相対論的３体散乱計算Relativistic Faddeev Scattering Calculations

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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

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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)

§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?

Phys. Rev. C 59, 3035 (1999)

Data

Only 2NF

Total cross section of the pd scattering

Phys. Rev. C 57, 2111 (1998)

§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 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.

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)

Two-body t-matrix

E - k’’2/m

)

nrnr

nrnr

^ ^

E)E)

Nonrelativistic LS eq.

Relativistic LS eq.

Boosted relativisitic LS eq.

Two-body t-matrix

E - k’’2/m

)

nrnr

nrnr

^ ^

E)E)

Nonrelativistic LS eq.

Relativistic LS eq.

Boosted relativisitic LS eq.

Two-body t-matrix

E - k’’2/m

)

nrnr

nrnr

^ ^

E)E)

Nonrelativistic LS eq.

Relativistic LS eq.

Boosted relativisitic LS eq.

Two-body t-matrix

)

^ ^

E - k’’2/m

nrnr

nrnr

E)E)

Nonrelativistic LS eq.

Relativistic LS eq.

Boosted relativisitic LS eq.

§３ Identification to therelativistic potential

Few-Body Syst. (2010) 48, 109

§３ Identification to therelativistic potential

: (pseudo) Relativistic potential

Phys. Rev. Lett. 80, 2457(1998)

“Scale-transform it from nonrelativity to relativity ”

Scale transformation

Type １

Two-body t-matrix

E - k’’2/m

)

nrnr

nrnr

^ ^

E)E)

Nonrelativistic LS eq.

Relativistic LS eq.

Coester-Pieper-Serduke (CPS)

(PRC11, 1 (1975))

Type ２

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

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

Convergence to the iteration

§４ Boost Correction

§４ Boost Correction

2 22 22 ( ) 2 ( ) qv k v q k q

Boosted Hamiltonian in 2N system

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

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

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

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

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

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

Deuteron Wave Function

S-Wave

Solid:Kharkov

Dotted: CDBonn

Deuteron Wave Function

D-Wave

Solid:Kharkov

Dotted: CDBonn

§５ Triton binding energy

Type 1

Type 2 Coester-Pieper-Serduke (CPS)

Type 0 no identification

“Scale-transform it from nonrelativity to relativity ” (ST)

Rel. Nonrel.

Phys. Rev. C66, 044010 (2002) 5ch calculation

Triton binding energies (Type 1)

MeV

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

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

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

CD Bonn

DS [CDBonn]

E=135MeV

Solid:Relativistic

Dashed: Nonrelativistic

Ay [CDBonn]

E=135MeV

Solid:Relativistic

Dashed: Nonrelativistic

iT11 [CDBonn]

E=135MeV

Solid:Relativistic

Dashed: Nonrelativistic

T20 [CDBonn]

E=135MeV

Solid:Relativistic

Dashed: Nonrelativistic

T21 [CDBonn]

E=135MeV

Solid:Relativistic

Dashed: Nonrelativistic

T22 [CDBonn]

E=135MeV

Solid:Relativistic

Dashed: Nonrelativistic

Results (elastic nd scattering)

from Kharkov potential

• Nonrelativistic Interpretation （Type2）

DS

E=13MeV

Solid:CDBonn

Dashed:Kharkov

Ay

E=13MeV

Solid:CDBonn

Dashed:Kharkov

iT11

E=13MeV

Solid:CDBonn

Dashed:Kharkov

T20

E=13MeVSolid:CDBonn

Dashed:Kharkov

T21

E=13MeV

Solid:CDBonn

Dashed:Kharkov

T22

E=13MeV

Solid:CDBonn

Dashed:Kharkov

DS

E=135MeVSolid:CDBonn[rel.]

Dashed:CDBonn[nonrel.]

Dotted:Kharkov

Ay

E=135MeV

Solid:CDBonn[rel.]

Dashed:CDBonn[nonrel.]

Dotted:Kharkov

iT11

E=135MeV

Solid:CDBonn[rel.]

Dashed:CDBonn[nonrel.]

Dotted:Kharkov

T20

E=135MeV

Solid:CDBonn[rel.]

Dashed:CDBonn[nonrel.]

Dotted:Kharkov

T21

E=135MeV

Solid:CDBonn[rel.]

Dashed:CDBonn[nonrel.]

Dotted:Kharkov

T22

E=135MeV

Solid:CDBonn[rel.]

Dashed:CDBonn[nonrel.]

Dotted:Kharkov

・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

・ 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