Study of bound and scattering states of few-body systems with the HH method

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Electron-Nucleus Scattering XIII, June 23-27, 214 Mini-symposium to honor Professor Sergio

1 Study of bound and scattering states of few-body systems with the HH method M. Viviani INFN, Sezione di Pisa & Department of Physics, University of Pisa Pisa (Italy) Electron-Nucleus Scattering XIII, June 23-27, 214 Mini-symposium to honor Professor Sergio Rosati on his 8th Birthday M. Viviani (INFN-Pisa) Few-body systems with the HH method Marciana Marina, 27/6/14 1 / 32

2 Outline 1 The rise (and fall...) of the Few-Body Pisa Group 2 3NF effects in few-nucleon systems p d scattering p 3 He scattering p 3 H scattering Collaborators S. Rosati, A. Kievsky & L.E. Marcucci - INFN & Pisa University, Pisa (Italy) L. Girlanda - INFN & Universitá del Salento, Lecce (Italy) R. Schiavilla, M. Piarulli, A. Baroni, F. Spadoni - Jefferson Lab. & ODU, Norfolk (VA, USA) S. Pastore - USC, Columbia (SC, USA) M. Viviani (INFN-Pisa) Few-body systems with the HH method Marciana Marina, 27/6/14 2 / 32

3 The rise (and fall...) of the Few-Body Pisa Group The Jastrow period ( ) B. Barsella and S. Rosati, On the Effect of n-p Tensor Forces in 3 H Λ, Nuovo Cimento 2, 914 (1961). J. Murphy and S. Rosati, A two-body method for the bound states of a three-body system, Nucl. Phys. 63, 625 (1965). M. Barbi and S. Rosati, Direct numerical solution of the three-body problem, Phys. Rev. 147, 73 (1966). S. Fantoni, L. Panattoni, and S. Rosati, Calculation on nuclear 3-body and 4-body systems with jastrow-type correlated wave functions, Nuovo Cimento 69, 8 (197). L. Lovitch and S. Rosati, Bound State Solution of the Two-Nucleon Schroedinger Equation with Tensor Forces, Comp. Phys. Com. 2, 353 (1971). Ψ = g(r 12 )g(r 13 )g(r 23 ) δ Ψ H E Ψ = Euler equation: for an S-state and for a central spin-independent potential v(r): 1 ( ) M 2 g(r)+ v(r)+w[g, r] g(r) = Eg(r) Non-linear iterative solution M. Viviani (INFN-Pisa) Few-body systems with the HH method Marciana Marina, 27/6/14 3 / 32

Interlude 1 The Many-body period (1972 1988) [S. Fantoni and S.

International Conference on Recent Progress in Many-Body Theories Odenthal-Altemberg, Germany, August 29 September 3, 1983 A

4 Interlude 1 The Many-body period ( ) [S. Fantoni and S. Rosati, Expansion Procedure for Jastrow-Type Correlated Wave Functions, Nuovo Cimento A 1, 145 (1972)] My first conference: Third International Conference on Recent Progress in Many-Body Theories Odenthal-Altemberg, Germany, August 29 September 3, 1983 A wonderful 2-days trip from Pisa to Altenberg (and back) with Sergio, Stefano, and Adelchi, crammed in Stefano s FIAT 127 M. Viviani (INFN-Pisa) Few-body systems with the HH method Marciana Marina, 27/6/14 4 / 32

5 A new start... Improvements of the wave functions and extension to realistic potentials ( ) From the CHO to the CHH expansion (selected papers) A. Kievsky, S. Rosati and M. Viviani, Euler and Correlated Harmonic Oscillator Wave Functions for Three-Nucleon Systems, Nucl. Phys. A 51, 53 (1989). Few-Body Systems 9, 1 (199). M. Viviani, A. Kievsky and S. Rosati, Correlated Hyperspherical Harmonic Calculations for Three- and Four-Body Systems, Nuovo Cimento A 15, 1473 (1992). A. Kievsky, M. Viviani and S. Rosati, The Three-Nucleon Bound-State with Realistic Soft and Hard Core Potentials, Nucl. Phys. A 551, 241 (1993). A. Kievsky, M. Viviani and S. Rosati, Variational Calculations for Scattering States in Three-Nucleon Systems, Few-Body Sys. Suppl. 7, 278 (1994). M. Viviani, A. Kievsky and S. Rosati, Calculation of the Alpha-Particle Ground-State, Few-Body Systems 18, 25 (1995). A. Kievsky, L. E. Marcucci, S. Rosati and M. Viviani, High-Precision Calculation of the Triton Ground State within the Hyperspherical Harmonics Method, Few-Body Systems 22, 1 (1997). M. Viviani, S. Rosati and A. Kievsky, Neutron- 3 H and Proton- 3 He Zero Energy Scattering, Phys. Rev. Lett. 81, 158 (1998). M. Viviani (INFN-Pisa) Few-body systems with the HH method Marciana Marina, 27/6/14 5 / 32

6 The goal: reach the accuracy achieved by other groups (Friar and Coll., Glöckle and Coll., Kamimura and Coll.,...) + be able to treat realistic interactions, also hard-core potentials full inclusion of the Coulomb interaction A y puzzle in p d elastic scattering Ψ = N c i=1,3α=1 { [ ] } ] f α(r jk )g(r ij )g(r ik ) Y (ˆx lα i)y (ŷ Lα i) [(s j s k ) Sα s i [(t j t k ) Tα t i F α(x i, y i ) ]Λα Σ α JJ T z α F α(x i, y i ) expanded first in either HO or HH basis CHH=correlated Hyperspherical harmonics expansion ( PHH if g = 1) Method B (MeV) T (MeV) P s (%) P D (%) P P (%) AV18 potential PHH Witala et al. (23) Hamada-Johnston (hard-core) potential CHH Delves & Hennel (1971) M. Viviani (INFN-Pisa) Few-body systems with the HH method Marciana Marina, 27/6/14 6 / 32

(FBXII) Vancouver, B.C., Canada, July 2-8, 1989 M.

7 Interlude 2 The first Few-Body conference: 12th International Conference on Few-Body Problems in Physics (FBXII) Vancouver, B.C., Canada, July 2-8, 1989 M. Viviani (INFN-Pisa) Few-body systems with the HH method Marciana Marina, 27/6/14 7 / 32

8 Many ideas... Some of the ideas produced by Sergio during the many discussions we had in those years... M. Viviani (INFN-Pisa) Few-body systems with the HH method Marciana Marina, 27/6/14 8 / 32

9 One-point integration... Idea: to integrate one needs to compute the integrand in just one point b I = dx f(x) = (b a) f(x ) a It would be very useful in case of multidimensional integration x x I = dx 1... dx n f(x 1,...,x n) = Volume f( x 1,..., x n) Unfortunately we could not succeed... M. Viviani (INFN-Pisa) Few-body systems with the HH method Marciana Marina, 27/6/14 9 / 32

10 The last period Extension of the method to non-local momentum space potentials (25 present) no correlation factor the interactions are softer & easy to trasform the wave function from coordinate to momentum space (and viceversa) (selected papers) M. Viviani, A. Kievsky and S. Rosati, Calculation of the Alpha Particle Ground State within the Hyperspherical Harmonic Basis, Phys. Rev. C 71, 246 (25) M. Viviani, L. E. Marcucci, S. Rosati, A. Kievsky and L. Girlanda, Variational Calculation on A=3 and 4 Nuclei with Non-Local Potentials, Few-Body Systems 39, 159 (26) A. Kievsky, S. Rosati, M. Viviani, L. E. Marcucci, L. Girlanda, A High-Precision Variational Approach to Three- and Four-Nucleon Bound and Zero-Energy Scattering States, J. Phys. G: Nucl. Part. Phys. 35, 6311 (28) M. Viviani, A. Deltuva, R. Lazauskas, J. Carbonell, A. C. Fonseca, A. Kievsky, L.E. Marcucci, and S. Rosati, Benchmark calculation of n- 3 H and p- 3 He scattering, PRC 84, 541 (211) M. Viviani (INFN-Pisa) Few-body systems with the HH method Marciana Marina, 27/6/14 1 / 32

11 Interlude 3 18th International Conference on Few-Body Problems in Physics (FBXII) Santos, SP, Brazil, August 21-26, 26 M. Viviani (INFN-Pisa) Few-body systems with the HH method Marciana Marina, 27/6/14 11 / 32

12 Applications EM and Weak transitions in light nuclei p, p d, and hep astrophysical factor Form factors of 3 H, 3 He, and 4 He Parity-violation Longitudinal polarization in n + 3 He p + 3 H Study of chiral effective field theory potentials and currents.5 S(E c.m. ) (ev b) LUNA Griffiths et al. Schmid et al E c.m. (kev) M. Viviani (INFN-Pisa) Few-body systems with the HH method Marciana Marina, 27/6/14 12 / 32

13 3NF effects in few-nucleon systems Nuclear Dynamics: the EFT approach Chiral effective field theory (χeft): N π interaction dictated by chiral symmetry [Weinberg (199), Bernard, Kaiser, & Meissner (1995), Ordonéz, Ray, & van Kolck (1996), Epelbaum, Meissner, & Epelbaum (1998),...] Pionless effective field theory (π/eft): low energy processes [Kaplan, Savage, & Wise (1998), van Kolck (1999),...] Study of A = 3, 4 reactions using χeft Test of the derived NN & 3N interactions Different accurate theoretical techniques FY/AGS equations in momentum space [Witala et al., ], [Deltuva & Fonseca, 27, 212] FY equations in coordinate space [Lazauskas & Carbonell, 29, 212] HH expansion + Kohn variational principle NCSM/RG method [Navratil, Quaglioni & Coll., ] A y puzzle, energy production (NIF), reactions of astrophysical interest, study of fundamental symmetries, etc. muon capture on d MUSUN [Marcucci, 211 ],... M. Viviani (INFN-Pisa) Few-body systems with the HH method Marciana Marina, 27/6/14 13 / 32

14 A y puzzle NN interaction Old models : Argonne V18, CD-Bonn, Nijmengen (χ 2 1) Derived from χeft J-N3LO [Epelbaum and Coll, ] N3LO5 & N3LO6 [Entem & Machleidt, 23 & 211] E cm =266 kev E cm = 431 kev E cm = 666 kev E p =2.25 MeV E p =4 MeV E p =5.54 MeV A y George Fisher N3LO5 AV18 Fisher Alley Alley A y E cm = 1.33 MeV E cm = 1.66 MeV E cm = 2. MeV θ cm θ cm θ cm A = 3 Delicate balance between 4 P j waves θ [c.m.] θ [c.m.] θ [c.m.] A = 4 Confirmed also for other NN potentials (Deltuva & Fonseca 27) dashed lines: N3LO5, dotted-dashed lines: N3LO5/UIX, solid lines N3LO5/N2LO5* M. Viviani (INFN-Pisa) Few-body systems with the HH method Marciana Marina, 27/6/14 14 / 32

15 Inclusion of the 3NF Models from EFT Illinois models Fujita Miyazawa S wave scattering pion rings [Pieper et al., 21] 3π exchanges and π rings Illinois-7: coefficients chosen to reproduce the A = 4 12 spectrum 3n force at N 2 LO: 2 unknonw LEC s c d & c E + cutoff Λ [Epelbaum et al., 22] N2LO5 & N2LO6: LECs fixed by L. Marcucci (see next slide) N 3 LO & N 4 LO pion exchanges: [Krebs, Epelbaum, et al., ( )] 1 Contact terms at N 4 LO: [Girlanda, Kievsky, MV, (211)] (a) (b) (c) (d) (e) (f) M. Viviani (INFN-Pisa) Few-body systems with the HH method Marciana Marina, 27/6/14 15 / 32

16 New fit of c D and c E Sensitivity to c D and c E c D enters also in β-decay processes [Gardestig & Phillips, 26], [Gazit et al., 29] d R = M c 4πf πg D + M A 3 (c 3 + 2c 4 )+ 1 6 d R (and thus c D ) can be fixed from the tritium Gamow-Teller m.e. GT EXPT =.955±.4 d R X cd Models under study: Model Λ [MeV] c D c E B( 4 He) [MeV] a 2 (n d) [fm] N3LO N3LO5/N2LO N3LO6/N2LO AV AV18/IL Expt ±.3±.7 Expt a 2 (n d): K. Schoen et al., 23 M. Viviani (INFN-Pisa) Few-body systems with the HH method Marciana Marina, 27/6/14 16 / 32

17 HH expansion Example: p 3 He elastic scattering Ω ± LS (A, B) = 1 N N perm.=1 [Y L (ŷ p ) [φ A φ B ] S ] JJ z ( f L (y G L(η, q AB y p) p) ± i F ) L(η, q AB y p) q AB y p q AB y p Ψ LS = n,[k] a LS,[K] n,[k] + Ω LS (p, 3 He) L S S LS,L S Ω+ L S (p, 3 He) n,[k] HH states S LS,L S = S-matrix a LS,[K] and S LS,L S computed using the Kohn variational principle For a review, see [J. Phys. G: Nucl. Part. Phys. 35, 6311 (28) ] M. Viviani (INFN-Pisa) Few-body systems with the HH method Marciana Marina, 27/6/14 17 / 32

18 HH expansion Example: p 3 He elastic scattering Ω ± LS (A, B) = 1 N N perm.=1 [Y L (ŷ p) [φ A φ B ] S ] JJ z ( f L (y G L(η, q AB y p) p) ± i F ) L(η, q AB y p) q AB y p q AB y p Ψ LS = n,[k] a LS,[K] n,[k] + Ω LS (p, 3 He) L S S LS,L S Ω+ L S (p, 3 He) n,[k] HH states S LS,L S = S-matrix a LS,[K] and S LS,L S computed using the Kohn variational principle For a review, see [J. Phys. G: Nucl. Part. Phys. 35, 6311 (28) ] M. Viviani (INFN-Pisa) Few-body systems with the HH method Marciana Marina, 27/6/14 17 / 32

19 HH expansion Example: p 3 He elastic scattering Ω ± LS (A, B) = 1 N N perm.=1 [Y L (ŷ p) [φ A φ B ] S ] JJ z ( f L (y G L(η, q AB y p) p) ± i F ) L(η, q AB y p) q AB y p q AB y p Ψ LS = n,[k] a LS,[K] n,[k] + Ω LS (p, 3 He) L S S LS,L S Ω+ L S (p, 3 He) n,[k] HH states S LS,L S = S-matrix a LS,[K] and S LS,L S computed using the Kohn variational principle For a review, see [J. Phys. G: Nucl. Part. Phys. 35, 6311 (28) ] M. Viviani (INFN-Pisa) Few-body systems with the HH method Marciana Marina, 27/6/14 17 / 32

20 Convergence n 3 H at E cm = 3 MeV, 3 P 2 wave 1 δ(k)-δ(k-2).1.1 AV18 N3LO-Idaho K Convergence with the grand-angular quantum number M. Viviani (INFN-Pisa) Few-body systems with the HH method Marciana Marina, 27/6/14 18 / 32

21 Benchmark test of 4N scattering calculations [PRC 84, 541 (211)] p 3 He elastic scattering dσ/dω [mb/sr] A y A y MeV Famularo 1954 Fisher 26 AGS 6 12,4,2 Fisher 26 George ,2,1 Daniels Mcdonald 1964 Fisher Fisher MeV 6 12 Daniels Alley MeV Mcdonald Alley 1993 Daniels N3LO5 potential AGS= Deltuva & Fonseca FY= Lazauskas & Carbonell M. Viviani (INFN-Pisa) Few-body systems with the HH method Marciana Marina, 27/6/14 19 / 32

22 Benchmark test of 4N scattering calculations [PRC 84, 541 (211)] p 3 He elastic scattering dσ/dω [mb/sr] A y A y MeV Famularo 1954 Fisher 26 AGS HH 6 12,4,2 Fisher 26 George ,2,1 Daniels Mcdonald 1964 Fisher Fisher MeV 6 12 Daniels Alley MeV Mcdonald Alley 1993 Daniels N3LO5 potential AGS= Deltuva & Fonseca FY= Lazauskas & Carbonell M. Viviani (INFN-Pisa) Few-body systems with the HH method Marciana Marina, 27/6/14 19 / 32

23 Benchmark test of 4N scattering calculations [PRC 84, 541 (211)] p 3 He elastic scattering dσ/dω [mb/sr] A y A y MeV Famularo 1954 Fisher 26 AGS HH FY 6 12,4,2 Fisher 26 George ,2,1 Daniels Mcdonald 1964 Fisher Fisher MeV 6 12 Daniels Alley MeV Mcdonald Alley 1993 Daniels N3LO5 potential AGS= Deltuva & Fonseca FY= Lazauskas & Carbonell M. Viviani (INFN-Pisa) Few-body systems with the HH method Marciana Marina, 27/6/14 19 / 32

24 Effect of the N 2 LO & Illinois 3NF in p d scattering p-d scattering at E p =3 MeV A y.2 it NN only NN+3N AV18/IL θ θ Bands: results obtained for Λ = 5 & Λ = 6 MeV NN: N3LO5 & N3LO6 NN+3N: N3LO5/N2LO5 & N3LO6/N2LO6 M. Viviani (INFN-Pisa) Few-body systems with the HH method Marciana Marina, 27/6/14 2 / 32

25 Study of the N 3 LO & N 4 LO 3NF in p d scattering [Krebs, Epelbaum, et al., 212,213] First study in n d scattering [Witala et al., 213].2 (a) (b) (c) (d) (e) (f) A y.1 E n =14.1 MeV d(n,n)d (1) (2) (3) (4) (5) (6) (7) (8) (9) (1) Θ cm (1) (2) (3) + many others diagrams... (4) green band: J-N3LO NN interaction only magenta band: + N 3 LO 3NF (no 2π contact terms) Very complicate implementation! M. Viviani (INFN-Pisa) Few-body systems with the HH method Marciana Marina, 27/6/14 21 / 32

26 3N contact terms at N 4 LO (1) [Girlanda, Kievsky, MV, PRC 84, 141 (211)] Two of the operators O = spatial derivatives (N O 1 N)(N O 2 N)(N O 3 N) 146 possible combinations, which can be reduced to 1 using Fierz transformation and other constraints O 1 O 2 O [1,τ 1 τ 2,τ 1 τ 3 ] σ 1 2 σ 2 [1,τ 1 τ 2,τ 1 τ 3 ] σ 2 2 σ 1 [1,τ 1 τ 2,τ 1 τ 3 ] σ 1 σ 2 [1,τ 1 τ 2,τ 1 τ 3 ] σ 1 2 σ 3 [1,τ 1 τ 2,τ 1 τ 3,τ 2 τ 3 ] σ 3 2 σ 1 [1,τ 1 τ 2,τ 1 τ 3,τ 2 τ 3 ] σ 1 σ 3 [1,τ 1 τ 2,τ 1 τ 3,τ 2 τ 3 ] σ 1 [τ 1 τ 2 τ 3 ] σ 3 [τ 1 τ 2 τ 3 ] σ 1 σ 2 σ 3 [τ 1 τ 2 τ 3 ] σ 2 1 σ 1 σ 3 [τ 1 τ 2 τ 3 ] M. Viviani (INFN-Pisa) Few-body systems with the HH method Marciana Marina, 27/6/14 22 / 32

27 3N contact terms at N 4 LO (2) Final form of the new 3NF terms V = i j k [ ] (E 1 + E 2 τ i τ j + E 3 σ i σ j + E 4 τ i τ j σ i σ j ) Z ij)+2 Z (r ij) r ij Z (r ik ) [ +(E 5 + E 6 τ i τ j )S ij Z ij) Z (r ] ij) Z (r ik ) r ij +(E 7 + E 8 τ i τ k )(L S) ij Z (r ij) r ij Z (r ik ) +(E 9 + E 1 τ j τ k )σ j ˆr ij σ k ˆr ik Z (r ij)z (r ik) Local (using a cutoff of the type F(k 2 j ;Λ)F(k 2 k ;Λ)) At N 4 LO there are new 1 LEC s Z (r;λ) = dp (2π) 3 eip r F(p 2 ;Λ) M. Viviani (INFN-Pisa) Few-body systems with the HH method Marciana Marina, 27/6/14 23 / 32

28 Fit of the LEC s Preliminary study Minimal model: AV18 + 3NF only from 3N contact terms Terms E 3, E 5, E 7 Terms E 5, E 7, E 1 4 Λ=2 MeV 2 χ /d.o.f = 4 c E =.654 a 2 =.652 fm E 3 =-.78 E 5 =-.143 E 7 =1.523 B( 3 H) = MeV.2 4 Λ=3 MeV 2 χ /d.o.f = 3.5 c E =.542 a 2 =.615 fm E 5 =-.262 E 7 =1.756 E 1 =-.649 B( 3 H) = MeV.2 dσ/dω 3 2 T dσ/dω 3 2 T θ (degrees) θ (degrees) θ (degrees) θ (degrees) T 2.1 T T 2.1 T θ (degrees) θ (degrees) θ (degrees) θ (degrees) i T 11 A y i T 11 A y θ (degrees) θ (degrees) θ (degrees) θ (degrees) M. Viviani (INFN-Pisa) Few-body systems with the HH method Marciana Marina, 27/6/14 24 / 32

29 3NF effect in p 3 He scattering (1) p 3 He phase-shift - Comparison PSA/Theory (PSA: [Daniels et al., 21]) N2LO 3NF only -2 3 phase-shift S NN only NN+3N TUNL 21 phase-shift P -6 phase-shift S1 ε(1 + ) phase-shift E p [MeV] E p [MeV] M. Viviani (INFN-Pisa) Few-body systems with the HH method Marciana Marina, 27/6/14 25 / 32

30 3NF effect in p 3 He scattering (1) p 3 He phase-shift - Comparison PSA/Theory (PSA: [Daniels et al., 21]) N2LO 3NF only -2 3 phase-shift S NN only NN+3N TUNL 21 AV18/IL7 phase-shift P -6 phase-shift S1 ε(1 + ) phase-shift E p [MeV] E p [MeV] M. Viviani (INFN-Pisa) Few-body systems with the HH method Marciana Marina, 27/6/14 25 / 32

31 3NF effect in p 3 He scattering (1) p 3 He phase-shift - Comparison PSA/Theory (PSA: [Daniels et al., 21]) N2LO 3NF only -2 3 phase-shift S NN only NN+3N TUNL 21 N3LO5/N2LO5 N3LO6/N2LO6 phase-shift P -6 phase-shift S1 ε(1 + ) phase-shift E p [MeV] E p [MeV] M. Viviani (INFN-Pisa) Few-body systems with the HH method Marciana Marina, 27/6/14 25 / 32

32 3NF effect in p 3 He scattering (2) p 3 He phase-shift - Comparison PSA/Theory PSA: [Daniels et al., 21] 4 2 phase-shift P1 NN only NN+3N TUNL 21 phase-shift ε(1 - ) 5 6 phase-shift P2 P1 5 phase-shift E p [MeV] E p [MeV] M. Viviani (INFN-Pisa) Few-body systems with the HH method Marciana Marina, 27/6/14 26 / 32

33 3NF effect in p 3 He scattering (2) p 3 He phase-shift - Comparison PSA/Theory PSA: [Daniels et al., 21] 4 2 phase-shift P1 NN only NN+3N TUNL 21 AV18/IL7 phase-shift ε(1 - ) 5 6 phase-shift P2 P1 5 phase-shift E p [MeV] E p [MeV] M. Viviani (INFN-Pisa) Few-body systems with the HH method Marciana Marina, 27/6/14 26 / 32

34 3NF effect in p 3 He scattering (2) p 3 He phase-shift - Comparison PSA/Theory PSA: [Daniels et al., 21] 4 2 phase-shift P1 NN only NN+3N TUNL 21 N3LO5/N2LO5 N3LO6/N2LO6 phase-shift ε(1 - ) 5 6 phase-shift P2 P1 5 phase-shift E p [MeV] E p [MeV] M. Viviani (INFN-Pisa) Few-body systems with the HH method Marciana Marina, 27/6/14 26 / 32

35 p 3 He observables at E p = 5.54 MeV (1) 4 NN band NN+3N band 3 dσ/dω [mb/sr].5.4 A y.2 A y M. Viviani (INFN-Pisa) Few-body systems with the HH method Marciana Marina, 27/6/14 27 / 32

36 p 3 He observables at E p = 5.54 MeV (1) 4 NN band NN+3N band AV18/IL7 3 dσ/dω [mb/sr].5.4 A y.2 A y M. Viviani (INFN-Pisa) Few-body systems with the HH method Marciana Marina, 27/6/14 27 / 32

37 p 3 He observables at E p = 5.54 MeV (1) 4 3 NN band NN+3N band N3LO5/N2LO5 N3LO6/N2LO6 dσ/dω [mb/sr].5.4 A y.2 A y M. Viviani (INFN-Pisa) Few-body systems with the HH method Marciana Marina, 27/6/14 27 / 32

38 p 3 He observables at E p = 5.54 MeV (2).2.2 A yy.2 A xx.1 A xz M. Viviani (INFN-Pisa) Few-body systems with the HH method Marciana Marina, 27/6/14 28 / 32

39 S = & S = 1 n 3 He scattering lenght [fm] Int. Method a (fm) a 1 (fm) AV18 HH 7.69 i i.77 RGM 7.79 i i.66 FY 7.71 i i.82 N3LO5 HH 7.57 i i.48 FY 3.56 i.7 AGS 7.82 i i.68 N3LO5/N2LO5* HH 7.61 i i.42 N3LO5/N2LO5 HH 7.67 i i.5 N3LO6/N2LO6 HH 7.92 i i.49 Exp.[ILL-1] 7.37(58) i4.448(5) 3.278(53) i.1(2) Exp.[ILL-2] 7.46(2) 3.36(1) Exp.[NIST] 7.57(3) 3.48(2) RGM: [Hofmann & Hale, PRC 77, 442 (28)] FY: [Lazauskas et al., PRC 83, 346 (211)] AGS: [Deltuva, priv. comm.] ILL-1: [Zimmer et al., EPJA 4, 1 (22)] ILL-2: [Ketter et al., EPJA 27, 243 (26)] NIST: [Huffman et al., PRC 7, 144 (24)] From neutron interferometry at NIST [Huber et al., PRL 13, (29)] Re(a 1 a ) EXPT = 4.2(3) fm Re(a 1 a ) N3LO5/N2LO5 = 4.24 fm Re(a 1 a ) N3LO5/N2LO5 = 4.29 fm M. Viviani (INFN-Pisa) Few-body systems with the HH method Marciana Marina, 27/6/14 29 / 32

40 Proton- 3 H scattering at low energies First monopole resonance of 4 He E = 8.2±.5 MeV, W = 27±5 kev [Walcher, 197] FY: [R. Lazauskas, 29] Monopole resonance: [Hiyama et al., 24 ] [Bacca et al., 213 ] Tension with the experimental resonance transition FF F M (q) F M 2 /4π * Koebschall et al. Frosch et al. Walcher Hiyama et al. AV18+UIX NN(N 3 LO) + 3NF(N 2 LO) q 2 [fm -2 ] 1 Balaskho (196) N3LO5/N2LO5 (only S-waves) p- 3 H elastic scattering 8 E p =.4 MeV E p =.6 MeV E p =.99 MeV dσ/dω [mb/sr] 6 4 PRELIMINARY PRELIMINARY M. Viviani (INFN-Pisa) Few-body systems with the HH method Marciana Marina, 27/6/14 3 / 32

41 Work in progress fix the contact N 4 LO 3NF in 3N systems and see their effect in A = 4 Extension of the HH technique to treat N d breakup (in collaboration with E. Garrido - CSIC, Madrid (Spain) comparison with the benchmark calc. of [Friar et al., (1995)] PRC (in press): p d breakup: effect of the Coulomb int. Complete the study of p 3 H, n 3 He & d d scattering Extension to A > 4 Extension of the HH technique to A > 4 systems (in collaboration with M. Gattobigio - INLN, Nice (France) ) [PRA 79, (29)], [PRC 83, 241 (211)] Use of Integral Relations to compute observables (in collaboration with E. Garrido - CSIC, Madrid (Spain) & C. Romero-Redondo - TRIUMF, Vancouver (Canada) ) No need of the asymptotic part [PRA 83, 2275 (211)], [PRC 85, 141 (212)] M. Viviani (INFN-Pisa) Few-body systems with the HH method Marciana Marina, 27/6/14 31 / 32

42 and... Thank you Sergio!!!! Australia, 1993, Kangaroo Island M. Viviani (INFN-Pisa) Few-body systems with the HH method Marciana Marina, 27/6/14 32 / 32

Few- Systems. Selected Topics in Correlated Hyperspherical Harmonics. Body. A. Kievsky

Few- Systems. Selected Topics in Correlated Hyperspherical Harmonics. Body. A. Kievsky Few-Body Systems 0, 11 16 (2003) Few- Body Systems c by Springer-Verlag 2003 Printed in Austria Selected Topics in Correlated Hyperspherical Harmonics A. Kievsky INFN and Physics Department, Universita