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Bound and scattering states with non-local potentials

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). ** Outline **. NN interaction HH method for non-local potentials Bound states Strange FF of 4 He

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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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