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 Relativistic Calculation §３ Identification to the relativistic potential §４ Boost Correction §５ Kharkov potential §６ Triton binding energy §７ Nd elastic scattering §８ Summary and 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:
①　a 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. Relativistic LS eq. ^ ^
E k’’2/m
Relativistic LS eq. ^ ^
Boosted relativisitic LS eq.

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

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

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

12. §３ Identification to the relativistic potential

13. “Realistic NN potential” ΧPT，AV18, CDBonn, Nijmegen etc start
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 to the relativistic potential
Few-Body Syst. (2010) 48, 109

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

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

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

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

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

21. Convergence to the iteration

25. §４ Boost Correction Boosted Hamiltonian in 2N system
Physics Letters B655, (2007), (nucl-th/ )

26. Real Part q=0 fm-1 q=10 fm-1 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 Kharkov　potential

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

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

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

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

33. Deuteron Wave Function
D-Wave
pψ(p)[fm ]
1/2
Solid:Kharkov
Dotted: CDBonn -1
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 (Type 1)
MeV
Rel.
Nonrel.
Phys. Rev. C66, (2002) 5ch calculation

36. Triton binding energies (Type 2)
MeV
Rel.
Nonrel.
-6.97
-8.22
-7.58
-7.90
-7.68
-7.59
-7.02
-8.33
-7.65
-8.00
-7.76
-7.66
0.05
0.11
0.07
0.10
0.08
5ch calculation
EPJ Web of Conferences 3, (2010)

37. Triton binding energies (Type 0) of Kharkov potential
Relativistic
Nonrelativistic
Difference
Kharkov(UCT1)　
-7.42
（-7.49）
0.07
AV18
（-7.59）
-7.66
CD-Bonn
（-8.22）
-8.33
0.11
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)
Regularization　
Relativistic
Nonrelativistic
Difference
χEFT　R=0.9
(-7.706)
-7.832
0.126
χEFT　R=1.0
(-7.748)
-7.867
0.119
χEFT　R=1.1
(-7.733)
-7.848
0.115
CD-Bonn
(-8.150)
-8.249
0.099
Kharkov(UCT1)
-7.460
(-7.528)
0.068
Kharkov（UCT2)
-7.981
(-8.099)
0.118
N4LO pot. : E. Epelbaum et al., Eur. Phys. J. A51, 53 (2015)
; E. Epelbaum et al., Phys. Rev. Lett. 115, (2015)
42ch calculation

39. Triton Binding Energy Type１ Type２ [MeV] 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／dΩ　differential 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
65MeV
135MeV

48. 5MeV
13MeV
65MeV
135MeV

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

50. Θc.m.＝180deg

51. Comparison II
CDBonn (rel.)
Kharkov (UCT1)
Kharkov (UCT2)

52. 5MeV
13MeV
65MeV
135MeV
CDBonn
UCT1
UCT2

53. 5MeV
13MeV
65MeV
135MeV
CDBonn
UCT1
UCT2

54. Ay puzzle Koike’s conjecture 3P0 3P1 3P2 NN phase shift (P-wave) 50MeV
Partial wave
Nijmegen
DATA
CDBonn
Bonn B
UCT1
UCT2
Effect on Ay (UCT1,UCT2)
3P0
10.70
10.79
12.24
12.16
11.78
↓,↓
3P1
-8.25
-8.23
-8.77
-8.58
-7.82
-,-
3P2
5.89
5.91
6.14
6.00
4.77
-, ↓
The NN phase shift is not well described yet !

55. 5MeV
13MeV
CDBonn
UCT1
UCT2
65MeV
135MeV

56. 5MeV
13MeV
CDBonn
UCT1
UCT2
65MeV
135MeV

57. 5MeV
CDBonn
UCT1
UCT2
13MeV
65MeV
135MeV

58. 5MeV
13MeV
65MeV
135MeV
CDBonn
UCT1
UCT2

59. §８　Summary and Outlook
・Kharkov potential: ⇒Kharkov potential gives relativistic potential directly. (No need identifications)
start
Relativistic potential?
Boost potential
Enter the relativistic Faddeev equation
Identification Type 1
Identification Type 2
yes no
end
Output :
Triton binding energy
Pd scattering
Kharkov
ΧPT，AV18, CDBonn, Nijmegen etc

60. §８　Summary and Outlook
・Triton binding energies: ⇒ Chiral potentials (N4LO) give similar results (-7.71～-7.73MeV) as CDBonn potential (-8.15MeV). ⇒ Kharkov potential (UCT1) needs not any identification and gives MeV. ⇒ Kharkov potential has a rather small difference between the relativistic binding energies and the nonrelativistic one. ⇒We need much 3 body force to reach data (8.48MeV).

61. §８　Summary and Outlook
・ Relativistic results for Nd elastic scattering: ⇒ The first Nd scattering calculation for Kharkov pot. ⇒ In the low energy region (<65MeV) the results of Kharkov potential reasonably agree with the CDBonn potential case except for Ay and iT11. ⇒ The phase shift of P-wave is not enough described. ⇒ Beyond the intermediate energy region (>65MeV) the prediction of Kharkov potential is getting to differ from the CDBonn potential case. However, it is difficult to distinguish whether the difference causes from relativistic effect or from its own parameterization.

62. §８　Summary and Outlook
・ Relativistic results for Nd elastic scattering: ⇒ 3body force was not included. ⇒ Kharkov potential gains less triton binding energy |Eb| than CDBonn potential case, therefore, we expect larger contribution from 3 body force.

63. CDBonn
Nonrel
Rel.
Nonrel+3NF
Rel.+3NF

65. dσ／d Ω
[mb/sr]
150
400
Θc.m.[deg]