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Physics Letters B 769 (2017) 227–232

Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

Resonance and continuum Gamow shell model with realistic nuclear

forces

Z.H. Sun, Q. Wu, Z.H. Zhao, B.S. Hu, S.J. Dai, F.R. Xu

∗

School of Physics, and State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, China

a r t i c l e i n f o a b s t r a c t

Article history:

Received

11 November 2016

Received

in revised form 21 March 2017

Accepted

21 March 2017

Available

online 30 March 2017

Editor:

J.-P. Blaizot

Keywords:

Realistic

nuclear forces

Gamow

shell model

Unstable

nuclei

Spectra

Resonance

Continuum

Starting from realistic nuclear forces, we have developed a core Gamow shell model which can describe

resonance and continuum properties of loosely-bound or unbound nuclear systems. To describe properly

resonance and continuum, the Berggren representation has been employed, which treats bound, resonant

and continuum states on equal footing in a complex-momentum (complex-k) plane. To derive the model-

space

effective interaction based on realistic forces, the full

Q -box folded-diagram renormalization has

been, for the ﬁrst time, extended to the nondegenerate complex-k space. The CD-Bonn potential is

softened by using the V

low-k

method. Choosing

O as the inert core, we have calculated sd-shell neutron-

rich

oxygen isotopes, giving good descriptions of both bound and resonant states. The isotopes

25,26

O are

calculated to be resonant even in their ground states.

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

Resonance is a general phenomenon happening in classic or

quantum systems. It plays a special role in weakly-bound or un-

bound

quantum systems. An unbound quantum system, such as

atomic cluster or unbound nucleus, can emerge in the form of in-

trinsic

resonance. Nuclear resonances are usually followed by par-

ticle

emissions as the Gamow quantum tunneling. Weakly-bound

or unbound nuclei are complex open quantum systems (OQS) in

which the coupling to the scattering continuum is crucial and

should be properly treated. A small uncertainty in modeling would

change the conclusions of physics.

the standard shell model (SM), harmonic oscillator (HO)

wave functions are always bound and localized, while a loosely-

bound

nucleus has small separation energy and large spatial

spread, in which the continuum plays a critical role. To over-

come

the shortcoming of the conventional SM, the continuum

shell model (CSM) [1–3] has been developed, taking into account

the continuum effect by projecting the model space onto the sub-

spaces

of bound and scattering states in a real-energy basis. The

continuum effect has also been well treated in the continuum cou-

pled

cluster [4] and the continuum-coupled shell model [5].

The

Gamow resonance is in fact a time-dependent problem, as-

sociated

with a decaying process. However, the exact treatment of

the time-dependent problem is diﬃcult. Berggren generalized the

Corresponding author.

E-mail

address: frxu@pku.edu.cn (F.R. Xu).

time-independent Schrödinger equation to a complex-k plane, giv-

ing

single-particle (SP) bound states, unbound resonant states and

nonresonant continuum states [6]. The three types of the SP states

construct a complete set of basis states, called the Berggren en-

semble

[6]. Using the Berggren basis, the so-called Gamow shell

model (GSM) has been advanced recently. With phenomenologi-

cal

interactions, the GSM has been successfully applied to nuclear

structure calculations [7–10].

is pursued currently to perform the ﬁrst-principles calcula-

tions

of nuclear structure. Such calculations require two funda-

mental

elements: (i) using realistic nuclear forces and (ii) rigor-

ously

treating many-body correlations. The shell model which is

beyond mean-ﬁeld approach provides a good platform to handle

many-body correlations. Starting from realistic forces, traditional

core shell model [11,12] or no core shell model (NCSM) calcula-

tions

[13] have obtained great success, while the GSM calculation

faces big challenges. For example, the wave functions of resonant

SP states are not square integrable, which complicates severely the

treatments of the complex non-Hermitian Hamiltonian and other

mechanical quantities. It is even more challenging to establish a

GSM based on realistic nuclear forces with an inert core. In this

case, one has to strive to build the model-space effective interac-

tion

in the complex-k basis. Nevertheless, the realistic core GSM

(CGSM) calculation has been proposed with limit to two- or three-

particle

systems [14,15]. The effective interaction was built by us-

ing

the degenerate

Q -box approach but neglecting folded diagrams

http://dx.doi.org/10.1016/j.physletb.2017.03.054

0370-2693/

SCOAP

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