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DMT model for πN scattering DMT model for πN scattering and pion e.m. productionand pion e.m. production

Shin Nan YangShin Nan YangNational Taiwan UniversityNational Taiwan University

EBAC discussion meeting, Jlab, May 24-26, 2010.

Dubna: KamalovMainz: Drechsel, TiatorTaipei: Guan Yeu Chen, SNY

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Motivation To construct a meson-exchange model forπN scattering and e.m. production of pion so that a consistent extraction of the resonance properties like, mass, width, and form factors, from both reactions can be achieved. Comparison with LQCD results requires reliable extraction. consistent extractions → minimize model dependence? The resonances we study are always of the type which results from dressing of the quark core by meson cloud. → understand the underlying structure and dynamics

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Taipei-Argonne πN model: meson-exchange N model below 400 MeV

0

N

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

B

G

B

N

N

0

0

0

,

with

(

) .

N

N

N N

T

B BB G BG

G

4

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

5

Choose to be given byNB

42

24 2 2

10

,

n

nF p

n m p

n

6C.T. Hung, S.N. Yang, and T.-S.H. Lee, Phys. Rev. C64, 034309 (2001)

7

DMT πN model:extension of Taipei-Argonne model

to energies ≦ 2 GeV

Inclusion of ηN channel in S11

Introduce higher resonances as indicated by the data

G.Y. Chen et al., Phys. Rev. C 76 (2007) 035206.

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Inclusion of ηN channel in S11

0 0 . .NR NRL ig R N ig RN h c

,ij ij ik k kjk

t E E E g E t E

Bij i

Rijj EE E

0†(0)

0,iR jRR

ij

R

h hE

E M

0 0

0 2

, ; , ;, ; .

2

i i i j j jRij

R R

f q E g g f q Eq q E

iE M E

9

0 0

0 2

2 : phenomenological effect

, ; , ;,

s of the coupling with

N chann

;

2

el

,i i i j j

j

R R

i

R

jR

E

E

f q E g g f q Eq q E

iE M

2 4 22 2

2 02 02 2

2

2

2

0

0

2

02

,

( ), the momentum of the 2 system

: pion or

,

( )

bital mo

,

mentum

l

RR

RR

R

lq

q

X q

X q

q E

q E Mq

l

Eq

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Introduction of higher resonances

If there are n resonances, then

How does one extract masses, widths et al. of the resonances?

1

, ; , ;n

NRR

ij ijn

q q E q q E

Bij i

Rijj EE E

,ij ij ik k kjk

t E E E g E t E

Coupled-Coupled-channels channels

equations can equations can be solvedbe solved

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How does one extract masses,

widths et al. of the resonances? Two schemes to separate the total

t- matrix into background and resonance contribution

1. Afnan et al. and Sato-Lee2. Dubna-Mainz-Taipei (DMT)

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Sato-Lee’s separation method

0 0

0 00

0

0

0

1,

where

,

.

,

,

i

i i i

i

i

i

i

i

i

i

B B B BN N N N

BR R R N

BR R

RN R R

B RN

R R

R

N N

N

t E h E h E

t E E g

h E h h g E t E

h E h t

t E

E g

E t

t E t E

E

E

E M

E h

Unitary with phase δB

Self-energySelf-energy

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Self-energy ΣR(E)

00

0 0 0 00 0 0

R R

R R

R

BR RN

h g E h E

h g E h h g gt E

E

h

0

0 Re ,

2 I

e

m

R 0,

.

R R R R

R R R R

R R

M M M

M

E M E

M

0

1RN R

R

R

R

t E h E h EE M E

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Extension of SL’s method to n resonances

1

i

NRB

N N Ni

t E t E t E

0

0

00

0 00

,

.

,

ere

1

wh

i i i

i i

i

i

i

i

i i

BR R R N

BR R N R

R

RN R R

R

t E

h E h h g E t E

h E

h

h t E g

E h EE

h

E

E

M

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DMT’s decomposition of bkg and reson.

0

0

where

,

,

.

B B BN N N N

R R RN N

BN N

N

RN

N

t E t E t

t E E g E t E

t E E E

E

g E t

0

0

0

, .....................

....... S

.. DMT

1LR

RRN

R

R R

R

RR

N

Rh Et E

E M E

h EE M

E

h

tE

E h E

Note that both tNote that both tBB and tand tR R have the have the same phase of same phase of

N

With only one With only one resoance,resoance,

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Extension of DMT’s method to n resonances

0

1

1

0

0

,

,

where

( )

,

( )

)

(

i

i

i

ii

N N

BN

nRB

N N Ni

B B BN N

N

N

R

N

N

i

R

RN

RN N

N

N

N

v v g E

t

t

E

v v v

t v v g E

v

E t E

t E

v

t

gt E

E

t E

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It can be shown,

0 0

0

0 2

0

0

00

1

0

where

,

,

,

.

,

ii

i i

i i

i

i

ii

i

i

i

i

i

i

j

BN

B

R

i i

i

RRN

R R

BN

NRB

N

R

R R

Nj i

R

BR

N

NR R

h h g E

h

E

E

t

ht E

E M E

g E t E

E

h E

h E

h

E

E

t

E

t

E g E h

E

E

contains contribution of Ri

excitation

iii RRR iEhEh exp

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Remark: the background in our separation,

1

0

0

where

,

,

( )

( ) i i

i

i

RN

RN

B BN N

R RN

N

Ni

N

N

B

N

N

BN v v g E

v v g E

t E

t

t E

t

E

t E

t

t E

" "BNt

already does contain some already does contain some resonance contributions and in the resonance contributions and in the calculation of the residues, the full t-calculation of the residues, the full t-matrix has to be employedmatrix has to be employed..

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To order e, the t-matrix for N → N is written as

Dynamical model for N → N

v, tN

0

where

transition potential,

-matrix,

1

( )( ) ( ) ,

( )

k

k

k

N

kk

k

k

v v

v

k

t

N t

E

t E E

H

g E

g E

t

two two ingredientsingredients

Both on- & off-shell

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• Multipole decomposition of gives the physical amplitude in channel =( , l , j), (with N intermediate states neglected)

where (), R() : N scattering phase shift and reaction matrix

in channel • k=| k|, qE : photon and pion on-shell momentum

( ) ( ) ( )

( ( )2( )

0

)

( , ; ) exp( )cos

' ( , '; ) ( ', )( , ) '

( ')N

E

EE

N

t q k E i i

q q q E q kq k P dq

v

E

R

E qv

t

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( )

background transition potential

contribution of a bare resonance

( ) )

(( )

(

If the transition potential consists ot two terms,

,

where

then one obtains

RB

R

B

B

v E

R

v E

v

t E

v E

v

t

v

)

( ( )

( )

) )

( ) (( )

(

)

,

with

B B

k k kk

R R

k

B

R

kk

R

k

v v g E

t E

t E

v v g E t E

E

E

E

t

t

both tB and tR satisfyFermi-Watson theorem,respectively.

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2 ( ) ,, 2

, ( ) ( ) 0

, 2

' ( , '; ) ( ', )( , ) ' ,

exp( )cos ( ')

( , ),

BN EB

BN

B

q R q q E v q kv W Q P dq DM

t i E E q

v W Q MAID

Bv

DMT Model

PV only

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Dubna-Mainz-Taipei (DMT)

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SL’s decomposition of bkg and reson.

0

0

,

where

,

( ) .

B B BN

R B RN

B

B

R

R

E g E t E

t

t E

t E

t E

t E

E

t

E

E

g t

dressed

DMT,

bare

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In DMT, we approximate the resonance contribution AR(W,Q2) by the

following Breit-Wigner form

with

f R = Breit-Wigner factor describing the decay of the resonance R

R (W) = total width

MR = physical mass

(W) = to adjust the phase of the total multipole to be equal to the corresponding N phase shift ().

Note that

2 22 2

( ) ( )( , ) ,( ) R R R RR i

R

R

R R

f W M f WA W Q e

M W iA

MQ

2 bare, DMT

dressed, MA D(

I)

RA Q

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2 22 2

( ) ( )( , ) ,( ) R R R RR i

R

R

R R

f W M f WA W Q e

M W iA

MQ

Efforts are being undertaken to use the Efforts are being undertaken to use the dressed propagators and vertices obtained dressed propagators and vertices obtained in DMT πN model to achieve consistency in in DMT πN model to achieve consistency in the analyses ofπN and π-production.the analyses ofπN and π-production.

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Results of DMT model near threshold,

28M. Weis et al., Eur. Phys. J. A 38 (2008) 27

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Photon Beam Asymmetry near ThresholdPhoton Beam Asymmetry near Threshold

Data: A. Schmidt et al., PRL 87 (2001) @ MAMIDMT: S. Kamalov et al., PLB 522 (2001)

D. Hornidge (CB@MAMI)private communication

D. Hornidge (CB@MAMI)private communication

D. Hornidge (CB@MAMI)private communication