Final state interaction effects in production near threshold

V.F. Dmitriev and A.I. Milstein

Budker Institute of Nuclear Physics

NN

Ratio of electric and magnetic formfactors In the threshold region.

2

1 2 2

1 2

,4

.

Ep

M

qG F F

m

G F F

21 2 p

FF q

m

qp (Ν)

-p (N)

Γ

_210 MeVpT

R.Bijker, F.Jachello, Phys.Rev. C69 (2004)

General features of the processq

p

-p1/ 2 0.1 fmpm

P P formation scale _

At distances 0.1 < r < 1.4 fm there is a strong interaction between slowly moving P and Pbar. This scale corresponds to the energies starting from few ten’s MeV. Strong final state interaction in this energy scale is very likely responsible for the energy dependence of the formfactors near threshold.

QUANTUM NUMBERS of P Pbar state

PHOTON J=1, P=-1, C=-1

P Pbar

1P ( 1) , C ( 1) , L L S J = L +S

3 31 1S DL - even S = 1

Isospin not fixed I=0,1

Elementary amplitude of N Nbar production

2 1

† †1 2 2 1

1 1( ) ( ) ( )( ) 3 ( )

3 3N NN N

EF FE ME F F E M F F

M E M

e e e e

2T p p p p

M – proton mass, E – nucleon energy,

1 2 and F F -Dirac formfactors responsible for short distance behaviour

1 2 and F F are smooth functions, nearly constants in threshold region

e - polarization of virtualphoton

Accounting for N Nbar interaction

† 3 †( )( ) ' ( ) ( )T d p pT p p' T p'

†( ) ( )

p r - N Nbar scattering state with asymptotic behaviour

†( ) * *( ) | .ipr

ir

ee

r

prp r

* † is the spin polarization vector for S=1 state of N Nbar pair

†( ) †( )ˆ( ) ( ) ( )H E M p pr r

2

ˆ ( ) ( )H U iWM

p

r r

Paris N Nbar potential, Phys.Rev. C 50 (1994) 2710

Real part - meson exchange for r > 0.84 - 1 fm and fit

2 3 20 1 2 3 0 1 2 or a a r a r a r b b r b r for r < 0.84 - 1 fm.

In both I=0,1 channels

p n

p n

Coupled channels

Imaginary part 0 (2 )( ) ~

K MrW r

r

Production amplitude becomes

*0 2 2

ˆ4 2 ,1 | 211 ( )mT G G m Y p

S-wave D-wave

Electromagnetic formfactors

211 01 21 M1 01

200 00 20 M0 00

1 0 1 0

2 , G2

2 , G2

,

E

E

Ep E E Mp M M

GG G G G

GG G G G

G G G G G G

1 0 1 0, En E E Mn M MG G G G G G

Results for | / |Ep MpG G

2

1

1Re

2

V

V

F

F

2

1

1Im

2

V

V

F

F

1

1

Re 1.12S

V

F

F

1

1

Im 1.20S

V

F

F

2

1

Re 0.96S

V

F

F

2

1

Im 1.15S

V

F

F

Summary

• We developed an approach to account for final state interaction (FSI) in P Pbar production near threshold.

• The effects of FSI are considerable; they lead to strong energy dependence near threshold.

• The approach is not a parameter free one. More data are necessary to check its validity.