Chiral three-nucleon force and continuum for dripline nuclei and beyond
Y. Z. Ma, F. R. Xu, L. Coraggio, B. S. Hu, J. G. Li, T. Fukui, L. De Angelis, N. Itaco, A. Gargano
CChiral three-nucleon force and continuum for dripline nuclei and beyond
Y. Z. Ma a , F. R. Xu a, ∗ , L. Coraggio b , B. S. Hu a , J. G. Li a , T. Fukui b , L. De Angelis b , N. Itaco b,c , A. Gargano b a School of Physics, and State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, China b Istituto Nazionale di Fisica Nucleare, Complesso Universitario di Monte S. Angelo, Via Cintia, I-80126 Napoli, Italy c Dipartimento di Matematica e Fisica, Universita‘ degli Studi della Campania Luigi Vanvitelli, viale Abramo Lincoln 5 - I-81100 Caserta, Italy
Abstract
Three-nucleon force and continuum play important roles in reproducing the properties of atomic nuclei around driplines. Thereforeit is valuable to build up a theoretical framework where both e ff ects can be taken into account to solve the nuclear Schr¨odingerequation. To this end, in this letter, we have expressed the chiral three-nucleon force within the continuum Berggren representation,so that bound, resonant and continuum states can be treated on an equal footing in the complex-momentum space. To reduce themodel dimension and computational cost, the three-nucleon force is truncated at the normal-ordered two-body level and limitedin the sd -shell model space, with the residual three-body term being neglected. We choose neutron-rich oxygen isotopes as thetest ground because they have been well studied experimentally, with the neutron dripline determined. The calculations have beencarried out within the Gamow shell model. The quality of our results in reproducing the properties of oxygen isotopes around theneutron dripline shows the relevance of the interplay between three-nucleon force and the coupling to continuum states. We alsoanalyze the role played by the chiral three-nucleon force, by dissecting the contributions of the 2 π exchange, 1 π exchange andcontact terms. Keywords:
Three-nucleon forces, Continuum, Gamow shell model, Unstable nuclei, Binding energy, Spectra
1. Introduction
One of the main challenges in nuclear theory today is todescribe weakly-bound and unbound isotopes which locatearound the driplines. These exotic nuclei are also focal pointsof current and next-generation rare-isotope-beam (RIB) facil-ities in experimental programs. Peculiar phenomena, such ashalo and narrow resonances, emerge close to the dripline re-gions, which make the exotic nuclei ideal laboratories to test ad-vanced many-body methods. The proton-magic oxygen chainis one of the best among these laboratories. This element is theheaviest one whose neutron dripline position has been pinneddown experimentally. The interest to study these nuclear sys-tems has been invigorated with the latest experiments on − O,which are placed beyond the dripline. O ground state has beenfound barely unbound with two-neutron separation energy ofonly −
18 keV [1], and an excited state in the unbound resonantisotope O has been observed in a recent experiment [2].Many theoretical works have shown the importance of takinginto account the three-nucleon force (3NF) in nuclear structurecalculations with realistic potentials to reproduce the positionof oxygen dripline [3, 4, 5, 6]. It is shown that 3NF can pro-vide a repulsive contribution to the binding energies of nuclei,which resolves the overbinding problem in this mass regions.Besides, it is worth pointing out that the repulsive e ff ect origi-nates mainly from the long-range 2 π exchange in 3NF [3]. Inweakly-bound and unbound nuclei, it is mandatory to consideralso the mixing with continuum states, which is crucial for the ∗ [email protected] understanding of the loose structures of these nuclei. These nu-clear systems have low particle-emission thresholds and hencestrong coupling to resonance and continuum. Resonant andnon-resonant continuum states usually have large spatial dis-tributions in their wave functions. Therefore the bound andspatially-localized harmonic oscillator (HO) basis is no longere ff ective in expressing the widely-spread resonant and contin-uum states. In the last two decades, the conventional shellmodel (SM) has been extended to include the continuum ef-fect, e.g., shell model embedded in the continuum [7, 8, 9, 10]and continuum-coupled shell model [11].An elegant treatment of the continuum e ff ect is basedon the Berggren method [12], which generalizes one-bodySchr¨odinger equation to a complex-momentum plane, creatingnaturally bound, resonant and continuum single-particle states.A many-body Hamiltonian may be written in the Berggren ba-sis and diagonalized with complex-energy many-body meth-ods. Following this approach, it has been developed the so-called Gamow shell model (GSM), which, starting from phe-nomenological interactions, has proved to be very successful[13, 14, 15, 16]. The realistic nuclear potential has also beenemployed to perform GSM calculations [17, 18], and withinthe no-core Gamow SM approach [19]. Moreover, also Gamowcoupled-cluster calculations have been successfully performedfor closed-shell nuclei and nearby [20].
2. Outline of calculations
In this work, we extend the GSM by introducing a chiral 3NFin the complex momentum Berggren space, where the contin-
Preprint submitted to Physics Letters B August 4, 2020 a r X i v : . [ nu c l - t h ] A ug um is included. More precisely, we start from the chiral two-nucleon force (2NF) derived at next-to-next-to-next-to-leadingorder (N LO) by Entem and Machleidt [21] and the three-nucleon force at next-to-next-to-leading order (N LO) whichhas been adopted in Refs. [22, 23]. Then, we derive an ef-fective Hamiltonian for a GSM-with-core calculation [18] toinvestigate the structure of oxygen isotopes.The chiral N LO 3NF consists of three components, namelythe two-pion (2 π ) exchange term V (2 π )3N , the one-pion (1 π ) ex-change plus contact term V (1 π )3N and the contact term V (ct)3N . Itshould be pointed out that the low-energy constants (LECs) c , c and c [21] which appear in V (2 π )3N are the same as in 2NFand have been fixed within the renormalization procedure ofthe two-body LECs. However, there are two LECs, c D and c E , which measure the one-pion exchange and contact terms,respectively. They cannot be constrained by two-body observ-ables, and need to be determined by reproducing observables insystems with mass number A >
2. We adopt the same values of c D = − c E = − .
34 as in Ref. [24].For the GSM calculation, we choose the doubly-magic Oas the inert core, and the A -body intrinsic Hamiltonian is brokenup into a one-body term H and a residual interaction H via theintroduction of an auxiliary one-body potential U : H = (cid:88) i < j ( p i − p j ) mA + ˆ V NN + ˆ V = A (cid:88) i = p i m + U + A (cid:88) i < j V ( i j )NN − U − p i mA − p i · p j mA + A (cid:88) i < j < k V ( i jk )3N = H + H , (1)where H = (cid:80) Ai = ( p i m + U ) describes the independent motionof nucleons. In the present calculations, we choose the Woods-Saxon (WS) potential as the auxiliary potential U , whose pa-rameters are reported in Ref. [18]. The 3NF matrix element V ( i jk )3N involves six (three initial and three final) states, whichincreases the dimension of the shell model drastically. Particu-larly for the GSM where continuum states are included, the di-mension can be terribly huge due to the inclusions of both 3NFand continuum. We perform the normal-ordering decomposi-tion [25] of the 3NF Hamiltonian in order to reduce the com-putational complexity. The final GSM Hamiltonian has a two-body form but includes the 3NF contribution. It has been shownthat the 3NF normal-ordered approximation with neglecting theresidual three-body term works well in nuclear structure calcu-lations [25]. The normal-ordered two-body term can be writtenas ˆ V (2B)3N = (cid:88) i jkl (cid:104) i j | V (2B)3N | kl (cid:105){ a † i a † j a l a k } = (cid:88) i jkl (cid:88) h ∈ core (cid:104) i jh | V | klh (cid:105){ a † i a † j a l a k } , (2)where (cid:104) i jh | V | klh (cid:105) and (cid:104) i j | V (2B)3N | kl (cid:105) are the antisymmetrized matrix elements of the 3NF and normal-ordered two-body term,respectively. a † i ( a i ) stands for the particle creation (annihila-tion) operator with respect to the nontrivial vacuum. The sym-bol { ... } means that the creation and annihilation operators inbrackets are normal-ordered.For the calculations of the sd -shell nuclei, we choose the dou-bly magic system O as the core, with its ground-state Slater-determinant as the reference state for the normal-ordering de-composition. Due to the huge computational cost of generat-ing the 3NF matrix elements, we do not let the states ( i , j , k , l )in Eq. (2) run as freely as in the 2NF calculation, but limitthem in the sd -shell model space. This limitation means that3NF e ff ects from high-lying orbits are missing. However, it hasbeen shown in Ref. [22], where the same truncation has beenadopted, that the e ff ect is not significant since the results ob-tained within this approximation agree quite reasonably withthose of an ab initio calculation. Besides the normal-orderedtwo-body term, there exist normal-ordered one and zero-bodyterms [25]. In the present work, we adopt the Woods-SaxonBerggren single-particle basis, which is expected to reproducethe experimental single-particle levels in O, thus absorbingthe normal ordered one-body term. The normal-ordered zero-body term, which is just a constant, can be absorbed in the coreenergy.We calculate antisymmetrized N LO 3NF matrix elements inthe Jacobi-HO basis in the momentum space and the detail canbe found in our previous paper [22]. Then the normal-ordered3NF two-body matrix elements are added to the N LO 2NFmatrix elements. The full matrix elements are transformed intothe Berggren basis by computing overlaps between the HO andBerggren basis wave functions (see [18] for details). We takea truncation with 22 shells for the HO basis and 20 discretiza-tion points for the Berggren continuum contour. These are thesame as in our previous work [18], and the convergence hasbeen well tested [18]. The complex-energy Berggren represen-tation [12] naturally produces bound, resonant and continuumstates, which are crucial in the descriptions of weakly-boundand unbound nuclear systems.Hamiltonian (1) is intrinsic in the center-of-mass (CoM)frame, but the wave functions are written in the laboratory coor-dinates. This means that one should still need to consider the ef-fect from the CoM motion. However, it has been observed thatthe CoM e ff ect can be neglected for low-lying states [18, 26].The Berggren basis is generated by the WS potential, includ-ing spin-orbit coupling [27]. We adopt the same WS parametersas in Ref. [18], obtaining bound 0 d / and 1 s / orbits at ener-gies − .
31 MeV and − .
22 MeV, respectively, and a resonant0 d / orbit with energy ˜ e = . − . i MeV (the eigen energyof a resonant state is written as ˜ e n = e n − i γ n /
2, with γ n beingthe resonance width). The high-lying orbits f / , / , ip / , / ( i ≥ g / , / , id / , / ( i ≥
1) and is / ( i ≥
2) are continuum par-tial waves. Since 0 d / is a narrow resonant state, playing a cru-cial role in the description of the sd -shell nuclear resonances,the coupling between the 0 d / resonance and d / continuumneeds to be treated explicitly. We choose { d / , 1 s / , 0 d / , d / -continuum } as the GSM model space with O as the core.We construct the e ff ective shell-model interaction in the2odel space { d / , 1 s / , 0 d / , d / -continuum } , within theframework of the many-body perturbation theory. More pre-cisely, we employ the so-called Q-box folded-diagram method,which was developed by Kuo and coworkers [28]. The ˆ Q -box folded-diagram method has been extended to the complex-momentum Berggren space, to derive the GSM realistic e ff ec-tive interaction that includes e ff ects from the continuum andcore polarization [18]. The Berggren model space is nondegen-erate. Therefore we use the extended nondegenerate KuoKren-ciglowa (EKK) method [29] for the ˆ Q -box calculation. Thedetails about the complex-energy ˆ Q -box folded-diagram calcu-lation can be found in Ref. [18]. The continuum e ff ect entersinto the model through both the e ff ective interaction and activemodel space.The complex symmetric non-Hermitian GSM Hamiltonian isdiagonalized in the { d / , 1 s / , 0 d / , d / -continuum } spaceby using the Jacobi-Davidson method in the m -scheme. Due tothe presence of the nonresonant continuum, the matrix dimen-sion grows dramatically when adding valence particles, whichis a challenge for diagonalizing in complex space. In the presentcalculations, we allow at most three valence particles in the con-tinuum, which can give converged results [30, 18]. The WS po-tential [18], based on the universal parameters [27], gives the0 d / energy lower than the experimental energy by 1 .
17 MeV.As discussed in [18], we shift it to reproduce the binding ener-gies of , O.
3. Results
As mentioned in the introduction, we hope that the inclu-sions of both 3NF and continuum can improve the calculationsof weakly-bound and unbound nuclei. We take neutron-richoxygen isotopes (up to beyond the neutron dripline) as the testground. Figure 1 shows the calculated − O ground-state en-ergies with respect to the O core. The comparison with theresults from other calculations shows that both 3NF and con-tinuum play important roles in reproducing the experimentalbinding energies, particularly in the vicinity of the dripline. Theconventional SM calculations with 3NF but without the contin-uum [3] reproduce the experimental neutron-dripline nucleus(i.e., O) correctly, but the calculated energies deviate from theexperimental data. The continuum coupled-cluster (CC) calcu-lations with a density-dependent e ff ective 3NF [31] provide asimilar result, reproducing the dripline qualitatively. The GSMcalculations with only 2NF cannot reproduce the observed neu-tron dripline, systematically overestimating the ground-stateenergies.To see the continuum and 3NF e ff ects within the presentmodel, we have performed the shell-model calculation withoutthe inclusions of the continuum and 3NF. In this calculation, theWS potential is solved in the HO basis (instead of the Berggrenbasis), which gives non-continuum discrete WS single-particlestates, meaning no continuum included. Within the discrete WSbasis and the shell model space { d / , s / , d / } as in GSM,the shell-model calculation (including Q -box folded diagrams)is performed. The results are shown in Fig. 1, indicated by Figure 1: Calculated − O ground-state energies with respect to the O core,compared with experimental data and other calculations: conventional SM(HO)with 3NF but without continuum [3] and continuum coupled cluster (CC) witha density-dependent e ff ective 3NF [31]. SM(HO) and SM(WS) stand for theshell-model calculations performed in the HO and WS bases, respectively. Ifnot specified, NN and 3N indicate N LO (NN) and N LO (NNN), respectively.The inset shows the 3NF contributions from two-pion exchange indicated by“2 π ”, one-pion exchange plus contact indicated by “1 π ” and contact term indi-cated by “contact”. SM(WS) to be distinguished from the SM in the HO basis in-dicated by SM(HO). Note that in the SM(HO) calculation ofRef. [3] 3NF is included. Also, it should be pointed out that inRef. [3] the two-body N LO potential has been renormalizedby the V low − k procedure, at variance with the present calcula-tions. We see that the continuum coupling lowers the ground-state energy while 3NF gives an opposite e ff ect. The recentexperiment observed that O (which is beyond the dripline)is barely unbound with a resonant ground-state energy of only18 keV above threshold [1]. The present calculation, obtainedby taking into account continuum and 3NF e ff ects, reproducesthe experimental observation. Table 1 lists the calculated two-neutron separation energies for the dripline nucleus O and be-yond. We see that the GSM with the inclusion of 3NF improvesthe calculations of the separation energies significantly, com-pared with data.
Table 1: Two-neutron separation energies S , calculated by GSM with onlytwo-body (NN) force at N LO and with the inclusion of three-body (3N) forceat N LO, compared with data [32, 1]. S n (MeV) NN NN +
3N Expt. O 9 .
110 6 .
924 6 . O 6 .
254 3 .
259 3 . O 3 . − . − . As shown in Fig. 1, the results of GSM calculations providea clear di ff erence in the reproduction of ground-state energieswhen including or excluding 3NF. This di ff erence enlarges withthe increase of the number of valence neutrons. The 3NF givesstrong repulsive contributions to the ground-state energies of3he nuclei in the vicinity of the dripline. Figure 2 shows theneutron e ff ective single-particle energies (ESPE), see [23] forthe definition of the ESPE. We see that the neutron ESPE’s cal-culated without 3NF (blue curves in Fig. 2) drop persistentlywith increasing the neutron number, while the 3NF pushes upthe single-particle orbitals (red curves). In particular, the 0 d / and 1 s / ESPEs are pushed up significantly by the 3NF, whilethe 3NF does not change so much the 0 d / ESPE for N ≤ N ≤
14 (i.e., lighter than O), the1 s / and 0 d / are almost empty. Hence their binding energiesare less a ff ected by the 3NF, see Fig. 1. From O ( N = s / and 0 d / orbitals start to be occupied. Therefore, the 3NFe ff ects become significant, which leads to remarkable improve-ments in the reproduction of the observed binding energies andtwo-neutron separation energies for − O (see Fig. 1 and Ta-ble 1). Moreover, in Fig. 2, we see that the 3NF enhances theneutron sub-shell closures at N =
14 and 16.In the inset of Fig. 1, we dissect the 3NF e ff ect in − O,where the 3NF e ff ects are significant. The two-pion exchange V (2 π )3N and the contact term V ( ct )3N supply repulsive contributionsto the binding energy, indicated by bars above the 2NF GSMenergy, while the one-pion exchange plus contact term V (1 π )3N produces an attractive contribution, displayed by a bar belowthe 2NF GSM energy. We also see that, besides the two-pionexchange V (2 π )3N which corresponds to the long-range 3NF in-teraction, the one-pion exchange and contact terms also havenon-negligible contributions. The V (1 π )3N and V ( ct )3N contributionsare very close in absolute value but opposite in signs. Conse-quently, their net e ff ect is almost canceled out, implying thatthe two-pion exchange is responsible for the 3NF strong repul-sive e ff ect. As can be seen in the inset of Fig. 1, the role of thetwo-pion exchange becomes more important when increasingthe neutron number, while the contributions from the one-pionexchange and contact components are almost unchanged. Figure 2: Neutron e ff ective single-particle energies in the oxygen chain. Thered solid and blue dot-dash curves are for the calculations with and without3NF, respectively. The numbers of N = ,
16 indicate the neutron sub-shellclosures.
The GSM can describe both bound and unbound states on equal footing. Figure 3 displays the calculated spectra of theneutron-rich bound isotopes − O. We see that 3NF improvesagreements with experimental data. The N =
14 sub-shell clo-sure at O is clearly seen with a large excitation energy of the2 + state. Figure 3: Calculated spectra for − O by the GSM with 2NF (NN) only andwith 3NF included (NN + − O. The experimental data are takenfrom [37, 2, 38, 1].
Our interests are in weakly-bound and unbound nuclei. Fig-ure 4 shows the spectroscopic calculations with and without the3NF e ff ects, compared with existing experimental observations,for the oxygen isotopes near the neutron dripline. In O, theobserved resonant excited 2 + and 1 + states are reproduced, andthe 3NF improves the calculations in both the excitation ener-gies and resonance widths. In O, the GSM calculation withthe N LO 2NF gives over-bound binding energy, while the in-clusion of the N LO 3NF describes well the unbound resonantproperty of the ground state with the resonance width agreeingwell with the experimental measurement. A new excited state4ith a possible configuration of J π = / + was reported re-cently in the experiment [2]. The present calculations supportthe experimental suggestion.As discussed above, the GSM calculation with 3NF can welldescribe the ground state of the observed unbound O beyondthe dripline. As regards the unbound O, in a recent exper-imental work, a (2 + ) state has been observed at an excitationenergy of 1 .
28 MeV, while the experimental resolution has notbeen able to establish the resonance widths. Our results with3NF provide, besides an unbound ground state with a resonancewidth of 15 keV, a 2 + state that is lower in energy than the ob-served one and whose resonance width is 97 keV. We also pre-dict a second 2 + state at ∼ O has been calculated in the frame-work of GSM by using a phenomenological two-body resid-ual interaction [39]. The results in Ref. [39] predicts a barelybound ground state and a 2 + state at E x ≈ .
08 MeV with a res-onance width of ∼
27 keV. Therefore, it is worth pointing outthat the above results evidence that the interplay between con-tinuum and 3NF e ff ects reveals itself crucial to reproduce theoxygen dripline correctly.
4. Conclusions
In conclusion, we have been successful in extending the chi-ral N LO three-body interaction to the complex-momentumBerggren space in which the resonance and continuum are in-cluded. To reduce the computation task, the 3NF is normal-ordered. With the chiral N LO 2NF and N LO 3NF, we haveperformed the Gamow shell-model calculations for neutron-rich oxygen isotopes as the test ground. The calculationswith the inclusions of both 3NF and continuum reproduce thedripline position and the unbound properties of the nuclei be-yond the dripline. The present calculations explain well theexperimental resonance widths of the O excited states andpredict the particle-emission widths for other resonant statesof the isotopes. The 3NF two-pion exchange and the contactterm have repulsive contributions to binding energies, while theone-pion exchange has an attractive contribution. The one-pionexchange and the contact term have similar strengths but oppo-site e ff ects (attractive and repulsive, respectively). The contri-bution of the two-pion exchange increases with increasing theneutron number in the oxygen chain, playing a crucial role inthe descriptions of the data. The 3NF significantly pushes upthe 0 d / and 1 s / orbits, which are heavily occupied in theisotopes beyond the dripline O. Therefore, 3NF becomes im-portant in the dripline region of the sd shell. The recent exper-imental observation of the barely unbound O is reproducedreasonably, with a small unbound two-neutron separation en-ergy, which is close to the experimental datum. The compar-ison with the other calculations has evidenced the importanceof including both 3NF and continuum e ff ects for the class ofisotopes under investigation.
5. Acknowledgements
Valuable discussions with Nicolas Michel, Simin Wangand Zhonghao Sun are gratefully acknowledged. This workhas been supported by the National Key R&D Program ofChina under Grant No.2018YFA0404401, the National Natu-ral Science Foundation of China under Grants No.11835001,No. 11921006, No. 11575007, and No.11847203; and theCUSTIPEN (China-US Theory Institute for Physics with Ex-otic Nuclei) funded by the US Department of Energy, O ffi ceof science under Grant No. DE-SC0009971. We acknowledgethe High-performance Computing Platform of Peking Univer-sity for providing computational resources. [1] Y. Kondo, T. Nakamura, R. Tanaka, R. Minakata, S. Ogoshi, N. A.Orr, N. L. Achouri, T. Aumann, H. Baba, F. Delaunay, P. Doornenbal,N. Fukuda, J. Gibelin, J. W. Hwang, N. Inabe, T. Isobe, D. Kameda,D. Kanno, S. Kim, N. Kobayashi, T. Kobayashi, T. Kubo, S. Leblond,J. Lee, F. M. Marqu´es, T. Motobayashi, D. Murai, T. Murakami, K. Muto,T. Nakashima, N. Nakatsuka, A. Navin, S. Nishi, H. Otsu, H. Sato,Y. Satou, Y. Shimizu, H. Suzuki, K. Takahashi, H. Takeda, S. Takeuchi,Y. Togano, A. G. Tu ff , M. Vandebrouck, K. Yoneda, Phys. Rev. Lett. 116(2016) 102503.[2] M. D. Jones, K. Fossez, T. Baumann, P. A. DeYoung, J. E. Finck,N. Frank, A. N. Kuchera, N. Michel, W. Nazarewicz, J. Rotureau, J. K.Smith, S. L. Stephenson, K. Stiefel, M. Thoennessen, R. G. T. Zegers,Phys. Rev. C 96 (2017) 054322.[3] T. Otsuka, T. Suzuki, J. D. Holt, A. Schwenk, Y. Akaishi, Phys. Rev. Lett.105 (2010) 032501.[4] H. Hergert, S. Binder, A. Calci, J. Langhammer, R. Roth, Phys. Rev. Lett.110 (2013) 242501.[5] S. K. Bogner, H. Hergert, J. D. Holt, A. Schwenk, S. Binder, A. Calci,J. Langhammer, R. Roth, Phys. Rev. Lett. 113 (2014) 142501.[6] G. R. Jansen, J. Engel, G. Hagen, P. Navratil, A. Signoracci, Phys. Rev.Lett. 113 (2014) 142502.[7] K. Bennaceur, F. Nowacki, J. Okołowicz, M. Płoszajczak, Nucl. Phys. A651 (3) (1999) 289.[8] J. OkoÅowicz, M. PÅoszajczak, I. Rotter, Phys. Rep. 374 (4) (2003) 271.[9] J. Rotureau, J. Okołowicz, M. Płoszajczak, Phys. Rev. Lett. 95 (2005)042503.[10] A. Volya, V. Zelevinsky, Phys. Rev. Lett. 94 (2005) 052501.[11] K. Tsukiyama, T. Otsuka, R. Fujimoto, Prog. Theor. Exp. Phys. 2015 (9)(2015) 093D01.[12] T. Berggren, Nucl. Phys. A 109 (2) (1968) 265.[13] R. Id Betan, R. J. Liotta, N. Sandulescu, T. Vertse, Phys. Rev. Lett. 89(2002) 042501.[14] N. Michel, W. Nazarewicz, M. Płoszajczak, K. Bennaceur, Phys. Rev.Lett. 89 (2002) 042502.[15] N. Michel, W. Nazarewicz, M. Płoszajczak, J. Okołowicz, Phys. Rev. C67 (2003) 054311.[16] N. Michel, W. Nazarewicz, M. P, T. Vertse, J. Phys. G 36 (1) (2008)013101.[17] K. Tsukiyama, M. Hjorth-Jensen, G. Hagen, Phys. Rev. C 80 (2009)051301.[18] Z. H. Sun, Q. Wu, Z. H. Zhao, B. S. Hu, S. J. Dai, F. R. Xu, Phys. Lett. B769 (2017) 227.[19] G. Papadimitriou, J. Rotureau, N. Michel, M. Płoszajczak, B. R. Barrett,Phys. Rev. C 88 (2013) 044318.[20] G. Hagen, D. J. Dean, M. Hjorth-Jensen, T. Papenbrock, Phys. Lett. B656 (4) (2007) 169.[21] D. R. Entem, R. Machleidt, Phys. Rev. C 66 (2002) 014002.[22] T. Fukui, L. De Angelis, Y. Z. Ma, L. Coraggio, A. Gargano, N. Itaco,F. R. Xu, Phys. Rev. C 98 (2018) 044305.[23] Y. Z. Ma, L. Coraggio, L. De Angelis, T. Fukui, A. Gargano, N. Itaco,F. R. Xu, Phys. Rev. C 100 (2019) 034324.[24] P. Navr´atil, V. G. Gueorguiev, J. P. Vary, W. E. Ormand, A. Nogga, Phys.Rev. Lett. 99 (2007) 042501.[25] R. Roth, S. Binder, K. Vobig, A. Calci, J. Langhammer, P. Navr´atil, Phys.Rev. Lett. 109 (2012) 052501.
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