A meson-exchange model for N scattering

up to energies s 2

GeV

Shin Nan YangNational Taiwan University

Collaborators: S. S. Kamalov (Dubna) Guan Yeu Chen (Taipei)

18th International Conference on “Few-body Problems in Physics”

August 20 – 26, 2006, Santos, Brazil 1

OutlineMotivation

Previous works

Meson-exchange N model below 400 MeV

Dubna-Mainz-Taipei (DMT) dynamical model for pion e.m. production

Extension to higher energies

Conclusion

2

Motivation

low energies ─ ChPT high energies, high momentum transfer─ pQCD medium energies ․LQCD ․Phenomenology : hadron models, reaction theory

QCD Hadronic phenomena

3

Aim: To construct a coupled-channel dynamical model to study N scattering and pion electromagnetic production

Aim: To construct a coupled-channel dynamical model to study N scattering and pion electromagnetic production

low energies:

e.m. threshold pion production, low energy theorems and compare with ChPT

medium energies:

N-transition form factors, resonance parameters

high energy and high momentum transfer:

transition to pQCD region?4

N

0

0

Bethe-Salpeter equation

,

where

sum of all irreducible two-particle Feynman amplitues

relativistic free pion-nuc

can be r

leon propagator

ewritten as

N N

N

N N

N

T B B G T

T

B

G

B B

N

N

0

00

,

with

( .)

N

NN N

T

B B B

G

B G G

Meson-exchange N model below 400 MeV

5

N 0

Find good

and B G

Three-dimensional reduction

0

0

0

Choose a such that

1. becomes

( , )

three-dimensio

2. can reproduce N elas

na

tic

l

cut

N NN NT B B T

G k P

G

G

Cooper-Jennings reduction scheme Cooper-Jennings reduction scheme

6

Choose to be given byNB

7

C.T. Hung, S.N. Yang, and T.-S.H. Lee, Phys. Rev. C64, 034309 (2001) 8

Both on- & off-shell

Dynamical model for electromagnetic pion production

9

( )

background transition potential

cont

(

ribution of a bare resonance

)

(

( )

(

)

the transition potential consists ot two terms,

,

where

then one obtai

f

ns

IB

B

R

B

Rv E

v

v E

R

t E

v E

v

t

v

0

0

( )

)

( ) ( )

( ) (

(

( )

)

)

,

with

B

N

B

R

N

E

t E v v g E t E

v v E t E

t E

t E g

both tB and tR satisfyFermi-Watson theorem,respectively.

10

Bv

DMT Model

PV only

11

12

13

14

Frolov et al., PRL, 1999

15

0

2 24.0

Q e

e e p

G V

p

16

Extension to higher energies

coupled , , 2 channels

Include resonances R’s with couplings to

, , 2 channels

coupled , , 2 channels

Include resonances R’s with couplings to

, , 2 channels

( ) ( ) ( ),

( , , , )

ij ij ik k kjk

t E v v g E t E

i j k

( ) ( ) ( )B Rij ij ijv E v E v E

17

Non-resonant background :

as given before,

0 .

Bij

B

B B Bv

v

v

v

v

(0) (0) h.c.I NR NRL ig R N ig R N ������������� �

18

1

( ) ( )(0) (0)

(0)

0, 2

2

( )

2

,

( )

2

0

(0)n

( , ; ),

( , ; ) ( , : )

( )1

2

,

Resonan :ce contribution

for N overlapping resonances.

bare mass of resonance R ,

n

n

n

NRR

ij ij

n

n n

i ji jR

ij

R

n

n

n

n

n

Rij

R

v v q q E

f q E g g f qv

E E

E

M i

q

q

v

M

2 4 22 2

0,

2 2

2

effects of N d e cay channel

l l

nX q

X q

19

2R

3R

4R

- - - - TB = VB + VBg0TB

20

S11

---------- 1R

2R

3R

No evidence for the 4th S11 resonance in

N -> N

No evidence for the 4th S11 resonance in

N -> N

21

Consistent with data from ELSAN N

Extraction of resonance parameters

1

1

( ) ,

wh

( )

( )

( )

ere

( ) ( ) ( ) ( ),

( ) ( ) ( ) ( ).

( )

(

/ )

)

2 (

n n

n

n

n

n n

R

R

N

n

B Bk k k

k

R Rk k k

R

R

B

R

B

k

t E

t E

t E

t E

v E v E g E t E

v E v E g E t E

E i

t

M

t E

E

22

23

full

BtRt

Need 4 S11

Resonances

2=64

2=3.5

24

(3,3)

25

S13

(3,3), red PDG, blue DMT

(3,3)

26

P11

(2,3)

27

P31

(2,2)

28

D15

(2,2)

29

F15

1st Resonance 2nd Resonance 3rd Resonance

N* Wp Γp Wp Γp Wp Γp

S111499

(1505, 1501)

67(170, 124)

1642(1660, 1673)

97(160, 82)

2065(2150±70)

223(350±100)

S311598

(1600, 1585)

136(115, 104)

1775(1870±40)

36(1850±50)

2012(2140±80)

148(200±80)

P111366

(1365, 1346)

179(210, 176)

1721(1720, 1770)

185(230, 378)

1869(2120±40,

1810)

238(240±80, 622)

P131683

(1700, 1717)

239(250, 388)

1846(not listed)

180(not listed)

P311729(1714)

70(68)

1896(1855, 1810)

130(350, 494)

2065 161

P331218

(1210, 1211)

90(100, 100)

1509(1600, 1675)

236(300, 386)

2149(1900,

1900±80)

400(300, 300±100)

Pole Positions and Residues in [ MeV ] (preliminary)

Red ： PDG04 　　　　　　　　 Blue ： Arndt95

Green ： Cutkovsky80 　　　　　 Orange ： Vrana00 30

1st Resonance 2nd Resonance 3rd Resonance

N* Wp Γp Wp Γp Wp Γp

D13 (3,3) 1516 (1510, 1515)

123(115, 110)

not seen(1680, not seen)

not seen(100, not seen)

1834(1880±100)

210(160±80)

D33 (2,2) 1609(1660, 1655)

133(200, 242)

2070(1900±100)

267(200±60)

D15 (2,2) 1657(1660, 1663)

132(140, 152)

2188(2100±60)

238(360±80)

D35 (2,1) 1992(1890, 1913)

270(250, 246)

not seen

(2400±60)

not seen

(400±150)

F15 (2,2) 1663(1670, 1670)

115(120, 120)

1931(not listed)

62 (not listed)

F35 (2,2) 1771(1830, 1832)

190(280, 254)

2218(2150±100,

1697)

219(350±100, 112)

F37 (2,1) 1860(1885, 1880)

201(240, 236)

2207(2350±100)

439(260±100)

Pole Positions and Residues in [ MeV ] (preliminary)

Red ： PDG04 　　　　　　　　 Blue ： Arndt95

Green ： Cutkovsky80 　　　　　 Orange ： Vrana00

31

Summary

The DMT coupled-channel dynamical model gives excellent description of the pion scattering and pion e.m. production data from threshold to first resonance region

The model has been extended up to c.m. energy 2 GeV in the

sector and resonance parameters extracted

and N N

32

Our analysis at W > 1750 GeV yields considerable strength, which can be explained by a third and a fourth S11 resonance with masses 1846(47) and 2113(70) MeV

33

The End

(3,4), red PDG, blue DMT

25

S11

(2,2)

35

(3,3)

37

(3,3)

38

(2,2)

40

(2,1)

41

(2,2)

43

(2,1)

44