A unified description of hadronic form factors

Preliminary Results

Qian WanCenter for Theoretical Physics,

Yale UniversityAugust 20, 2004

Introduction

N’

N

Electromagnetic form factors (helicity amplitudes) are closely related to the structure of hadrons

2 2 2 2 2 21/2 3/2 1/2: (Q ), (Q ), (Q ) A (Q ), A (Q ), S (Q )E M CBaryons G G G

Related Processes: Elastic Electron-Nucleon Scattering, Meson Electroproduction, Photoproduction, …

• High Q2 region: pQCD gives out asymptotic behavior

• Low Q2 region: Only phenomenological models

• Any consistent theory for all hadrons in low Q2 region?

Two component Model

Intrinsic structure (algebraic model of baryons)

Meson cloud (vector meson dominance)

Nonstrange baryons: (7) (6) (3) sf cU SU SU

N’

N

N’

N

N’

N

,, +=

(A) (B)

2

2 2 Propagator:

m

m Q

R. Bijker, F. Iachello and A. Leviatan,

Ann. Phys. 236, 69 (1994)

F. Iachello, A.D. Jackson and A. Lande,

Phys. Lett. 43B, 191 (1973)

Algebraic Structure of Baryons, , , ,1 2

1 2 3

3

1, 0

bylinear products

1 , , ,( )2

1

( 2 )6

(7)

Complete algebraic structure: (6) (3) (7)

S sym

m m m m

sf c

b b b br r

m

r r r s s

U

SU SU U

metry need to be imposed for three identical quarks

1

2

3

Basis States:

(7) (3) (4) (3) (3) (3) (3) (3) (2)

U U U U U SO SO SO SO

N n

Two coupled 3D H.O.

or

(7) (3) (4) (3) (4) (3) (3) (3) (2)

Ln L L L M

U U U U SO SO SO SO SO

N n L L L 3D H.O. + 3D Morse Osc.

LM

Algebraic Structure of Baryons

, , ,

0 3 3,

/ / /3 0, ,

Operator for EM transition:

1 6 / [ ]

where ( )2

Operator for strong transition:

1

(2

z D z D z Dik D X ik D X ik D Xm m m

D

H k e ks U Tg

im kU e and T D e e D

X

H

, , ,

'3 3, 3, 3, 3,3/ 2 1/ 2

0

/ / /3 0, ,

1 16 { [ ( ) ] ( )}

) (2 ) 6 2

where ( )2

z D z D z D

Mz z z

ik D X ik D X ik D Xm m m

D

X gks U hs T P P U h s T s Tk

im kU e and T D e e D

X

1, 2

2 1( );

2 3

Collective string model:

dim{ (3)} [dim{ (6)}, ]

Distributed string :

( ) / 2

S Pf J sf v v K

a

SU SU L

g e a

2 2 2

2 2 2 2 2 2

1, 0

(1 )

2,

(1 ) 3(1 )

Nucleon:

p nE E

p nM M

G Gk a

G Gk a k a

21/ 2 2 2 2

0

23/ 2 2 2 2

0

21/ 2 2 2 2

0

2 2 1( )

3 (1 )

2 6 1( )

3 (1 )

2 1( ) 2 3

(1 )

Resonance:

s

A Q kk a k

A Q kk a k

S Q Ak a k

2 2 22 2

2 2 2

( ), is the photon momentum in Breit frame

2( )N

N

W mk Q k

W m Q

1

1

0

0.13

1.259

0.2913

GeV

a GeV

k GeV

Nucleon EM Form Factors 2

2 21 1 2 2 1 1 2 22

22 2

1 1 2 2 1 1 2 22

222 2

1 2 2 2 2

2 21

( ) ( ) ( ) ( ) ( ) ( )4

( ) ( ) ( ) ( ) ( ) ( )4

1( ) ( ) (1 )

2

1( ) ( ) (1 )

2

p v s v s p v s v sM E

n s v s v n s v s vM E

s

QG Q F F F F G Q F F F F

M

QG Q F F F F G Q F F F F

M

mmF Q g Q

m Q m Q

mF Q g Q

2

2 2

222 2

2 2 2 2 2

22 2

2 2 2

1( ) ( ) ( 0.120 )

2

1( ) ( ) 3.706

2

s

m Q

mmF Q g Q

m Q m Q

mF Q g Q

m Q

2

We use same parameters as those obtained

in . . 43, 545 (1973) :

0.672, 1.102, 0.112,

0.052, and 0.25 ( / )

Phys Lett

GeV c

Nucleon EM Form Factors

22 2

Space-like region:

1( )

(1 )g Q

Q

2

2 2

Time-like region:

1( )

(1 )ig Q

e Q

2 2

2 2 23

22 2 2 2 2 2 2

2

2

2 2 22 22 2

2

2 2 22 22 2

2

8 /

( 1)4

(4 ) ( ) / ( 4 ) 4

4

442where ( ) ln[ ] 0

2

442ln[ ] 4

2

m m m

m Q Qm

m Q m Q Q m i Q m mQm

m Q Qm QQ if Q

Q m

Q m QQ mif Q m

Q m

Spacelike Region

Timelike Region

Nucleon EM Form Factors The model’s asymptotic behavior agrees with p-QCD. The coupling constants determined by a 1973 calculation

indicate significant contributions from meson cloud.

Results agree perfectly with the experiments using the recoil polarization technique.

Calculations are in perfect agreement with spacelike proton data, but deviate drastically from spacelike neutron data at

An analytic continuation of the original model is in excellent agreement with timelike data for both proton and neutron.

2 21( / )Q GeV c

There might be a discrepancy between spacelike data and timelike data of neutron according to the properties of analytic functions.

EM Form Factors2

2 '1/ 2 1 1 2 2 22 2

0 2 2

2

2 '3/ 2 2 2 2 2 22 2

0 2 2

2

21/ 2 1 2 22 2

0 2 2

82 2 1

( ) [ ] ,(4 ) ( )3 (1 )

82 6 1

( ) [ ](4 ) ( )3 (1 )

82 1

( ) 2 3 [(4(1 )

T

T

s

mm

A Q k b qm k kk k

m km

mm

A Q k b qm k kk k

m km

mm

S Q A r rmk k

m k

2 2

2 2 22 2

2 2 2

2 2 22 2 21/ 2 3/ 2

]) ( )

( )where

2( )

SLAC's experiments: ( ) ( ) ( )

T

N

N

T

k k

m

W mk Q

W m Q

A Q A Q A Q

Helicity Amplitudes:

EM Form Factors*

2 2 21/ 2 3/ 2

2*

1 2 2 2 22 20 2 2

* 3/ 2 1/ 2 3/ 21 1 1

2 3/ 21 1/ 21

( ) [ ( ) 3 ( )]

82 2 1

( ) [ ](4 ) ( )3 (1 )

1Re( ) Im( ) 3

Im( ) 3

Ash N NM

cm

N N

Tcm

EEM

M

m mG Q A Q A Q

e q m

mmm m

k p pm k ke q m k k

m km

A AE M E A

RM A A AM

2

1 2 2 2 22 2

23/ 2

1 2 2 2 22 2

2

1 2 2 2 22 2

* 3/ 21/ 2 1/ 21 1 1

2 3/ 21 1/ 2 3/ 21

1 2

8

(4 ) ( )

8

(4 ) ( )

8

(4 ) ( )

Re( ) Im( ) 9

Im( ) 2 23

T

T

T

SMM

mm

q qm k k

m kmm

mp p

m k km k

m

mm

r rm k k

m kS S mS M S

RM A kA AM

p p

2

2 2 22 2

8

(4 ) ( )T

mm

m k km k

m

Observables:

1 1 2

' '2 1 2

1 1 2

' '2 1 2

1

2

Parameters:

3

3

p b b

p q q

q b b

q q q

r

r

EM Form Factors

Data:

2( )AshMG Q

Frolov (TJNAF 1999)

Stein (SLAC 1975)

Alder (DESY 1972)

Batzner (BONN 1972)

Bartel (DESY 1968)

PDG 2002

Joo (TJNAF CLAS 2001)

Pospischil (MAMI 2000)

Mertz (MIT BATES 1998)

Kamalov (TJNAF 1999)

Frolov (TJNAF 1999)

Beck (DAPHNE 1997)

Blanpied (LEGS 1997)

Burkert (DESY 1979)

Alder (DESY 1972)

Siddle (DNPL 1971) and/or EM SMR R

2 2 22 2 21/ 2 3/ 2

2 22 2 22 2 2

2

In SLAC's experiments:

( ) ( ) ( )

( ) ( ) ( ) ( )

2

T

N NT T

A Q A Q A Q

m m mF Q G Q A Q

Q

Haidan (DESY 1979)

E133 (SLAC 1992)

E89 (SLAC)

E19 (SLAC)

Brasse 1976

2( )TA Q

GAshM(Q2)

2 2 2Results from the simultaneous fit of ( ), ( ) and ( ) :AshM EM SMG Q R Q R Q

REM(Q2)2 2 2Results from the simultaneous fit of ( ), ( ) and ( ) :Ash

M EM SMG Q R Q R Q

RSM(Q2)2 2 2Results from the simultaneous fit of ( ), ( ) and ( ) :Ash

M EM SMG Q R Q R Q

AT(Q2)

2

2 '1/ 2 1 1 2 2 22 2

0 2 2

2

2 '3/ 2 2 2 2 2 22 2

0 2 2

2

21/ 2 1 2 22 2

0 2 2

82 2 1

( ) [ ] ,(4 ) ( )3 (1 )

82 6 1

( ) [ ](4 ) ( )3 (1 )

82 1

( ) 2 3 [(4(1 )

T

T

s

mm

A Q k b qm k kk k

m km

mm

A Q k b qm k kk k

m km

mm

S Q A r rmk k

m k

2 2 ]) ( )Tk k

m

Parameters for

3 '1 1

3 '2 2

3 41 2

8.78 10 , 1.52;

8.74 10 , 1.66;

3.50 10 , 8.89 10 ;

0.350

b q

b q

r r

• There is significant contribution from meson cloud

• b1=b2 confirms the SU(6) symmetry of the intrinsic part of form factors

EM Form Factors

Fitting results can perfectly describe experimental data.

The importance of meson cloud is further confirmed by the fitted parameters.

Results show a slightly larger quark core with size parameter = 0.350 (GeV)-1 than the results for nucleon.

At high Q2, the model predicts REM →Const., RSM

→0, although current parameters doesn’t give REM →1.

Future direction

Understand MBB’ vertex better Explore the choice of other mesons Use algebraic methods to minimize the

number of parameters in the coupling constants

A unified description of EM form factors of all hadrons

Conclusion

EM form factors of baryons may be described by a two component model in terms of an intrinsic q3

structure and a meson cloud. If the Distributed String Model is used for the intrinsic

part and vector mesons are included as the meson cloud, results on N and agree with experimental data respectively.

It is very likely that a unified description of all EM form factors of hadrons can be obtained based on this two component model.

Thank You!