Relativistic Faddeev Calculations for Elastic Nd Scattering with Kharkov Potential

1. Relativistic Faddeev calculations for elastic Nd scattering with Kharkov potential H. Kamada (Kyushu Institute of Technology, Japan)H. Witala , J. Golak, R. Skibinski (Jagiellonian University, Poland) O. Shebeko , A. Arslanaliev (Kharkov Institute of Physics and Technology, NAS of Ukraine, Kharkiv, Ukraine) Ｎ APFB 2017 2017 –AUG-24~30, Guilin, CHINA Ｎ Ｐ

2. Outline • §１　Motivation • §２　RelativisticCalculation • §３　Identificationto the relativistic potential • §４　BoostCorrection • §５　Kharkovpotential • §６　Tritonbinding energy • §７　Ndelastic scattering • §８　Summaryand Outlook

3. §１　Motivation The nonrelativistic theoretical prediction of the Nd scattering backward cross section beyond 200MeV/u is getting to be poor even including the 3-body force (FM type).What is missing?

4. Phys. Rev. C 57, 2111 (1998)

5. §２　Ｒｅｌａｔｉｖｉｓｔｉｃ　Ｃａｌｃｕｌａｔｉｏｎ§２　Ｒｅｌａｔｉｖｉｓｔｉｃ　Ｃａｌｃｕｌａｔｉｏｎ • There are essentially two different approaches to relativistic three-nucleon calculation: • ①a manifestly covariant scheme linked to a field theoretical approach. • ② a scheme based on relativistic quantum mechanics on spacelike hypersurfaces (including the light front) in Minkowski space.

6. B. Bakamjian, L.H. Thomas, Phys. Rev. 92, 1300 (1952). • Within the second scheme② the relativistic Hamiltonian for on-the-mass-shell particles consists of relativistic kinetic energies and two- and many-body interactions including their boost corrections, which are dictated by the Poincare algebra. ’

7. What is the boost correction? A potential in an arbitrary moving frame (q≠0)is different, which enters a relativistic Lippmann-Schwinger equation. q≠０ Ｖnr Ｖnr (q=0) (q=0)≠(q≠0)

8. Two-body t-matrix Nonrelativistic LS eq. nr nr E) nr nr E) E - k’’2/m Relativistic LS eq. ^ ^ Boosted relativisitic LS eq.

9. Two-body t-matrix Nonrelativistic LS eq. nr nr E) nr nr E) E - k’’2/m Relativistic LS eq. ^ ^ Boosted relativisitic LS eq.

10. Two-body t-matrix Nonrelativistic LS eq. nr nr E) nr nr E) E - k’’2/m Relativistic LS eq. ^ ^ Boosted relativisitic LS eq.

11. Two-body t-matrix Nonrelativistic LS eq. nr nr E) nr nr E) E - k’’2/m Relativistic LS eq. ^ ^ Boosted relativisitic LS eq.

12. §３　Identification totherelativistic potential

13. start “Realistic NN potential” Relativistic potential? ΧPT，AV18, CDBonn, Nijmegen etc no Identification Type 1 Identification Type 2 yes Boost potential Enter the relativistic Faddeev equation Output : Triton binding energy Pd scattering end

14. §３　Identification totherelativistic potential Few-BodySyst.(2010)48,109

15. Type　１ :(pseudo) Relativistic potential “Scale-transformation from nonrelativity to relativity ” Scale transformation Phys. Rev. Lett. 80, 2457(1998)

16. Two-body t-matrix Nonrelativistic LS eq. nr nr E) nr nr E) E - k’’2/m Relativistic LS eq. ^ ^

17. Type　２ Coester-Pieper-Serduke (CPS) (PRC11, 1 (1975))

19. nr Sandwiching it between <k | and |k’>, we get

20. Iteration Method Physics Letters B655, 119-125 (2007),(nucl-th/0703010)

21. Convergence to the iteration

25. §４　BoostCorrection Boosted Hamiltonian in 2N system Physics Letters B655, 119-125 (2007),(nucl-th/0703010)

26. q=0 fm-1 q=10 fm-1 Real Part q=20 fm-1 CD-Bonn potential 1S0 partial wave E=350MeV Half-shell t-matrix

27. Imaginary Part q=20 fm-1 q=10 fm-1 CD-Bonn potential 1S0 partial wave E=350MeV Half-shell t-matrix q=0 fm-1

28. §５ Kharkovpotential

29. Kharkov Potential I. Dubovyk, O. Shebeko, Few-Body Sys. 48, 109 (2010).

30. Two-body t-matrix Nonrelativistic LS eq. nr nr E) nr nr E) E - k’’2/m Relativistic LS eq. ^ ^ Boosted relativisitic LS eq.

31. start Relativistic potential? ΧPT，AV18, CDBonn, Nijmegen etc no Identification Type 1 IdentificationType 2 yes Kharkov Boost potential Enter the relativistic Faddeev equation Output : Triton binding energy Pd scattering end

32. Deuteron Wave Function S-Wave Solid:Kharkov Dotted: CDBonn 1/2 -1 pψ(p)[fm ] p[ｆｍ ]

33. Deuteron Wave Function D-Wave Solid:Kharkov Dotted: CDBonn 1/2 -1 pψ(p)[fm ] p[ｆｍ ]

34. §６ Triton Binding Energy • Type 1 • Type 2 Coester-Pieper-Serduke (CPS) • Type 0 no identification “Scale-transform it from nonrelativity to relativity ” (ST)

35. Triton binding energies (Type1) MeV Rel. Nonrel. Phys. Rev. C66, 044010 (2002) 5ch calculation

36. Triton binding energies (Type2) MeV Rel. Nonrel. 0.05 0.11 0.07 0.10 0.08 0.07 -6.97 -8.22 -7.58 -7.90 -7.68 -7.59 -7.02 -8.33 -7.65 -8.00 -7.76 -7.66 5ch calculation EPJ Web of Conferences 3, 05025 (2010)

37. Triton binding energies (Type0)of Kharkov potential Type 2 5ch calculation H.Kamada, O. Shebeko, A. Arslanaliev, Few-Body Syst. 58 (2017), 70.

38. Triton binding energies (Type2)of N4LO and Kharkov pot. (MeV) N4LO pot. : E. Epelbaum et al., Eur. Phys. J. A51, 53 (2015) ; E. Epelbaumet al., Phys. Rev. Lett. 115, 122301 (2015) 42ch calculation

39. Triton Binding Energy [MeV] Type１ Type２ Kharkov(UCT1) Kharkov(UCT2) ←Exp. CDBonn [MeV]

40. §７　Ｅｌａｓｔｉｃ　Ｎｄ　ｓｃａｔｔｅｒｉｎｇ§７　Ｅｌａｓｔｉｃ　Ｎｄ　ｓｃａｔｔｅｒｉｎｇ

41. Comparison I • Relativistic and non-relativistic (without 3NF) • Kharkov potential (UCT1) • dσ／dΩdifferential cross section • Ay proton vector polarization • iT11 deuteron vector polarization • T20, T21, T22 deuteron vector polarization • Ep=5,13,65,135 MeV

42. 5MeV 13MeV 65MeV 135MeV

43. 5MeV 13MeV 65MeV 135MeV

44. 13MeV 5MeV 65MeV 135MeV

45. 13MeV 5MeV 65MeV 135MeV

46. 5MeV 13MeV 65MeV 135MeV

47. 5MeV 13MeV 135MeV 65MeV

48. 5MeV 13MeV 65MeV 135MeV

49. Phys. Rev. C 57, 2111 (1998)

50. Θc.m.＝180deg