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

H. Witała, R. Skibiński, J. GolakJagiellonian UniversityW. GloeckleRuhr 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

2 2 ( )p dpq dq pq pq pq pq

Faddeev equation: 0T tP tPG T

12 23 13 23P P P P P - free 3N propagator

- initial state: deuteron and momentum state of the proton

0G

Standard momentum space basis:

1 1( ) ( ) ( ) ( )

2 2pq pq ls j I jI J t T

2 2p dpq dq pq pq

max:pq j j

max:pq j j

- nuclear VN and the pp screened Coulomb VcR interaction is acting (in t=1 states)

- only VcR is acting in the pp subsystem

2 20

'

2 20

'

' ' ' ' ' ' ' ' ' '

' ' ' ' ' ' ' ' ' '

RN c

RN c

RN c

pq T pq t P

pq t PG p dp q dq p q p q T

pq t PG p dp q dq p q p q T

Projecting Faddeev equation on the states pq and pq :

2 20

'

2 20

'

' ' ' ' ' ' ' ' ' '

' ' ' ' ' ' ' ' ' '

Rc

Rc

Rc

pq T pq t P

pq t PG p dp q dq p q p q T

pq t PG p dp q dq p q p q T

The term 0~ ' ' ' ' ' 'R Rc cpq t PG p q p q t

A direct calculation of its isospin part shows that it vanishes.

Inserting pq T into Eq. for pq T one gets:

0

2 20

'

2 20

'

2 20 0

'

2 20

' ' ' ' ' ' ' ' ' '

' ' ' ' ' ' ' ' ' '

' ' ' ' ' ' ' ' ' '

' ' ' ' ' ' ' '

R R RN c N c c

R RN c c

RN c

R RN c c

RN c

pq T pq t P pq t PG t P

pq t PG p dp q dq p q p q t P

pq t PG p dp q dq p q p q T

pq t PG t PG p dp q dq p q p q T

pq t PG p dp q dq p q p

0'

2 2

"

' '

" " " " " " " " " "

Rcq t PG

p dp q dq p q p q T

- 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 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

The t-matrix tN+cR is generated by the interactions VN+Vc

R.

For |> and |’> states with t=1 its matrix element

is a linear combination of the tpp+cR and tnp:

23( ) ' '

4RN cp t E q p

m

1 1 1 21 ' 1 '

2 2 3 3

3 3 2 11 ' 1 '

2 2 3 3

1 3 21 ' 1 ' ( )

2 2 3

3 1 21 ' 1 ' ( )

2 2 3

R RN c np pp c

R RN c np pp c

R RN c np pp c

R RN c np pp c

t T t t T t t

t T t t T t t

t T t t T t t

t T t t T t t

For t=0:1 1

0 ' 0 '2 2

RN c npt T t t T t

The amplitudes pq T provide transition amplitude for elastic scattering:

10' 'U PG PT

and for breakup:

0 0 0 (1 )U P T with 0 1 1 2 2 3 3pqm m m

Namely:

2 2

'

2 2

'

2 20

'

2 2

"

2 20

'

' ' ' ' ' ' ' ' ' '

' ' ' ' ' ' ' ' ' '

' ' ' ' ' ' ' ' ' '

" " " " " " " " " "

' ' ' ' ' ' ' ' ' '

Rc

Rc

R Rc c

pq T pq p dp q dq p q p q T

pq p dp q dq p q p q t P

pq p dp q dq p q p q t PG

p dp q dq p q p q T

pq t P pq t PG p dp q dq p q p q T

The screening limit

The screening limit of pq T is governed by 23

( ) ' '4

RN cp t E q p

m

For pd elastic scattering amplitude one needs for off-shell p,q values:pq T

223

4

pq E

m m

The breakup amplitude requires for on-shell p,q values:pq T

223

4

pq E

m m

off-shell t-matrices

half-shell t-matrices

- do not acquire a phase factor

- acquire an infinitely oscilating

phase factor ( )Ri pe

For exponential screening:

( )( )

nrR RcV r e

r

phase ( ) [ln(2 ) ]R p pRn

0.5772 - Euler number

2

m

p

- the Sommerfeld parameter

Procedure to follow:

1) Solve Faddeev equations for off-shell elastic scattering amplitude

2) Using them determine on-shell

223

:4

ppq T q E

m m

22 20

0 max

3 3:

4 4

pp q T q E q

m m m

breakup amplitude

0 0 0 0

2 20 0

'

2 20 0

'

2 20 0 0

'

2 20 0

' ' ' ' ' ' ' ' ' '

' ' ' ' ' ' ' ' ' '

' ' ' ' ' ' ' ' ' '

' ' ' '

R R RN c N c c

R RN c c

RN c

R RN c c

RN c

p q T p q t P p q t PG t P

p q t PG p dp q dq p q p q t P

p q t PG p dp q dq p q p q T

p q t PG t PG p dp q dq p q p q T

p q t PG p dp q dq p

0'

2 2

"

' ' ' ' ' '

" " " " " " " " " "

Rcq p q t PG

p dp q dq p q p q T

Notice: in linear combinations RN ct use renormalized pp+c half-shell t-matrices

Results:

-simple dynamical model: NN interaction taken as CD Bonn active only in states 1S0 and 3S1-3D1

- 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