Core momentum distribution in two-neutron halo nuclei

The core momentum distribution of a weakly-bound neutron-neutron-core exotic nucleus is computed within a renormalized zero-range three-body model, with interactions in the s-wave channel. The halo wave-function in momentum space is obtained by using as inputs the two-body scattering lengths and the two-neutron separation energy. The core momentum densities are computed for $^{11}$Li, $^{14}$Be $^{20}$C and $^{22}$C. The model describes the experimental data for $^{11}$Li, $^{14}$Be and to some extend $^{20}$C. The recoil momentum distribution of the $^{20}$C from the breakup of $^{22}$C nucleus is computed for different two-neutron separation energies, and from the comparison with recent experimental data the two-neutron separation energy is estimated in the range $100\lesssim S_{2n}\lesssim 400$ KeV. The recoil momentum distribution depends weakly on the neutron-$^{20}$C scattering length, while the matter radius is strongly sensitive to it. The expected universality of the momentum distribution width is verified by also considering excited states for the system.


Introduction
The core recoil momentum distribution of radioactive two-neutron halo nuclei close to the drip line, extracted from breakup reactions at few hundreds 20 C and 22 C [9], as obtained by the halo breakup on nuclear targets (see also Ref. [10]). The approach is the above described three-body model, which we found appropriate for the analysis of low-binding energy systems as these ones. In the particular cases of 11 Li, 14 Be and the carbon systems 20 C and 22 C, we consider that the neutron-neutron and the neutron-core interactions are dominated by s−wave states. The calculations of core momentum distributions are performed within a renormalized zero-range three-body model, with the halo nucleus described as two neutrons with an inert core (n − n − c) [7,11]. The detailed expressions for the momentum distribution are given in [12], within an approach that requires as inputs one two-body (n − c) and one three-body (n − n − c ) observable, given that the other two-body observable is fixed to the well-known virtual-state energy of the n − n system. Usually, within such approach it is appropriate to consider the corresponding two-body scattering lengths (positive, for bound, and negative for virtual state systems); with a three-body scale given by the two-neutron separation energy, S 2n . Therefore, in a more general description of low-energy three-body physics with two distinguished particles (α − α − β), an appropriate universal scaling function is given (see e.g. [7]), where only three low-energy inpus are enough to determine any other relevant low-energy observable of the system. Within our study on the momentum distributions of the core in halo nuclei the observable that we are concerned is the variance of the momentum distribution, given by σ 2 (associated with the normal one), which is universally correlated to the two possible scattering lengths and S 2n . One obtains σ from the Full Width at Half Maximum (FWHM) of the momentum distribution, such that on can find that FWHM = 2 √ 2 ln 2 σ. Once this quantity is known experimentally, one can use the scaling function to estimate the value of S 2n or, eventually, to constraint some other poorly known low-energy observable, such as a subsystem energy, or scattering length. The natural units for σ in halo physics is MeV/c. As we are interested in scaling properties of observables, it is convenient to introduce the dimensionless ratio σ/ √ S 2n m n , where m n is the neutron mass. By taking m n as the mass unit, a scaling function can be defined, with a general form given by where the + and − signs refer to the bound and virtual subsystem energies, respectively. The core mass number is A ≡ m c /m n . The corresponding energies, E nn and E nc , are positive defined quantities, with a nn and a nc being the respective two-body scattering lengths. In our specific case of the two-neutron halo nuclei the above scaling function (1) has E nn fixed to the n − n virtual state. In the next, our units are such that the Planck constant and the velocity of light c are set to one. All masses are taken in units of m n .
For the momentum distribution width, the scaling function (1) is the limit cycle of the correlation function associated with σ as a function of E nn , E nc and S 2n , when the three-body ultraviolet (UV) cut-off is driven to infinite in the three-body integral equations, or equally the scattering lengths driven to zero with a fixed UV cut-off. Similar procedure is performed within a renormalized zero-range three-body model, in the subtracted integral equations, where the subtraction energy is fixed and the two-body scattering lengths are driven towards infinite. In practice, both procedures provides very close results, as shown in Ref. [13]. In the exact Efimov limit (E nn = E nc = 0), the width is a universal function of the mass number A, σ/ √ S 2n = S c (0, 0, A) , which is associated to a limit cycle. Already in the first cycle it approaches the results of the renormalized zero-range three-body model (see e.g. [7]), namely given by the subtracted Skorniakov and Ter-Martirosian equations for mass imbalanced systems [14].
For the analysis of the core momentum distribution, we consider data for 11 Li [1], 14 Be [8] and 20 C [9] as the low-energy parameters, which are the inputs of our renormalized zero-range model. This procedure allows us to verify the utility of such "bare" formula (1), which does not include distortion effects from the scattering, to analyse the actual breakup data for those systems, taken at few-hundred MeV/A.
As an application of our model, we study in more detail the two neutron halo of the Borromean nuclei 22 C, in an attempt to extract information of the halo properties, by using the correlation between observables expressed in Eq.(1), namely the width of the core recoil distribution as a function of S 2n and the energy of the s−wave virtual state of 21 C. From the experimental point of view the two-neutron separation energy of 22 C is not well constrained, with a value of 0.42 ± 0.94 MeV given by systematics [15] and from a mass measurement, it was found S 2n = -0.14(46) MeV [16]. There is an indirect evidence that 22 C could be bound by less than 70 keV [17]. Other independent information on the binding energy of this nucleus can be obtained from the matter radius. Tanaka and collaborators [18] extracted a root-mean-square (rms) matter radius of 5.4 ± 0.9 fm from the analysis of the large reaction cross sections of 22 C on liquid hydrogen target at 40A MeV, using a finite-range Glauber calculation under an optical-limit approximation. Furthermore, the two-valence neutrons occupy preferentially one s 1/2 orbital in their analysis. Such rms matter radius, taken together with the corresponding one of 20 C (2.98(5) fm [19]), suggest a halo neutron orbit with rms radius of 15 ± 4 fm in 22 C, which is constraining the S 2n to be below 100 keV [20]. This value is consistent with results obtained from a shell-model approach [21] and results from effective field theory with contact interaction [22,23]. The estimated 22 C quantities should be compared with the fairly small value of S 2n = 369.15(65) keV for 11 Li in the nuclear scale [24], and with the neutron-neutron (n − n) average separation distances R nn in 11 Li around 6-8 fm, which is obtained from the n − n correlation function measured by the breakup cross-section on heavy nuclei [25,26]. However, Riisager [2] pointed out that a comparison of experimental data obtained for the core recoil momentum distributions of 11 Li [1] and 22 C [9] suggests similar neutron halo sizes for these nuclei, which could indicate an overestimation of the matter radius of this carbon isotope.
Our present work can give more insights in resolving the issue of the size of the two neutron halo in 22 C. The constraints in the parameters associated with the 22 C halo structure and two-neutron separation energy provided by the scaling formula for the width of the core recoil momentum distribution are discussed on the basis that corresponding data, fitted to three-body model calculations. The particular case of 22 C is interesting considering that the corresponding observables are probably dominated by the tail of the three-body wave function in an ideal s−wave three-body model. That ideal structure was already considered in Ref. [27], within a Borromean n − n− 20 C configuration for 22 C, where all two-body subsystems, n− 20 C and n − n are not bound.
As it will be shown in the following, the recent experimental results for 20 C and 22 C [9] allow us, in principle, to constraint S 2n and the matter radius of 22 C, even considering that the scattering length of the subsystem neutron-20 C is not well known. From the experimental analysis performed in Ref. [17], the associated s−wave virtual-state energy of 21 C is found to be about 1 MeV.
The present study on the constraint for S 2n are relying on the applicability of the renormalized three-body zero-range model and scaling function (1) derived for the width of the core recoil momentum distribution. In the case of 22 C, this is obtained by fitting this distribution to the experimental breakup crosssection data given in Ref. [9]. For our estimative of S 2n is also essential that the scaling function given in (1) has a weak dependence of the E nc /S 2n ratio.
One of the sources of information on the sizes of unstable neutron-rich nuclei, is the n − n correlation function obtained from Coulomb breakup experiments with neutron rich projectile on heavy nuclei [28,25,26]. The experimental results for the n − n correlation function for Borromean nuclei 11 Li and 14 Be are found quite consistent with the corresponding computed quantities obtained within a subtracted renormalized zero-range model [29], unless an unexpected theoretical minimum before the correlation function approach unity for large relative momentum. Data from the experiments are showing a monodic decrease of the correlation function with momentum; however, the accessible data goes only up to the predicted minima region.
Next, we present the basic formalism. In section 3 we have the main results, followed by the section 4 where we summarize our conclusions.

Model formalism
In the following, we briefly sketch the formalism, based on the renormalized zero-range three-body model, leading to the core recoil momentum distribution formula, which is used in our data analysis of the halo nuclei systems 11 Li, 14 Be, 20 C and 22 C.
The renormalized zero-range model which we are considering to describe the halo wave-function has been explained in detail in the review [7]. In order to built the s−wave three-body wave function for the n − n − c system, one needs to solve a coupled integral equation for the independent spectator functions χ nn (q) and χ nc (q). Within the zero range model, a regularization is needed, which can be implemented with a cutoff momentum parameter, such as in Ref. [30], or by considering the subtraction procedure used in [31], which we follow in the present approach. Therefore, the present subtractive regularization approach for the spectator functions is performed at a given energy scale µ 2 , by the following coupled equations: The above set of coupled equations can also be derived from a renormalized Hamiltonian as shown in [7], where the associated renormalization group properties are also discussed. The minus (−) sign refers to a bound state subsystem and the plus sign (+) to a virtual state subsystem. Therefore, within the perspective of a more general α − α − β system, the following cases can be described by the above coupled integral equations: all-bound configuration, when there is no unbound subsystems; Borromean configuration, when all the subsystems are unbound; tango configuration [32,33,35], when we have two unbound and one bound subsystems; and samba configuration [31], when just one of the two-body subsystems is unbound. In the present case, as we are concerned with n−n−c halo nuclei system, only samba and Borromean configurations are possible, once we take that n − n is unbound with a virtual-state energy of about 143 keV. This implies that only the sign + is to be considered for τ nn in Eq. (3).
One can further simplify Eq. (2), for numerical purpose, by having an uncoupled integral equation for χ nc : The corresponding s−wave three-body wave-function can be written in terms of the spectator functions χ nn (q) and χ nc (q) as: where {| q c p c } is the relative Jacobi momentum basis, with q c the relative momentum of the core to the center-of-mass of the n − n system, and p c the relative momentum between the two neutrons. Note that, as we are going to present results corresponding to the limit cycle, namely, when all involved energies tends to zero with respect to the subtraction or regularization scale, we have dropped the regularization term in the denominator of the wavefunction, which was introduced in Ref. [31]. The configuration space halo wavefunction, which is given by the Fourier Transform of the momentum wavefunction, is an eigenstate of the free Hamiltonian, except when two particles are at the same point, such that in our model the two halo-neutrons are always found in the classically forbidden region. This model can represent a real halo state as long as the neutrons have a large probability to be found outside of the potential range and of the core.
From the wave-function, given in momentum space by Eq. (6), we can define the core momentum distribution for the n − n − c system as with normalization such that d 3 q c n(q c ) = 1. In the context of cold atoms the large momentum behaviour of the above momentum density has been studied in detail for three-bosons in [34] and for mass imbalanced systems in [12]. The log-periodic solution of the spectator equations (2) in the ultraviolet limit, when µ → ∞, is the key property to derive asymptotic formulas for the one-body momentum densities. Furthermore, it was verified in [12] how the solutions of (2) approaches the log-periodic form for the higher Efimov excited states. In addition, it was shown that the density properties at low momentum behaviour are universal, namely, approach the limit-cycle already for the ground state with finite µ, and depend on the three-body binding energy and scattering lengths.

Results and discussion
The solution for the set of integral equations (2) provides the spectator functions and ultimately the momentum probability density (7). We start by showing results for E nn = E nc = 0, in order to study the limit-cycle for the core momentum distribution in the context of the two-neutron halo nuclei. To illustrate this limit we show in Fig. 1 the corresponding scaling function (1) for σ, in terms of the dimensionless ratio σ/ √ S 2n as a function of the core mass number. Results for the ground and two excited states in Fig. 1 show that the limit-cycle is universal and in practice found for the ground state. We compare with the experimental values of σ/ √ S 2n obtained for 11 Li, σ = 21(3)MeV/c, coming from the halo breakup reaction 11 Li +C → 9 Li + X at 800 MeV/A [1], and for 14 Be, which has a FWHM= 92.7 ± 2.7MeV/c for the core recoil momentum distribution [8] and S 2n =1.337 MeV [15]. The flattening of the scaling function for large A reaching an asymptotic value can be understood by inspecting the set of coupled equations (2) and the wave-function (6) by noticing that the limit A → ∞ can be performed, where all dependences on A are cancelled out. One has to remind that even for A → ∞ the dependence of the core momentum distribution on the relative momentum q c just reflects the momentum distribution of the center of mass the two halo-neutrons in the nucleus. On the other hand, for A → 0, the momentum distribution tends be concentrated at small momentum as one can easily check that the relevant contribution to the integral equation for the spectator function comes from small momentum and σ → 0. Naively, the light particle explores large distances, as the characteristic momentum is of the order of S 2n /A, and therefore σ ∼ √ A for A → 0.
The dependence of σ on the subsystems energies E nn and E nc is investigated by considering the results presented in Figs with respect to √ S 2n , as is seen when the values at the origin of these figures are compared to Fig. 1. It means that the halo shrinks as the virtual state energy increases in absolute value. This effect was found in [31], namely for a given S 2n the size of the halo shrinks when going from all-bound configuration to the Borromean one. This behaviour happens because the interaction becomes less attractive, such that to keep the three-body binding energy the state has to become smaller. This effect is also observed as the value of E nc increases in for 11 Li (left-frame) and 14 Be (right-frame), as shown in Fig. 2 Fig. 2, and from that we could say roughly that s−wave virtual state of 13 Be E nc is less than 1 MeV, which is consistent with 0.2 MeV that is the known value (see e.g. [31]). We note that the dependence on E nc is very mild and by changing it from 0 to S 2n a variation of σ/ √ S 2n of only 10% is found in our model, which puts a constraint in the error in the experimental ratio σ/ √ S 2n in order to be useful to extract information on the neutron-core virtual state energy.
In Fig. 3, we show results for the scaling plots for the core momentum distribution σ in 20 C (left-frame) and 22 C (right-frame). In the left frame, the subsystem n− 18 C forming the s−wave one neutron halo 19 C is bound with energy E nc ≡ S 1n = 580 keV [15], where S 1n is the one neutron separation en-   Fig. 3. Scaling plots for the core momentum distribution σ for 20 C (left-frame) and 22 C (right-frame), for a fixed E nn = 143 keV (virtual-state energy). In the left frame, for 20 C, we use S 2n = 3.5 MeV [15]. In the right frame, for 22 C, we use three values for S 2n : 100 keV (dashed line), 250 keV (dotted line) and 400 keV (solid line).
ergy. Although, all halo low-energy scales are known for 20 C, we allow variation of the ratio E nc /S 2n to illustrate how the width of the momentum distribution varies in the case of halo nuclei with bound n − c subsystem. The width decreases as E nc /S 2n increases as the bound state energy becomes closer to the lowest scattering threshold, and consequently the neutron distance to core increases leading to the sudden drop of σ to zero, when E nc /S 2n goes to unity. In the right-frame of the figure, we present results for σ/ √ S 2n as a function of (E nc /S 2n ) 1/2 for 22 C computed with different values of S 2n , 100 keV, 250 keV and 400 keV. We observe in the figure that while σ exhibits a strong dependence on S 2n with E nn and E nc kept constant, the variation of σ with the ratio E nc /S 2n for S 2n constant shows a quite weak sensitivity, as one could expect for the Borromean case. In that sense, as already recognized, the value of σ gives a good constraint for S 2n in this case.  11 Li are extracted from [1] and for 20 C from [9]. For 11 Li the experiment detected the 9 Li transverse momentum to the beam and for 20 C the inclusive parallel momentum of 18 C. The distribution for 20 C was folded with the experimental resolution of σ = 28 MeV/c. After our discussion of the general scaling properties of the with of the momentum distribution, we show in Fig. 4 our calculations of the core recoil momentum distribution for 11 Li (left-frame) and 20 C (right-frame) compared to actual results from halo breakup experiments obtained reactions with carbon target at 800 MeV/A [1] and at 240 MeV/A [9], respectively. For 11 Li, a wide distribution with σ =80 MeV/c is added to the computed narrow one, which has σ = 22 MeV/c. We remark that all three inputs to compute the narrow distribution are fixed to known values of S 2n = 369 keV [24], the s−wave virtual state energy of 10 Li, E nc =50 keV, and the singlet n − n virtual state, E nn =143 keV. The wide momentum distribution is beyond our model, which is more concerned on the halo neutrons. That contribution should be associated with inner part of the halo neutron orbits, close to the core region.
In the comparison with the experimental data, the normalisations of the wide and narrow distributions are fitted to the data. After that, we find a fair reproduction of the experimental data as shown in the figure. This procedure confirms that our approach is a viable tool to extract information on the large two-neutron halo properties from the core momentum distribution.
The right-frame of Fig. 4 presents the core momentum distribution for 20 C.
The calculations were performed with S 2n = 3.5 MeV and with 19 C oneneutron separation energy equal to 580 keV [15]. The model is compared to data obtained from [9], after folding with the experimental resolution of σ = 28 MeV/c. We observe that a wide distribution is somewhat missing to fit the experimental results in this case.
The model results for the core recoil momentum distribution in 22 C, with twoneutron separation energies of 100 and 400 KeV, is presented in the left-frame of Fig. 5, and compared to data obtained from [9]. The singlet virtual n − n energy is fixed to 143 keV, with the virtual-state energy of n− 20 C chosen as 0 and 1 MeV [17]. The narrow theoretical distribution is folded to the experimental resolution of σ = 27 MeV/c and added to a wide one with σ = 89.6 MeV/c. The results presented in this figure illustrate the weak sensitivity of the core recoil momentum distribution to the variation of the virtual-state energy of 21 C, which is taken between 0 and 1 MeV, as it was shown by the results with S 2n = 100 keV. The difference between the distributions obtained with S 2n = 100 and 400 keV, computed with E nc = 1 MeV is not enough to discriminate S 2n in view of the experimental data error. The model sensitivity to the physical inputs, in the interesting case of 22 C is further explored, in the right-frame of the figure, where the scaling plot for σ/ √ E nn as a function of E nc /E nn is shown for three values of S 2n . The weak sensitivity to the s−wave virtual state energy of 21 C is seen and one could consider to obtain an upper limit to the experimental relative error in order to extract information on the n− 20 C scattering length. However, a variation of the ratio E nc /E nn between 0 and 9 gives 10% variation of σ (E nn is fixed), which is surmounted by a variation of about 50% in S 2n . Therefore, it is required an independent source of information to constrain the n− 20 C scattering length, which we can find from the matter radius of 22 C [18]. To close our discussion of 22 C, we computed the matter radius starting with the rms radius of the halo neutrons (r n ) with respect to the center-of-mass, which is obtained from the configuration space n − n − c wave-function, which is obtained by considering the Fourier transform of the corresponding momentum wave-function (6). For details on this procedure, see [31,20]. The corresponding formula of the matter radius is given by . The 20 C matter radius is r m [ 20 C] = 2.98(5) fm [19]. The plot of Fig. 6 shows the theoretical values of the rms matter radius of 22 C as a function of S 2n for a fixed s−wave virtual state energies of the singlet n − n pair (143 keV) and 21 C (1 MeV [17] and 0) compared to data from [18]. In the figure, the limits for the extracted matter radius [18] are shown, and we can make some remarks analysing the consistence between the different available data and our model, considering that 100 keV S 2n 400 keV: (i) for E nc = 0, one finds that 3.5 fm r 2 m [ 22 C] 1/2 4.5 fm; and, (ii) for E nc = 1 MeV, we have r 2 m [ 22 C] 1/2 3.5 fm. Only if E nc ∼ 0 we obtain a region for S 2n close to 100 keV, consistent with rms matter radius of 22 C within one standard deviation and in the lower bound of the radius, namely ∼ 4.5 fm. The value of E nc = 1 MeV for the virtual state of 21 C and S 2n ∼ 100 keV is compatible with two standard deviation; from that, r 2 m [ 22 C] 1/2 3.5 fm. This combined analysis for 22 C of the core recoil momentum distribution, with rms matter radius and virtual state energy of n− 20 C, suggests that such virtual-state energy and matter radius are overestimated. Independent new data on the S 2n for 22 C could help in clarifying the tension between data analysis with the present universal model.

Conclusions
In summary, by considering the renormalized zero-range model applied to the case of core recoil momentum distributions of 11 Li and 14 Be, we found a fair consistency with experimental data, just by using the known low-energy parameters. Relying on the fact that such simplified model gives already a valid description of the two-neutron s-wave halo, we proceed with a combined analysis of recent experimental data on the core momentum distribution in 22 C, which is given by Kobaiashi et al. [9], the corresponding rms matter radius and the 21 C virtual state energy. Our conclusion is that, with the value of the two-neutron separation energy of 22 C given in the interval from 100 to 400 keV, the rms matter radius of 22 C will be within two standard deviations if the virtual state energy of 21 C is close to 0. By considering the 21 C with a virtual-state energy between 0 and 1 MeV, the matter rms radius should be between 3.5 and 4.5 fm. To reconcile a virtual-state energy with E nc ∼1 MeV, a matter radius of 5.4±0.9 fm and 100 keV S 2n 400 keV, the possibility is S 2n ∼ 100 keV and E nc < 1 MeV, implying that r 2 m [ 22 C] 1/2 ∼ 4.5 fm. A refined analysis of the core momentum distribution, beyond the Serber model [36], is desirable, of course. However, the comparison of results obtained by the present model for 11 Li and 14 Be with corresponding data suggests small corrections to the distribution verified for 22 C.
We thank partial support from the Brazilian agencies FAPESP, CNPq and CAPES.