✬ ✩ Two-body Momentum Distributions in 2 ≤ A ≤ 40 nuclei M. Alvioli National Research Council Research Institute for Geo-Hydrological Hazards Perugia, Italy M. Alvioli MIT 2016 1 ✫ ✪
✬ ✩ CONTENTS 0. Introduction & basic quantities - Few-body nuclei, “ exact ” wave functions - Many-body nuclei, cluster expansion - Additional slides � 1. Two-Body Momentum Distributions: - factorization into n rel ( k rel ) and n c.m. ( K c.m. ) - Scaling to n D ( k ) - Definition of the scaling coefficients C pn A - Relationship between one- and two-body momentum dis- tribution - Relationship between K c.m. -integrated and K c.m. =0 2BMD 2. Comparison with experimental data MIT 2016 2 ✫ ✪
✬ ✩ 0. Comprehensive review of experimental results � 1 1 1 1 1 Counts 5.9 GeV/c, 98 0.8 0.8 0.8 0.8 0.8 8.0 GeV/c, 98 9.0 GeV/c, 98 0.6 0.6 0.6 0.6 0.6 p 40 5.9 GeV/c, 94 0.4 0.4 0.4 0.4 0.4 7.5 GeV/c, 94 γ p 30 0.2 0.2 0.2 0.2 0.2 cos γ 0 0 0 0 0 20 -0.2 -0.2 -0.2 -0.2 -0.2 -0.4 -0.4 -0.4 -0.4 -0.4 10 -0.6 -0.6 -0.6 -0.6 -0.6 -0.8 -0.8 -0.8 -0.8 -0.8 0 -1.00 -0.98 -0.96 -0.94 -0.92 -0.90 γ -1 -1 -1 -1 -1 cos 0.05 0.05 0.05 0.05 0.05 0.1 0.1 0.1 0.1 0.1 0.15 0.15 0.15 0.15 0.15 0.2 0.2 0.2 0.2 0.2 0.25 0.25 0.25 0.25 0.25 0.3 0.3 0.3 0.3 0.3 0.35 0.35 0.35 0.35 0.35 0.4 0.4 0.4 0.4 0.4 0.45 0.45 0.45 0.45 0.45 0.5 0.5 0.5 0.5 0.5 0.55 0.55 0.55 0.55 0.55 p n (GeV/c) A ( e, e ′ p ) X A ( p, 2 p ) Tang et al., PRL90 (2003) A ( p, ppn ) Aclander et al., PLB (1999) theory: Ciofi et al., PRC53 (1996) theory: Piasetzky et al., PLB (1999) triple coincidence A ( e, e ′ pN ) X : Carbon Helium heavier targets Subedi et al., Science, 320 (2008) Korover et al., PRL 113 (2014) Hen et al., Science, 356 (2014) Shneor et al., PRL99 (2007) M. Alvioli 3 ✫ ✪
✬ ✩ 0. Nuclear Hamiltonian • The non-relativistic nuclear many-body problem: H = − ℏ 2 i + 1 ∇ 2 ∑ ˆ ∑ ˆ ˆ H Ψ n = E n Ψ n , v ij + ... ˆ 2 m 2 i ij • Exact ground-state wave functions obtained by various methods are avail- able for light nuclei ( A ≤ 12); ⇒ calculations will be shown using 2 H , 3 He , 4 He WFs; = • Variational wave functions of nuclei can be obtained with approximated methods; usually difficult to use/generalize = ⇒ we developed an easy-to-use cluster expansion technique for the cal- culation of basic quantities of medium-heavy nuclei , 12 C , 16 O , 40 Ca ; • SRCs implemented MC generator for nuclear configurations for nuclei from 12 C to 238 U for the initialization of pA and AA collisions simulations http://sites.psu.edu/color M. Alvioli MIT 2016 4 ✫ ✪
✬ ✩ 0. Calculation of basic quantities • one- and two-body densities: A ∫ ∫ d r j Ψ o † ρ ST N ( r 1 , r ′ P ST pN Ψ o A ( x ′ ∑ ∏ A ( x 1 , ..., x A ) ˆ 1 ) = d r 1 1 , x 2 , ..., x A ) j =3 A ∫ ρ (2) d r j Ψ o † pN ( r 1 , r 2 ; r ′ 1 , r ′ P pN Ψ o A ( x ′ 1 , x ′ A ( x 1 , ..., x A ) ˆ ∑ ∏ 2 ) = 2 , x 3 , ..., x A ) j =3 • one- and two-body momentum distributions: 1 ∫ 1 e − k 1 · ( r 1 − r ′ n ST d r 1 d r ′ 1 ) ρ ST N ( r 1 , r ′ N ( k 1 ) = 1 ) (2 π ) 3 1 ∫ 2 e − k 1 · ( r 1 − r ′ 1 ) e − k 2 · ( r 2 − r ′ n (2) 2 ) · d r 1 d r ′ 1 d r 2 d r ′ pN ( k 1 , k 2 ) = (2 π ) 6 · ρ (2) → n (2) pN ( r 1 , r 2 ; r ′ 1 , r ′ 2 ) ← pN ( k rel , K CM ) M. Alvioli MIT 2016 5 ✫ ✪
✬ ✩ 0. Two-Body Momentum Distributions k rel ≡ k = 1 r ′ = r ′ 1 − r ′ 2 ( k 1 − k 2 ) r = r 1 − r 2 2 R = 1 R ′ = 1 2 ( r ′ 1 + r ′ K CM ≡ K = k 1 + k 2 2 ( r 1 + r 2 ) 2 ) n (2) ( k , K ) = n (2) ( k rel , K CM , Θ) = 1 ∫ d r d r ′ d R d R ′ e − i K · ( R − R ′ ) e − i k · ( r − r ′ ) ρ (2) ( r , r ′ ; R , R ′ ) = (2 π ) 6 1 ∫ ∫ d r d r ′ d R e − i k · ( r − r ′ ) ρ (2) ( r , r ′ ; R , R ) n (2) ( k ) = d K n ( k , K ) = (2 π ) 3 ∫ 1 ∫ d r d R d R ′ e − i K · ( R − R ′ ) ρ (2) ( r , r ; R , R ′ ) n (2) ( K ) = d k n ( k , K ) = (2 π ) 3 1 ∫ d r d r ′ d R d R ′ e − i k · ( r − r ′ ) ρ (2) ( r , r ′ ; R , R ′ ) n (2) ( k , K = 0) = (2 π ) 6 K CM = 0 corresponds to k 2 = − k 1 , i.e. back-to-back nucleons M. Alvioli MIT 2016 6 ✫ ✪
✬ ✩ 0. Using Realistic WFs of large nuclei: Cluster Expansion • Cluster Expansion is a technique to reduce the computational effort in f ( n ) many many-body calculations; we use: Ψ o = ˆ n ˆ F Φ o = ∏ ∑ ij Φ o ij • Expectation value over Ψ o of any one- or two-body operator ˆ Q : = ⟨ ∏ ˆ Q ∏ (1 + ˆ F † ˆ f † ˆ ⟨ Ψ o | ˆ = ⟨ ˆ Q ˆ Q ˆ = ⟨ ˆ Q | Ψ o ⟩ F ⟩ f ⟩ η ) ⟩ ⟨ ∏ (1 + ˆ = ⟨ ∏ ˆ ⟨ ˆ ⟨ Ψ o | Ψ o ⟩ F 2 ⟩ f 2 ⟩ η ) ⟩ Q (1 + ∑ ˆ η + ∑ ˆ Q ∑ ˆ = ⟨ ˆ ≃ ⟨ ˆ Q ⟩ + ⟨ ˆ η ˆ η + . . . ) ⟩ η ⟩ ⟨ (1 + ∑ ˆ η + ∑ ˆ 1 + ⟨ ∑ ˆ = η ˆ η + . . . ) ⟩ η ⟩ [ ] ( ) ⟨ ˆ Q ⟩ + ⟨ ˆ ∑ ∑ ≃ ⟨ ˆ Q ⟩ + ⟨ ˆ ∑ ≃ Q η ⟩ ˆ 1 − ⟨ η ⟩ + ... ˆ Q η ⟩ L ˆ [ f 2 − 1 ] ⟩ is the small expansion parameter ; ⟨ ˆ Q ⟩ ≡ ⟨ Φ o | ˆ ˆ • ⟨ ˆ η ⟩ = ⟨ Q | Φ o ⟩ • we end up with linked clusters; up to 4b diagrams needed for 2B density , each involving the square of: ˆ n f n ( r ij ) ˆ f = ∑ O n ( ij ) M. Alvioli MIT 2016 7 ✫ ✪
✬ ✩ 0. Using Realistic WFs of large nuclei: Cluster Expansion correlation functions: Central , Spin-Isospin , Tensor f(r) 0.10 0.08 1 - f (/10) 4 - 0.06 c 5 - S 2 - 0.04 6 - S 3 - 0.02 0.00 -0.02 16 O - AV8' -0.04 0 1 2 3 4 r [ fm ] 5 [ ] f 2 − 1 ˆ • ⟨ ˆ η ⟩ = ⟨ ⟩ is the small expansion parameter • we end up with linked clusters; up to 4b diagrams needed for 2B density , each involving the square of: ˆ n f n ( r ij ) ˆ f = ∑ O n ( ij ) M. Alvioli MIT 2016 8 ✫ ✪
✬ ✩ • at first order of the η − expansion, the full correlated one-body mixed density matrix expression is as follows: 1 ) = ρ (1) 1 ) + ρ (1) 1 ) + ρ (1) ρ (1) ( r 1 , r ′ o ( r 1 , r ′ H ( r 1 , r ′ S ( r 1 , r ′ 1 ) , with ∫ [ ] ρ (1) H ( r 1 , r ′ H D ( r 12 , r 1 ′ 2 ) ρ (1) o ( r 1 , r ′ 1 ) ρ o ( r 2 ) − H E ( r 12 , r 1 ′ 2 ) ρ (1) o ( r 1 , r 2 ) ρ (1) o ( r 2 , r ′ 1 ) = d r 2 1 ) ∫ [ ] ρ (1) S ( r 1 , r ′ d r 2 d r 3 ρ (1) H D ( r 23 ) ρ (1) o ( r 2 , r ′ 1 ) ρ o ( r 3 ) − H E ( r 23 ) ρ (1) o ( r 2 , r 3 ) ρ (1) o ( r 3 , r ′ 1 ) = − o ( r 1 , r 2 ) 1 ) and the functions H D and H E are defined as: 6 f ( p ) ( r ij ) f ( q ) ( r kl ) C ( p,q ) D ( E ) ( r ij , r kl ) − C (1 , 1) ∑ H D ( E ) ( r ij , r kl ) = D ( E ) ( r ij , r kl ) p,q =1 with C ( p,q ) D ( E ) ( r ij , r kl ) proper functions arising from spin-isospin traces; ( Alvioli, Ciofi degli Atti, Morita, PRC72 (2005)) M. Alvioli MIT 2016 9 ✫ ✪
✬ ✩ • at first order of the η − expansion, the full correlated two-body mixed density matrix expression is as follows: 2 ) = ρ (2) 2 ) + ρ (2) 2 ) + ρ (2) 2 ) + ρ (2) ρ (2) ( r 1 , r 2 ; r ′ 1 , r ′ SM ( r 1 , r 2 ; r ′ 1 , r ′ 2b ( r 1 , r 2 ; r ′ 1 , r ′ 3b ( r 1 , r 2 ; r ′ 1 , r ′ 4b ( r 1 , r 2 ; r ′ 1 , r ′ 2 ) with: ρ (2) SM ( r 1 , r 2 ; r ′ 1 , r ′ 2 ) = C D ρ o ( r 1 , r ′ 1 ) ρ o ( r 2 , r ′ 2 ) − C E ρ o ( r 1 , r ′ 2 ) ρ o ( r 2 , r ′ 1 ) 2 ) = 1 2 ) − 1 ρ (2) 2b ( r 1 , r 2 ; r ′ 1 , r ′ η ( r 12 , r 1 ′ 2 ′ ) ρ o ( r 1 , r ′ 1 ) ρ o ( r 2 , r ′ η ( r 12 , r 1 ′ 2 ′ ) ρ o ( r 1 , r ′ 2 ) ρ o ( r 2 , r ′ 2 ˆ 2 ˆ 1 ) ∫ ρ (2) 3b ( r 1 , r 2 ; r ′ 1 , r ′ η ( r 13 , r 1 ′ 3 ) [ ρ o ( r 1 , r ′ 1 ) ρ o ( r 2 , r ′ 2 ) = d r 3 ˆ 2 ) ρ o ( r 3 , r 3 ) + − ρ o ( r 1 , r ′ 1 ) ρ o ( r 2 , r 3 ) ρ o ( r 3 , r ′ 2 ) + + ρ o ( r 1 , r 3 ) ρ o ( r 2 , r ′ 1 ) ρ o ( r 3 , r ′ 2 ) + − ρ o ( r 1 , r 3 ) ρ o ( r 2 , r ′ 2 ) ρ o ( r 3 , r ′ 1 ) + + ρ o ( r 1 , r ′ 2 ) ρ o ( r 2 , r 3 ) ρ o ( r 3 , r ′ 1 ) + − ρ o ( r 1 , r ′ 2 ) ρ o ( r 2 , r ′ 1 ) ρ o ( r 3 , r 3 ) ] 2 ) = 1 ∫ ρ (2) 4b ( r 1 , r 2 ; r ′ 1 , r ′ d r 3 d r 4 ˆ η ( r 34 ) · 4 ( − 1) P [ ρ o ( r 1 , r P 1 ′ ) ρ o ( r 2 , r P 2 ′ ) ρ o ( r 3 , r P 3 ) ρ o ( r 4 , r P 4 ) ] ∑ · P∈C ( Alvioli, Ciofi degli Atti, Morita, PRC72 (2005)) ( Alvioli, Ciofi degli Atti, Morita, PRL100 (2008)) M. Alvioli MIT 2016 10 ✫ ✪
✬ ✩ 1 1’ one-body , non-diagonal 2 2 − ρ ( r 1 , r ′ ← 1 ) diagrams 1 1’ 1 1’ 2 3 2 3 1 1’ 1 1’ a two-body , diagonal ρ (2) ( r 1 , r 2 ) diagrams − → M. Alvioli 11 ✫ ✪
✬ ✩ 0. One-Body Mom distrs: Few- and Many-Body nuclei 0 3 10 0 He 10 4 Full He Full (ST) -1 AV18 (ST) 10 -1 10 AV8' (10) (10) ] (00) -2 ] 3 -2 10 (00) 10 3 (k) [fm (k) [fm (01) (01) -3 -3 10 10 (11) (11) -4 -4 10 10 n(ST) (ST) 4 -5 -5 3 10 10 n n -6 10 -6 10 0 1 2 3 4 5 0 1 2 3 4 5 -1 k [fm ] -1 k [fm ] 16 40 O Full Ca Full -1 -1 10 10 (ST) (ST) AV8' AV8' (10) (10) -2 -2 10 10 ] ] (00) 3 3 (00) (k) [fm (k) [fm (01) (01) -3 -3 10 10 (11) (11) -4 -4 10 10 (ST) (ST) 16 40 -5 n -5 n 10 10 -6 -6 10 10 0 1 2 3 4 5 0 1 2 3 4 5 -1 -1 k [fm ] k [fm ] M. Alvioli et al., PRC87 (2013); IntJModPhys E22 (2013) M. Alvioli MIT 2016 12 ✫ ✪
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