6 Li bound state and d + α scattering Motivation EST separable expansion Faddeev-AGS equations Summary and outlook Three-body approach to d + α scattering and bound state using realistic forces in a separable or non-separable representation L. Hlophe NSCL/FRIB (Collaborators: Jin Lei, Ch. Elster, F. M. Nunes, A. Nogga) 1 / 29
6 Li bound state and d + α scattering Motivation EST separable expansion Faddeev-AGS equations Summary and outlook Importance of ( d, p ) -reactions • Probing single-particle structure of nuclei • Extracting neutron-capture rates relevant for astrophysics 2 / 29
6 Li bound state and d + α scattering Motivation EST separable expansion Faddeev-AGS equations Summary and outlook Importance of ( d, p ) -reactions • Probing single-particle structure of nuclei • Extracting neutron-capture rates relevant for astrophysics 130 Sn ( d, p ) 131 Sn 130 Sn ( n, γ ) 131 Sn − → Kozub et. al PRL 109 , 172501 (2012) 2 / 29
6 Li bound state and d + α scattering Motivation EST separable expansion Faddeev-AGS equations Summary and outlook Three-Body Model for ( d, p ) Reactions The many-body problem • The deuteron ( d ) + target ( A ) system consists of A + 2 nucleons • Solutions not feasible for reactions involving heavy targets Isolating relevant degrees of freedom r r R R • Formulation of three-body problem by Faddeev • Momentum space formulation: Faddeev-AGS equations 3 / 29
6 Li bound state and d + α scattering Motivation EST separable expansion Faddeev-AGS equations Summary and outlook The Effective Three-Body Hamiltonian np-system • High precision NN potentials with χ 2 ≈ 1 , e.g., CD-Bonn [R. Machleidt, Phys. Rev. C63, 024001 (2001)] • NN potentials derived from chiral EFT nA system • Phenomenological fits of elastic scattering data to Woods-Saxon form, e.g. � 1 � d V 0 � + V so � l · σ v ( r ) = − � r − R 0 r dr � r − Rso 1+exp 1+exp a 0 aso • Microscopically computed, e.g., J. Rotureau, Phys. Rev. C 95, 024315 (2017) pA system • Similar to nA but with the Coulomb repulsion 4 / 29
6 Li bound state and d + α scattering Motivation EST separable expansion Faddeev-AGS equations Summary and outlook Solving the Faddeev-AGS Equations Challenges 1. Non-trivial singularities in the kernel of multivariate integral equations 2. Treatment of the Coulomb interaction in momentum space Remedy: 1. Employing separable two-body interactions ( i.e. v ( r, r ′ ) = h 1 ( r ) λ 11 h 1 ( r ′ ) + h 1 ( r ) λ 12 h 2 ( r ′ ) + ... ) • Reduces the Faddeev-AGS equations into coupled integral equations in one variable 2. Formulation of Faddeev-AGS equations in the Coulomb basis (A. Mukhamedzhanov, et al. Phys.Rev. C86 , 034001 (2012).) – based on separable two-body potentials 5 / 29
6 Li bound state and d + α scattering Motivation EST separable expansion Faddeev-AGS equations Summary and outlook Objectives 1. Construct separable expansions for: • High precision NN interactions • Effective nA and pA potentials 2. Benchmark for the three-body problem: Faddeev-AGS equations with (1) original three-body Hamiltonian and (2) its separable expansion: (a) 3-body bound state: • Compare 3-body binding energies and momentum distributions (b) Benchmark for d + A scattering: • Compare angular distributions for elastic scattering as well as transfer and deuteron breakup reactions 6 / 29
6 Li bound state and d + α scattering Motivation EST separable expansion Faddeev-AGS equations Summary and outlook Separable expansion for 2-Body potentials: EST scheme • Start from potential V , solve for eigenstates of Hamiltonian H 0 + V at energies E i : H | ψ i � = E i | ψ i � rank • Separable expansion: v sep = � V | ψ i � λ ij � ψ i | V ij [ λ − 1 ] ij = � ψ i | V | ψ j � • Momentum space: | ψ i � = | p i � + G (+) ( E i ) V | ψ i � 0 • Physical solutions: p i = √ 2 µE i • To accelerate convergence of observables: include off-shell solutions with independent p i and E i • Notation: t -matrix t ( E i ) | p i � = V | ψ i � ≡ | h i � • Matrix elements given as v sep ( p ′ , p ) = � h i ( p ) λ ij h j ( p ) ij [Ernst et al. , Phys.Rev. C9, 1780 (1974)] 7 / 29
6 Li bound state and d + α scattering Motivation EST separable expansion Faddeev-AGS equations Summary and outlook The np t -matrix for J = S = 1 with CD-Bonn potential 0 (a) -0.1 l np = l’ np = 0 -0.2 -1 -0.3 CDBonn: p’= 0.3 fm 2 ] -0.4 -1 t l’ np l np (p’, p, E=-50) [fm CDBonn: p’= 0.8 fm 0.012 (b) -1 rank-6: p’= 0.3 fm l np = l’ np = 2 0.008 -1 rank-6: p’= 0.8 fm 0.004 0 -0.004 l np = 2, l’ np = 0 (c) 0.15 0.1 0.05 0 0 1 2 3 4 5 6 7 8 -1 ] p [fm • Support points: { E m , p m } = {− 60 , 0 . 4 } , {− 60 , 1 . 1 } , {− 60 , 2 . 5 } , {− 5 , 0 . 4 } , {− 5 , 1 . 1 } , {− 5 , 2 . 5 } • Shape of potential in p -space determines location of support momenta 8 / 29
6 Li bound state and d + α scattering Motivation EST separable expansion Faddeev-AGS equations Summary and outlook Removing Pauli-Forbidden States • S 1 / 2 partial wave supports Pauli-forbidden state | φ � → ˜ • To project out the state | φ � : V − V = V + lim Γ →∞ | φ � Γ � φ | • Corresponding t -matrix: | φ �� φ | ˜ t ( E ) = t ( E ) − ( E − H 0 ) ( E − E b )[1 − ( E − E b ) / Γ] ( E − H 0 ) • Γ limit can be taken analytically t ( p ′ , p ; E ) = t ( p ′ , p ; E ) − ( E − E p ′ ) φ ( p ′ ) φ ( p ) ˜ E − E b ( E − E p ) 9 / 29
6 Li bound state and d + α scattering Motivation EST separable expansion Faddeev-AGS equations Summary and outlook Removing Pauli-Forbidden States • S 1 / 2 partial wave supports Pauli-forbidden state | φ � → ˜ • To project out the state | φ � : V − V = V + lim Γ →∞ | φ � Γ � φ | • Corresponding t -matrix: | φ �� φ | ˜ t ( E ) = t ( E ) − ( E − H 0 ) ( E − E b )[1 − ( E − E b ) / Γ] ( E − H 0 ) • Γ limit can be taken analytically t ( p ′ , p ; E ) = t ( p ′ , p ; E ) − ( E − E p ′ ) φ ( p ′ ) φ ( p ) ˜ E − E b ( E − E p ) Separable Expansion • Separable expansion of V also supports bound state | φ � , must be removed • Convenient approach: expand ˜ V instead of V • Advantages: (1) straightforward implementation and (2) does not increase rank 9 / 29
6 Li bound state and d + α scattering Motivation EST separable expansion Faddeev-AGS equations Summary and outlook Jacobi Coordinates: 3 different particles, 3 arrangement channels p n n p i q j q k p p n p k q i p j A A A ( i ) ( j ) ( k ) • Pair momenta: p i , p j , p k , spectator momenta q i , q j , q k • Notation: V i ≡ V np , V j ≡ V pA , V k ≡ V nA ⇒ 2-body potentials • Free Hamiltonian H 0 = p 2 i / 2 µ i + q 2 i / 2 M i • 3-Body Hamiltonian: H 3 b = H 0 + V i + V j + V k 10 / 29
6 Li bound state and d + α scattering Motivation EST separable expansion Faddeev-AGS equations Summary and outlook Faddeev equations for a three-body bound state: • Three-body wavefunction | Ψ � = | ψ i � + | ψ j � + | ψ k � • Faddeev components have definition | ψ i � ≡ G 0 ( E 3 ) V i | Ψ � � � Coupled equations: | ψ i � = G 0 ( E 3 ) t i ( E 3 ) | ψ j � + | ψ k � • Two-body t -matrix: t i ( E 3 ) = V i + V i G 0 ( E 3 ) t ( E 3 ) • Explicit momentum space representation � 2 t α i α ′ � dp ′ i p ′ ( p i , p ′ ψ i ( p i q i α i ) = G 0 ( E q i , p i ) i ; E q i ) i i i α i ′ ψ j ( p ′ i q i α ′ i ) + ψ k ( p ′ i q i α ′ � � × i ) � Coupled integral equations in two variables: p i and q i 11 / 29
6 Li bound state and d + α scattering Motivation EST separable expansion Faddeev-AGS equations Summary and outlook Bound state Faddeev equations with separable potentials • Separable potential ⇒ t -matrix elements have form rank t α i α ′ mα i ( p i ) τ α i α ′ � ( p i , p ′ h i mn ( E q i ) h i i ( p ′ i i ; E q i ) = i i ) i nα ′ mn • Faddeev components become separable, e.g., if rank=1: ψ i ( p i q i α i ) = h i ( p i ) F ( i ) α i ( q i ) • Task is reduced to solving for functions F ( i ) ( q i ) which fulfill � 2 Z ( ij ) dq j ′ q ′ j ) F ( j ) � j ; E 3 b ) τ α j α ′ F ( i ) j ( q i , q ′ j ( q ′ j ( q ′ α i ( q i ) = j ) j α i α ′ α ′ α j α ′ j � 2 Z ( ik ) k ) F ( k ) � dq ′ k q ′ k ( q i , q ′ k ; E 3 b ) τ α k α ′ k ( q ′ k ( q ′ + k ) k α i α ′ α ′ α k α ′ k 12 / 29
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