1

Study of resonances withStudy of resonances with

Dubna-Mainz-Taipei (DMT) dynamical Dubna-Mainz-Taipei (DMT) dynamical modelmodel

Shin Nan YangShin Nan Yang

National Taiwan UniversityNational Taiwan University

INT Program on Lattice QCD studies of excited resonances and multi-hadron systems

July 30 - August 31, 2012

Dubna: KamalovMainz: Drechsel, TiatorTaipei: GY Chen, CT Hung, CC Lee, SNY

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

3

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

What is the low-lying excited spectrum of mesons and baryons?

Experimentally, how are they extracted ?

SCIENTIFIC GOALS of This Program

5

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

6

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

7

Choose to be given byNB

42

24 2 2

10

,

n

nF p

n m p

n

tree approximation of a chiral effective Lagrangian

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

9

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.

Inclusion of ηN channel in S11

,ij ij ik k kjk

t E E E g E t E B

ij iRijj EE E

0†(0)

0, ,0 if or = iR jRR

ij

R

Bij

h hE

E Mv i j

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

(if only one R)

Effects of is taken into account with the introduction of instead of including channels

N2 ( )R E

( , N, )N

11

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

(in the case of only one (in the case of only one resonance) resonance)

12

0

0

Resonance mass and width

Re ,

2 Im .

0

,

Re ,

R R

R R R R

R R R R

R R

E M E

M M M

M M

M

MR, ΓR

depend on B

Nt

0

0

RRRN

RR

h Et E

E M

h

EE

0 00N RR

BRh g Eh tE E h

0 0 0 00 0 0

BNRR R R Rh g E h h t Eg g hE

Note that both

the dressed vertex

and self energy

depend on

R

R

BN

h

t

13

Introduction of higher resonances

If there are n resonances, then

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

results of our fits to the SAID s.e. partial results of our fits to the SAID s.e. partial waveswaves

nonresonant backgroundbare resonances

single-energy pw analysis from SAID

requires 4 resonances in S11

15

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

Dynamical model for N → N

0

00

where

transition potential,

( ) ( )

-matrix,

1

( )

,

( )

N

N N

t Et E

t

E H

g Ev v

v

t

g E

two two ingredientsingredients

Both on- & off-shell (gauge invariance?)

, Nv t

16

• Multipole decomposition

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

Off-shell rescatterings

17

( )

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

0

0

( ))

( ( ) ( )

( )) ( )(

)

,

with

R

R R

B B

N

N

R

B v v g E

t E

t E v v

t E

g

t

E t E

E

E

both tB and tR satisfyFermi-Watson theorem,respectively.

18

2 ( ) ,, ( ) ( ) , 2

0

' ( , '; ) ( ', )exp( )cos [ ( , ) ' ]

( ')

BN EB B

N

q R q q E v q kt i v W Q P dq

E E q

Bv

DMT Model

PV only

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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 refers to a bare vertex

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( ) R

A Q

20

Results of DMT model near threshold, (depends only on )

and B BNv t

21M. Weis et al., Eur. Phys. J. A 38 (2008) 27

22

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)

Sensitive w.r.t. multipole 1 (1/ 2)M

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

Dashed (dotted) curves are results for

including (excluding) the principal value integral contribution.

Bt

( ) ( ) , 2

2 ( ) ,

0

,

' ( , '; ) ( ', )'

(

exp( )cos { ( , )

')}

BN E

B

N

B

q R q q

i v W

E v q kP dq

t

E E q

Q

△ deformation

(hyperfine qq interaction → D-state component in the △ )

REM=-2.4%

A1/2(10-3GeV-1/2)

A3/2(fm2)

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, and with experiments and Sato-Lee’s prediction. The numbers within the parenthesis in red correspond to the bare values. Small bare value of indicate that bare Delta is almost spherical.

NQ

NQ

N

N

NQ

Resonance masses, widths, and pole positions

bare and physical resonance masses, total widths, bare and physical resonance masses, total widths, branching ratios and background phases branching ratios and background phases

for N* resonances (I=1/2)for N* resonances (I=1/2)

bare phys

our analysisPDG

our analysisPDG

baremass

physical mass

additional res.

additional res.

additional res.

resonance parametersresonance parametersfor for resonances (I=3/2) resonances (I=3/2)

baremass

physical mass

our analysisPDG

resonance parametersresonance parametersfor for resonances (I=3/2) resonances (I=3/2)

baremass

physical mass

our analysisPDG

additional res.

Results for resonance pole positions

S11 pole positions S11 pole positions

P11 pole positions P11 pole positions

P33 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 2JEBAC(MeV)

DMTPDG

(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 － 76i1364 － 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

(1910) ****

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

Two poles feature of P11(1440)

Appearance of π△ complex branch cut

pole A: unphys. sheetpole 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.

0

0

0

0

( 2 )1 0

1

0

0

2

0

Two sheets: let , then

0 2 ,

: 2 4 .

If we write , 0 2 , t

sheet I:

II

are both poles of

( )( ) , ( ) : anal

hen

ytica

, and .

Residues (

( )

l

i

i

i

i

g zf z z g z

z z

z r e

z re

z

f z

re

g

r

R

z e

0 0( 2 )1 0 2 2 0

unle ( ) ss contains a factor like .

) ( )

i iz r e R g z e

g z z

r

2.

0

0

0

20

0

20

two Riemann sheets

appearing in one of the

( ) ,

Pole

Again, there are and, there

is

depending on

0 (s

( ) will appear

two Riemann sheets.

in sheet I or II

he

only one pole

et I)

ip

af z z

z z

z r e z

0 or 2 (sheet II)

Analytic functions with two cuts of squared root nature

• product form a cut can be drawn by connecting za and zb

two Riemann sheets are needed.

additive form four Riemann sheets are required to make

function of this kind single-valued. This is the case with pion-nucleon

scattering amplitude

( )( )a bz z z z

a bz z z z

Two classes of functions with two additive branch points

(a) two cuts starting from za a and zand zbb

4 Riemann sheets4 Riemann sheets

pole zpole z00 appears in all 4 sheetsappears in all 4 sheets, with equal residues if g(z) is , with equal residues if g(z) is

analyticalanalytical

(b) (b) 3 branch points z3 branch points zaa, z, zbb, and z, and z00, but only 4 Riemann sheets are , but only 4 Riemann sheets are

neededneeded

to make fto make f22(z) single-valued(z) single-valued

polepole appears only in 2 sheetsappears only in 2 sheets with different residues even if with different residues even if

h(z) is analyticalh(z) is analytical

10

( )( ) a b

g zf z z z z z

z z

2

0

( )( ) .a b

h zf z z z z z

z z

20z

SNY, Chinese J. Phys. 49 (2011) 1158.

N. Suzuki, T. Sato, T.-S.H. Lee, Phys. Rev. C82 (2010) 045206

0

0

RRRN

RR

h Et E

E M

h

EE

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. Two-loop contributions small. For electroproduction, ChPT might need to go to O(p4).

• DMT predicts = 3.514 N, =-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

N NQ

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

57

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.