Borromean Feshbach resonance in 11Li studied via 11Li(p,p')

A dipole resonance of 11Li is newly found by a 9Li + n + n three-body model analysis with the complex-scaling method. The resonance can be interpreted as a bound state in the 10Li + n system, that is, a Feshbach resonance in the 9Li + n + n system. As a characteristic feature of the Feshbach resonance of 11Li, the 10Li + n threshold is open above the 9Li + n + n one, which reflects a distinctive property of the Borromean system. A microscopic four-body reaction calculation for the 11Li(p,p') reaction at 6 MeV/nucleon is performed by taking into account the resonance and nonresonant continuum states of the three-body system. The angular distribution of the elastic and inelastic scattering as well as the breakup energy spectrum recently observed are reproduced well.

Elucidation of resonances, which are omnipresent in different hierarchies in nature, is one of the most important subjects in physics. For example, the tetraquark and pentaquark baryons in hadron physics [1] as well as the so-called Efimov resonance [2,3] of ultracold atoms in atomic physics have attracted the attention of many experimentalists and theorists. In nuclear physics, various resonances have been discovered and investigated in detail. Studies of resonances in nuclear physics will be characterized by the diversity. Nuclei, a self-organized strongly interacting system, show a wide variety of structures as the atomic number, the mass number, and the excitation energy change. From a different point of view, we have better knowledge on the basic interaction that forms many-nucleon systems than in hadron physics. Various types of resonances, e.g., single-particle resonances, gas-like α cluster states, and giant resonances have therefore been investigated on the solid basis. Nowadays resonant structures for nuclei near and even beyond the neutron dripline have intensively been proceeded. Furthermore, a recent experiment suggested that four neutrons form a resonance, that is, the so-called tetraneutron [4].
Very recently, measurement of the 11 Li(p, p ′ ) reaction at 6 MeV/nucleon with a high statistic and high resolution has been performed [23], and a low-lying excited state of 11 Li has clearly been identified. In the analysis, the authors adopted a macroscopic model for the transition of 11 Li combined with the distorted wave Born approximation (DWBA); a form factor of the isoscalar electric dipole (E1) excitation is assumed. The macroscopic model, however, does not describe the Borromean nature of 11 Li and a microscopic approach to the structure of the low-lying continuum states of 11 Li is eagerly desired. On the reaction side, the applicability of DWBA in the energy region of our interest is quite questionable. In other words, if the reaction observable suffers from higher-order processes, it is not trivial at all to relate the observable and a response of a nucleus to a specific transition operator. Furthermore, there is no guarantee that a single operator is responsible for the proton inelastic scattering measured at backward angles. The purpose of this Letter is to analyze the 11 Li(p, p ′ ) cross sections at 6 MeV/nucleon with a sophisticated reaction model, that is, the microscopic four-body continuumdiscretized coupled-channels method (CDCC) [24][25][26][27][28]. A complete set of the 9 Li + n + n three-body wave functions in a space relevant to the 11 Li(p, p ′ ) reaction is implemented in CDCC and thereby the validity of the continuum structure of 11 Li is examined. Classification of the three-body wave functions with the complex-scaling method (CSM) [29][30][31] suggests a low-lying three-body Feshbach resonance [32] of 11 Li, which is the principal finding of the present study.
For 11 Li, we adopt a 9 Li + n + n three-body model, with assuming for simplicity that 9 Li is a spinless and inert particle that has a naïve shell-model configuration. This simplified model has been applied to analyses of some reactions of 11 Li [15][16][17]. Three-body wave functions Φ I π ν , where I π represents the spin-parity and ν is the index of eigenenergy, of 11 Li are obtained by diagonalizing the three-body Hamiltonian: Here, K r and K y are the kinetic energy operators for the Jacobi coordinates r and y shown in Fig. 1 in Ref. [25], respectively. V nn (V cn ) is a two-body interaction between the two neutrons ( 9 Li and a neutron), and V cnn is a phenomenological three-body force (3BF). Φ I π ν is explicitly antisymmetrized for the exchange between the two valence neutrons, whereas the exchange between each valence neutron and a nucleon in 9 Li is approximately treated by the orthogonality condition model [33]. For understanding properties of the three-body continuum of 11 Li in more detail, we employ CSM, in which the radial part of each Jacobi coordinate is transformed as r → re iθc , y → ye iθc (2) with the scaling angle θ c , and h is rewritten as h θc accordingly. As a result of diagonalization of h θc , eigenstates ϕ θc I π ν that have complex eigenenergies ε θc I π ν are obtained. A resonance is identified as an eigenstate on the complex-energy plane isolated from other nonresonant states; the real and imaginary parts of the eigenenergy represent the resonant energy ε R and a half of the decay width Γ/2, respectively.
The total wave function Ψ of the p + 11 Li reaction system is obtained by solving the Schrödinger equation where K R is the kinetic energy operator regarding the coordinate R between the center-of-mass of 11 Li and p. The nuclear interaction between p and the ith nucleon in 11 Li is denoted by v 0i . V C is the Coulomb interaction between p and the centerof-mass of 11 Li; we thus ignore the Coulomb breakup process. In CDCC, Ψ (+) is approximately expanded in terms of a finite number of Φ I π ν as where γ = (I π , ν) and χ (+) γ is the relative wave function regarding R. Inserting Eq. (4) into Eq. (3) leads to a set of coupled equations for χ (+) γ : with ε γ = Φ I π ν |h|Φ I π ν . For coupling potentials U γ ′ γ , we adopt a microscopic folding model [40][41][42] with transition densities of 11 Li for 11 nucleons, which can be calculated with Φ I π ν and a ground state density of 9 Li for 9 nucleons [16]. Equation (5) is solved under the standard boundary condition; details are shown in Ref. [24]. By solving Eq. (5), one obtains a transition matrix element from which a cross section to the ground state or a discretizedcontinuum state of 11 Li can be evaluated. To obtain a continuous breakup energy spectrum, we employ the smoothing method based on CSM proposed in Ref. [28]. Consequently, the double differential breakup cross section with respect to the energy ε of the 9 Li + n + n system measured from the three-body threshold and the solid angle Ω of the center-ofmass of the three particles, d 2 σ/(dεdΩ), is obtained. As shown by Eq. (21) in Ref. [28], the breakup energy spectrum is given by an incoherent sum of the contributions from the eigenstates of h θc . This property is crucial to clarify the role of a resonance in describing breakup observables.
As for the numerical input, we take the Minnesota force [34] for V nn and the interaction used in Ref. [35] is adopted as V cn . The V cn generates a resonance of 10 Li in the 0p 1/2 state with the resonant energy (decay width) of 0.46 MeV (0.36 MeV). This resonance is denoted by 10 Li below for simplicity. This value of the resonant energy is in good agreement with the latest experimental data [36]. For V cnn , we adopt the volume-type 3BF [37] given by a product of Gaussian functions for the two Jacobi coordinates; the range parameter for each coordinate is set to 2.64 fm and the strength is determined to optimize the ground state energy ε 0 = −0.369 MeV [38] of 11 Li. We employ the Jeukennd-Lejeune-Mahaux (JLM) effective nucleon-nucleon interaction [39] as v 0i . As in the preceding works [40][41][42], a normalization factor N I for the imaginary part of the JLM interaction is introduced; N I is determined to be 0.55 so as to reproduce both the elastic and breakup cross section data around 100 • , where the breakup cross section data exist. Note that we do not include any other adjustable parameters. Eigenstates of h and h θc are obtained by the Gaussian expansion method (GEM) [43], where we adopt the parameter set II for h and set III for h θ c shown in Table I in Ref. [28]. In CSM, the scaling angle θ c is set to 20 • . In CDCC calculation, we select the Φ I π ν with ε < 5 MeV and the resulting number of states is 93, 111, and 131 for I π = 0 + , 1 − , and 2 + , respectively. The model space gives good convergence of the elastic and breakup cross sections. FIG. 1: Eigenenergies for 1 − states calculated with CSM on the complex-energy plane measured from the 9 Li + n + n threshold. The scaling angle θc is taken to be 20 • , and the cross mark shows the 10 Li-n threshold on the complex plane.
We first discuss the structure of the continuum states of 11 Li (the 9 Li + n + n system). For this purpose in Fig. 1 we plot the eigenenergies of h θc with I π = 1 − on the complex-energy plane. The solid square represents the three-body resonance of 11 Li with ε R = 0.42 MeV and Γ/2 = 0.14 MeV, where ε R is consistent with the value obtained in Ref. [18]. The open circles represent three-body nonresonant continuum states of the 9 Li + n + n system, whereas the closed circles indicate two-body continuum states between the valence neutron and 10 Li. One may find that the three-body resonance is located near the 10 Li-n threshold and the energy of the valence neutron is negative with respect to 10 Li. This indicates that the dipole resonance of 11 Li is a Feshbach resonance [32] in a three-body system. We will return to this point later.
FIG. 2: Angular distribution of the differential elastic cross section for the 11 Li + p scattering at 6 MeV/nucleon [23]. The solid and dotted lines represent results of the microscopic four-body CDCC and without breakup channels, respectively.
Next we discuss how the continuum structure of 11 Li affects reaction observables. Figure 2 shows the angular distribution of the elastic scattering of 11 Li on p at 6 MeV/nucleon. The solid and dotted lines show the results with and without breakup effects; the former corresponds to the microscopic four-body CDCC calculation. The solid line which is adjusted to the data around 100 • with setting N I = 0.55, agree well with the data at forward angles. One can see that breakup effects represented by the difference between the dotted and solid lines are significant for the elastic scattering. The solid line deviates from the data around the dip of the cross section. It is known that in the region a spin-orbit part of the distorting potential, which is disregarded in the present study, plays an important role. It should be noted also that the JLM is applicable to nucleon scattering above 10 MeV [39]. Considering these things, we conclude that the agreement between the solid line and the experimental data is satisfactory.
In Fig. 3(a) we show the angular distribution of the breakup cross section; d 2 σ/(dεdΩ) is integrated over ε from 0 MeV to 1.13 MeV so as to cover well the peak structure of the cross section in Fig. 3(b). The thick solid line represents the result of the microscopic four-body CDCC; it reproduces the experimental data around 100 • , as N I is chosen so. The slight deviation of the solid line from the data around 80 • will come from the same reason as for the elastic cross section. The dotted, dashed and dot-dashed lines represent the breakup cross sections to the 0 + , 1 − and 2 + states, respectively. One sees that the breakup cross section to the 1 − state is dominant but the 0 + and 2 + components are not negligible in the region where the experimental data exist. In other words, a model that assumes a pure dipole transition of 11 Li will not explain the measured cross sections unless an unrealistic structural model of 11 Li is adopted. Furthermore, since the transition potential adopted in the present calculation cannot be written as a simple functional form, to use a single transition operator can not be justified. Our final remark on Fig. 3(a) is the importance of the coupled-channel effects. The thin solid line shows the result of a one-step calculation that severely overestimates the thick solid line by about one-order at middle angles. We therefore conclude that DWBA is not applicable to the 11 Li(p, p ′ ) at 6 MeV/nucleon. In Fig. 3(b), we show the breakup cross section with respect to the three-body energy ε after breakup, which is obtained by integrating d 2 σ/(dεdΩ) over θ c.m. from 115 • to 124 • . Here, we have taken into account the energy resolution of the experimental data. The total breakup cross section represented by the thick solid line reproduces the experimental data up to ε ∼ 1.0 MeV including a low-lying peak. One sees that the contribution from the dipole resonance of 11 Li shown by the thin solid line dominates the low-lying peak. Although the calculated resonant width Γ = 0.28 MeV is narrow compared with the evaluation Γ = 1.15 ± 0.06 MeV in Ref. [23], the measured dσ/dε spectrum is reproduced by taking into account the 0 + and 2 + non-resonant components. It can be concluded therefore that the nonresonant components should be properly evaluated and subtracted from the measured spectrum to extract reliable information on the resonance. The calculated cross section undershoots the data for ε 1.0 MeV, which will be due to some other degrees of freedom that are not taken into account in the present calculation, for example, a transition to higher spin states and a core excitation in 9 Li.
Thus we have shown through CDCC calculation that the three-body structure of 11 Li both in the bound and continuum states including the dipole resonance of 11 Li shown in Fig. 1 is consistent with the measured cross sections. We here discuss the property of the 11 Li resonance. As mentioned above, the resonance is located near the 10 Li-n threshold. A certain connection between the 11 Li resonance and 10 Li is therefore expected, as suggested in Ref. [18]. To clarify this, we change the strength of V cn with multiplied by λ nc and see the positions of the two resonances on the complex-energy plane; the result is shown in Table I. One sees that the 11 Li resonance, if exists, always shows up near the 10 Li-n threshold. For λ nc less than 0.96, the resonant pole of 11 Li cannot be found clearly. The property of the 11 Li resonance thus strongly depends on that of 10 Li, and ε R of 11 Li is always below ε R of 10 Li. Moreover, to investigate the existence probability of 10 Li in 11 Li continuum states, we calculate an overlap function between ϕ θc 1 − n and a complex-scaled wave function of 10 Li, φ θc 1 2 − , defined by α θc n = 2 φ θc 1 − n |φ θc where a factor of 2 means to exist two pairs of the 9 L-n system in 11 Li. In general, α θc n becomes complex, and its real part can be interpreted as a probability [44]. In fact, for the 9 Li + n + n nonresonant continuum states shown by the open circles in Fig. 1, which are expected not to contain 10 Li, he real part of α θc n is almost 0. On the other hand, the real part of α θc n is larger than 0.9 for the 10 Li + n continuum states shown by the closed circles in Fig. 1. For the 11 Li resonance, the real part of α θc n exceeds 0.9 as the 10 Li + n continuum states. It should be noted that since ε R of 10 Li is higher than ε R of the 11 Li resonance, one can interpret that the 11 Li resonance is a bound state of the 10 Li + n system. The 10 Li-n relative wave function can be regarded as an s-wave because I π of the 11 Li resonance is 1 − and that of 10 Li is 1/2 − . From the point of view of a bound state embedded in the 9 Li + n + n three-body continuum, the 11 Li resonance can be interpreted as a Feshbach resonance [32]. In Fig. 4, we summarize properties of the complex-scaled states shown in Fig. 1. In the three-body Feshbach resonance, the 9 Li + n + n threshold energy is lower than the 10 Li-n threshold, which is a distinctive character of the Borromean system. We thus refer to this resonance as a Borromean Feshbach resonance. 9 Li+n+n threshold In conclusion, we have found a dipole resonance in 11 Li at 0.42 MeV with the width of 0.28 MeV in a 9 Li + n + n three-body model calculation with CSM. The continuum structure of the three-body system including the resonance has been validated by the good agreement between the results of the microscopic four-body CDCC calculation and the recently measured 11 Li(p, p ′ ) data at 6 MeV/nucleon for both the angular distribution and the breakup energy spectrum. Important remarks on the comparison with the experimental data are i) contribution of not only the resonance but also the nonresonant continuum states are important, ii) a one-step calculation (DWBA) does not work at all, and iii) the transition operator cannot be written in a simple form as assumed in preceding studies.
The 11 Li resonance is interpreted as a bound state of the valence neutron with respect to 10 Li, that is, a Borromean Feshbach resonance. It should be noted that the 10 Li-n threshold is above the 9 Li + n + n three-body threshold, which is a distinctive character of a Borromean system. The ordinary Feshbach resonance has intensively been discussed mainly in atomic physics. The finding of the Borromean Feshbach resonance in the present study will be characterized by its appearance in a Borromean system that is unique in the nucleonic system. Another important feature is that we have some pieces of information on the interactions between the constituents of 11 Li. This allows one to carry out realistic studies on the 11 Li resonance. Nevertheless, more information on the n-9 Li interaction will be desired to make our understanding on the continuum structure of 11 Li more profound and complete. Inclusion of the intrinsic spin of 9 Li as well as the excitation of the 9 Li core will also be very important.
the support by the Hirao Taro Foundation of the Konan University Association for Academic Research. This work has been supported in part by Grants-in-Aid of the Japan Society for the Promotion of Science (Grants No. JP16K05352).