Three-Body Scattering Without Partial Waves
Hang Liu Charlotte Elster Walter Glöckle
1U
Elastic Process sum of Projectile-Target interactions
10U
3
2
11
23
Breakup process: sum of interaction within “all” pairs
3B Scattering
dth EE2
3
kinematically allowed if
3B Transition Amplitude
The Faddeev Equation in momentum space by using Jacobi Variables
)(2
1
3
22
1
321
32
kkkq
kkp
qp
Traditionally solved by Partial Wave Decomposition scheme
At low energy (<250MeV): practically works well
At high energy(>250MeV): more and more PWs are needed
algebraically and computationally difficult
Three Dimensional Calculation:Solving Faddeev equation directly in vector momentum space
Applicable to high energy case in order to explore few-body dynamics at short distance, especially the inclusion of 3BF and relativity.
Variables invariant under rotation:
freedom to choose coordinate system for numerical convenience
q system : z || q
q0 system : z || q0
Angular Relations in q system
Working in q system simplified the structure of moving pole
3D integral equation with 5 variables Solved iteratively by Pade summation
The position of the pole in three-body propagator depends on q, q” and x”
Moving singularity
Fixed “deuteron” pole
Moving Singularity In x” Integration
220 4
3
2qQ
220 4
3
2qQ
max
max
1
2 20 00
1 1
2 2 2 20 0 0 00 1 1
( , )
( )( )
1 ( , ) 1 ( , )
( ) ( ) ( ) ( )
-1
q
q
F q xdq dx
x x i q q i
F q x F q xdq dx dq dx
q q x x i x x q q i
0
0
1ln ln " ln "1
ln " ln "
x q
q q q qx q
qq q q q
q
max max max1 1 10
00 0 00 1 0 1 0 1
( '', ") 1 ( ", ) ( ", )" ( ", )" " "
q q q
F q x F p x F p xdq dx dq F p x dx dq dx
x x i x x i x x
00
0 11
1ln xi
x
x
The integral with logarithmic singularity carried out by Spline based method. More efficient !
qqxqFdqq
q
"ln,''" 0
Logarithmic Singularity q” Integration
F(q”,x0) approximated by cubic splines and the integration carried out analytically
The Self Consistency Check
Solution of Faddeev equation and coordinate independence
MeV0.3labE
mb
mb
1384.2561
7364.25610
q
q
mb
mb
5359.2561
6736.25610
q
q
Results from Pade summationResults from reinseration
The Unitarity non trivial, nonlinear relation
5.1%
mb28.11251.4878.63
mb87.106Im3
)2(4
MeV500
000
2
breltot
dd
lab
qUqq
m
E
0.5%
mb52.189475.7377.1820
mb98.1904Im3
)2(4
MeV10
000
2
breltot
dd
lab
qUqq
m
E
Can be improved with finer grid
Rescattering Effects
Elab =200 MeV
1. Forward: 103 mb
2. Backward: 100-101 mb
3. Adding term by term:
monotonically toward convergence
4. Rescattering terms are important
1st rescattering
Rescattering Effects
Elab=500MeV
1. Forward: 102-103 mb
2. Backward: 10-2 – 10-1mb
3. Adding term by term:
alternatively toward convergence
4. Rescattering terms are still important
1st rescattering
Summary & Outlook
• The property of rescattering effects has strong energy dependence.
• Study convergence of multiple scattering series through breakup observables at different energy regions and kinematic configurations.
• Check “typical” “high energy” approximations (Glauber)
Computation Model
Multileveled by MPI+OpenMP
Well suited on cluster machine with ideal scalability and high performance.