Dineutron correlations in knockout reactions with Borromean halo nuclei

. We study dineutron correlations in proton-target knockout reactions induced by Borromean two-neutron halo nuclei. Using a core + n + n three-body model for the projectile and a quasifree sudden reaction framework, we focus on the correlation angle as a function of the intrinsic neutron momentum. Our results indicate that the correlations are strong in a range of neutron momenta associated to the nuclear surface. We also discuss on the role of core excitations for such correlations.


Introduction
Borromean halo nuclei are loosely bound systems formed by a compact core and two valence neutrons, such that the core + n subsystem is unbound [1,2].The diffuse matter distribution arising from this particular arrangement has deep implications for their dynamics [3,4] and provides information about the limits of nuclear stability [5].A two-neutron halo wast first observed in the seminal work by Tanihata et al., where a relatively large interaction cross section for 11 Li was reported [6].Other cases have been studied over the years, including 6 He, 14 Be, 17,19 B, 22 C or 29 F [7][8][9][10][11][12][13][14].The Borromean nature of these systems implies that the correlations between the valence neutrons are a key aspect to understand their properties [15].These correlations tend to favour a compact neutron-neutron structure typically referred to as dineutron configuration [16], the extent of which is somewhat related to the amount of mixing between different-parity orbitals [17].Coulomb dissociation and knockout reactions have been employed to access these spatial correlations, for instance in 11 Li, typically linked to a small angle between the neutrons in coordinate space and a large opening angle in momentum space [8,18,19].Recently, a similar approach has been introduced via quasifree knockout of the halo neutrons with proton targets, (p, pn), which provides spectroscopic information of the projectile ground state while probing the unbound binary subsystems [10,13,20].Interestingly, this enables the exploration of such correlations as a function of the neutron density by analyzing the average correlation angle as a function of the intrinsic neutron momentum [21,22].For 11 Li, it was reported that the dineutron correlations are localized at the surface of the nucleus [23], an idea already suggested in Ref. [16] and predicted by more recent theories [24,25] In this contribution we recall a quasifree sudden reaction framework to describe proton-target (p, pn) knockout reactions that incorporates the three-body structure of the Borromean core + n + n projectile and the unbound nature of the core + n residue in a consistent way.The method, which was succesfully applied to 11 Li, is now extended to 14 Be including core excitations.First, the theoretical approach is briefly discussed.Then, some results for the angular and intrinsic momentum distributions of 14 Be, as well as the correlation between them, are presented.

Theoretical description
As discussed in Refs.[21][22][23], intermediate-energy proton-target knockout reactions can be described within a quasifree sudden reaction framework.The model assumes a zero-range V pn interaction, and absorption and distortion effects are included through an eikonal S -matrix between the projectile and the proton.In the case of a valence-neutron knockout from a Borromean halo nucleus, the transition amplitude takes the form where {x, y} are the usual Jacobi coordinates depicted in Fig. 1, and k x , k y are the conjugated intrinsic momenta.
Here Φ g.s.(x, y) represents the ground state of the Borromean core + n + n projectile, and ϕ c−n (k x , x) is a continuum state describing the unbound core + n subsystem after knockout, which accounts for final state interactions.If no absorption is included (i.e., S = 1) and final state interactions are ignored (i.e., replacing ϕ c−n (k x , x) by a plane wave in the x coordinate), the above amplitude T becomes a simple Fourier transform of the ground-state wavefunction Φ g.s.(x, y).Therefore, Eq. ( 1) can be understood as a distorted Fourier transform.More details about the derivation of this equation can be found in Ref. [22], where the method was then applied to 11 Li(p, pn).Continuum states ϕ c−n (k x , x) are calculated by solving the two-body problem with standard scattering boundary conditions.The three-body ground-state wave function Φ gs (x, y) can be obtained within the hyperspherical formalism [1], provided that reliable core-n and n-n interactions are available.In particular, the analytical transformed harmonic oscillator method is adopted [26,27] to build the wave function in the so-called Jacobi-T representation, where the two valence neutrons are related by the x coordinate.Then, a transformation to the Jacobi-Y coordinates shown in Fig. 1 is performed.For consistency, the same core-n interaction is employed to compute ϕ c−n and Φ gs .Following the notation in Ref. [22], the ground state in Jacobi-Y coordinates can be written as where η = l x , j x , I, j 1 , l y , j 2 is a set of quantum numbers involving all relative angular momenta and spins that follows the coupling scheme (3) Here, s = 1/2 is the spin of the neutros and I represents the spin of the core.This means that j x = l x + s is the singleparticle angular momentum of a neutron with respect to the core, so that j x + I = j 1 is the total angular momentum of the core + n subsystem.With this coupling order, the distorted Fourier transform in Eq. ( 1) can be given the same form, where now the angles refer to that of the conjugate momenta.Here, the w η functions are obtained as a modified Fourier transform of the different partial waves of the overlaps between Φ g.s. and ϕ c−n .It can be shown that where ξ η are the corresponding radial overlaps (i.e., obtained by integrating over x).Note that these are formally the same structure overlaps used in DWBA and Transferto-the-continuum (TC) calculations presented in Refs.[28] (for transfer) and [29] (for knockout), respectively.It it also worth noting that these overlaps may involve different configurations for a given j 1 of the core + n system.This is particularly important if different core states I are allowed.In this work, core excitations are included by deforming the core + n interaction, in such a way that different core states can be coupled by in a rotational model [10,30,31].
From the previous equations, the cross section for the knockout process can be written as where P L are Legendre polynomials.The z-axis has been defined in such a way that θ is precisely the angle between k x and k y , and the quantities C ηη ′ and D (L) ηη ′ contain spin factors, phases and 6j symbols (see the Appendix of Ref. [22]).Equation ( 6) implies that an asymmetric θ distribution requires mixing between different-parity states in the core-n subsystem, such that odd L values are allowed [17,23].It is worth noting that the above cross section can be separated in contributions from different core states to assess the relevance of core excitations in the description of the relevant overlaps.

Results
In this work we apply the sudden formalism to describe the 14 Be(p, pn) knockout reaction.We use the 14 Be three-body wave function corresponding to model P2 in Ref. [10], which contains about ∼ 60% p-wave content mostly linked to a a low-lying p-wave resonance in 13 Be.The model includes core excitations in a rotational picture, using an effective quadrupole deformation parameter β 2 = 0.8 [31], in such a way that ∼ 20% of the norm corresponds to 12 Be in its first 2 + excited state.With a two-neutron separation energy of 1.3 MeV, the computed matter radius is 3.01 fm, which is consistent with Ref. [32].The ground-state probability density in Jacobi-T representation exhibits a clear maximum at small r nn distances, as shown in Fig. 2, associated to the dineutron configuration.With this structure input, we computed the intrinsic momentum and opening angle distributions by direct integration of Eq. ( 6).This requires an eikonal S -matrix for the p-core absorption.As in Ref. [22], we employ the tρ prescription, using a Hartree-Fock density with the SkX interaction [33] for the core.This introduces absorptive effects for small p-core distances, and thus for small p-n distances due to the zero-range approximation for V pn .As a result, the effect of absorption in the calculations is more important at large intrinsic momenta [22].
Figure 3 presents the intrinsic momentum distribution obtained by integrating over all angles θ and specific k x windows, which correspond to different 12 Be + n relative energies E rel = ℏ 2 k 2 x /(2µ).The main contributions to the distribution are shown with labels j 1 [l x j x ⊗ I], i.e., corresponding to the coupling order given by Eq. ( 3) for the  core + n subsystem.At small relative energies the distributions are governed by knockout from a p 1/2 orbital, whereas at higher energies the effect of the d 5/2 becomes more important.The top panel shows also the total distribution without absorption, indicating that absorption effects tend to narrow the distribution as it occurs at small distances and thus large momenta.Figure 4 shows the opening angle distribution, integrated for all relative energies, and compared with the result in a specific region of intrinsic k y momenta around the maximum of Fig. 3.
The asymmetry in the distribution, associated to the mixing between different-parity components, is more apparent in the restricted k y window, where the amplitudes of s-, pand d-waves interfere and lead to θ > 90 deg in momentum space.
In Fig. 5, the average opening angle θ as a function of the intrinsic momentum k y is presented.The correlation plot exhibits a clear maximum around k y ≃ 0.2 fm −1 , corresponding to ⟨θ⟩ > 90 deg, where the opposite-parity sand p-wave contributions of the momentum distribution in Fig. 3 are comparable.Within the present model, this mixing is also responsible for the maximum of the probability in Fig. 2 at small interneutron distances r nn , which happens at the nuclear surface (not the interior, not the halo tail).Note that, at large intrinsic momenta, the results for ⟨θ⟩ are strongly influenced by absorption effects [22], so the region for k y ≳ 0.5 fm −1 is not reliable to extract nuclear structure information within the present formalism.The low-k y maximum, however, is robust under changes in absorption effects.Figure 5 shows also the contribution of each core state (0 + ground state, 2 + excited state) to the correlation angle.The weight of each contribution can be easily deduced from the intrinsic momentum distribution.The results indicate that core excitations, while small within the adopted three-body model, produce a reduction of the average angle in momentum space, effectively reducing the dineutron correlations.Therefore, the inclusion of those channels may be crucial to improve the description.The comparison of the present calculations with experimental data is in progress and will be presented elsewhere [34], together with results for 11 Li and 17 B.

Figure 1 .
Figure 1.Schematic representation of the proton-target knockout reaction.

Figure 5 .
Figure 5. Average opening angle as a function of the intrinsic neutron momentum.The solid line is the complete calculation, while the dashed and dot-dashed lines correspond to the core left in its 0 + ground state or first 2 + excited state, respectively.