Transcript "Study of resonances with Dubna-Mainz-Taipei(DMT)dynamical model"

Study of resonances with
Dubna-Mainz-Taipei (DMT) dynamical model
Shin Nan Yang
National Taiwan University
Dubna: Kamalov
Mainz: Drechsel, Tiator
Taipei: GY Chen, CT Hung,
CC Lee, SNY
INT Program on
Lattice QCD studies of excited resonances and multi-hadron systems
July 30 - August 31, 2012
1
Outline
• Motivation
• DMT πN model
• DMT model for electromagnetic production
of pion
• Results for the resonances
• Multiple poles feature of resonances in the
presence Riemann sheet
• Summary
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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SCIENTIFIC GOALS of This Program
What is the low-lying excited
spectrum of mesons and
baryons?
Experimentally, how are they extracted ?
Taipei-Argonne πN model:
meson-exchange  N model below 400 MeV
Bethe-Salpeter equation
T N  B N  B N G0T N ,
where
B N  sum of all irreducible two-particle Feynman amplitues
G0  relativistic free pion-nucleon propagator
can be rewritten as 
T N  B N  B N G 0T N ,
with
B N  B N  B N (G0  G 0 ) B N .
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Three-dimensional reduction
Choose a G 0 (k , P ) such that
1. T N  B N  B N G 0T N becomes
three-dimensional
2. G 0 can reproduce  N elastic cut
Cooper-Jennings reduction scheme
6
Choose B N to be given by
tree approximation of a chiral effective Lagrangian
 
F p2

n 4


 n 4  m2  p 2


n


 ,
2


n  10
7
8
C.T. Hung, S.N. Yang, and T.-S.H. Lee, Phys. Rev. C64, 034309 (2001)
DMT πN model:
extension of Taipei-Argonne model
to energies ≦ 2 GeV
 Inclusion of ηN channel and
effects of ππN channel in S11
 Introducing 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
tij  E   ij  E   ik  E  g k  E  tkj  E ,
k
ij  E   iBj  E   ijR  E 
vijB  0 if i or j =  ,
ijR  q, q; E  
ijR  E  
(if only one R)
hiR†(0) hjR0
 0
,
fi  i , q; E gi 0 g j0 f j  j , q; E
.


E  MR

i
E  M R 0   2R  E 
2
Effects of  N is taken into account with
the introduction of  2R ( E ) instead of including
channels ( N ,  N, )
decomposition of bkg and reson.
(in the case of only one resonance)
t N  E   tBN  E   tRN  E  ,
where
tBN  E   BN  E   BN g 0  E  t N  E  ,
tRN  E   RN  E   RN g0  E  t N  E  .
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tRN  E  
h R  E  h 0R
E  M R 0  E    R  E 
h R  E   h 0R  tBN  E  g0  E  h 0R
 0
 0
 0
 0
B
E

h
g
E
h

h
g
t
E
g
h






R
R 0
R
R 0 N
0 R
Note that both
the dressed vertex h R
and self energy  R
Resonance mass M R and width  R ,
depend on tBN
E  M R 0  Re  R  E   0,
M R  M R 0  Re  R  M R ,
 R  M R   2 Im  R  M R .
MR, ΓR
depend on
tBN
12
Introduction of higher resonances
If there are n resonances, then
N
ijR  q, q; E   ijR  q, q; E 
n
n 1
ij  E   iBj  E   ijR  E 
Coupledchannels
equations can
be solved
tij  E   ij  E   ik  E  g k  E  tkj  E ,
k
13
results of our fits to the SAID s.e. partial waves
bare resonances
nonresonant background
requires 4 resonances in S11
single-energy pw analysis from SAID
Dynamical model for  N →  N
To order e, the t-matrix for  N →  N is written as
t ( E )  v  v g 0 ( E )t N ( E ),
two ingredients
where
v  transition potential,
t N   N t -matrix,
g0 ( E ) 
1
E  H0
v , t N
Both on- & off-shell
(gauge invariance?)
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• Multipole decomposition
Off-shell
rescatterings
t( ) (qE , k ; E  i )  exp(i ( ) ) cos  ( )

 ( )
q '2 R(N) (qE , q '; E ) v( ) (q ', k ) 
 v (qE , k )  P  dq '

E

E
(
q
')


N
0
where
• (), R() :  N scattering phase shift and
reaction matrix in channel 
• k=| k|, qE : photon and pion on-shell
momentum
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If the transition potential v consists ot two terms,
v ( E )  vB ( E )  vR ( E ),
where
vB  background transition potential
vR  contribution of a bare resonance R
then one obtains
t ( E )  tB ( E )  tR ( E ),
with
tB ( E )  vB  vB g 0 ( E )t N ( E )
tR ( E )  vR  vR g 0 ( E )t N ( E )
both tB and tR satisfy
Fermi-Watson theorem,
respectively.
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DMT Model




B
v  





 PV only 

tB ,  exp(i ( ) ) cos  ( ) [vB , (W , Q 2 )  P  dq '
0









q '2 R(N) (qE , q '; E )vB , (q ', k )
E  E N (q ')
18
]
In DMT, we approximate the resonance contribution AR(W,Q2) by the
following Breit-Wigner form
R
A (W , Q )  A (Q )
R
2
2
f R (W ) R M R f R (W )
M R2  W 2  iM R  R
ei ,
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
R
A (Q 2 )
refers to a bare vertex
19
Results of DMT model near threshold,
B
and tBN )
(depends only on v
20
M. Weis et al., Eur. Phys. J. A 38 (2008) 27
21
Photon Beam Asymmetry near Threshold
Data: A. Schmidt et al., PRL 87 (2001) @ MAMI
DMT: S. Kamalov et al., PLB 522 (2001)
Sensitive w.r.t. multipole
M1 (1/ 2)
22
D. Hornidge (CB@MAMI)
private communication
D. Hornidge (CB@MAMI)
private communication
D. Hornidge (CB@MAMI)
private communication
MAID
DMT
Results on resonance properties
• Δ deformation
• Masses, widths, poles positions of the
resonances
△ deformation
(hyperfine qq interaction → D-state component in the △ )
REM=-2.4%
Dashed (dotted)
curves are results for tB
including (excluding)
the principal value
integral contribution.
tB,  exp(i ( ) ) cos  ( ) {vB, (W , Q 2 )

 P  dq '
0
q '2 R(N) (qE , q '; E )vB , (q ', k )
E  E N (q ')
}
A1/2
(10-3GeV-1/2)
A3/2
QN 
(fm2)
 N 
PDG
-135
-255
-0.072
3.512
LEGS
-135
-267
-0.108
3.642
MAINZ
-131
-251
-0.0846
3.46
DMT
-134
(-80)
-256
(-136)
-0.081
(0.009)
3.516
(1.922)
SL
-121
(-90)
-226
(-155)
-0.051
(0.001)
3.132
(2.188)
Comparison of our predictions for the helicity amplitudes, QN  and  N 
with experiments and Sato-Lee’s prediction. The numbers within the
parenthesis in red correspond to the bare values. Small bare value of QN 
indicate that bare Delta is almost spherical.
Resonance masses, widths,
and pole positions
bare and physical resonance masses, total widths,
N branching ratios and background phases
for N* resonances (I=1/2)
bare
phys
our analysis
PDG
bare
mass
physical
mass
our analysis
PDG
additional res.
additional res.
additional res.
resonance parameters
for  resonances (I=3/2)
bare
mass
physical
mass
our analysis
PDG
resonance parameters
for  resonances (I=3/2)
bare
mass
physical
mass
our analysis
PDG
additional res.
Results for resonance
pole positions
S11 pole positions
P11 pole positions
P33 pole positions
What do the LQCD results correspond to the
quantities extracted from experiments?
Comparison with EBAC’s results
Features of EBAC vs. DMT
1. 8 coupled-channels
(γN, πN, ηN, σN, ρN, πΔ, KΛ, KΣ)
2. different treatments in
a. derivation of background potential
b. prescription in maintaining
gauge invariance
N* poles from EBAC-DCC vs. DMT
L2I 2J
EBAC
(MeV)
DMT
PDG
(MeV)
S11(1535)
1540 － 191i
1449 － 34i
(1490 ~ 1530) － ( 45 ~ 125)i
(1650)
1642 － 41i
1642 － 49i
(1640 ~ 1670) － ( 75 ~ 90)i
S31(1620)
1563 － 95i
1598 － 68i
(1590 ~ 1610) － ( 57 ~ 60)i
P11(1440)
1356 － 76i
1364 － 105i
1366 － 90i
(1350 ~ 1380) － ( 80 ~ 110)i
(1710)
1820 － 248i
1721 － 93i
(1670 ~ 1770) － ( 40 ~ 190)i
P13(1720)
1765 － 151i
1683 － 120i
(1660 ~ 1690) － ( 57 ~ 138)i
P31(1750)
1771 － 88i
1729 － 35i
1748 － 262i
Not found
1896 － 65i
(1830 ~ 1880) － (100 ~ 250)i
P33(1232)
1211 － 50i
1218 － 45i
(1209 ~ 1211) － ( 49 ~ 51)i
D13(1520)
1521 － 58i
1516 － 62i
(1505 ~ 1515) － ( 52 ~ 60)i
D15(1675)
1654 － 77i
1657 － 66i
(1655 ~ 1665) － ( 62 ~ 75)i
D33(1700)
1604 － 106i
1609 － 67i
(1620 ~ 1680) － ( 80 ~ 120)i
F15(1680)
1674 － 53i
1663 － 58i
(1665 ~ 1680) － ( 55 ~ 68)i
F35(1905)
1738 － 110i
1771 － 95i
(1825 ~ 1835) － (132 ~ 150)i
F37(1950)
1858 － 100i
1860 － 100i
(1870 ~ 1890) － (110 ~ 130)i
(1910)
****
Two poles feature of P11(1440)
Appearance of π△ complex branch cut
pole A:  unphys.
sheet
pole B:  phys.
sheet
Group
Arndt et al. (85)
CMB(90)
EBAC(11)
1st pole
(1359,-100)
(1370,-114)
(1356,-76)
2nd pole
(1410,-80)
(1360,-120)
(1364,-105)
Multiple poles in the presence of
Riemann sheets
A. One branch cut of squared root nature only
1.
g ( z)
f ( z)  z 
, g ( z ) : analytical
z  z0
Two sheets: let z  rei , then
sheet I: 0    2 ,
II: 2    4 .
If we write z0  rei0 , 0   0  2 , then
z1  r0 ei0 , and z2  r0 ei (0  2 ) are both poles of f ( z ).
Residues R1  g ( z1  r0 ei0 )  R2  g ( z2  r0 ei (0  2 ) )
unless g ( z ) contains a factor like
z.
2. f ( z )  z 
a0
,
z  z0
Again, there are two Riemann sheets and, there
is only one pole appearing in one of the two Riemann sheets.
Pole z p  (r0 ei0 ) 2  z02 will appear in sheet I or II depending on
0   0   (sheet I) or    0  2 (sheet II)
Analytic functions with two cuts
of squared root nature
( z  za )( z  zb )
• product form
a cut can be drawn by connecting za and zb two
Riemann sheets are needed.
z  za  z  zb
 additive form
four Riemann sheets are required to make
function of this kind single-valued.
This is the case with pion-nucleon scattering
amplitude
Two classes of functions with two
additive branch points
(a)
g ( z)
f1 ( z )  z  za  z  zb 
z  z0
two cuts starting from za and zb
4 Riemann sheets
pole z0 appears in all 4 sheets, with equal residues if g(z) is
analytical
(b)
h( z )
f 2 ( z )  z  za  z  zb 
.
z  z0
3 branch points za, zb, and z0, but only 4 Riemann sheets are needed
to make f2(z) single-valued
2
pole z 0 appears only in 2 sheets with different residues even if
h(z) is analytical
SNY, Chinese J. Phys. 49 (2011) 1158.
tRN  E  
h R  E  h 0R
E  M R 0  E    R  E 
N. Suzuki, T. Sato, T.-S.H. Lee, Phys. Rev. C82 (2010) 045206
Summary
 The DMT coupled-channel dynamical model gives
excellent description of the pion scattering and pion
photoproduction data from threshold up to W≦
2GeV
• Excellent agreement with 0 threshold production data. Twoloop contributions small. For electroproduction, ChPT might
need to go to O(p4).
• DMT predicts  N  = 3.514 N, QN  =-0.081 fm2, and
REM=-2.4%, all in close agreement with the experiments.
 dressed  is oblate
• Bare  is almost spherical. The oblate deformation of
the dressed  arises almost exclusively from pion cloud
 The resonance masses, width, and pole positions
extracted with DMT agree, in general, with PDG
numbers.
The most serious discrepancy appears in S11 channel
●
●
four resonances are found, instead of three listed on PDG
Pole position for S11(1535): (1449,-34i ) (DMT), (1510 ± 20, -85±40i) (PDG),
(154,-191i ) (EBAC),
 Two poles are found corresponding to P11(1440) in
several analyses, including SAID(85), CMB(90),
GWU(06), Juelich(09), and EBAC(11), where a
complex branch cut associated with the opening ofπΔ
channel is included in the analysis.
We demonstrate that multiple poles structure is a
common feature of all resonances when additional
Riemann sheet appears.
END
R
A (W , Q )  A (Q )
R
2
2
f R (W ) R M R f R (W )
M R2  W 2  iM R  R
ei ,
Efforts are being undertaken to use the
dressed propagators and vertices obtained in
DMT πN model to achieve consistency in the
analyses ofπN and π-production.
57