Amplitude small

6. R.J. Liotta et al., PLB 367, 1 (1996)… used Berggren basis to describe single-particle resonance in nuclei;
later for two-particle resonance (Betan et al., PRL 89, (2002)
using phenomenological potential
For many-body systems

7. 4He core Michel, Nazarewicz, Ploszajczak, Rotureau et al., 2003--
phenomenological potential
Michel, Nazarewicz, Płoszajczak, Vertse,
JPG 36 (2009)
4He core

8. Hagen, Hjorth-Jensen et al., PRC 73, 064307 (2006): Core GSM
with realistic forces, but neglecting Q-box, for two-particle systems (e.g., 18O)
later Tsukiyama Hjorth-Jensen, Hagen, PRC 80, (R) (2009):
improving by using Q-box but no folded-diagrams.
Papadimitriou et al. , Phys. Rev. C 88, (2013): realistic forces
Ab initio no-core Gamow shell model for light nuclei
Tritium [`trɪtɪəm], hydrogen

9. Gamow shell model with an inert core
Start from realistic forces;
Take a double magic core
Q-box + folded diagrams （MBPT）
3. Calculate resonance spectra
CD-Bonn

10. Chiral EFT 2N forces 3N forces 4N forces Leading Order
Next-to Leading Order
Power counting ：
Next-to-Next-to Leading Order
Next-to-Next-to-Next-to Leading Order

11. Realistic nuclear forces Gamow shell model calculations
CGSM based on realistic nuclear forces
Realistic nuclear forces Gamow shell model calculations
Taking a doubly closed core
𝑽 𝒍𝒐𝒘 𝒌 or SRG
To remove hard core,
but still keep good descriptions of NN scattering phase shifts
Bare forces:
Strong repulsion，slow convergence Q
Non-degenerate extended Kuo-Krenciglowa folded-diagram method (EKK) by Takayanagi, NPA 852, 61 (2011);
See Tsunoda’s talk on Monday

12. For each given partial wave
Model space ……
d3/2 channel
g9/2 …
Continuum
1p1/2, 2p1/2…
Continuum
f5/2 …
1p3/2, 2p3/2…
f7/2…
1d3/2, 2d3/2 …
For each given partial wave
0d3/2
0d3/2
Berggren Completeness Relation
1s1/2
1s1/2
0d5/2
0d5/2
Bound
Q-box folded diagrams in complex-k basis
0p1/2
0p3/2 2
0s1/2

13. Convergence against discretization number N_L
discretization [dis’kr:tai’zesen]

14. Convergences of spectroscopic calculations

15. CD-Bonn CGSM, compared with conventional H.O. SM
Hard cutoff Λ=2.6 fm-1 to reduce 3NFs

16. (Λ=2.6 fm-1) ?
Y. Kondo et al., PRL 116, (2016)

17. Binding energies, one-neutron separation energies
3NFs
0d3/2 i i
e=0.0
1s1/2 i i i
0d5/2
i (MeV)
Expt. SPE’s
WS SPE’s
[Extracted from 17O,
by Michel et al., PRC 67, (2003)]

18. 3NFs are important for binding energy calculations
Bogner et al., PRL 113, (2014): In-medium SRG, N3LO, 3NF=induced / initial (N2LO)
Otsuka et al., PRL 105, (2010): SM, N3LO, N2LO 3NF
G. Hagen et al., PRL 108, (2012): Continuum CC, N3LO, N2LO 3NF

19. 22O core (N=14 closed shells ) 3NF effects
One can adjust WS parameters by fitting 23O s.p. energies

20. Realistic nuclear forces (CD Bonn)
Summary
Realistic nuclear forces (CD Bonn)
We have developed the full Q-box folded diagrams to nondegenerate complex-k space, which includes contributions from core polarization and excluded space.
Renormalization by Vlow-k
Many-body solutions by CGSM
Successfully applied to excited-state spectra of weakly-bound or unbound oxygen isotopes.

21. Also thanks for discussions with:
Group members
Also thanks for discussions with:
J. Vary, Rup Machleidt, Bruce Barrett, Luigi Coraggio, Thomas Papenbrock,
Herbert Muether, B.A. Brown, Witek Nazarewicz, R. Liotta, J.C. Pei ……

22. Stockholm June 7, 2017 22