Structure and decays of nuclear three-body systems: the Gamow coupled-channel method in Jacobi coordinates
S. M. Wang N. Michel W. Nazarewicz F. R. Xu
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FRIB/NSCL Laboratory, Michigan State University, East Lansing, Michigan 48824, USA

FRIB Laboratory, Michigan State University, East Lansing, Michigan 48824, USA

Department of Physics and Astronomy and FRIB Laboratory, Michigan State University, East Lansing, Michigan 48824, USA

School of Physics, Peking University, Beijing 100871, China







  Background
Weakly bound and unbound
nuclear states appearing around particle thresholds are prototypical open quantum systems. Theories of such states must take into account  configuration mixing effects in the presence of strong coupling to the particle continuum space.

  Purpose
To describe structure and decays of three-body systems, we developed a Gamow coupled-channel (GCC) approach in Jacobi coordinates by employing the complex-momentum formalism. We benchmarked the new framework  against the complex-energy Gamow Shell Model (GSM).

  Methods
The GCC formalism is expressed in Jacobi coordinates, so that the center-of-mass motion is automatically eliminated. To solve 
the coupled-channel equations, we use hyperspherical harmonics to describe the angular wave functions while the radial wave functions   are expanded  in the Berggren ensemble, which includes bound, scattering and Gamow states. 

  Results
We show that the GCC method is both accurate and robust. 
Its results for energies, decay widths, and nucleon-nucleon angular correlations are in good agreement with the GSM results. 

  Conclusions
We have demonstrated that a three-body GSM formalism explicitly constructed in
cluster-orbital shell model coordinates provides similar results to a GCC framework expressed in  Jacobi coordinates, provided that a large configuration space is employed. 
Our calculations for A=6 systems and  ^26O  show that nucleon-nucleon angular correlations are 
sensitive to the valence-neutron interaction. 
The new GCC technique has many attractive features
when applied to bound and unbound states of three-body systems: it is precise, efficient, and can  be extended  by introducing a microscopic model of the core.



Structure and decays of  nuclear three-body systems: the Gamow coupled-channel method in Jacobi coordinates
    F.R. Xu (许甫荣)
    December 30, 2023
===========================================================================================================





§ INTRODUCTION


Properties of rare isotopes that inhabit remote regions of the nuclear landscape at and beyond the particle driplines are in the forefront of nuclear structure and reaction research <cit.>.
The next-generation of rare isotope beam  facilities will provide unique data on dripline systems that  will test  theory, highlight shortcomings,  and  identify areas for improvement. 
The challenge for nuclear  theory is to develop methodologies to reliably calculate and understand the properties and dynamics of new physical systems  with different properties due to large neutron-to-proton asymmetries  and low-lying reaction thresholds. 
Here, dripline systems are of particular interest as they can exhibit exotic radioactive  decay modes such as two-nucleon emission <cit.>. Theories of such nuclei must take into account their open quantum nature.
 
Theoretically, a powerful suite of A-body
approaches based on inter-nucleon interactions provides a quantitative description of light and medium-mass nuclei and  their reactions <cit.>.
To unify nuclear bound states with resonances and scattering continuum within
one consistent framework, advanced   continuum shell-model approaches have been introduced <cit.>. Microscopic models of exotic nuclear states have been supplemented by a suite of powerful, albeit more phenomenological  models,  based on effective degrees of freedom such as cluster structures. While such models provide a  “lower resolution" picture of the nucleus, they can be extremely useful when interpreting experimental data, providing guidance for future measurements, and provide guidance for more microscopic approaches.

The objective of  this work is to develop a new three-body method to describe both reaction and structure aspects of two-particle emission. A prototype system of interest  is
the two-neutron-unbound ground state of ^26O
<cit.>. According to theory,  ^26O 
 exhibits the dineutron-type  correlations
<cit.>. To describe such a system, nuclear model should be based on a fine-tuned interaction capable of describing particle-emission thresholds, a sound many-body method, and a capability to treat simultaneously bound and unbound states. 


If one considers bound three-body systems, few-body models are very useful <cit.>, especially models based  on the Lagrange-mesh technique <cit.> or cluster-orbital shell model (COSM) <cit.>. However, for the description of  resonances, the outgoing wave function in the asymptotic region need to be treated very carefully. For example, one can divide the coordinate space into internal and asymptotic regions, where the R-matrix theory <cit.>,
microscopic cluster model <cit.>, and the diagonalization of the Coulomb interaction <cit.> can be used. Other useful techniques include the Green function method <cit.> and the complex scaling  <cit.>. 

Our strategy is to construct a precise three-body framework to  weakly bound and unbound systems
similar to that of the GSM <cit.>. The attractive feature of the GSM is that – by employing the Berggren ensemble <cit.> – it treats bound, scattering, and outgoing Gamow states on the same footing. Consequently,  energies and decay widths are obtained simultaneously as the real and imaginary parts of the complex eigenenergies of the shell model Hamiltonian <cit.>. In this study, we develop a three-body Gamow coupled-channel (GCC) approach in Jacobi coordinates with the Berggren basis. Since the Jacobi coordinates allow for the exact treatment
of  nuclear wave functions in both nuclear and asymptotic regions, and as the Berggren basis explicitly takes into account continuum effects, a comprehensive description of weakly-bound three-body systems can be achieved. As the GSM is based on the  COSM coordinates, 
a recoil term appears due to the center-of-mass motion. 
Hence, it is of interest to compare Jacobi- and COSM-based frameworks for the description of weakly bound and resonant nuclear states. 

This article is organized as follows. 
Section <ref>  contains the description of  models and  approximations. In particular, it lays out the new GCC approach and GSM model used for benchmarking, and defines the configuration spaces used. 
The results for A=6 systems and  ^26O are contained in Sec. <ref>. Finally, the summary and outlook are given in Sec. <ref>.





§ THE MODEL





 §.§ Gamow Coupled Channel approach


In the three-body GCC model, the nucleus is described in terms of a core and two valence nucleons (or clusters). The GCC Hamiltonian  can be written as:

    Ĥ = ∑^3_i=1p̂⃗̂^2_i/2 m_i +∑^3_i>j=1 V_ij(r⃗_ij)-T̂_ c.m.,

where V_ij is the interaction between clusters i and j, including central, spin-orbit and Coulomb terms, and  T̂_ c.m. stands for  the kinetic energy of the center-of-mass.

The unwanted feature  of three-body models is the appearance of Pauli forbidden states arising from the lack of antisymmetrization between core and valence particles.
In order to eliminate the Pauli forbidden states, we implemented the orthogonal projection method <cit.> by adding to the GCC Hamiltionan the Pauli operator

    Q̂= Λ∑_c |φ ^j_c m_c⟩⟨φ ^j_c m_c|,

where Λ is a constant and | φ^j_c m_c⟩ is a 2-body state involving forbidden s.p. states of core nucleons. At large values of Λ,
Pauli-forbidden states appear at high  energies, so that they are effectively suppressed.



 In order to describe three-body asymptotics and to eliminate the spurious center-of-mass motion exactly, we express  the GCC model in the relative  (Jacobi) coordinates <cit.>:

    x⃗   = √(μ _ij) (r⃗_i - r⃗_j),
    y⃗   = √(μ _(ij)k)(r⃗_k - A_ir⃗_i + A_jr⃗_j/A_i + A_j),

where r⃗_i is the position vector of the i-th cluster, A_i is  the i-th cluster mass number, and μ _ij and μ _(ij)k are the reduced masses associated with x⃗ and y⃗, respectively:

    μ _ij   = A_iA_j/A_i+A_j,
    μ _(ij)k   = (A_i+A_j)A_k/A_i+A_j+A_k.

As one can see in Fig. <ref>, Jacobi coordinates can be expressed as  T- and Y-types, each associated with  a complete basis set. In practice, it is convenient to calculate the matrix elements of the two-body interaction individually in T- and Y-type coordinates, and then transform them to one single Jacobi set. To describe the transformation between different types of Jacobi coordinates, it is convenient to introduce
the basis of hyperspherical harmonics (HH) <cit.>.
The hyperspherical coordinates  are constructed from a five-dimensional hyperangular coordinates Ω_5 and a hyperradial coordinate ρ=√(x^2 + y^2).
The transformation between different sets of Jacobi coordinates is given by the Raynal-Revai coefficients <cit.>.

Expressed in HH, the total wave-function can be written as <cit.>:
  
    Ψ ^JMπ (ρ, Ω_5) = ρ ^-5/2∑_γ Kψ ^Jπ_γ K(ρ) 𝒴 ^JM_γ K (Ω_5),

where K is the hyperspherical quantum number and γ = {s_1,s_2,s_3,S_12,S,ℓ_x,ℓ_y,L} is a set of quantum numbers other than K. The quantum numbers s and ℓ stand for spin and orbital angular momentum, respectively, ψ ^Jπ_γ K(ρ) is the hyperradial wave function, and 𝒴 ^JM_γ K (Ω_5) is the hyperspherical harmonic. 

The resulting  Schrödinger equation for the hyperradial wave functions can be written as a set of coupled-channel equations:
  
    [ -ħ^2/2m(d^2/dρ^2 - (K+3/2)(K+5/2)/ρ^2)-Ẽ] ψ ^Jπ_γ K(ρ)   
    
       + ∑_K'γ' V^Jπ_K'γ', Kγ(ρ) ψ ^Jπ_γ'K'(ρ)
          +∑_K'γ'∫_0^∞ W_K'γ', Kγ(ρ,ρ')ψ ^Lπ_γ'K'(ρ')dρ'=0,


 where
  
    V^Lπ_K'γ', Kγ(ρ) = ⟨𝒴 ^JM_γ' K'|  ∑^3_i>j=1 V_ij(r⃗_ij)| 𝒴 ^JM_γ K⟩

and 
  
    W_K'γ', Kγ(ρ,ρ') = ⟨𝒴 ^JM_γ' K' | Q̂  |𝒴 ^JM_γ K⟩

is the  non-local potential generated by the Pauli projection operator (<ref>).

In order to treat the positive-energy continuum space precisely, we use the Berggren  expansion technique for the hyperradial wave function:

    ψ ^Jπ_γ K(ρ) = ∑_ n C^Jπ M_γ n Kℬ ^Jπ_γ n(ρ),

where ℬ ^Jπ_γ n(ρ) represents a s.p. state belonging to to the Berggren  ensemble <cit.>.
The Berggren ensemble defines a basis in the complex momentum plane, which includes bound, decaying, and scattering states.
The completeness relation for the Berggren ensemble can be written as:
  
    ∑_ n∈ b,dℬ_ n(k_ n,ρ)ℬ_ n(k_ n,ρ^')  +    ∫_L^+ℬ(k,ρ)ℬ(k,ρ^')dk  
        = δ(ρ-ρ^'),

where b are bound states and  d are  decaying resonant (or Gamow)  states lying between the real-k momentum axis in the fourth quadrant of the  complex-k plane, and the     L^+ contour representing the complex-k scattering continuum.
 For numerical purposes,  L^+  has to be discretized, e.g., by adopting the Gauss-Legendre quadrature <cit.>. 
In principle, the contour L^+  can be chosen arbitrarily as long as it encompasses the resonances of interest. If the contour L^+ is chosen to lie along the real k-axis, the Berggren completeness relation reduces to the Newton completeness relation <cit.> involving bound and real-momentum scattering states.

To calculate radial matrix elements  with the Berggren basis,
we employ  the exterior complex scaling <cit.>, where integrals are calculated along a complex radial path:
    	
    ⟨ℬ_ n|   V(ρ)|ℬ_ m⟩ =∫_0^Rℬ_ n(ρ)V(ρ)ℬ_ m(ρ)dρ
       + ∫_0^+∞ℬ_ n(R+ρ e^iθ)V(R+ρ e^iθ)ℬ_ m(R+ρ e^iθ)dρ.

For potentials that decrease as O(1/ρ^2) (centrifugal potential) or faster (nuclear potential), 
R should be sufficiently large to bypass all singularities and the scaling angle θ is  chosen so that the integral converges, see  Ref. <cit.> for details. As the Coulomb potential is not square-integrable, its matrix elements diverge when  k_n = k_m. A practical solution is provided by the so-called “off-diagonal method" proposed in Ref. <cit.>. Basically, a small offset ±δ k is added to the linear momenta k_n and k_m of involved scattering wave-functions, so that the resulting diagonal Coulomb matrix element converges.



 §.§ Gamow Shell Model


In the GSM, expressed in  COSM coordinates, one deals with the center-of-mass motion by adding a recoil term (p̂⃗̂_1·p̂⃗̂_2/m_nA_ core) <cit.>.
The GSM Hamiltonian is diagonalized in a basis of Slater determinants built from the
one-body Berggren ensemble. In this case, it is convenient  to deal with the Pauli principle
by eliminating spurious excitations  at a level of the s.p. basis. In practice, one just needs to construct a valence s.p.  space that does not contain the orbits occupied in the core. It is equivalent to the projection technique used in GCC wherein the Pauli operator (<ref>) expressed   in Jacobi coordinates has a two-body character.
The treatment of the interactions is the same in  GSM and GCC. In both cases,  we use the  complex scaling method to calculate  matrix elements <cit.> and the “off-diagonal method" to deal with the Coulomb potential <cit.>. 

The  two-body  recoil term is  treated in GSM  by expanding it  in a truncated basis of harmonic oscillator (HO). The HO basis depends on the oscillator length b and the number of states used in the expansion. As it was demonstrated in Refs. <cit.>, GSM eigenvalues and eigenfunctions converge for a sufficient number of HO states, and the dependence of the results on b is very weak.

Let us note in passing that  one has to be careful when using  arguments based on the variational principle when comparing  the performance of GSM with GCC. Indeed,  the treatment of the Pauli-forbidden states is slightly different in the two approaches. Moreover,
the recoil effect in the GSM is not removed exactly. (There is no recoil term in GCC as the center-of-mass motion is eliminated through the use of Jacobi coordinates.)





 §.§ Two-nucleon correlations


In order to study the correlations between the two valence nucleons, we  utilize the  two-nucleon density <cit.> ρ_nn'(r,r',θ) = ⟨Ψ|δ(r_1-r)δ(r_2-r^')δ(θ_12 - θ)|Ψ⟩, where r_1, r_2, and θ_12 are defined in Fig. <ref>(a).
In the following, we apply the normalization convention of Ref. <cit.> in which the
Jacobian 8π^2 r^2 r'^2 sinθ is incorporated into the definition of ρ_nn', i.e., it does not appear explicitly.
The angular density of the two valence nucleons is obtained  by integrating
ρ_nn'(r,r',θ) over radial coordinates:
 
    ρ(θ) = ∫ρ_nn'(r,r^',θ) dr dr^'.

The angular density is normalized to one: 
∫ρ(θ) dθ = 1.

While it is straightforward to calculate ρ_nn' with COSM coordinates, the angular density cannot be calculated directly with the Jacobi T-type coordinates used to diagonalize the GCC Hamiltonian.
Consequently, one can either calculate the density distribution ρ_ T(x,y,φ) in T-type coordinates and then transform it to  ρ(r_1,r_2,θ_12) in COSM coordinates by using the geometric relations of Fig. <ref>(a),
or – as we do in this study – one can apply the  T-type-to-COSM 
coordinate transformation. This transformation <cit.>, provides an analytical relation between hyperspherical harmonics in COSM coordinates 𝒴 ^JM_γ^' K^' (r⃗_1^', r⃗_2^' ) and the T-type Jacobi coordinates 𝒴 ^JM_γ K (x⃗^', y⃗^' ), where r⃗_1^', r⃗_2^', x⃗^' and y⃗^' are:

    r⃗_1^'   = √(A_i)r⃗_1,
    r⃗_2^'   = √(A_j)r⃗_2,
    x⃗^'   = x⃗= √(μ_ij)(r⃗_1-r⃗_2),
    y⃗^'   = √(A_i+A_j/μ_(ij)k)y⃗ = A_ir⃗_1+A_jr⃗_2/√(A_i+A_j).




 §.§ Model space and parameters

 
In order to compare approaches  formulated in   Jacobi and COSM coordinates, we consider model spaces defined by the cutoff value ℓ_ max, which is the maximum orbital angular momentum associated with (r⃗_1, r⃗_2) in GSM and  (x⃗, y⃗) in GCC. 
The remaining truncations come from the Berggren basis itself.

The nuclear two-body interaction between valence nucleons has been approximated by the  finite-range Minnesota force with the original parameters of Ref. <cit.>. For the core-valence Hamiltonian, we took a Woods-Saxon (WS) potential  with  parameters  fitted to the resonances of the core+n system. The one- and two-body Coulomb interaction has been considered when valence protons are present.

In the case of GSM, we use the Berggren basis for the spd partial waves and a HO basis for the channels with  higher orbital angular momenta. 
For ^6He, ^6Li and ^6Be we assume the ^4He core.
For ^6He and ^6Be, GSM we took a complex-momentum contour defined by the segments  k= 0 → 0.17-0.17i → 0.34 → 3 (all in  fm^-1)  for the  p_3/2 partial wave,  and 
0 → 0.5 → 1 → 3 fm^-1 for the remaining  spd partial waves. For ^6Li, we took the contours
0 → 0.18-0.17i → 0.5 → 3 fm^-1 for p_1/2; 0 → 0.15-0.14i → 0.5 → 3 fm^-1 for p_3/2;  and 0 → 0.25 → 0.5 → 3 fm^-1 for the sd partial waves. Each segment was discretized with  10 points. This is sufficient for the energies and most of other physical quantities, but one may need more points to describe wave functions precisely, especially for the unbound resonant states that are  affected by Coulomb interaction. Hence, we choose 15 points for each segment to calculate the two-proton angular correlation of the unbound  ^6Be. 
The  HO basis  was defined through the oscillator length b = 2 fm and the maximum radial quantum number n_ max=10.
The WS parameters  for the A = 6 nuclei are: the depth of the central
term V_0= 47 MeV;   spin-orbit strength V_ s.o. = 30 MeV; diffuseness a=0.65 fm; and the WS (and charge) radius  R=2 fm.  With these  parameters we predict  the 3/2^- ground state (g.s.) of ^5He at E=0.732 MeV (Γ=0.622 MeV), and its   first excited   1/2^- state at E=2.126 MeV (Γ=5.838 MeV). 

For ^26O, we consider the  ^24O  core <cit.>. In the GSM variant, we used the contour  0 → 0.2-0.15i → 0.4 → 3 fm^-1 for d_3/2,  and 0 → 0.5 → 1 → 3 fm^-1 for the remaining spd partial waves. For the HO basis we took b = 1.75 fm and n_ max=10. The  WS potential for ^26O  has fitted in Ref. <cit.> to the resonances of ^25O. Its parameters are:
V_0= 44.1 MeV, V_ s.o.= 45.87 MeV, a= 0.73 fm, and R = 3.6 fm.


The GCC calculations have been carried out with the maximal hyperspherical quantum number K_ max = 40, which is sufficient for all the physical quantities we  study. We checked that the calculated energies  differ by as little as   2 keV when varying  K_ max from 30 to 40. Similar as in  GSM, in GCC we used the Berggren basis for the K ⩽ 6 channels and the HO basis for the higher angular momentum channels. The complex-momentum contour of the Berggren basis is defined as:  k  = 0 → 0.3-0.2i → 0.5 → 0.8 → 1.2 → 4 (all in fm^-1), with each segment discretized with   10 points. We took the  HO basis with  b = 2 fm and  n_ max = 20. As k_ρ^2 = k_x^2 + k_y^2,  the energy range covered by the GCC basis is roughly doubled as compared to that of GSM.

For the one-body Coulomb potential, we use 
the dilatation-analytic form <cit.>: 

    U^(Z)_ c(r) = e^2Z_ c erf(r/ν_ c)/r,

where ν_c=4R_0/(3√(π)) fm,  R_0 is the radius of the WS potential, and Z_ c is the number of core protons.

We emphasize that the large continuum space, containing states of both parities, is essential for the formation of the dineutron structure in nuclei such as ^6He or ^26O  <cit.>. In the following, we shall study  the effect of including positive and negative parity continuum shells on the stability of threshold configurations.




§ RESULTS




 §.§ Structure of A=6 systems


We begin with  the GCC-GSM benchmarking for the A=6 systems.
Figure <ref> shows the convergence rate for the g.s. energies of  ^6He, ^6Li, and ^6Be with respect to ℓ_ max. 
(See Ref. <cit.> for a similar comparison between GSM and complex scaling results.)
While  the g.s. energies of ^6He and ^6Be are in a reasonable  agreement with experiment, ^6Li is overbound. This is because the  Minnesota interaction does not explicitly
separate the T = 0 and T = 1 channels.  The structure of ^6He and ^6Be is given by the T=1 force, while  the T=0  channel that is crucial for ^6Li has not been optimized. This is of minor importance for this study, as our goal is to benchmark GCC and GSM  not to provide quantitative predictions.  As we use different coordinates in GCC and GSM,
their  model spaces are manifestly different. Still for ℓ_ max=10 both approaches provide very similar results, which is most encouraging.  

 

One  can see in Fig.  <ref> that  
the calculations done with Jacobi coordinates converge  faster than those  with COSM coordinates. This comes from the attractive character of the nucleon-nucleon interaction,  which 
results in   the presence of a di-nucleon  structure (see discussion below). Consequently, as T-type Jacobi coordinates 
well describe  the  di-nucleon cluster,  they are able to capture  correlations in a more efficient way than COSM coordinates. This is in agreement with the findings of Ref. <cit.> based on  the complex scaling  method with COSM coordinates, who obtained  the g.s. energy ^6He  that was
slightly less bound as compared to results of Ref. <cit.> using  Jacobi coordinates. In any case,
our calculations have  demonstrated that one obtains very similar results in GCC and GSM when  sufficiently large model spaces are considered. As shown in Table <ref>, the  energy difference between  GCC and GSM predictions for A=6 systems is very small, around 20 keV for majority of states.
The maximum deviation of ∼70 keV is obtained for the 3^+ state of ^6Li. However,
because of the attractive character of the T=0 interaction,
the GSM calculation for this  state has not fully converged at ℓ_ max = 10.





Motivated by the discussion in Ref. <cit.>,
we have also studied the effect of   the ℓ-dependent core-nucleus potential. To this end, we  changed the WS strength V_0 from 47 MeV to 49 MeV for the ℓ=1 partial waves while keeping the standard strength for the remaining ℓ values. As seen in Fig. <ref>, the convergence behavior obtained with  Jacobi and COSM coordinates is fairly  similar to that shown in Fig. <ref>, where the WS  strength V_0 is the same for all partial waves. 
For ℓ_ max=12, the difference between GSM and GCC energies of ^6He 
becomes very small. 
This result is consistent  with the findings of  Ref. <cit.> that the 
recoil  effect can indeed 
be  successfully eliminated using COSM coordinates  at the expense of reduced convergence.



In order to see whether the difference between the model spaces of GCC and GSM can be compensated by renormalizing
the effective  Hamiltonian,
we slightly readjusted the depth  of the WS potential in GCC calculations  to reproduce the g.s. GSM energy of ^6He at model space ℓ_ max=7. As a result, the strength V_0 changed from 47 MeV to 46.9 MeV. Except for the 2^+ state of ^6He, the GSM and  GCC energies for A=6 systems got significantly closer as a result of such a renormalization. This indicates that the differences between Jacobi coordinates and COSM coordinates can be partly accounted for by refitting interaction parameters, even though model spaces and asymptotic behavior are different.

GCC is also in rough agreement with GSM when comparing decay widths, considering that they are very sensitive to the asymptotic behavior of the wave function, which is treated differently with Jacobi and COSM coordinates. Also, the presence of the recoil term in GSM, which is dealt with by means of the HO expansion, is expected to impact the GSM results for  decay widths.


In order to check  the precision of decay widths  calculated with GCC, we adopted the current expression <cit.>:
  
    Γ  =i  ∫( Ψ^† Ĥ Ψ - Ψ Ĥ Ψ^† )  dx⃗dy⃗/∫|Ψ|^2 dx⃗dy⃗,

which can be expressed in  hyperspherical coordinates as <cit.>:
  
    Γ  =i ħ^2/m. ∫ dΩ_5  Im[ψ∂/∂ρψ^†]|_ρ=ρ_ max/∫^ρ_ max_0 |ψ|^2 dρ dΩ_5,

where ρ_ max  is larger than the nuclear radius (in general, the decay width should not depend on the choice of ρ_ max). 
By using the current expression, we obtain Γ=42 keV for 2^+ state of ^6He and Γ=54 keV for 0^+ state of ^6Be, which are practically the same as the GCC values of Table <ref> obtained  from the direct diagonalization.








We now discuss the angular correlation  of the two valence neutrons in the g.s. of  ^6He. Figure <ref> shows GSM and GCC results for  model  spaces defined by  different values of ℓ_ max. 
The distribution ρ(θ) shows two maxima <cit.>. The higher peak, at a small opening angle, can be associated with a dineutron configuration. 
The second maximum, found in the region of large angles, represents the cigarlike configuration.
The GCC results for ℓ_ max=2 and 10 are already very close. This is not the case for the GSM, which shows sensitivity to the cutoff value of ℓ.  
This is because the large continuum space, including states of positive and negative parity is needed in the COSM picture to describe dineutron correlations <cit.>.
Indeed, as ℓ_ max increases, the angular correlations obtained in GSM and GCC are very similar. This indicates that Jacobi and COSM descriptions of ρ(θ) are  essentially equivalent  provided that the model space is sufficiently large.

In order to benchmark GCC and GSM calculations for the valence-proton case, in Fig. <ref> we compare two-nucleon angular correlations for A = 6 nuclei ^6He, ^6Li, and ^6Be. 
Similar to Refs. <cit.>, we find that the  T=1 configurations have a dominant 
S = 0 component, in which the two neutrons in ^6He or two protons in ^6Be are in the spin singlet state. The amplitude of the S = 1 density component  is  small.
For all nuclei, GCC and GSM angular correlations are close.

Similar to ^6He, the two peaks in ^6Be indicate diproton and cigarlike configurations <cit.> (see also Refs. <cit.>).
It is to be noted  that the dineutron peak in ^6He is slightly higher than the diproton maximum in ^6Be. This is  due to the repulsive character of the Coulomb interaction between valence protons. 
The large maximum at small opening angles seen in ^6Li corresponds to the deuteron-like structure. As discussed in Ref. <cit.>, this peak is larger that the the dineutron correlation in ^6He.  Indeed, the valence proton-neutron pair in ^6Li is very strongly correlated because the T=0 interaction is much stronger than the T=1 interaction.
The different features in the two-nucleon angular correlations  in the three  A=6 systems shown in Fig. <ref>
demonstrate that the angular correlations  contain useful information on the effective interaction between valence nucleons.





 §.§ Structure of unbound ^26O


After benchmarking GSM and GCC for A=6 systems,
we apply both models to ^26O, which is believed to be a threshold dineutron structure
<cit.>.



It is a theoretical challenge to reproduce the resonances in ^26O as both continuum and high partial waves must be considered.  As ^24O can be 
 associated with the  subshell closure in which the 0d_5/2 and 1s_1/2 neutron shells
are occupied <cit.>, it can be used as core in our three-body model. 

Figure <ref> illustrates the convergence  of the g.s. of ^26O with respect to  ℓ_ max in GSM and GCC calculations. It is seen that in the GCC approach the energy converges 
nearly exponentially  and that the stable result is practically reached at ℓ_ max=7. 
While slightly higher in energy, the GSM results are quite satisfactory, as they differ only  by about  30 keV from the GCC benchmark. Still, it is clear that ℓ_ max=12 is not sufficient to reach the full convergence in GSM.


The  calculated energies and widths  of g.s. and 2^+ state of ^26O are displayed  in Table <ref>; they are both  consistent with the most recent  experimental values <cit.>. 



The amplitudes of dominant configurations listed in  Table <ref>  illustrate the importance of considering partial waves of different parity in the GSM description of a dineutron g.s. configuration in ^26O <cit.>.



The g.s. wave function of ^26O  computed in GCC is shown in  Fig. <ref> in the Jacobi coordinates. The corresponding angular distribution is displayed in Fig. <ref>.



Three pronounced peaks associated with the dineutron, triangular, and cigarlike configurations <cit.> can be identified. 
In GCC, the (ℓ_x, ℓ_y) = (s, s), (p, p)  components dominate the g.s. wave function of  ^26O; this is consistent with a sizable clusterization of the two neutrons.
In COSM coordinates,  it is the (ℓ_1, ℓ_2) = (d,d) configuration that dominates, but the the negative-parity (f, f) and (p, p) channels contribute with ∼20%.
Again, it is encouraging to see that with ℓ_ max=10 both approaches predict very similar two-nucleon densities.



In Table <ref>  we also display the predicted structure of the excited 2^+ state of ^26O . The predicted energy is close to experiment <cit.> and other theoretical studies, see, e.g., <cit.>. 
We obtain a small width for this state, which is consistent with the GSM+DMRG calculations of Ref. <cit.>.
The GCC occupations of Table <ref> indicate that the wave function of the 2^+ state is spread  out in space, as the main three configurations, of cluster type, only contribute to the wave function with only 65%.
When considering the GSM wave function, the (d, d) configuration  dominates.  The  corresponding two-neutron angular correlation shown in Fig. <ref>(b)  exhibits a broad distribution with a maximum  around 
90^∘. 
This situation is fairly similar to what has been  predicted for the 2^+ state of ^6He <cit.>.


Finally, it is interesting to study  how the neutron-neutron interaction impacts the angular correlation.
To this end, Fig. <ref>(a) shows ρ(θ) obtained with  
the Minnesota neutron-neutron interaction whose strength has been reduced by 50%. 
While there are still three peaks present, the distribution becomes more uniform and 
the dineutron component no longer dominates. We can this conclude that the nn angular correlation
can be used as an indicator of the  interaction between valence nucleons.




§ CONCLUSIONS


We developed a Gamow coupled-channel approach in Jacobi coordinates with the Berggren basis to describe structure and decays of three-body systems. We benchmarked  the performance of the new approach against the Gamow Shell Model. Both methods are capable of considering large continuum spaces but differ in their treatment of  three-body asymptotics, center-of-mass motion, and Pauli operator.
In spite of these differences, we demonstrated that the Jacobi-coordinate-based framework (GCC) and COSM-based framework (GSM) can produce fairly similar results, provided that the continuum space is sufficiently large.

For benchmarking and illustrative examples we choose  ^6He, ^6Li, and ^6Be,   and ^26O – all viewed as a core-plus-two-nucleon systems. We discussed the spectra, decay widths, and nucleon-nucleon angular correlations in these nuclei.
The Jacobi coordinates capture  cluster correlations (such as dineutron and deuteron-type) more efficiently; hence, the convergence rate of GCC is faster than that of GSM. 

For  ^26O, we demonstrated the sensitivity of nn angular correlation to the valence-neutron interaction. It will be interesting to investigate this aspect further to provide 
guidance for future experimental investigations of di-nucleon correlations in bound and unbound states of dripline nuclei.

In summary, we developed an efficient approach to structure and decays of three-cluster systems. The GCC method is based on a Hamiltonian involving a two-body interaction between valence nucleons and a
one-body field representing the core-nucleon potential. The advantage of the model is its ability to correctly describe the three-body asymptotic behavior and the efficient treatment of the continuum space, which is of particular importance for the treatment of threshold states and narrow resonances. The model can be easily extended along the lines of the resonating group method   by introducing a microscopic picture of the core <cit.>. Meanwhile, it can be used to elucidate  experimental findings on  dripline systems, and to provide finetuned  predictions to guide  A-body approaches.



We thank Kévin Fossez, Yannen Jaganathen,  Georgios Papadimitriou, and Marek Płoszajczak for useful discussions. 
This material is
based upon work supported by the U.S. Department of Energy, Office of
Science, Office of Nuclear Physics under  award numbers 
DE-SC0013365 (Michigan State University),   DE-SC0008511 (NUCLEI
SciDAC-3 collaboration), DE-SC0009971 (CUSTIPEN: China-U.S. Theory Institute for Physics with
Exotic Nuclei),
and also supported in part by Michigan State University through computational resources provided by the Institute for Cyber-Enabled Research.




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