Ab-initio no-core Gamow shell model calculations of multi-neutron systems
Jian Guo Li, Nicolas Michel, Bai Shan Hu, Wei Zuo, Fu Rong Xu
AAb-initio no-core Gamow shell model calculations of multi-neutron systems
J.G. Li, N. Michel,
2, 3
B.S. Hu, W. Zuo,
2, 3 and F.R. Xu ∗ School of Physics, and State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, China Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China School of Nuclear Science and Technology, University of Chinese Academy of Sciences, Beijing 100049, China (Dated: November 19, 2019)The existence of multi-neutron systems has always been a debatable question. Indeed, both inter-nucleon correlations and a large continuum coupling occur in these states. We then employ the ab-initio no-core Gamow shell model to calculate the resonant energies and widths of the trineu-tron and tetraneutron systems with realistic interactions. Our results indicate that trineutron andtetraneutron are both unbound and bear broad widths. The calculated energy and width of tetra-neutron are also comparable with recent experimental data. Moreover, our calculations suggestthat the energy of trineutron is lower than that of tetraneutron, while its resonance width is alsonarrower. This strongly suggests that trineutron is more likely to be experimentally observed thantetraneutron. We thus suggest experimentalists to search for trineutron at low energy.
I. INTRODUCTION
Few-body multi-neutron systems are located close tothe neutron drip line. Thus, they provide us with aunique laboratory to understand nuclear properties atdrip lines [1–3] and nuclear forces in the absence ofCoulomb interaction [4].Tremendous efforts have been made during the lastfew decades to understand few-nucleon systems, in par-ticular few-neutron systems [5–7]. Earlier experimentsfailed to find positive evidence for the existence of multi-neutron systems (see a short summary in Ref.[5]). In2002, Marqu´es et al. reported that a bound tetraneu-tron was observed in a breakup reaction of the Be → Be+4n channel [7]. After that experiment, several the-oretical attempts were performed to examine the possi-ble existence of bound tetraneutron, but all the calcula-tions failed to reproduce experimental data [8, 9]. Theinterest in multi-neutron systems has been resurrectedthrough an experiment where a candidate for resonanttetraneutron was observed, whose resonant energy andwidth were measured in the doubly charge-exchange reac-tion He( He, Be) [6]: E r = 0 . ± . stat ) ± . syst )MeV and Γ ≤ ab-initio frameworks in recent years [8, 9, 13–19]. The ex-trapolation of no-core shell model (NCSM) calculationsutilizing the realistic two-body JISP16 interaction witha harmonic oscillator (HO) basis showed that tetraneu-tron is located near 0.8 MeV above threshold and thatits width is about 1.4 MeV [13]. Recently, NCSM calcu- ∗ [email protected] lations performed in larger model spaces provided withtwo resonant states of tetraneutron, at 0.3 MeV and 0.8MeV, of widths about 0.85 MeV and 1.3 MeV, respec-tively [20]. Calculations have also been done in the frameof no-core Gamow shell model (NCGSM), using the den-sity matrix renormalization group (DMRG) method andnatural orbitals (n.o.) [15]. It was speculated that theresonance width of tetraneutron is larger than 3.7 MeV,so that the formation of a nucleus therein is precluded.However, the calculations were incomplete, as they couldbe performed only in truncated model spaces or with un-physically overbinding interactions, so that no definiteconclusion could be made [15].The quantum Monte Carlo (QMC) framework has beenused to calculate multi-neutron systems based on localchiral interactions via an extrapolation of bound states(obtained with a confining auxiliary potential) to thephysical domain of unbound multi-neutron systems. Thecalculations predicted that the trineutron is located at1.11 MeV, below the energy of tetraneutron, at 2.12MeV [14, 21]. The Faddeev method has also been em-ployed for that matter [16–19]. The latter calculationssuggested that the existence of multi-neutron resonancescan only be obtained by strongly modifying standard nu-clear forces, which is incompatible with our current un-derstanding of nucleon-nucleon interactions [19]. The cal-culations of Ref.[13–15] showed that multi-neutron sys-tems are sensitive neither to various realistic interactions,in the presence or absence of three-body forces, nor tothe momentum cutoff present in renormalization groupmethods. In this paper, we will then present a nearlyexact study of trineutron and tetraneutron systems em-ploying ab-initio NCGSM [15, 22]. Indeed, NCGSM com-prises both a consistent treatment of many-body cor-relations and coupling to the continuum by using theBerggren ensemble [23]. Both the resonant energies andwidths of the trineutron and tetraneutron systems willbe presented. a r X i v : . [ nu c l - t h ] N ov II. THE METHOD
A resonance state, which represents a decaying pro-cess, is time-dependent. However, the exact treatmentof time-dependence is difficult to handle theoretically inmany-body system (see Ref.[24] for examples of time-dependent shell model calculations in light nuclei). Inorder to be able to consider many-body resonance statesin a time-independent framework, it is convenient to re-formulate the Schr¨odinger equation in the complex mo-mentum plane. For this, one uses the Berggren basis,comprising bound, resonant and scattering single-particle(s.p.) states. The time dependence is then taken into ac-count by removing the hermitian character of the Hamil-tonian, so that eigenvalues become complex. The imag-inary part of the energy is indeed proportional to thedecay width [23].The completeness relation, introduced by Berggren[23], reads, (cid:88) n | φ nj (cid:105)(cid:104) (cid:101) φ nj | + 1 π (cid:90) L + | φ j ( k ) (cid:105)(cid:104) φ j ( k ∗ ) | dk = 1 , (1)where φ nj are the Berggren-basis pole states of boundand decaying character, and φ j ( k ) is the scattering statebelonging to the L + contour of complex momenta. Inpractical calculations, the integral in Eq.(1) is discretizedby means of an appropriate quadrature rule (the Gauss-Legendre quadrature in our case).In the present work, we employ the NCGSM methodto calculate multi-neutron systems. Many-body statesare the linear combinations of the Slater determinants | SD n (cid:105) = | u , ...., u A (cid:105) , where u k is a bound, resonant ornon-resonant (scattering) state. The Hamiltonian ma-trix in the NCGSM is complex symmetric and bearscomplex eigenvalues. Coupling to the continuum is thenpresent at basis level, and many-body correlations occurthrough configuration mixing [15, 22, 25–27]. The widthsof the eigenstates obtained in NCGSM take all particle-emission channels into account, so that they are totaldecay widths. Conversely, other methods, as those usingFaddeev-Yakubovsky equations [17] and NCSM calcula-tions [13], need to consider all partial decay processes tocalculate the total decay width.The initial realistic Hamiltonian reads, H = 1 A A (cid:88) i ( p i − p j ) m + A (cid:88) i In the present work, the s.p. Berggren basis is gener-ated by a finite-depth WS potential including spin-orbitcoupling. The parameters of the WS potential read R = 1.9 fm for its radius, a = 0.67 fm for the diffuseness, V ls = 9.5 MeV for its spin-orbit strength and V = − s / and 0 p / orbitals are bound states, and thenon-resonant scattering partial waves consist of the L + contours (in the complex momentum space) defined bythe coordinate points (0 , , (0 . , − . , (0 . , . 0) and(4 . , . 0) (all in fm − ). Each segment of the contours L + is discretized with 15 points. We employ the chi-ral two-body interaction N LO [4], renormalized by theway of the V low- k method using a momentum cutoff Λ= 2.1 fm − . The results for multi-neutron systems havebeen checked to be sensitive neither to the realistic nu-clear forces used nor to the cutoff of the renormalizationmethod [14, 15]. This is partly due to the dilute den-sity of multi-neutron systems, as noticed in Refs.[14, 15].We have also checked the basis-dependence by changing V for trineutron. The trineutron energy varies by onlyabout 50 keV if V changes by a few MeV. Therefore, ba-sis dependence is negligible in our calculations. We takea model space composing of the s / and p / partialwaves using the Berggren basis, while the other partialwaves, i.e. p / , d, f and g / , are represented by HO ba-sis states. All HO orbitals satisfy N max = 2 n + l ≤ b = 2 fm. Ithas been checked in Ref.[15] that calculations are almostindependent of the length parameter and of the inclusionof other partial waves.A complete diagonalization of the Hamitonian fortrineutron can be done using the model space describedabove using NCGSM. However, it is not possible fortetraneutron because of the huge model dimension andstrong coupling to continuum basis states. The full modelspace is then called the large space. Calculations inNCGSM-3p3h truncated model spaces, i.e. with threeneutrons occupying scattering basis states at most, canbe done. Thus, in order to obtain nearly exact (NCGSM-4p4h) results for tetraneutron, we first did calculations ina model space bearing fewer high lying orbitals, definedby taking all the s / and p / Berggren basis states, twoHO shells for p / , d and one HO shell 0 f and 0 g / forthe remaining partial waves. Calculations without trun-cations can then be effected in this model space. Thismodel is deemed as the small space. By comparing theresults obtained with NCGSM-3p3h and NCGSM-4p4hin the small and large spaces, using n.o. with V aux < − . V aux (cid:39) V aux . Consequently, in order to estimatethe energy and width of tetraneutron in the large space,we calculate the energy and width of tetraneutron in thesmall space without truncations and remove 360 keV forenergy and 380 keV for width from the latter values, re-spectively. We can then assume that the exact energyand width are nearly the same as with this ansatz.For the four-neutron system, when the strength of theconfining WS potential gradually weakens, the widths ofresonances increase quickly. Moreover, the differences of energies using NCGSM-3p3h and NCGSM-4p4h modelspaces also become gradually larger therein. These re-sults suggest that, in general, resonances bear a strongcoupling to the non-resonant continuum, with the in-duced configuration mixing becoming more and more im-portant when the width increases. Thus, coupling to con-tinuum should be considered properly in the treatmentof nuclei close to drip-line [25, 29–31]. -5 -4 -3 -2 -1 0 V aux (MeV) -4-20246 E n e r g y ( M e V ) 4n NCGSM-3p3h (small)4n NCGSM-3p3h (large)4n NCGSM-4p4h (small)4n NCGSM-4p4h (n.o) (large)3n NCGSM (large) FIG. 1. Evolution of the energies and widths of trineutronand tetraneutron as function of the depth of the auxiliary WSpotential in the small and large spaces with NCGSM-3p3hand NCGSM-4p4h truncations, using either the Berggren orn.o. basis (see text for definitions). n.o. has been used inNCGSM-4p4h space for V aux < − . The final results for the three-neutron and four-neutron systems, i.e., for which V aux = 0, and with tetra-neutron energies and widths modified by 360 keV and380 keV, respectively, following the method explainedabove, as well as experimental data for tetraneutron, areshown in Fig. 2. Our calculations provide with neutron-unbound trineutron and tetraneutron, whose energies are1.29 MeV and 2.64 MeV, respectively. We also predictthe widths of trineutron and tetraneutron to be Γ n =0.91 MeV and Γ n = 2.38 MeV. The calculated energyand width of tetraneutron are within the range of the ex-perimental error [6]. Our calculations predict the energyof the trineutron to be lower than that of tetraneutronand that the width of the trineutron is smaller than thatof tetraneutron. Consequently, we suppose that trineu-tron should be more likely to be observed experimentallythan tetraneutron. We also compared our calculations -101234 E n e r g y ( M e V ) E x p t T h i s w o r k G F M C N C S M N C S M TetraneutronTrineutron FIG. 2. Calculated energies of trineutron and tetraneutronby different models. The corresponding references are QMC:[14], NCSM2016: [13] and NCSM2018: [20]. The experimen-tal datum is from Ref.[6]. with other ab-initio predictions for multi-neutron sys-tems, see Fig.2. The energies of trineutron and tetra-neutron in our calculations are close to the extrapolatedresults of the QMC calculations [14]. Moreover, our cal-culated energy for tetraneutron is higher than that ofNCSM calculations, which is clearly due to the neglectof continuum coupling in NCSM. The present predictionof energy for tetraneutron is similar to that of K. Fossez et al. [15]. Indeed, the extrapolation of energy aris-ing from the DMRG calculation in Ref.[15] (where notruncations are made) provides more or less with thesame tetraneutron energy as in our calculations. How-ever, the predicted width for tetraneutron of Ref.[15] isnot consistent with ours. It was speculated therein [15]that the width of tetraneutron is larger than 3.7 MeV, whereas our calculation predicts a width of about 2.38MeV. We explain this discrepancy from the fact that thepredicted width from K. Fossez et al. [15] was basedon a calculation effected in a NCGSM-2p2h truncatedspace. Our present calculations have shown that the cal-culated width of tetraneutron decreases by consideringmore correlations, i.e., by using the NCGSM-3p3h andNCGSM-4p4h spaces. IV. SUMMARY In this paper, we utilized the ab - initio NCGSM frame-work along with the realistic chiral two-body N LO nu-clear force to calculate few-body multi-neutron systems.In the NCGSM framework, inter-nucleon correlationsand continuum coupling are taken into account. Our cal-culations predict both trineutron and tetraneutron sys-tems to be broad resonances, with energies equal to 1.29and 2.64 MeV, and widths equal to 0.91 and 2.38 MeV,respectively. The calculated energy and width of tetra-neutron are both within the range of the experimen-tal error [6]. Our results also show that the energy oftrineutron is lower than that of tetraneutron, and widthspresent a similar tendency. As trineutron has smallerenergy and width than tetraneutron, one can supposethat trineutron would more likely to be observed thantetraneutron. We then encourage more experiments tobe carried out to study the trineutron resonance state. ACKNOWLEDGMENTS Valuable discussions with W. Nazarewicz, M.P(cid:32)loszajczak, Dean Lee, J. P. Vary, Z.H. Yang, S.M.Wang, J.C. Pei and G.C. Yong are gratefully acknowl-edged. This work has been supported by the Na-tional Key R&D Program of China under Grant No.2018YFA0404401; the National Natural Science Founda-tion of China under Grants No. 11835001, No. 11921006,No. 11575007, No. 11847203, No. 11435014 and No.11975282; China Postdoctoral Science Foundation underGrant No. 2018M630018; and the CUSTIPEN (China-U.S. Theory Institute for Physics with Exotic Nuclei)funded by the U.S. Department of Energy, Office of Sci-ence under Grant No. de-sc0009971. We acknowledge theHigh-Performance Computing Platform of Peking Uni-versity for providing computational resources. [1] I. Tanihata, H. Hamagaki, O. Hashimoto, Y. Shida,N. Yoshikawa, K. Sugimoto, O. Yamakawa,T. Kobayashi, and N. Takahashi, Phys. Rev. Lett. , 2676 (1985).[2] Y. Kondo, T. Nakamura, R. Tanaka, R. Minakata,S. Ogoshi, N. A. Orr, N. L. Achouri, T. Aumann,H. Baba, F. 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