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相対論的3体散乱計算Relativistic Faddeev Scattering Calculations
P
N
核力に基づく核構造、核反応物理の展開
2017-03-27 — 2017-03-29
YITP・京都
N
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)
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§1 Motivation
The nonrelativistic theoretical prediction of the Nd scattering cross section beyond 200MeV/u is getting to be poor even including the 3-body force (FM type).
What is missing?
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Phys. Rev. C 59, 3035 (1999)
Data
Only 2NF
Total cross section of the pd scattering
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Phys. Rev. C 57, 2111 (1998)
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§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.
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B. Bakamjian, L.H. Thomas,
Phys. Rev. 92, 1300 (1952).
• Within the second scheme 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.
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What is the boost correction?
A potential in an arbitrary moving frame (q≠0) is
different, which enters a relativistic Lippmann-
Schwinger equation.
Vnr
Vnr
(q=0) (q=0)≠ (q≠0)
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Two-body t-matrix
E - k’’2/m
)
nrnr
nrnr
^ ^
E)E)
Nonrelativistic LS eq.
Relativistic LS eq.
Boosted relativisitic LS eq.
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Two-body t-matrix
E - k’’2/m
)
nrnr
nrnr
^ ^
E)E)
Nonrelativistic LS eq.
Relativistic LS eq.
Boosted relativisitic LS eq.
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Two-body t-matrix
E - k’’2/m
)
nrnr
nrnr
^ ^
E)E)
Nonrelativistic LS eq.
Relativistic LS eq.
Boosted relativisitic LS eq.
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Two-body t-matrix
)
^ ^
E - k’’2/m
nrnr
nrnr
E)E)
Nonrelativistic LS eq.
Relativistic LS eq.
Boosted relativisitic LS eq.
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§3 Identification to therelativistic potential
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Few-Body Syst. (2010) 48, 109
§3 Identification to therelativistic potential
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: (pseudo) Relativistic potential
Phys. Rev. Lett. 80, 2457(1998)
“Scale-transform it from nonrelativity to relativity ”
Scale transformation
Type 1
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Two-body t-matrix
E - k’’2/m
)
nrnr
nrnr
^ ^
E)E)
Nonrelativistic LS eq.
Relativistic LS eq.
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Coester-Pieper-Serduke (CPS)
(PRC11, 1 (1975))
Type 2
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nr
Sandwiching it between <k | and |k’>, we get
3
3
3
1( , ') 2 ( ) ( , ') 2 ( , ') ( ') ( , '') ( '', ') ''
4
1 1( ) ( ') ( , ') ( , '') ( '', ') ''
2 4
namely,
1 1( , ') 2 ( , ') ( , '') ( '', ') '' .
( ) ( ') 2
nr
nr
V k k k v k k v k k k v k k v k k d km
k k v k k v k k v k k d km m
v k k mV k k v k k v k k d kk k
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Physics Letters B655, 119-125 (2007), (nucl-th/0703010)
(0)
( 1) ( ) ( ) 3
1( , ') 2 ( , '),
( ) ( ')
1 1( , ') 2 ( , ') ( , '') ( '', ') ''
( ) ( ') 2
nr
n n n
nr
v k k mV k kk k
v k k mV k k v k k v k k d kk k
31 1( , ') 2 ( , ') ( , '') ( '', ') '' .
( ) ( ') 2nrv k k mV k k v k k v k k d k
k k
Iteration Method
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Convergence to the iteration
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§4 Boost Correction
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§4 Boost Correction
2 22 22 ( ) 2 ( ) qv k v q k q
Boosted Hamiltonian in 2N system
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boosted
2 2 2 2 2 2
0ˆ( 4( ) ) 4( ) (2)qm k q v m k q
2 2 2 2 2 2 21 ˆ ˆ( 4( ) 4( ) ) (4)4
nr q q qV v m k q m k q v vm
2 2 2 2 2 2 2 2 2 2
2 2 2
0
ˆ ˆ ˆ(4( ) 4( ) 4( ) )
4( ) (3)
q q qm k q v m k q m k q v v
m k q
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2 22 ( ) 2 ( ) q k k q
31 1( , ') 2 ( , ') ( , '') ( '', ') '' .
( ) ( ') 2
q nr q q
q q
v k k mV k k v k k v k k d kk k
31 1( , ') 2 ( , ') ( , '') ( '', ') '' .
( ) ( ') 2
nrv k k mV k k v k k v k k d k
k k
Non-boosted
Iteration method
Boosted
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CD-Bonn potential
1S0 partial wave
E=350MeV
Half-shell t-matrix
Q=0 fm-1
Q=20 fm-1
Q=10 fm-1
Real Part
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CD-Bonn potential
1S0 partial wave
E=350MeV
Half-shell t-matrix
Q=0 fm-1
Q=20 fm-1
Q=10 fm-1
Imaginary Part
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Relativistic
potential?
Boost potential
Enter the relativistic
Faddeev equation
Identification Type 1 Identification Type 2
yes
no
Output :
Triton binding energy
Kharkov
ΧPT,AV18, CDBonn, Nijmegen etc
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I. Dubovyk, O. Shebeko, Few-Body Sys. 48, 109 (2010).
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Deuteron Wave Function
S-Wave
Solid:Kharkov
Dotted: CDBonn
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Deuteron Wave Function
D-Wave
Solid:Kharkov
Dotted: CDBonn
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§5 Triton binding energy
Type 1
Type 2 Coester-Pieper-Serduke (CPS)
Type 0 no identification
“Scale-transform it from nonrelativity to relativity ” (ST)
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Rel. Nonrel.
Phys. Rev. C66, 044010 (2002) 5ch calculation
Triton binding energies (Type 1)
MeV
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Triton binding energies (Type 2)
Rel. Nonrel.
-6.97-8.22-7.58-7.90-7.68-7.59
0.050.110.070.100.080.07
-7.02-8.33-7.65-8.00-7.76-7.66
5ch calculation EPJ Web of Conferences 3, 05025 (2010)
MeV
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Triton binding energies (Type 0)
of Kharkov potential
5ch calculation
potential Relativistic Nonrelativistic Difference
Kharkov -7.42 (-7.49) 0.07
AV18 (-7.59) -7.66 0.07
CD-Bonn (-8.22) -8.33 0.11Type 2
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Triton binding energies (Type2)
of N4LO
42ch calculation
regularization Relativistic Nonrelativistic Difference
R=0.9 -7.706 -7.832 0.126
R=1.0 -7.748 -7.867 0.119
R=1.1 -7.733 -7.848 0.115
CD-Bonn -8.150 -8.249 0.099
Kharkov -7.461 -7.528 0.067
N4LO pot. : E.Epelbaum et al.,Eur.Phys. J. A51, 53 (2015)
; E.Epelbaum et al.,Phys. Rev. Lett. 115, 122301 (2015)
MeV
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CD Bonn
![Page 41: 相対論的3体散乱計算 Relativistic FaddeevScattering …naoyuki.itagaki/abin/abin-k...相対論的3体散乱計算 Relativistic FaddeevScattering Calculations P N 核力に基づく核構造、](https://reader036.vdocuments.net/reader036/viewer/2022071020/5fd3b50cd0dfd769a300f33b/html5/thumbnails/41.jpg)
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![Page 43: 相対論的3体散乱計算 Relativistic FaddeevScattering …naoyuki.itagaki/abin/abin-k...相対論的3体散乱計算 Relativistic FaddeevScattering Calculations P N 核力に基づく核構造、](https://reader036.vdocuments.net/reader036/viewer/2022071020/5fd3b50cd0dfd769a300f33b/html5/thumbnails/43.jpg)
DS [CDBonn]
E=135MeV
Solid:Relativistic
Dashed: Nonrelativistic
![Page 44: 相対論的3体散乱計算 Relativistic FaddeevScattering …naoyuki.itagaki/abin/abin-k...相対論的3体散乱計算 Relativistic FaddeevScattering Calculations P N 核力に基づく核構造、](https://reader036.vdocuments.net/reader036/viewer/2022071020/5fd3b50cd0dfd769a300f33b/html5/thumbnails/44.jpg)
Ay [CDBonn]
E=135MeV
Solid:Relativistic
Dashed: Nonrelativistic
![Page 45: 相対論的3体散乱計算 Relativistic FaddeevScattering …naoyuki.itagaki/abin/abin-k...相対論的3体散乱計算 Relativistic FaddeevScattering Calculations P N 核力に基づく核構造、](https://reader036.vdocuments.net/reader036/viewer/2022071020/5fd3b50cd0dfd769a300f33b/html5/thumbnails/45.jpg)
iT11 [CDBonn]
E=135MeV
Solid:Relativistic
Dashed: Nonrelativistic
![Page 46: 相対論的3体散乱計算 Relativistic FaddeevScattering …naoyuki.itagaki/abin/abin-k...相対論的3体散乱計算 Relativistic FaddeevScattering Calculations P N 核力に基づく核構造、](https://reader036.vdocuments.net/reader036/viewer/2022071020/5fd3b50cd0dfd769a300f33b/html5/thumbnails/46.jpg)
T20 [CDBonn]
E=135MeV
Solid:Relativistic
Dashed: Nonrelativistic
![Page 47: 相対論的3体散乱計算 Relativistic FaddeevScattering …naoyuki.itagaki/abin/abin-k...相対論的3体散乱計算 Relativistic FaddeevScattering Calculations P N 核力に基づく核構造、](https://reader036.vdocuments.net/reader036/viewer/2022071020/5fd3b50cd0dfd769a300f33b/html5/thumbnails/47.jpg)
T21 [CDBonn]
E=135MeV
Solid:Relativistic
Dashed: Nonrelativistic
![Page 48: 相対論的3体散乱計算 Relativistic FaddeevScattering …naoyuki.itagaki/abin/abin-k...相対論的3体散乱計算 Relativistic FaddeevScattering Calculations P N 核力に基づく核構造、](https://reader036.vdocuments.net/reader036/viewer/2022071020/5fd3b50cd0dfd769a300f33b/html5/thumbnails/48.jpg)
T22 [CDBonn]
E=135MeV
Solid:Relativistic
Dashed: Nonrelativistic
![Page 49: 相対論的3体散乱計算 Relativistic FaddeevScattering …naoyuki.itagaki/abin/abin-k...相対論的3体散乱計算 Relativistic FaddeevScattering Calculations P N 核力に基づく核構造、](https://reader036.vdocuments.net/reader036/viewer/2022071020/5fd3b50cd0dfd769a300f33b/html5/thumbnails/49.jpg)
Results (elastic nd scattering)
from Kharkov potential
• Nonrelativistic Interpretation (Type2)
![Page 50: 相対論的3体散乱計算 Relativistic FaddeevScattering …naoyuki.itagaki/abin/abin-k...相対論的3体散乱計算 Relativistic FaddeevScattering Calculations P N 核力に基づく核構造、](https://reader036.vdocuments.net/reader036/viewer/2022071020/5fd3b50cd0dfd769a300f33b/html5/thumbnails/50.jpg)
DS
E=13MeV
Solid:CDBonn
Dashed:Kharkov
![Page 51: 相対論的3体散乱計算 Relativistic FaddeevScattering …naoyuki.itagaki/abin/abin-k...相対論的3体散乱計算 Relativistic FaddeevScattering Calculations P N 核力に基づく核構造、](https://reader036.vdocuments.net/reader036/viewer/2022071020/5fd3b50cd0dfd769a300f33b/html5/thumbnails/51.jpg)
Ay
E=13MeV
Solid:CDBonn
Dashed:Kharkov
![Page 52: 相対論的3体散乱計算 Relativistic FaddeevScattering …naoyuki.itagaki/abin/abin-k...相対論的3体散乱計算 Relativistic FaddeevScattering Calculations P N 核力に基づく核構造、](https://reader036.vdocuments.net/reader036/viewer/2022071020/5fd3b50cd0dfd769a300f33b/html5/thumbnails/52.jpg)
iT11
E=13MeV
Solid:CDBonn
Dashed:Kharkov
![Page 53: 相対論的3体散乱計算 Relativistic FaddeevScattering …naoyuki.itagaki/abin/abin-k...相対論的3体散乱計算 Relativistic FaddeevScattering Calculations P N 核力に基づく核構造、](https://reader036.vdocuments.net/reader036/viewer/2022071020/5fd3b50cd0dfd769a300f33b/html5/thumbnails/53.jpg)
T20
E=13MeVSolid:CDBonn
Dashed:Kharkov
![Page 54: 相対論的3体散乱計算 Relativistic FaddeevScattering …naoyuki.itagaki/abin/abin-k...相対論的3体散乱計算 Relativistic FaddeevScattering Calculations P N 核力に基づく核構造、](https://reader036.vdocuments.net/reader036/viewer/2022071020/5fd3b50cd0dfd769a300f33b/html5/thumbnails/54.jpg)
T21
E=13MeV
Solid:CDBonn
Dashed:Kharkov
![Page 55: 相対論的3体散乱計算 Relativistic FaddeevScattering …naoyuki.itagaki/abin/abin-k...相対論的3体散乱計算 Relativistic FaddeevScattering Calculations P N 核力に基づく核構造、](https://reader036.vdocuments.net/reader036/viewer/2022071020/5fd3b50cd0dfd769a300f33b/html5/thumbnails/55.jpg)
T22
E=13MeV
Solid:CDBonn
Dashed:Kharkov
![Page 56: 相対論的3体散乱計算 Relativistic FaddeevScattering …naoyuki.itagaki/abin/abin-k...相対論的3体散乱計算 Relativistic FaddeevScattering Calculations P N 核力に基づく核構造、](https://reader036.vdocuments.net/reader036/viewer/2022071020/5fd3b50cd0dfd769a300f33b/html5/thumbnails/56.jpg)
DS
E=135MeVSolid:CDBonn[rel.]
Dashed:CDBonn[nonrel.]
Dotted:Kharkov
![Page 57: 相対論的3体散乱計算 Relativistic FaddeevScattering …naoyuki.itagaki/abin/abin-k...相対論的3体散乱計算 Relativistic FaddeevScattering Calculations P N 核力に基づく核構造、](https://reader036.vdocuments.net/reader036/viewer/2022071020/5fd3b50cd0dfd769a300f33b/html5/thumbnails/57.jpg)
Ay
E=135MeV
Solid:CDBonn[rel.]
Dashed:CDBonn[nonrel.]
Dotted:Kharkov
![Page 58: 相対論的3体散乱計算 Relativistic FaddeevScattering …naoyuki.itagaki/abin/abin-k...相対論的3体散乱計算 Relativistic FaddeevScattering Calculations P N 核力に基づく核構造、](https://reader036.vdocuments.net/reader036/viewer/2022071020/5fd3b50cd0dfd769a300f33b/html5/thumbnails/58.jpg)
iT11
E=135MeV
Solid:CDBonn[rel.]
Dashed:CDBonn[nonrel.]
Dotted:Kharkov
![Page 59: 相対論的3体散乱計算 Relativistic FaddeevScattering …naoyuki.itagaki/abin/abin-k...相対論的3体散乱計算 Relativistic FaddeevScattering Calculations P N 核力に基づく核構造、](https://reader036.vdocuments.net/reader036/viewer/2022071020/5fd3b50cd0dfd769a300f33b/html5/thumbnails/59.jpg)
T20
E=135MeV
Solid:CDBonn[rel.]
Dashed:CDBonn[nonrel.]
Dotted:Kharkov
![Page 60: 相対論的3体散乱計算 Relativistic FaddeevScattering …naoyuki.itagaki/abin/abin-k...相対論的3体散乱計算 Relativistic FaddeevScattering Calculations P N 核力に基づく核構造、](https://reader036.vdocuments.net/reader036/viewer/2022071020/5fd3b50cd0dfd769a300f33b/html5/thumbnails/60.jpg)
T21
E=135MeV
Solid:CDBonn[rel.]
Dashed:CDBonn[nonrel.]
Dotted:Kharkov
![Page 61: 相対論的3体散乱計算 Relativistic FaddeevScattering …naoyuki.itagaki/abin/abin-k...相対論的3体散乱計算 Relativistic FaddeevScattering Calculations P N 核力に基づく核構造、](https://reader036.vdocuments.net/reader036/viewer/2022071020/5fd3b50cd0dfd769a300f33b/html5/thumbnails/61.jpg)
T22
E=135MeV
Solid:CDBonn[rel.]
Dashed:CDBonn[nonrel.]
Dotted:Kharkov
![Page 62: 相対論的3体散乱計算 Relativistic FaddeevScattering …naoyuki.itagaki/abin/abin-k...相対論的3体散乱計算 Relativistic FaddeevScattering Calculations P N 核力に基づく核構造、](https://reader036.vdocuments.net/reader036/viewer/2022071020/5fd3b50cd0dfd769a300f33b/html5/thumbnails/62.jpg)
・Triton binding energies:
→ Chiral potentials (N4LO) give similar results
(-7.71~-7.73MeV) as CDBonn potential (-8.15MeV).
→ Kharkov potential needs not any identification and gives -7.46MeV.
→ Kharkov potential has a rather small difference between the relativistic binding energies and the nonrelativistic one.
§6 Summary and Outlook
![Page 63: 相対論的3体散乱計算 Relativistic FaddeevScattering …naoyuki.itagaki/abin/abin-k...相対論的3体散乱計算 Relativistic FaddeevScattering Calculations P N 核力に基づく核構造、](https://reader036.vdocuments.net/reader036/viewer/2022071020/5fd3b50cd0dfd769a300f33b/html5/thumbnails/63.jpg)
・ Nd elastic scattering results:
→ In the low energy region (<65MeV) the prediction of Kharkov potential is reasonably agree with the CDBonn potential case.
→ 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 if the difference causes from relativistic property or from its own parameterization.
§6 Summary and Outlook