A novel approach to include pp Coulomb force into the 3N Faddeev calculations

1. A novel approach to include pp Coulomb force into the 3N Faddeev calculations H. Witała, R. Skibiński, J. Golak Jagiellonian University W. Gloeckle Ruhr Universitaet Bochum

2. Possible method to include pp Coulomb force in 3N calculations: - screening- in nature Coulomb force is always screened - screening allows to use standard methods developed for finite range forces - to get final predictions follow numerically the limit to the unscreened situation • Big problem when working with partial waves: • looking for the screening limit requires to increase • the screening radius. The number of partial waves required • to reproduce the screened pp Coulomb t-matrix increases • drastically with the screening radius. This leads to the explosion • of the number of partial waves in the 3N system. • solution: treatment of the screened pure Coulomb part without • relying on partial wave decomposition – 3-dimensional LS Keep the pp Coulomb force in the proper coordinate

5. Faddeev equation: - initial state: deuteron and momentum state of the proton - free 3N propagator Standard momentum space basis: - nuclear VN and the pp screened Coulomb VcR interaction is acting (in t=1 states) - only VcR is acting in the pp subsystem

6. and Projecting Faddeev equation on the states : The term A direct calculation of its isospin part shows that it vanishes.

7. Inserting into Eq. for one gets: - it is coupled set of integral equations in the space of the states |a> only - it incorporates the contributions of the pp Coulomb force from all waves up to infinity The leading term and the kernel term must be calculated with the 3-dimensional pp screened Coulomb t-matrix Details of formulation: see nucl-th 0903.1522, 0906.3226

8. The t-matrix tN+cR is generated by the interactions VN+VcR. For |a> and |a’> states with t=1 its matrix element is a linear combination of the tpp+cR and tnp: For t=0:

9. The amplitudes provide transition amplitude for elastic scattering: and for breakup: with Namely:

10. The screening limit is governed by The screening limit of For pd elastic scattering amplitude one needs for off-shell p,q values: off-shell t-matrices - do not acquire a phase factor The breakup amplitude requires for on-shell p,q values: half-shell t-matrices • acquire an infinitely oscilating • phase factor

11. For exponential screening: phase - Euler number - the Sommerfeld parameter

19. Procedure to follow: 1) Solve Faddeev equations for off-shell elastic scattering amplitude 2) Using them determine on-shell breakup amplitude Notice: in linear combinations use renormalized pp+c half-shell t-matrices

20. Results: • simple dynamical model: NN interaction taken as • CD Bonn active only in states 1S0 and 3S1-3D1 - exponential screening with n=1

29. Summary and conclusions: • novel approach to include the pp Coulomb force into the • momentum space 3N Faddeev calculations • it is based on a standard formulation for finite range forces • it relies on a screening of the long-range Coulomb interaction • we apply directly the 3-dimensional pp screened Coulomb • t-matrix • we treat the pp Coulomb force in its proper coordinate • for a simple dynamical model feasibility of the approach was • demontrated • physical pd elastic scattering amplitude has a well defined • limit and does not require renormalization • to get breakup amplitude on-shell 3N amplitudes are required • and renormalized pp half-shell screened t-matrices must be • used