1 DMT model for πN DMT model for πN scattering and pion e.m. scattering and pion e.m. production production Shin Nan Yang Shin Nan Yang National Taiwan University National Taiwan University EBAC discussion meeting, Jlab, May 24-26, 2010. Dubna: Kamalov Mainz: Drechsel, Tiator Taipei: Guan Yeu Chen, SNY
32
Embed
1 DMT model for πN scattering and pion e.m. production Shin Nan Yang National Taiwan University DMT model for πN scattering and pion e.m. production Shin.
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
1
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
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
3
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
Introduce higher resonances as indicated by the data
G.Y. Chen et al., Phys. Rev. C 76 (2007) 035206.
8
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
10
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
11
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)
12
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
13
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
14
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
15
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,
16
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
17
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
18
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..
19
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
20
• 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
21
( )
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.
22
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
23
Dubna-Mainz-Taipei (DMT)
24
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
25
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
26
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.
27
Results of DMT model near threshold,
28M. Weis et al., Eur. Phys. J. A 38 (2008) 27
29
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)