1. The off-shell Coulomb-like amplitude in momentum space
Takashi Watanabe ,
Collaborator : Shinsho Oryu, Yasuhisa Hiratsuka,
Yoshio Togawa, Akio Kodama, Masayuki Takeda
Department of Physics, Faculty of Science and Technology,
Tokyo University of Science, Yamazaki, Noda City, Chiba Japan
The Seventh Asia-Pacific Conference on Few-Body Problems in Physics
(APFB 2017) is to be held on Aug. 25 (Fri.) – 30 (Wed.), 2017 in Guilin

2. The New Horizon in the Exact Coulomb treatment !
We have a long history of Coulomb problem more
than a half century!
Nobody finished the Coulomb problems except for
some approximations.
Faddeev himself could not solve the problem, and
finally his wife Veselova tried it !

3. Do you think, the Coulomb force in the nuclear and particle physics is supplementary?

4. Do you think, the Coulomb force in the nuclear and particle physics is supplementary?
No, in low energy, in heavy ions, in the threshold behavior, etc., the Coulomb treatment is important.

5. I. The next New Horizon can be discovered
in the Exact Coulomb treatment !
Do you think, the Coulomb force in the nuclear and particle physics is supplementary?
No, in low energy, in ｈｅａｖｙ ｉｏｎｓ，in ｔｈｒｅｓｈｏｌｄ　ｂｅｈａｖｉｏｒ, etc. , the Coulomb treatment is important.
Do you think that the Coulomb interaction is well investigated in nuclear physics?

6. I. The next New Horizon can be discovered
in the Exact Coulomb treatment !
Do you think, the Coulomb force in the nuclear and particle physics is supplementary?
No, in low energy, in ｈｅａｖｙ ｉｏｎｓ，in ｔｈｒｅｓｈｏｌｄ　ｂｅｈａｖｉｏｒ, etc. , the Coulomb treatment is important.
Do you think that Coulomb interaction is well investigated in nuclear physics?
In two-body Coulomb, which is analytically known in r-space, Yes. But in p-space No!

7. 3) Do you know the Coulomb off-shell amplitude in r-space?
I. The next New Horizon can be discovered
in the Exact Coulomb treatment !
Do you think, the Coulomb force in the nuclear and particle physics is supplementary?
No, in low energy, in ｈｅａｖｙ ｉｏｎｓ，in ｔｈｒｅｓｈｏｌｄ　ｂｅｈａｖｉｏｒ, etc. , the Coulomb treatment is important.
Do you think that Coulomb interaction is well investigated in nuclear physics?
In two-body Coulomb, which is analytically known
in r-space, Yes. But in p-space No!
3) Do you know the Coulomb off-shell amplitude
in r-space?

8. Nobody knows ! I. The next New Horizon can be discovered
in the Exact Coulomb treatment !
Do you think, the Coulomb force in the nuclear and particle physics is supplementary?
No, in low energy, in ｈｅａｖｙ ｉｏｎｓ，in ｔｈｒｅｓｈｏｌｄ　ｂｅｈａｖｉｏｒ, etc. , the Coulomb treatment is important.
Do you think that Coulomb interaction is well
investigated in nuclear physics?
In two-body Coulomb, which is analytically known
in r-space, Yes. But in p-space No!
3) Do you know the Coulomb off-shell amplitude in r-space?
Nobody knows !

9. important in the three-body calculation.
I. The next New Horizon can be discovered
in the Exact Coulomb treatment !
Do you think, the Coulomb force in the nuclear and particle physics is supplementary?
No, in low energy, in ｈｅａｖｙ ｉｏｎｓ，in ｔｈｒｅｓｈｏｌｄ　ｂｅｈａｖｉｏｒ, etc. , the Coulomb treatment is important.
Do you think that Coulomb interaction is well
investigated in nuclear physics?
In two-body Coulomb, which is analytically known
in r-space, Yes. But in p-space No!
3) Do you know the Coulomb off-shell amplitude in r-space?
Nobody knows !
You can find that the off-shell amplitude is
important in the three-body calculation.

10. Can you obtain the exact solution in p-space by the “Fourier transformation” ?

11. No! the Fourier transformation is done by a plane wave.
Can you obtain the exact solution in p-space by
the “Fourier transformation” ?
No! the Fourier transformation is done
by a plane wave.

12. a) the partial wave expansion of Coulomb potential diverges.
Can you obtain the exact solution in p-space by
the “Fourier transformation” ?
No! the Fourier transformation is done by a plane wave.
In p-space,
a) the partial wave expansion of Coulomb
potential diverges.

13. the Coulomb potential cannot be satisfied.
Can you obtain the exact solution in p-space by
the “Fourier transformation” ?
No! the Fourier transformation is done by a plane wave.
In p-space,
a) the partial wave expansion of Coulomb potential 　　
　　　　diverges
b) Lippmann-Schwinger (LS) equation for
the Coulomb potential cannot be satisfied.

14. Can you obtain the exact solution in p-space by
the “Fourier transformation” ?
No! the Fourier transformation is done by a plane wave.
In p-space,
a) the partial wave expansion of Coulomb potential
　　　　diverges
b) LS equation for the Coulomb potential cannot be
　　　　satisfied
In r-space,
can you analytically solve the three-body Faddeev equation with Coulomb potential ?

15. Can you obtain the exact solution in p-space by
the “Fourier transformation” ?
No! the Fourier transformation is done by a plane wave.
In p-space,
a) the partial wave expansion of Coulomb potential diverges
b) LS equation for the Coulomb potential cannot be satisfied
In r-space,
can you analytically solve the three-body Faddeev equation with Coulomb potential ?
No! you will find that any r-space calculations contain some approximations.

16. If you use any screened Coulomb potential in p-space, two- and three-body calculations are difficult because of the cancellation of significant digits, when you increase the screening range.

17. Motivation & Goal

20. Motivation: 1) In the Coulomb plus nuclear interactions,
the boundary conditions are mixed.

21. Motivation: 1) In the Coulomb plus nuclear interactions,
the boundary conditions are mixed.
2) Traditional screened Coulomb phase shift
doesn’t converge when the range is
increased.

22. Motivation: Goal: 1) In the Coulomb plus nuclear interactions,
the boundary conditions are mixed.
2) Traditional screened Coulomb phase shift
doesn’t converge when the range is
increased.
Goal:
In order to unify the boundary conditions,
we construct a screened Coulomb potential (SCP) with long range information.

23. 　　　Calculated results

26. Proton-proton case: MeV
R= fm
No. of Gauss points　1400
No. of digits　　　 250
Theory　 deg.
Numerical　96.9 deg.
deg
R=10000 fm
Gauss points　630
No. of digits 200
R=900 fm
Gauss points　500
No. of digits 100
R=800　fm
Gauss points　350
No. of digits 100
Proton-proton case: MeV

31. η　　　 　　　　E

37. Black bands are short range best fit
Red bands are given by the analytic range function
Red and Black lines coincide when

43. However, we confirmed that the Alt’s renormalization method is correct in some sense !!

52. Off-shell auxiliary amplitude
TΦR0(p,p’;E) [fm-1]
Off-shell auxiliary amplitude
p=p’
E=1.0 MeV
k= fm-1
p=0.062 fm-1
p’ [fm-1]

53. [fm-1]
E=1.0 MeV
k= fm-1
p=0.062 fm-1
on-shell
p [fm-1]

54. These results show
We can not neglect the off-shell auxiliary term.
Off-shell part could interfere with many terms in the three-body equation.

55. Concrusion

57. Thank you very much for your attention!

60. A New Horizon of Few-Body Problems
was found by L.D. Faddeev in 1960

61. I. The next New Horizon can be discovered
in the Exact Coulomb treatment !
We have a long history of Coulomb problem more
than a half century!
Nobody finished the Coulomb problems except for
some approximations.
Faddeev himself could not solve the problem, and
finally his wife Veselova tried it !

65. m=1 case (Yukawa-type) 4-digit accuracy Number of Gaus
Number of digits 　

66. m=5 case
4-digit accuracy
Number of Gaus
Number of digits 　

67. m=5 case 3-digit accuracy H E C D B A G F I R=4.50×105 fm
E=1.0×10-3 MeV
σ0=96.8 deg.
m=5 case
3-digit accuracy
R=4 fm
E=100 MeV
σ0= deg.
R=7.7×104 fm
E=1.0×10-2 MeV
σ0=-16.3 deg.
R=8 fm
E=50.0 MeV
σ0= deg.
R=35 fm
E=10.0 MeV
σ0=-1.17 deg. H
R=3700 fm
E=0.1 MeV
σ0=-10.7 deg. E C D B A G F
R=17 fm
E=20.0 MeV
σ0= deg.
R=620 fm
E=1.0 MeV
σ0=-3.66 deg.
R= 6.2×106fm
E=1.0×10-4 MeV
σ0=949.9 deg. I

74. The renormalization method: Schrödinger equations,
Asymptotic wave functions
Define an auxiliary function

79. Photo by Prof. E. KOGANOVA in 2003

84. If you use any screened Coulomb potential in p-space,
two- and three-body calculations are difficult because of the cancellation of significant digits, when you increase
the screening range.
Some recent examples:
Alt-Sandhas-Ziegelmann’s (ASZ) “phase-shift renormalization method”: (1976~1985)

85. If you use any screened Coulomb potential in p-space,
two- and three-body calculations are difficult because of the cancellation of significant digits, when you increase
the screening range.
Some recent examples:
Alt-Sandhas-Ziegelmann’s (ASZ) “phase-shift renormalization method”: (1976~1985)
It is almost impossible to calculate by using the method “less than 1010 digits accuracy”.

86. If you use any screened Coulomb potential in p-space,
two- and three-body calculations are difficult because of the cancellation of significant digits, when you increase
the screening range.
Some recent examples:
Alt-Sandhas-Ziegelmann’s (ASZ) “phase-shift renormalization method”: (1976~1985)
It is almost impossible to calculate by using the method
“less than 1010 digits accuracy”.
Deltuva-Fonseca-Sauer (DFS) p-d calculation, by “a short-range cut-off screened Coulomb potential with ASZ’s method”. (2005)

87. It has almost same problems which were seen in ASZ.
If you use any screened Coulomb potential in p-space,
two- and three-body calculations are difficult because of the cancellation of significant digits, when you increase
the screening range.
Some recent examples:
Alt-Sandhas-Ziegelmann’s (ASZ) “phase-shift renormalization method”: (1976~1985)
It is almost impossible to calculate by using the method
“less than 1010 digits accuracy”.
Deltuva-Fonseca-Sauer (DFS) p-d calculation, by “a short-range cut-off screened Coulomb potential with ASZ’s method”. (2005)
It has almost same problems which were seen in ASZ.

88. History of the renormalization method:
1) W. Tobocman, M. H. Kalos PR 97, 55 (1955)

89. History of the renormalization method:
1) W. Tobocman, M. H. Kalos PR 97, 55 (1955)
2) A. M. Veselova, Teor. Mat. Fiz 3, 326 (1970)~(1978)

90. History of the renormalization method:
1) W. Tobocman, M. H. Kalos PR 97, 55 (1955)
2) A. M. Veselova, Teor. Mat. Fiz 3, 326 (1970)~(1978)
Photo by Prof. E. Kalganova in St. Petersburg 2004

91. History of the renormalization method:
1) W. Tobocman, M. H. Kalos PR 97, 55 (1955)
2) A. M. Veselova, Teor. Mat. Fiz 3, 326 (1970)~(1978)
3) E. O. Alt, W. Sandhas, H. Zankel, H. Ziegelmann, PRL. 37, 1537 (1976)~(1985).

92. 4) A. Deltuva, A. C. Fonseca, P. U. Sauer, PRC 71 054005 (2005).
History of the renormalization method:
1) W. Tobocman, M. H. Kalos PR 97, 55 (1955)
2) A. M. Veselova, Teor. Mat. Fiz 3, 326 (1970)~(1978)
3) E. O. Alt, W. Sandhas, H. Zankel, H. Ziegelmann,
PRL. 37, 1537 (1976)~(1985)
4) A. Deltuva, A. C. Fonseca, P. U. Sauer,
PRC (2005).
Deltuva et al. used Alt-method, and also used the generalized potential and the renormalization phase with m=5 !

93. r-space calculation for p-d with a Coulomb approximation:
History of the renormalization method:
1) W. Tobocman, M. H. Kalos PR 97, 55 (1955)
2) A. M. Veselova, Teor. Mat. Fiz 3, 326 (1970)~(1978)
3) E. O. Alt, W. Sandhas, H. Zankel, H. Ziegelmann,
PRL. 37, 1537 (1976)~(1985)
4) A. Deltuva, A. C. Fonseca, P. U. Sauer, PRC (2005).
r-space calculation for p-d with a Coulomb approximation:
T. Sasakawa, T. Sawada , Phys. Rev. C 20, 1954
(1979).
A. Kievsky, S. Rosati, M. Viviani, PRL 82, 3759 (1999).
　S. Ishikawa , Few-Body Syst. 32: 229 (2003).

95. The1950s as an era before daybreak (dawn) which was a chaotic age in few-body problem in hadron physics !

96. The1950s as an era before daybreak (dawn) which was a chaotic age in few-body problem in hadron physics !
1) The perturbation method in hadron physics?, which was denied, because of the coupling constant is larger than 1, therefore it didn’t converge!

97. The1950s as an era before daybreak (dawn) which was a chaotic age in few-body problem in hadron physics !
The perturbation method in hadron physics?, which was denied, because of the coupling constant is larger than 1.
Variational calculation ? , which needs many parameters, and didn’t converge!

98. The1950s as an era before daybreak (dawn) which was a chaotic age in few-body problem in hadron physics !
1) The perturbation method in hadron physics?, which was denied, because of the coupling constant is larger than 1.
2) Variational calculation ? , which needs many parameters,
but didn’t converge!
3) Optical potential ? which was phenomenological !

99. The1950s as an era before daybreak (dawn) which was a chaotic age in few-body problem in hadron physics !
1) The perturbation method in hadron physics?, which was denied, because of the coupling constant is larger than 1.
2) Variational calculation ? , which needs many parameters,
but didn’t converge!
Optical potential ? which was phenomenological !
4) Nuclear potentials? which were phenomenological too ! Sigma-meson was not found ! Since many mesons were found, then Yukawa meson is not everything. Therefore, nuclear potential could not be perfectly illustrated.

100. The1950s as an era before daybreak (dawn) which was a chaotic age in few-body problem in hadron physics !
1) The perturbation method in hadron physics?, which was denied, because of the coupling constant is larger than 1.
2) Variational calculation ? , which needs many parameters,
but didn’t converge!
3) Optical potential ? which was phenomenological !
4) Nuclear potential? which were phenomenological too !
Sigma-meson was not found ! Since many mesons were found,
then Yukawa meson is not everything.
Therefore, nuclear potential could not be perfectly illustrated.
5) The dispersion theory was discussed only as a framework.

101. The1950s as an era before daybreak (dawn) which was a chaotic age in few-body problem in hadron physics !
1) The perturbation method in hadron physics?, which was denied, because of the coupling constant is larger than 1.
2) Variational calculation ? , which needs many parameters,
but didn’t converge!
3) Optical potential ? which was phenomenological !
4) Nuclear potential? which were phenomenological too !
Sigma-meson was not found ! Since many mesons were found,
then Yukawa meson is not everything.
Therefore, nuclear potential could not be perfectly illustrated.
5) The dispersion theory was discussed only as a framework.
6) Not few-body, but, there were some good results.
Group theory by Wigner, Shell-model, etc.

102. The1950s as an era before daybreak (dawn) which was a chaotic age in few-body problem in hadron physics !
1) The perturbation method in hadron physics?, which was denied, because of the coupling constant is larger than 1.
Variational calculation ? , which needs many parameters,
but didn’t converge!
Optical potential ? which was phenomenological !
Nuclear potential? which were phenomenological too !
Sigma-meson was not found ! Since many mesons were found,
then Yukawa meson is not everything.
Therefore, nuclear potential could not be perfectly illustrated.
5) The dispersion theory was discussed only as a framework.
Not few-body but, there were some good results.
Group theory by Wigner, Shell-model, etc.
7) Few-body reactions: pick-up reaction, knock-on , stripping, heavy particle striping, DWBA, etc. which are unified by the Faddeev equation.

103. The1950s as an era before daybreak (dawn) which was a chaotic age in few-body problem in hadron physics !
1) The perturbation method in hadron physics?, which was denied, because of the coupling constant is larger than 1.
Variational calculation ? , which needs many parameters,
but didn’t converge!
Optical potential ? which was phenomenological !
Nuclear potential? which were phenomenological too !
Sigma-meson was not found ! Since many mesons were found,
then Yukawa meson is not everything.
Therefore, nuclear potential could not be perfectly illustrated.
5) The dispersion theory was discussed only as a framework.
Not few-body but, there were some good results.
Group theory by Wigner, Shell-model, etc.
7) Few-body reactions: heavy particle striping, knock-on, pick-up, stripping reactions, DWBA, etc. which are unified by the Faddeev equation.
Therefore, a discovery of the Faddeev equation was spectacular!