Continuum and three-nucleon force in Borromean system: The 17Ne case
Y. Z. Ma, F. R. Xu, N. Michel, S. Zhang, J. G. Li, B. S. Hu, L. Coraggio, N. Itaco, A. Gargano
CContinuum and three-nucleon force in Borromean system: The Ne case
Y. Z. Ma a , F. R. Xu a, ∗ , N. Michel b,c , S. Zhang a , J. G. Li a , B. S. Hu a , L. Coraggio d , N. Itaco d,e , A. Gargano d a School of Physics, and State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, China b Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China c School of Nuclear Science and Technology, University of Chinese Academy of Sciences, Beijing 100049, China d Istituto Nazionale di Fisica Nucleare, Complesso Universitario di Monte S. Angelo, Via Cintia, I-80126 Napoli, Italy e Dipartimento di Matematica e Fisica, Universita‘ degli Studi della Campania Luigi Vanvitelli, viale Abramo Lincoln 5 - I-81100 Caserta, Italy
Abstract
Starting from chiral two-nucleon (2NF) and chiral three-nucleon (3NF) potentials, we present a detailed study of Ne, a Borromeansystem, with the Gamow shell model which can capture continuum e ff ects. More precisely, we take advantage of the normal-ordering approach to include the 3NF and the Berggren representation to treat bound, resonant and continuum states on equalfooting in a complex-momentum plane. In our framework, 3NF is essential to reproduce the Borromean structure of Ne, whilethe continuum is more crucial for the halo property of the nucleus. The two-proton halo structure is demonstrated by calculating thevalence proton density and correlation density. The astrophysically interesting 3 / − excited state has its energy above the thresholdof the proton emission, and therefore the two-proton decay should be expected from the state. Keywords:
Continuum, Three-nucleon force, Gamow shell model, Borromean system, Halo structure
1. Introduction
One of the main challenges in modern nuclear theory is to un-derstand nuclei close to proton and neutron driplines, which isalso the focal point of current and next-generation rare-isotope-beam (RIB) facilities in experimental programs. Due to ex-treme proton-to-neutron imbalance, the nuclei exhibit exoticphenomena such as halo and Borromean structures, makingthem become ideal laboratories to study modern nucleon in-teractions, e.g., nuclear forces from chiral e ff ective field theory[1, 2, 3] and to test the reliability of existing many-body meth-ods. One may think of the neutron halos in , He [4, 5], Be[6] and Li [7]. The neutron dripline has been studied from theexperimental point of view in the A (cid:39)
20 region, as it has beenreached in the chains of fluorine and neon isotopes [8]. Oneis therein at the limit of experimental possibilities, as the neu-tron dripline is out of reach for heavier isotopes. Conversely,because of the Coulomb repulsion, the proton dripline is not sofar from the stability line compared with the neutron dripline,and has been reached experimentally up to Z ≈
90 more than adecade ago [9]. However, when focusing on light proton-richnuclei where the Coulomb barrier is not so high and the contin-uum e ff ect is prominent, interesting phenomena (like halo andBorromean structures) may emerge, similarly to the neutron-rich side of the nuclear landscape. Among these nuclei, Neis particular and attracts a lot of interest, both theoretically[10, 11, 12] and experimentally [13, 14, 15].The Ne nucleus is located on the proton dripline and is aBorromean nucleus with an unbound subsystem F. Ne maybear a similarity to the two-neutron halo nucleus He which ∗ [email protected] can be described in a cluster picture of the three-body He +
2n system. The question about two-proton halo in Ne wasformulated in Ref [16] but still not confirmed [10, 13, 11].This problem is enhanced by the prediction that Ne mightbe the only candidate to unveil the two-proton halo structure,as the Coulomb interaction should impose stronger confine-ment in heavier proton-rich systems [14]. Moreover, the two-proton decay from Ne low-lying states to O (which is a“waiting point” in the astrophysical rp-process) has a strongconnection to the hot Carbon-Nitrogen-Oxigen (CNO) cycle[17, 18, 11, 14], so that the theoretical study of Ne is alsointeresting for astrophysics. Within this context, the radioactiveproperties of Ne have been studied, using a three-body model[11, 12] and in the shell model embedded in the continuum [15].In theoretical studies of weakly-bound and unbound nuclei,due to the low particle-emission thresholds, it is crucial to con-sider the strong coupling to resonance and continuum. Cou-pling to the continuum is indeed responsible of the bound char-acter of Borromean nuclei, for example. An elegant treatmentof resonance and continuum e ff ects is based on the Berggren ba-sis [19]. The Berggren basis generalizes one-body Schr¨odingerequation to a complex- k plane and generates bound, resonantand non-resonant continuum single-particle (SP) states natu-rally. By representing the nuclear Hamiltonian in the Berggrenbasis, one can conveniently calculate the many-body states ofweakly bound and unbound nuclei, as they become the eigen-states of a complex symmetric matrix in the Berggren formal-ism. This is the object of the Gamow shell model (GSM) whichhas been developed to study the nuclear structure of weaklybound and resonance nuclei. Phenomenological interactionshave been used along with GSM [20, 21, 22, 23, 24, 25]. Morerealistic approaches have been introduced in GSM, e.g., the Preprint submitted to Physics Letters B August 6, 2020 a r X i v : . [ nu c l - t h ] A ug o-core GSM [26], the core GSM with realistic nuclear forces[27, 28, 29, 30, 31]. The Berggren basis has also been appliedin the coupled cluster (CC), named the complex CC framework,to study helium [32] and neutron-rich oxygen [33] isotopes, aswell as the positive-parity states of F [34]. In Ref. [35], theBerggren basis was applied in the in-medium similarity renor-malization group (Gamow IM-SRG).Meanwhile, many theoretical works [36, 37, 38, 39, 40] haveshown the importance of taking into account the three-nucleonforce (3NF) in nuclear structure calculations with realistic po-tentials. It is worth pointing out that the chiral perturbationtheory (ChPT) generates nuclear two-, three- and many-bodyforces on an equal footing, since most interaction vertices ap-pear in 3NF also occur in 2NF. One significant e ff ect from3NF is that it can provide a strong repulsive contribution [37]which can resolve the long-standing over-binding problem inheavy nuclei with most realistic two-body forces. 3NF hasbeen recently introduced in GSM and applied to to the studyof neutron-rich nuclei [31]. We have recognized the necessityof combining continuum and 3NF together for understandingnuclei around drip line, in order to get reliable results and pre-dictions for weakly-bound Borromean systems.The aim of this work is then to study the Ne isotope inGSM, for which both chiral two-nucleon force (2NF) and 3NFwill be used in the Hamiltonian. 2NF is considered at next-to-next-to-next-to-leading order (N LO), whereas 3NF is cal-culated at next-to-next-to-leading order (N LO). We can thenproperly investigate the combined roles of 2NF, 3NF and con-tinuum degrees of freedom in the halo formation occurring in Ne.
2. Outline of calculations
We start from the chiral N LO potential derived by En-tem and Machleidt [3] as the 2NF and a chiral N LO 3NF asthe 3NF. The chiral N LO 3NF consists of three components,namely the two-pion (2 π ) exchange V (2 π )3N , the one-pion (1 π ) ex-change plus contact V (1 π )3N and the contact term V (ct)3N . It shouldbe pointed out that the low-energy constants (LECs) c , c and c appearing in V (2 π )3N are the same as those in 2NF, so their val-ues are already fixed during the construction of the N LO two-nucleon potential. However, there are still two LECs c D and c E characterizing one-pion exchange and contact term, whichcannot be constrained by two-body observables and need to bedetermined by reproducing observables in systems with massnumber A >
2. We adopt the same values of c D = − c E = − .
34 given in Refs. [41, 42, 31].For the GSM calculation, similar to realistic shell model(RSM) [43, 44], an auxiliary one-body potential U is introducedinto Hamiltonian, which decomposes the intrinsic Hamiltonianof an A-nucleon system into a one-body term H and a residual interaction H , as follows, 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)with H = (cid:80) Ai = ( p i m + U ) having a one-body form and describingthe independent motion of the nucleons. In the present calcu-lations, U is taken as the Woods-Saxon (WS) potential of the O core. Due to the explosive dimension of the shell modelwith full inclusion of 3NF, particularly when continuum statesare included, we employed the normal-ordering approximation[45, 41, 31] to introduce the contribution of 3NF into our cal-culations. It has been shown that the 3NF normal-ordering ap-proximation with neglecting the residual three-body term workswell in nuclear structure calculations [45]. The normal-orderedtwo-body term can be written asˆ 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 antisymmetrizedmatrix elements of the 3NF and the normal-ordered two-bodyterm of the 3NF, respectively. a † i ( a i ) stands for the particlecreation (annihilation) operator with respect to the nontrivialvacuum. The symbol { ... } means that the creation and annihila-tion operators in brackets are normal-ordered. For the calcula-tion of Ne, the closed-shell nucleus O is chosen as the core,with its ground-state Slater determinant as the reference statefor the normal-ordering decomposition. We firstly calculate an-tisymmetrized N LO 3NF matrix elements in the Jacobi-HO(harmonic oscillator) basis in the momentum space, and thenthe normal-ordered 3NF two-body matrix elements are addedto the N LO 2NF matrix elements. The full matrix elementsare transformed into the Berggren basis by computing overlapsbetween the HO and Berggren basis wave functions. More de-tails can be found in Ref. [31]. The maximum shell number N shell = n + l + =
23 with a limit of l ≤ n and l standingfor the HO node and orbital angular momentum, respectively) istaken as the truncation in the HO basis, and 40 discretized scat-tering states are used in Berggren continuum contours. Conver-gence has been tested in Ref.[29]. In principle, one should alsoconsider the e ff ect of center-of-mass (CoM) motion because thewave functions are written in the laboratory coordinates, thoughHamiltonian (1) is purely intrinsic. However, it has been ob-served that the CoM e ff ect can be neglected for low-lying states[29, 35].2ince we choose the closed-shell O as core, the proton andneutron SPs in the model space actually correspond to the en-ergy spectra of F and O. The well bound 0 p / , d / , s / and 0 d / orbitals are selected to be the active neutron modelspace. Due to the unbound character of the ground state of F,whose ground 1 / + and first 5 / + excited states are respectivelydepicted by the 1 s / and 0 d / proton orbitals, it is necessaryto include the 1 s / and 0 d / orbitals, and associated scatter-ing states of the same partial waves in the proton model space.As the SP excited 3 / + state of F is high in energy, it is notnecessary to include the proton d / partial wave in the modelspace.Usually the Berggren basis is generated by the WS poten-tial including a spin-orbit coupling. In this work, the WS pa-rameters are based on the ‘universal’ one given in Ref. [46].However, to obtain the inverse positions of the proton 1 s / and 0 d / orbitals in F, we reduce the spin-orbital couplingstrength to 2 MeV and reduce the depth parameter by 2 MeVfor the proton WS potential. In physics, the need to re-duce the spin-orbit coupling strength of the WS potential isdue to the fact that 3NF contributes largely to the spin-orbitcomponent of the residual interaction (see comments in Refs.[41, 42]). Then, we obtain proton single-particle energies:˜ e = . − . i MeV for 1 s / , and ˜ e = . − . i MeV for 0 d / (˜ e n = e n − i γ n /
2, and γ n standing for the res-onance width), which are close to experimental single-particlelevels in F. Therefore, the choice of the model space is theneutron bound ν { p / , d / , s / , d / } and the proton res-onances π { s / , d / } plus continua π { s / , d / } . To speed upthe convergences of many-body calculations, we soften the 2NFby using V low- k method [47], with a cuto ff Λ = . − sameas in Ref. [29]. Next, we construct the e ff ective interactionin the model space, using the many-body perturbation theory.More precisely, we exploit the ˆ Q -box folded diagrams [48]to the complex- k space with the extended Kuo-Krenciglowa(EKK) method [49], during which contributions from the corepolarization and other continuum partial waves are taken intoaccount [29, 31]. Due to the explosion of the model dimen-sion when continuum included, the perturbative expansion ofthe Q -box is at the second-order level (computationally pro-hibitive to go to third order). At last, the complex symmetricnon-Hermitian GSM Hamiltonian is diagonalized in the GSMspace given above, by using the Jacobi-Davidson method in them-scheme.
3. Results
The calculated low-lying spectra of Ne and its isotone Fare presented in Fig. 1, with respect to the O ground-state en-ergy. The O ground-state energy actually is the ν p / posi-tion with respect to the O core. In our framework, the valenceproton SP states are all unbound, which means that only thenaive occupation of protons in model space can not give bound Ne and its bound nature should come from the correlationof valence particles. The comparison of the blue and red linesin Fig. 1 shows that 3NF lifts the whole spectra of Ne and
Figure 1: Calculated spectra of Ne along with its isotone F, with respectto O. Blue and red lines are the GSM calculations with only 2NF interactionand 2NF + F, making F unbound and Ne with a Borromean charac-ter. This Borromean structure can be seen more clearly in Table1, where the calculations with 3NF provide an unbound F anda bound Ne. In our previous work [31], we have found that all
Table 1: Calculated ground-state energies of Ne and F (with respect to O) with and without 3NF. The experimental data are taken from Ref. [50].
E(MeV) 2NF only 2NF + Ne − . − . − . F − . + . + . of the three components, 2 π -exchange V (2 π )3N , 1 π -exchange V (1 π )3N and contact V (ct)3N , have significant contributions but they behavedi ff erently. The repulsive contact term V (ct)3N has a similar ab-solute value as the attractive 1 π -exchange V (1 π )3N and their nete ff ect is almost cancelled out, leaving the long-range repulsivetwo-pion exchange V (2 π )3N dominant [31]. Moreover, 3NF has amore remarkable repulsive e ff ects in Ne than in F, whichmakes a smaller gap of ground-state energies between the twoisotones. As well known, the 3NF contribution increases withincreasing the number of valence particles [31]. The di ff erencesseen in Fig. 1 and Table 1 between calculations and data may bedue to the lack of correlations coming from higher-order contri-butions of the Q-box perturbative expansion. In a strict single-particle formulation of the shell model, there should not existexplicitly three-nucleon e ff ects in F with only one neutron andone proton in the model space. However, when we constructthe e ff ective interaction in the model space, the “core” is notfrozen and the correlation from nucleons in the core space canbe “folded” into two-body interaction by the folded-diagramprocedure. That means that the 3NF e ff ect in F can be tracedback to the nucleons in the core. Besides, we found strong con-figuration mixing in the ground state Ne, with a 54% s -wavecomponent of π s / ⊗ ν p / . This is consistent with the result3iscussed in Ref. [12]. Figure 2: The Ne density in the valence space, calculated by GSM with 2NFonly (blue dot-dashed line) and 2NF + + O density (black dash line) in valence space is also displayed forcomparison.
The small two-proton separation energy and the large weightof the proton s -wave configuration in the ground state is a typ-ical signature of the presence of a halo in Ne. In order tocorroborate this assumption, we calculated the one-body den-sity of Ne by the GSM (shown in Fig. 2). By comparing thedensity of Ne with that of O, we find that the Ne densityhas a long “tail” which is a direct evidence to support the halonature of Ne. The comparison between calculations with andwithout 3NF shows that 3NF does not have significant influ-ence on the density distribution. Furthermore, to underline thecontinuum e ff ect within the present model, we have performedthe shell-model calculation without the inclusions of the con-tinuum. In this calculation, the WS potential is solved in theHO basis (instead of the Berggren representation), which givesnon-continuum discrete WS single-particle states, see [31] formore details. We see from Fig. 2 that the calculated Ne den-sity (with or without 3NF) decreases rapidly at large distancewhen excluding the continuum. This can be explained in twoaspects: 1) the π s / ⊗ ν p / configuration is reduced to about20% when excluding the continuum; 2) the s -wave expandedby the HO basis can not carry enough density information atlarge distance due to the spatially-localized property of the HObasis. Therefore, the continuum e ff ect is crucial to describe thehalo structure of Ne.In order to see the inner structure of the halo, we calculatethe correlation densities of Ne and illustrate in Fig. 3. Thedefinition of the correlation density is defined as [24, 25], ρ ( r , θ ) = (cid:104) Ψ | r δ (cid:0) r − r (cid:48) (cid:1) δ (cid:0) r − r (cid:48) (cid:1) δ (cid:0) θ − θ (cid:48) (cid:1) | Ψ (cid:105) , (3)where r (cid:48) ( r (cid:48) ) is the radial coordinate of the first (second) nu-cleon, θ (cid:48) represents the angle between the two nucleons rela- Figure 3: Calculated proton-proton and proton-neutron correlation densitiesof the Ne ground state (a) with θ equal to 30 ◦ , 90 ◦ , 120 ◦ by dot-dashed,solid and dot lines, respectively. The two-dimension proton-proton correlationdensity ρ ( r , θ ) /ρ (normalized with the maximum value ρ of the correlationdensity) is also plotted for the 1 / − ground state (b) and 3 / − excited state (c). tive to the origin of coordinates, and | Ψ (cid:105) is the wave functionof the state. The Jacobian induced by angular dependence isimplicitly included in ρ ( r , θ ), see Ref. [25]. In panel (a) ofFig. 3, we show the proton-proton and proton-neutron corre-lation densites ρ ( r , θ ) in three di ff erent θ angles (30 ◦ , 90 ◦ and 120 ◦ ). It is clearly seen that the proton-proton correlationdensity has a “long tail”, which supports the two-proton halostructure of the Ne ground state. From panel (b) of Fig. 3,one can see that the two-proton correlation density of the 1 / − ground state concentrates mainly in an area close to the core at r ∼ θ ∼ ◦ between the two protons,while the rest of the distribution is around r ∼ θ ∼ ◦ . The two-proton decay of the excited J π = / − state is an important issue for solving the bypass problem ofthe O waiting point in nuclear astrophysics [14], since the ra-dioactive absorption of two protons is known to be a possiblebypath for this waiting point [17]. Then we also demonstratethe two-proton correlation density of the J π = / − state in thepanel (c) of Fig. 3. The two-proton correlation density of the3 / − excited state concentrates mainly around r ∼ θ ∼ ◦ , meaning that the two protons in the excitedstate stay in a farther area from the core than in the ground state.The 3 / − excited state is above the threshold of the two-protondecay. We may expect a two-proton decay from the Ne 3 / − excited state.
4. Conclusions
In conclusion, starting from a chiral 2NF and a chiral 3NF,we have presented the results of the GSM calculations for Neand investigated the roles played by the continuum and 3NFin several aspects. More specifically, we take the advantageof the normal-ordering approach to include 3NF e ff ects and theBerggren representation to treat bound, resonant and continuumstates on equal footing in a complex- k plane. From the calcu-lated results, we find that both continuum and 3NF are crucialto describe the Borromean halo nucleus, Ne. In particular,3NF is essential for binding energy and the continuum is morecrucial for density. The repulsive 3NF raises the energy of F4ver the threshold of the proton emission, which leads to a Bor-romean structure of Ne. The nucleus is further investigatedby calculating one-body density and two-nucleon correlationdensity, showing a two-proton halo structure. The 3 / − excitedstate lies above the threshold of the proton emission, therefore atwo-proton decay should be possible, which is interesting alsoin astrophysics.
5. Acknowledgements
This work has been supported by the National Key R&DProgram of China under Grant No. 2018YFA0404401; theNational Natural Science Foundation of China under GrantsNo. 11835001, No. 11921006, No. 11975282 and No.11435014; China Postdoctoral Science Foundation under GrantNo. BX20200136; the State Key Laboratory of NuclearPhysics and Technology, Peking University under Grant No.NPT2020ZZ01; the Strategic Priority Research Program ofChinese Academy of Sciences, Grant No. XDB34000000; andthe CUSTIPEN (China- U.S. Theory Institute for Physics withExotic Nuclei) funded by the U.S. 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. ReferencesReferences [1] S. Weinberg, Phys. Lett. B251 (1990) 288–292.[2] E. Epelbaum, H.-W. Hammer, U.-G. Meißner, Rev. Mod. Phys. 81 (2009)1773–1825.[3] D. R. Entem, R. Machleidt, Phys. Rev. C 66 (2002) 014002.[4] L.-B. Wang, P. 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