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Bound and scattering states with non-local potentials

This study explores the characteristics of bound and scattering states in nuclear systems, particularly focusing on the application of non-local potentials. The work employs the Hyperspherical Harmonic method to investigate nucleon-nucleon interactions using various realistic potentials like Argonne V18 and CD Bonn. Key questions include understanding the structure of complex nuclei, predicting nuclear reaction rates, and the significance of individual versus collective motion. The research also includes an examination of strange form factors in Helium-4 and scattering states involving proton and helium-3 nuclei.

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Bound and scattering states with non-local potentials

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  1. Bound and scattering states with non-local potentials FB18 Santos (Brazil) M.Viviani INFN - Pisa (Italy) Collaborators A. Kievsky, L.E. Marcucci, S. Rosati, L. Girlanda (post-doc)

  2. ** Outline ** • NN interaction • HH method for non-local potentials • Bound states • Strange FF of 4He • Scattering states • p-3He & p-3H scattering FB18 August 2006

  3. Basic questions • How are complex nuclei built from their constituents? • How well can we predict nuclear reaction rates (fusion/fission/astrophysical/neutron/…)? • What is the relative importance of individual/ collective motion? • Study of A=3,4 nuclear systems • Hyperspherical Harmonic Method FB18 August 2006

  4. NN interaction • Realistic (phenomenological) potentials • Argonne V18 [Wiringa et al, 1995] • CD Bonn [Machleidt, 2001] • Nijmegen [Stoks et al, 1994] • Doleshall [Doleshall et al, 2000] • Effective field theory based on chiral symmetry • [Weinberg 1991, van Kolck 1994] • N3LO potential[review: Epelbaum, 2005] • “Julich” [Epelbaum et al, 2004] • “N3LO” [Emtem & Machleidt, 2003 ] • Low-momentum potentials • V low-k[Bogner, Kuo & Schwenk, 2003] • UCOM [Roth et al, 2004] FB18 August 2006

  5. CSB NN interaction • Isospin symmetry breaking • np singlet scattering length –23.74±0.02 fm • pp singlet scattering length –17.3±0.4 fm (Coulomb corrected) • nn singlet scattering length –18.5±0.4 fm • They come ultimately from u-d different charge & mass • In the “modern” NN potentials: • Mass difference between +, - and 0 • Fitted to pn and pp scattering data sets • Important for • Strange FF of 4He • Reaction d+d 4He+ 0 [Gardestig & Phillips, 2005] [Miller et al, nucl-ex/0602021] CSB from PT: [Epelbaum & Meissner, 2005] FB18 August 2006

  6. 3N force 4NF from PT: [Epelbaum, 2006] • “Old models” • Brazil & Tucson Melbourne [Friar et al, 1999] • Urbana [Pudliner et al, 1997] • New proposed models • Illinois (3 exchanges)[Pieper et al, 2001] • Chiral symmetry • [Friar et al, 1999] • [Epelbaum et al, 2002] • CSB:  and  exchange • [Kaiser, 2006] FB18 August 2006

  7. AV18 NN potentials in p-space FB18 August 2006

  8. AV18 NN potentials in p-space CD BONN  very long tail FB18 August 2006

  9. AV18 AV18 N3LO NN potentials in p-space CD BONN CD BONN  very long tail VN3LO(k,k’)0 for k,k’>5 fm-1 FB18 August 2006

  10. AV18 N3LO NN potentials in p-space CD BONN  very long tail VN3LO(k,k’)0 for k,k’>5 fm-1 Vlow-k(k,k’)=0 for k,k’>2.1 fm-1 Vlow-k FB18 August 2006

  11. Deuton wave function • In r-space: 3S1 wave FB18 August 2006 0 2.5 5.0 7.5 10.0 12.5 15.0 r (fm)

  12. Deuton wave function • In p-space: 3S1 wave FB18 August 2006

  13. NN potentials in r-space FB18 August 2006

  14. V(r,r’)=<3S1|V(r,r’)| 3S1 > NN potentials in r-space: N3LO FB18 August 2006

  15. V(r,r’)=<3S1|V(r,r’)| 3S1 > NN potentials in r-space: Vlow-k FB18 August 2006

  16. 4 3 2 HH method (1) Fabre de la Ripelle, 1983 • Rainal-Revai coefficients • Matrix elements of the interaction 1 FB18 August 2006

  17. HH method (2) • Fourier transform • Usual choice: Lagrange polynomials FB18 August 2006

  18. r B A HH method (3) • Bound state • Rayleigh-Ritz variational principle • Boundary conditions • Scattering states • Kohn variational principle • Boundary conditions FB18 August 2006

  19. Technical methods Faddeev-Yakubovsky Equations Green Function Monte Carlo (GFMC) Variational methods Gaussian basis RGM Effective methods No-core Harmonic Oscillator method EIHH Other methods • Reviews: • [Carlson & Schiavilla, RMP 70, 743 (1998)] • [Gloeckle et al, Phys. Rep 274, 107 (1996)] • See also • [Kamada et al, PRC64, 044001 (2001)] FB18 August 2006

  20. Convergence FB18 August 2006

  21. Binding energy (MeV) 3H binding energy F: Nogga et al, PRC65, 054003 (2002); Deltuva et al, PRC68, 024005 (2003) FB18 August 2006 NCSM: Navratil & Barret, PRC59, 014311 (2004)

  22. 4He binding energy FY: Nogga et al, PRC 65, 054003 (2002) FB18 August 2006 NCSM: Navratil & Barret, PRC 59, 014311 (2004) HH: MV, Marcucci, Kievsky & Rosati, FBS, in press

  23. Parity violating scattering e-4He • Probe the strange-quark “content” of the nucleon • The left-right asymmetry ALR sensitive to  Z0 e N e N FB18 August 2006

  24. Musolf et al, 1994 The experiments • Experiments on nucleon: • Jefferson Lab (USA): HAPPEX & G0 • MIT-Bates (USA): SAMPLE • Mainz: A4 • Sensitive to an admixture of GE(s) and GM(s) • HAPPEX 2005; G0 2005; SAMPLE 2004; A4 2004 • In the case of a target (J,T)=(0+,0), at low Q2: • HAPPEX-He @ Jlab (2006-preliminary, K.Aniol QNP06) • ALR = +6.43  0.23 (stat)  0.22 (syst) ppm FB18 August 2006

  25. Strange 4He FF & T>0 components (1) • Vector neutral current • Three contributions • Left-right asymmetry FB18 August 2006

  26. Strange 4He FF & T>0 components (2) • At low Q2: MEC and spin-orbit contribution in J=0 are small and then • One needs to know FB18 August 2006

  27. Strange 4He FF & T>0 components (3) • Previous estimates of RT=1: • [Ramavataram et al, 1994] – based on an approx 4He w.f. • PT=10.0007% : RT=1 was estimated to be negligible • Current estimates of PT=13 to 5 times larger FB18 August 2006

  28. Strange 4He FF & T>0 components (4) Q2=0.0772 GeV2 q1.4 fm-1 PRELIMINARY FB18 August 2006

  29. K. Aniol, QNP06 Madrid June 2006 Strange 4He FF & T>0 components (5) Preliminary HAPPEX estimate @ Q2=0.0772 GeV2 (q1.4fm-1): ALR = +6.43  0.23 (stat)  0.22 (syst) ppm Rs-1.08 RT=1= 0.009  0.03 PRELIMINARY FB18 August 2006 In agreement with recent lattice calculations GEs= 0.001 0.004 [Leinweber et al, hep-lat/0601025]

  30. A=4 scattering • p-3He, p-3H, d-d,… • 3N force effect? • Fusion • Theoretical methods still under development • Faddeev-Yakubovsky [Lazauskas & Carbonell, 2004] [Fonseca, 1999, Deltuva & Fonseca, work in progress] • Variational – HH [MV et al, 2006] • Resonating Group Model[Pfitzinger, Hofmann & Hale, 2001] FB18 August 2006

  31. Very small effects from the (standard) 3N forces p-3He elastic scattering Unpolarized cross section AV18 FB18 August 2006 AV18+UIX MV, Marcucci, Kievsky & Rosati, PRC in press

  32. Very small effects from the (standard) 3N forces p-3He elastic scattering Analyzing power AV18 FB18 August 2006 AV18+UIX MV, Marcucci, Kievsky & Rosati, PRC in press

  33. Very small effects from the (standard) 3N forces p-3He elastic scattering Analyzing power AV18 FB18 August 2006 AV18+UIX MV, Marcucci, Kievsky & Rosati, PRC in press

  34. p-3He elastic scattering Double analyzing power AV18 FB18 August 2006 AV18+UIX MV, Marcucci, Kievsky & Rosati, PRC in press

  35. A=4 scattering with the N3LO potential • n-t scattering lenghts [fm] • Experimental situation • Theoretical calculations FY: Lazauskas & Carbonell, 2004 FB18 August 2006 PRELIMINARY

  36. n-t scattering lengths (expt) =1.700.03 b [Phillips et al, 1980] Coherent scattering length ac=3.590.02 fm [Rauch et al, 1985] ac=3.6070.017 fm [Hale et al, 1990] FB18 August 2006

  37. n-t scattering lengths (expt) =1.700.03 b [Phillips et al, 1980] Coherent scattering length ac=3.590.02 fm [Rauch et al, 1985] ac=3.6070.017 fm [Hale et al, 1990] AV18UIX FB18 August 2006

  38. as at n-t scattering lengths Rauch et al, 1985 (I) PRELIMINARY Hale et al, 1990 FB18 August 2006

  39. as at n-t scattering lengths Rauch et al, 1985 (I) PRELIMINARY Hale et al, 1990 FB18 August 2006

  40. p-t scattering at low energies (1) Isospin state T=1/2,Tz=+1/2 Isospin state T=1/2,Tz=-1/2 r 3H p The “internal” part contains T=0 and T=1 isospin channels FB18 August 2006

  41. p-t scattering at low energies (2) • Triplet phase shift [deg] • Ecm=0.1 MeV N3LO FY: Lazauskas & Carbonell, 2004 PRELIMINARY PRELIMINARY FB18 August 2006

  42. Summary • Developments of calculation method for non-local potentials • Bound state: o.k. • Scattering states: under progress, first results • Full inclusion of the Coulomb interaction • N3LO potentials: preliminary A=4 scattering phase shifts close FB18 August 2006

  43. Future work • Test of chiral and low-k potentials in A=3,4 scattering • Under progress: CDBonn, Doleshall, Vlow-k,…. • “hard” potentials vs. “soft” potentials • p- 3H, n-3He, d-d scattering & 4He resonances • Test of new models of 3N interaction • Nd scattering: mainly T=1/2 • test of T=3/2 3N force in A=4 scattering • Development of relativistic formalism (Bakamjan-Thomas) • Fusion reaction(dd,…) FB18 August 2006

  44. THE END… We’d like to invite everybody to the EUROPEAN FEW BODY CONFERENCE XX PISA (Italy) 10-15 September 2007 FB18 August 2006

  45. FB18 August 2006

  46. 4 3 2 HH method (1) • Rainal-Revai coefficients • Fourier transform 1 FB18 August 2006

  47. Energy gain 20 keV Origin of the mixing FY-Q calculation: Nogga et al, 2002 Isospin T>0 components in the 4He w.f. HH: MV, Marcucci, Kievsky & Rosati, 2005 FB18 August 2006

  48. 4He Form Factor - MEC Musolf et al, 1994 AV18 CDBonn N3LO PRELIMINARY Viviani & Schiavilla, 2006 FB18 August 2006

  49. FB18 August 2006

  50. Hyperspherical Harmonics Expansion of the wave functions usual expansion (HH) Correlated expansion (CHH) Jacobi vectors Hyperangular variables Kinetic energy HH & CHH method (1) FB18 August 2006

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