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

qq

220 4

3

2qQ

qq

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.