1. OSC 2nd Annual Graduate Student HPC Workshop 1 Three-Body Bound State Calculations with Two and Three-Body Forces Hang Liu Charlotte Elster Department of Physics & Astronomy Ohio University

2. OSC 2nd Annual Graduate Student HPC Workshop 2 Research Objectives Develop reliable computational procedure to calculate the three-nucleon (3N) bound state with – two-body and – three-body forces Novel aspect: calculations are carried out without traditionally employed angular momentum decomposition. Goal: Explore the dynamics of 3N forces in 3N System. At present: 3N bound state with 2NF and 3NF

3. OSC 2nd Annual Graduate Student HPC Workshop 3 Faddeev Equation for 3N bound state Faddeev component Total wave function Three-body free propagatorPermutation operator Two-body transition operatorThree-body force

4. OSC 2nd Annual Graduate Student HPC Workshop 4 Jacobi Coordinates 2 3 1 3 1 2 1 2 3 Different pairs can be represented by the permutation operators acting on one specific pair

5. OSC 2nd Annual Graduate Student HPC Workshop 5 Explicit Representation in Jacobi Coordinates Permutation operators Faddeev component and total wave function Three-body free propagator and two-body transition operator Three-body nucleon force

6. OSC 2nd Annual Graduate Student HPC Workshop 6 Permutation Operators BACK

7. OSC 2nd Annual Graduate Student HPC Workshop 7 Faddeev component and total wave function q p back

8. OSC 2nd Annual Graduate Student HPC Workshop 8 Three-body propagator: Two-body transition matrix: Lippmann-Schwinger Equation Off-shell energy of t matrix back

9. OSC 2nd Annual Graduate Student HPC Workshop 9 312 12 3 231 ++ back

10. OSC 2nd Annual Graduate Student HPC Workshop 10 Solving Faddeev Equation Standard eigenvalue problem Adjusting E to get the solution ofwith

11. OSC 2nd Annual Graduate Student HPC Workshop 11 Numerical Algorithm The eigenvalue problem is solved by Iterated Orthnormal Vector (IOV) method The basic operation is the iteration kernel acting on iterated state. Integration: Gaussian quadrature. Cubic spline multidimensional interpolation: permutation operators shift the arguments of iteration kernel and iterated state

12. OSC 2nd Annual Graduate Student HPC Workshop 12 The 3N Force Iteration

13. OSC 2nd Annual Graduate Student HPC Workshop 13 X-Z plane q p Z direction q p p q p q

14. OSC 2nd Annual Graduate Student HPC Workshop 14 3 dimensional interpolation to shift the total wave function from triton grid to integration grid 3 dimensional interpolation to build up the total wave function on triton grid

15. OSC 2nd Annual Graduate Student HPC Workshop 15 8.176258.188630.4832638.7654231.05753 44 8.176138.187560.4823438.7651731.05740 48 8.176098.182330.4751438.7651131.05737 52 8.176078.182350.4751738.7650831.05736 56 8.176088.182000.4744938.7650931.05738 64 3NF p, q dependence test

16. OSC 2nd Annual Graduate Student HPC Workshop 16 Bound state wave function of the 3N system  (p,q,x) at x=cos(  )=1 with the unit of

17. OSC 2nd Annual Graduate Student HPC Workshop 17 Wave function contour slice at 2.4 with the unit of Wave function contour slice at 0.001 with the unit of