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

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, 011304(R) (2005) N. Michel, W. Nazarewicz, J. Okolowicz, M. Ploszjczak, Nucl. Phys. A 752, 335c (2005)

P Q Core A time-dependent many-body problem

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

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

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, 042601 (2002) using phenomenological potential For many-body systems

4He core Michel, Nazarewicz, Ploszajczak, Rotureau et al., 2003-- phenomenological potential Michel, Nazarewicz, Płoszajczak, Vertse, JPG 36 (2009) 013101 4He core

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, 051301 (R) (2009): improving by using Q-box but no folded-diagrams. Papadimitriou et al. , Phys. Rev. C 88, 044318 (2013): realistic forces Ab initio no-core Gamow shell model for light nuclei Tritium [`trɪtɪəm], hydrogen

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

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

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

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

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

Convergences of spectroscopic calculations

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

(Λ=2.6 fm-1) ? Y. Kondo et al., PRL 116, 102503 (2016)

Binding energies, one-neutron separation energies 3NFs 0d3/2 1.06-0.089i 0.94-0.048i e=0.0 1s1/2 -3.22-0.00i -3.27-0.00i -4.14-0.00i 0d5/2 -5.31-0.00i (MeV) Expt. SPE’s WS SPE’s [Extracted from 17O, by Michel et al., PRC 67, 054311 (2003)]

3NFs are important for binding energy calculations Bogner et al., PRL 113, 142501 (2014): In-medium SRG, N3LO, 3NF=induced / initial (N2LO) Otsuka et al., PRL 105, 032501 (2010): SM, N3LO, N2LO 3NF G. Hagen et al., PRL 108, 242501 (2012): Continuum CC, N3LO, N2LO 3NF

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

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.

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 ……

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