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

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

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

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

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

The t-matrix t N+c R is generated by the interactions V N +V c R. For |  > and |  ’> states with t=1 its matrix element is a linear combination of the t pp+c R and t np : For t=0:

The amplitudesprovide transition amplitude for elastic scattering: and for breakup: with Namely:

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

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

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

Results: -simple dynamical model: NN interaction taken as CD Bonn active only in states 1 S 0 and 3 S D 1 - exponential screening with n=1

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