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

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

§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?

Phys. Rev. C 57, 2111 (1998)

§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.

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. ’

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)

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

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

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

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

§３ Identification to the relativistic potential

“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

§３ Identification to the relativistic potential   Few-Body Syst. (2010) 48, 109

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

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

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

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

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

Convergence to the iteration

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

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

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

§５ Kharkov　potential

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

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

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

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

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

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

Triton binding energies (Type 1) MeV Rel. Nonrel. Phys. Rev. C66, 044010 (2002) 5ch calculation

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, 05025 (2010)

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.

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, 122301 (2015) 42ch calculation

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

§７ Ｅｌａｓｔｉｃ Ｎｄ ｓｃａｔｔｅｒｉｎｇ §７　Ｅｌａｓｔｉｃ　Ｎｄ　ｓｃａｔｔｅｒｉｎｇ

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

5MeV 13MeV 65MeV 135MeV

5MeV 13MeV 65MeV 135MeV

13MeV 5MeV 65MeV 135MeV

13MeV 5MeV 65MeV 135MeV

5MeV 13MeV 65MeV 135MeV

5MeV 13MeV 65MeV 135MeV

5MeV 13MeV 65MeV 135MeV

Phys. Rev. C 57, 2111 (1998)

Θc.m.＝180deg

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

5MeV 13MeV 65MeV 135MeV CDBonn UCT1 UCT2

5MeV 13MeV 65MeV 135MeV CDBonn UCT1 UCT2

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 !

5MeV 13MeV CDBonn UCT1 UCT2 65MeV 135MeV

5MeV 13MeV CDBonn UCT1 UCT2 65MeV 135MeV

5MeV CDBonn UCT1 UCT2 13MeV 65MeV 135MeV

5MeV 13MeV 65MeV 135MeV CDBonn UCT1 UCT2

§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

§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 -7.460MeV. ⇒ 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).

§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.

§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.

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

dσ／d Ω [mb/sr] 150 400 Θc.m.[deg]