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Nucleon-to-resonance transition form factors in a vector-meson-dominance model
Nikolay Volchanskiy
Research Institute of Physics Southern Federal University

27 September 2011


Introduction Symmetries of the RS field Symmetries of the free RS field Consequences of symmetry breaking Point and gauge invariant N R-interactions N R-Lagrangian for J = 3/2 N R-Lagrangians for J 3/2 Multi-pole vector-mesonídominance model FFs as dispersionlike expansions pQCD constraints on the model Data analysis "Scaling" of the N (1232)-FF ratios Conclusion


Introduction Symmetries of the RS field Symmetries of the free RS field Consequences of symmetry breaking Point and gauge invariant N R-interactions N R-Lagrangian for J = 3/2 N R-Lagrangians for J 3/2 Multi-pole vector-mesonídominance model FFs as dispersionlike expansions pQCD constraints on the model Data analysis "Scaling" of the N (1232)-FF ratios Conclusion


Experimental data on N R-transition FFs
The exp eriments have b een carried out in JLab, MAMI, ELSA... Exp erimentalists study the electropro duction of the resonances with spins 1/2, 3/2 and 5/2: (1232), N (1440), N (1520), N (1535), N (1680)... The available data: inclusive: for (1232) up to Q2 = 9 GeV2 , N (1535) up to Q2 = 20 GeV2 ; exclusive: for (1232) up to Q = 8 GeV , for N (1440), N (1520), N (1535) up to Q2 = 4 GeV2 , for N (1680) up to Q2 = 1.5 GeV2 . Exp ected data: new measurements up to Q time-like FFs (SLAC).
p G znau A ID okee N (1 5 2 0 )
p N (1 5 3 5 )
-1 /2

1 .0 0 .8

p



( 1 2 3 2 )

) /(3 G
0 .6 0 .4 0 .2 0 -1 0 -2 0 -3 0
A S J F ro p u z l Es lo v 1 a rv e r lia -D i naur ner 2 y
D

F ro Sp Az Sp

lo a rv nau a rv

v1 er r er

9 is ya is

99 2005 n 2009 2008

K a m a lo v 2 0 0 1 S ta v e 2 0 0 8 V illa n o 2 0 0 9

G

*

M

R R
is a 0
SM

EM

2

2

(% )

2

= 12 16 GeV

2

(JLab);

0 .1

999 2005 z 2008 an 2009 06

K Ke St V

am l av ill

a ly 2 e an

lo v 0 20 o

2001 05 08 2009

R

EM

,R

SM

Q
2

(G e V
2

1

)

10

-1 /2

150 100 50 0 -5 0 -1 0 0 0

A

3 /2

p

G eV

PD A M M



ry a n 2 0 0 9 07 v 2009

160
-1 /2



150 100 50 0

A

3 /2

p

p



G eV

(1 0

(1 0

S

,S

1 /2

1 /2

p

(1 0

80 40 0 -4 0 0

1 /2

-3

-3

-3

G eV

120

A S
p

p

PDG A z n a u ry a n 2 0 0 9 M A ID 0 7 S to le r 1 9 9 3

M A ID 2 0 0 7 JL a b -M S U U IM PDG

N (1 6 8 0 )

)

)

)

1

A

,S

1 /2

p

,A

1 /2

1 /2 ,

1 /2

1 /2

A S

3 /2 ,

1 /2

3 /2

-5 0 0 .0

1 /2 p

p

p

p

p

p

A

S

1 /2

Q
2

(G e V
2

2

)

3

4

5

Q
2

10

(G e V
2

)

15

20

1 /2

p

0 .5

Q
2

(G e V

1 .0
2

)

1 .5

A

A

A


Two subproblems

1. It is necessary to find a mathematically consistent Lagrangian of nucleon interactions with higher-spin baryon resonances, photons, pions, and vector mesons. The Lagrangian and FFs should be defined unambiguously by symmetry conditions. This should determine polynomial Q2 -dependencies in the observables.


Two subproblems

1. It is necessary to find a mathematically consistent Lagrangian of nucleon interactions with higher-spin baryon resonances, photons, pions, and vector mesons. The Lagrangian and FFs should be defined unambiguously by symmetry conditions. This should determine polynomial Q2 -dependencies in the observables.

2. The Q2 -dependencies of the FFs should be reproduced in some particular dynamical model.


Introduction Symmetries of the RS field Symmetries of the free RS field Consequences of symmetry breaking Point and gauge invariant N R-interactions N R-Lagrangian for J = 3/2 N R-Lagrangians for J 3/2 Multi-pole vector-mesonídominance model FFs as dispersionlike expansions pQCD constraints on the model Data analysis "Scaling" of the N (1232)-FF ratios Conclusion


Covariant constraints for the RS fields
3 Nucleon resonances with spin J = + 1 2 2 are describ ed as the symmetric tensor-spinor Rarita Schwinger (RS) fields ²1 ...² . The spin content of the RS field ²1 ...² :

J,

J-

1 2

‘

, ...

1 2

‘

.


Covariant constraints for the RS fields
3 Nucleon resonances with spin J = + 1 2 2 are describ ed as the symmetric tensor-spinor Rarita Schwinger (RS) fields ²1 ...² . The spin content of the RS field ²1 ...² :

J,

J-

1 2

‘

, ...

1 2

‘

.

To eliminate redundant degrees of freedom, subsidiary conditions are imposed on the field: ² ²

2 2

...² ...²

=0 =0

transversality tracelessness


Free spin-3/2 field
The free-field Lagrangian for the vector-spinor field ² : ï Lff (A) = ² i² (A) - M ² (A) , ² (A) = g² - A ² g - A g² + 32 1 |A| - Re A + 2 2 ,

² (A) = g² - 3|A|2 - 3 Re A + 1 ² , where A =
1 2

is a complex parameter.

Free field equation and constraints: (i - M )² = 0, = 0 = .


Symmetries of the free spin-3/2 field
The free-field Lagrangian for the vector-spinor field ² : ï Lff (A) = ² i² (A) - M ² (A) .


Symmetries of the free spin-3/2 field
The free-field Lagrangian for the vector-spinor field ² : ï Lff (A) = ² i² (A) - M ² (A) . Point invariance of the equivalent class of the Lagrangians: ² = (A,A ) , ² (A,A ) = g² + ² Lff (A) = Lff (A ). A -A ² , 2 (2A - 1)


Symmetries of the free spin-3/2 field
The free-field Lagrangian for the vector-spinor field ² : ï Lff (A) = ² i² (A) - M ² (A) . Point invariance of the equivalent class of the Lagrangians: ² = (A,A ) , ² (A,A ) = g² + ² Lff (A) = Lff (A ). A -A ² , 2 (2A - 1)


Symmetries of the free spin-3/2 field
The free-field Lagrangian for the vector-spinor field ² : ï Lff (A) = ² i² (A) - M ² (A) . Point invariance of the equivalent class of the Lagrangians: ² = (A,A ) , ² (A,A ) = g² + ² Lff (A) = Lff (A ). Gauge invariance of the RS field in the massless limit: ² = ² + ² (x) for ² ² (x) = 0, Lff (A) = Lff (A). A -A ² , 2 (2A - 1)


Constraints for the interacting RS fields
Minimal EM coupling ï Lint = -eA ² ² breaks the free-field symmetries, modifies the constraints reducing their number, which leads to the excitation of unphysical DsOF and different pathologies.
G. Velo, D. Zwanziger, Phys. Rev. 186 (1969) 1337; K. Johnson, E.C.G. Sudarshan, Ann. Phys. (N.Y.) 13 (1961) 126; V. Pascalutsa, Phys. Rev. D 58 (1998) 096002.

For the trilinear nucleon-resonance interactions with pions, photons, and vector mesons we can solve all consistency problems by requiring invariance of the Lagrangian under the point and gauge transformations ² = ² + ² 1 (x) + ² 2 , i.e. the vector-spinor source J² should be traceless and transversal, J = 0 = J .


Introduction Symmetries of the RS field Symmetries of the free RS field Consequences of symmetry breaking Point and gauge invariant N R-interactions N R-Lagrangian for J = 3/2 N R-Lagrangians for J 3/2 Multi-pole vector-mesonídominance model FFs as dispersionlike expansions pQCD constraints on the model Data analysis "Scaling" of the N (1232)-FF ratios Conclusion


Nucleon-resonance interaction Lagrangian
N RV -Lagrangian (spin-3/2) consists of invariants such as ïï ², [
² ][ ] ïï

N , V

ï



, where ² = ² - ² ,

, are multi-indicies. ïï


Nucleon-resonance interaction Lagrangian
N RV -Lagrangian (spin-3/2) consists of invariants such as ïï ², [
² ][ ] ïï

N , V

ï



, where ² = ² - ² ,

, are multi-indicies. Symmetry properties of the coupling ïï matrices: [
² ][ ] ïï

= -[

²][ ] ïï

= -[

² ][ ] ïï

,


Nucleon-resonance interaction Lagrangian
N RV -Lagrangian (spin-3/2) consists of invariants such as ïï ², [
² ][ ] ïï

N , V

ï



, where ² = ² - ² ,

, are multi-indicies. Symmetry properties of the coupling ïï matrices: [ ² [
² ][ ] ïï

= -[

²][ ] ïï

= -[

² ][ ] ïï

,

² ][ ] ïï

= 0 (tracelessness of the coupling matrix).


Nucleon-resonance interaction Lagrangian
N RV -Lagrangian (spin-3/2) consists of invariants such as ïï ², [
² ][ ] ïï

N , V

ï



, where ² = ² - ² ,

, are multi-indicies. Symmetry properties of the coupling ïï matrices: [ ² [
² ][ ] ïï

= -[

²][ ] ïï

= -[

² ][ ] ïï

,

² ][ ] ïï

= 0 (tracelessness of the coupling matrix).

3 The coupling matrix for the spin- 2 :



[² ][ ]

1 = - (² + 3 ² ) . 6


N RV -Lagrangian (JR = 3/2)

L=

ig1 ï ² 2 [² ][ ] R N V 2 MN ig3 ï ², [² ][ + 2 2 2MN MR - [



+

g2 ï ², [ 2 2 MN MR - [
² ][ ] g

² ][ ] R

NV



+

] g

+

[² ][ ] g

² ][ ] g

- [

² ][ ] g

R N , V


N RV -Lagrangian (JR = 3/2)

L=

ig1 ï ² 2 [² ][ ] R N V 2 MN ig3 ï ², [² ][ + 2 2 2MN MR - [



+

g2 ï ², [ 2 2 MN MR - [
² ][ ] g

² ][ ] R

NV



+

] g

+

[² ][ ] g

² ][ ] g

- [

² ][ ] g

R N , V

The consequences of the point and gauge invariance: all terms of the Lagrangian are defined uniquely;


N RV -Lagrangian (JR = 3/2)

L=

ig1 ï ² 2 [² ][ ] R N V 2 MN ig3 ï ², [² ][ + 2 2 2MN MR - [



+

g2 ï ², [ 2 2 MN MR - [
² ][ ] g

² ][ ] R

NV



+

] g

+

[² ][ ] g

² ][ ] g

- [

² ][ ] g

R N , V

The consequences of the point and gauge invariance: all terms of the Lagrangian are defined uniquely; there is only one universal matrix as a symmetry "carrier";


N RV -Lagrangian (JR = 3/2)

L=

ig1 ï ² 2 [² ][ ] R N V 2 MN ig3 ï ², [² ][ + 2 2 2MN MR - [



+

g2 ï ², [ 2 2 MN MR - [
² ][ ] g

² ][ ] R

NV



+

] g

+

[² ][ ] g

² ][ ] g

- [

² ][ ] g

R N , V

The consequences of the point and gauge invariance: all terms of the Lagrangian are defined uniquely; there is only one universal matrix as a symmetry "carrier"; classification of the Lagrangian vertexes in terms of differential order.


N RV -Lagrangian (JR = 3/2)

L=

ig1 ï ² 2 [² ][ ] R N V 2 MN ig3 ï ², [² ][ + 2 2 2MN MR - [



+

g2 ï ², [ 2 2 MN MR - [
² ][ ] g

² ][ ] R

NV



+

] g

+

[² ][ ] g

² ][ ] g

- [

² ][ ] g

R N , V

The consequences of the point and gauge invariance: all terms of the Lagrangian are defined uniquely; there is only one universal matrix as a symmetry "carrier"; classification of the Lagrangian vertexes in terms of differential order.


N R -vertex (JR = 3/2)

N R -Lagrangian is also defined uniquely by the symmetry: L= f ï[ 2 m MR
² ],

[

² ][ ] R 5

N , ;

V. Shklyar, H. Lenske, Phys. Rev. C 80 (2009) 058201


Field tensor-spinors for arbitrarily high spin
Gauge-invariant RS field tensor spinors: =1: =2: [
²1 1 ] ²1
1

([

= ²1 1 - 1 ²1 ; 1 ][²2 2 ]) = ²1 ²2 1 2 1 2
-1



2

- ²1 2
1 2

1 ²2

-
²
2

- 1 ²2 ² :
([²1 1 ][²2 2 ]...[² ])

- 1 2 ²1
2 ...

;

=

( ²1

²2

§ § § ² 1

+ ...).


Field tensor-spinors for arbitrarily high spin
Gauge-invariant RS field tensor spinors: =1: =2: [
²1 1 ] ²1
1

([

= ²1 1 - 1 ²1 ; 1 ][²2 2 ]) = ²1 ²2 1 2 1 2
-1



2

- ²1 2
1 2

1 ²2

-
²
2

- 1 ²2 ² :
([²1 1 ][²2 2 ]...[² ])

- 1 2 ²1
2 ...

;

=

( ²1

²2

§ § § ² 1

+ ...).

Some multi-indices: Aa = [²a a ] ï A = ([²1 1 ][²2 2 ] . . . [² ]), ï Aa = ([²1 1 ] . . . [²a-1 a-1 ][²a+1 a+1 ] . . . [² ]).


The couplings for arbitrary spin
The expansion of the traceless coupling: A = AB R ïï ïï
() ï B ï

+ SB + T[ ï

ï

ï ]B

ï

.


The couplings for arbitrary spin
The expansion of the traceless coupling: A = AB R ïï ïï
() ï B ï

+ SB + T[ ï

ï

ï ]B

ï

.

Recurrence relation for the coupling matrix (arbitrary ):
()

AB = ïï

3 2(2 + 1)

2 a,b=1

( + 1)

( -1) (1) ï ï Aa B b Aa B

b

+
( -1) ï Aa B (1) bB

+ ( - 1)Aa ï

( -1) (1) ï Bb BbA

a

+
b=c=1

Aa ï

bc

B

c

.


N RV -Lagrangian for an arbitrary
L1 = L2 = L3 = i3 +2 g1 ï A, ï 3 2MN -1
2

...

AB R N ïï
2

()

,2 ...

V

1 1

, V
1 1

i3 +3 g2 ï A, ï 3 -1 2 MN MR i3 +2 g3 ï A, ï 3 -1 2 2 MN MR + ½N V

... ( ) AB R ïï ... ()

N

,2 ...

,

2

AB ïï
1

()

1

g 1 1

- AB ïï
()
1

()

1

g 1 1 ½

() g- ïï AB 1 [1 ] 1 ,2 ... 1 1

AB ïï

g [1 ] 1

- AB ïï

g [1 ] 1

.


N RV -Lagrangian for an arbitrary
L1 = L2 = L3 = i3 +2 g1 ï A, ï 3 2MN -1
2

...

AB R N ïï
2

()

,2 ...

V

1 1

, V
1 1

i3 +3 g2 ï A, ï 3 -1 2 MN MR i3 +2 g3 ï A, ï 3 -1 2 2 MN MR + ½N V

... ( ) AB R ïï ... ()

N

,2 ...

,

2

AB ïï
1

()

1

g 1 1

- AB ïï
()
1

()

1

g 1 1 ½

() g- ïï AB 1 [1 ] 1 ,2 ... 1 1

AB ïï

g [1 ] 1

- AB ïï

g [1 ] 1

.

The consequences of the point and gauge invariance: all terms of the Lagrangian are defined uniquely; there is only one universal matrix as a symmetry "carrier"; classification of the Lagrangian vertexes in terms of differential order;


N RV -Lagrangian for an arbitrary
L1 = L2 = L3 = i3 +2 g1 ï A, ï 3 2MN -1
2

...

AB R N ïï
2

()

,2 ...

V

1 1

, V
1 1

i3 +3 g2 ï A, ï 3 -1 2 MN MR i3 +2 g3 ï A, ï 3 -1 2 2 MN MR + ½N V

... ( ) AB R ïï ... ()

N

,2 ...

,

2

AB ïï
1

()

1

g 1 1

- AB ïï
()
1

()

1

g 1 1 ½

() g- ïï AB 1 [1 ] 1 ,2 ... 1 1

AB ïï

g [1 ] 1

- AB ïï

g [1 ] 1

.

The consequences of the point and gauge invariance: all terms of the Lagrangian are defined uniquely; there is only one universal matrix as a symmetry "carrier"; classification of the Lagrangian vertexes in terms of differential order; unified Lagrangian structure for an arbitrary high resonance spins.


Helicity amplitudes
Q2 ‘ ² ‘ M F1 + ²‘ MR F2 - Q2 + ²‘ M
N

A A

3/2

= =-

N

N

R

F3 ,

1/2

N ²‘ MR F1 + Q2 ‘ ²‘ M +2

F

2

²‘ MN F3 ,

S1/2 =

2 2 Q2 + M R + M N N Q+ Q- F1 - F2 + F3 , 2 2( + 2) 2MR

where N (Q2 ) =

2 ( !)2 Q‘ (2 )!M
4 +1 N

2( -1)

Q2

2 2 ( MR - MN )

,

² ‘ = MR ‘ MN ,

Q‘ =

Q2 + ²2 . ‘

The upper and bottom signs correspond to J P = (3/2)‘ , (5/2) , ...


pQCD constraints

A

3/2

(Q2 )

1 Q5 Ln1 S (Q )
2

,

A

1/2

(Q2 )

1 Q
3 Ln2

(Q2 )

,

1/2

(Q2 )

1 , Q3 Ln3 (Q2 )

where Lnf (Q2 ) = lnnf (Q2 /2 ) and n2 - n3 2 (Idilbi 2004, Carlson 1986). For J = + 1/2 we get: F1 (Q2 ) Q6+2 1 , Ln1 (Q2 ) F2 (Q2 ) 1 . Q6+2 Ln3 (Q2 ) Q
4+2

1 , Ln2 (Q2 )

F3 (Q2 )


pQCD constraints

It is interesting that the following asymptotic identities hold: A A S for n3 > n1 .
3/2 1/2

(Q2 ) = (Q2 ) = - (Q2 ) =

N (Q2 )Q2 F1 (Q2 ), N (Q2 ) 2 Q F2 (Q2 ), +2 N (Q2 ) Q4 2 2 F3 (Q ) 2( + 2) 2MR

1/2


Elastic nucleon FFs
The Lagrangian: L =e
V

ï g1(V ) N ² V ² N -

i 2M

N

ï g2(V ) N ² V
V

²

N.

Sachs FFs, the Dirac and Pauli FFs: GM (Q2 ) = F1 (Q2 ) + F2 (Q2 ), Ff (Q2 ) =
V

GE (Q2 ) = F1 (Q2 ) - gf (V ) m2 V . 2 + m2 Q V

Q2 2 2 F2 (Q ); 4MN

High-Q2 asymptotic relations: GM (Q2 ) = F1 (Q2 ), F1 (Q2 )
4 2

GE (Q2 ) =

1 , Q ln (Q2 /2 )

Q2 2 2 F2 (Q ), 2 MN 1 F2 (Q2 ) 6 . Q


Introduction Symmetries of the RS field Symmetries of the free RS field Consequences of symmetry breaking Point and gauge invariant N R-interactions N R-Lagrangian for J = 3/2 N R-Lagrangians for J 3/2 Multi-pole vector-mesonídominance model FFs as dispersionlike expansions pQCD constraints on the model Data analysis "Scaling" of the N (1232)-FF ratios Conclusion


Multi-pole vector-mesonídominance model

: Ff
(p,n)

(Q2 ) =

1 2

K k=1

kf (Q2 )m Q2 + m

( )

2 ( )k

K

2 ( )k

‘
k=1

kf (Q2 )m2)k ( Q2 + m2 (
)k

()

.

-meson families (PDG): (770) (782), (1450) (1420), (1700) (1650),

(1900) (1960),

(2150) (2205).


pQCD constraints on the model

Q2 -dependence of meson-baryon couplings is chosen a universal phenomelogical logarithmic function
( , ) kf

(Q )

2

( , ) (0) f ( , ) Lf (Q2 )

k

,


( ) f ( , )

L

= 1 + bf

Q2 ln 1 + 2

+a

( , ) f

Q2 ln 1 + 2

2

nf 2

.


pQCD constraints on the model

Ff (Q2 ) =

1 Lf (Q2 ) =

K k=1

kf (0)m2 k = Q2 + m 2 k
K

1 Lf (Q2 )

kf (0) -
k=1

1 Q2

K

m2 kf (0) + . . . , k
k=1

The superconvergence relations for the meson parameters
K n m2, ( k=1 )k

k

( , ) f

(0) = 0,

n = 2, 3, . . . 4 + f = 1, 3 n = 2, . . . 3 + f = 2 J = + 1 . 2


N (1232)-transition
+ (1232): J P = 3 2

1 .0 0 .8

p



( 1 2 3 2 )

) /(3 G
0 .6 0 .4 0 .2 0 -1 0 -2 0 -3 0
S F ro p u z l lo a rv J lia A na E sne v1 er D ur r2 9 is ia z ya 0
D

Magnetic FF:
3 MN 2 (MR 2 - MN ) MN )2 Q2 - 1/2

GM = -



2 (MR + ½ A1/2 + 3A3/2 ,

½

F ro Sp Az Sp

lo a rv nau a rv

v1 er r er

9 is ya is

99 2005 n 2009 2008

K a m a lo v 2 0 0 1 S ta v e 2 0 0 8 V illa n o 2 0 0 9

G

*

M

R R
SM

EM

The ratio of the electric and amplitudes to the magnetic one: A1/2 - A3/2 / 3 REM = , A1/2 + 3A3/2 2S1/2 RSM = . A1/2 + 3A3/2

Coulomb

(% )

0 .1

99 2005 2008 n 2009 06

K Ke St V

am l a il

al ly 2 ve 2 la n o

ov 2001 005 008 2009

R

EM

,R

SM

Q
2

(G e V
2

1

)

10

Magnetic FF and the amplitude ratios for the transition p (1232) (2 /D O F = 1.51)


The N (1520) and N (1680)
- N (1520): J P = 3 2
150 100 50 0 -5 0 -1 0 0 0
p N (1 5 2 0 )
-1 /2

+ N (1680): J P = 5 2

A

3 /2

-1 /2

p



G eV

G eV

PDG A z n a u ry a n 2 0 0 9

150 100 50 0

A

3 /2

M A ID 2 0 0 7 JL a b -M S U U IM PDG

)

-3

(1 0

,S

S

1 /2

1 /2

1 /2

p

(1 0

-3

)

A S

p

A
1

1 /2

3 /2 ,

p

,A

A

1 /2 ,

1 /2

1 /2

S
-5 0

p

Q
2

2

(G e V
2

)

3

4

0 .0

1 /2

3 /2

p

0 .5

Q
2

(G e V

1 .0
2

)

1 .5

A

A
F
2 p

1 .5 1 .0 0 .5 0 .0 - 0 .5 - 1 .0 0

p



N (1 5 2 0 )

0 .4 0 .2

F
2

M A ID 2 0 0 7 JL a b -M S U U IM PDG
3

F
3

F -F
1

,F

3

p

p

2

1

F

-F
1

p

PDG A z n a u ry a n 2 0 0 9

F
- 0 .2

1,

p

2,

F
0 .0

,F

p

F

3

1

Q
2

(G e V
2

2

)

3

4

0 .0

0 .5

Q
2

(G e V
2

1 .0

)

1 .5

2 .0

Helicity amplitudes and p oint and gauge invariant FFs for the transition p N+ (1520) (2 /D O F = 1.05)

Helicity amplitudes and p oint and gauge invariant FFs for the transition p N+ (1680) (2 /D O F = 0.87)


The N (1440) and N (1535)
Helicity amplitudes: A1/2 (Q2 ) = 2N Q2 F1 (Q2 ) + MN (MR ‘ MN ) F2 (Q2 ) , Q+ Q- S1/2 (Q2 ) = ‘ N (MR ‘ MN )F1 (Q2 ) - MN F2 (Q2 ) , 2MR [Q2 + (MR MN )2 ] where N = . 5 2 2 M N ( MR - MN ) Upper signs are for J P = 1 , bottom ones are for J 2 pQCD scaling (n1 - n2 2): F1 (Q2 ) 1 Q6 Ln1 (Q )
2 + P

=

1- 2

.

,

F2 (Q2 )

1 Q
6 Ln2

(Q2 )

.

Asymptotic relations: A
1/2

(Q2 )



2N Q2 F1 (Q2 ),

S

1/2

Q2 (Q2 ) ‘ N F2 (Q2 ). 2 MR


The N (1440) and N (1535)
+ N (1440): J P = 1 2
100 50 0

- N (1535): J P = 1 2

A

1 /2

p

p



N (1 4 4 0 )

160
-1 /2

p



N (1 5 3 5 )

120 80 40 0 -4 0 0

)

(1 0

S

1 /2

p

-3

G eV

A S
p

1 /2

p

PDG A z n a u ry a n 2 0 0 9 M A ID 0 7 S to le r 1 9 9 3

-1 0 0 0

A

1 /2

p

,S

-5 0 1 2
2

PDG A z n a u ry a n 2 0 0 9 A z n a u ry a n 2 0 0 5

1 /2

p

1 /2

Q

(G e V
2

)

3

4

5

Q
2

10

(G e V
2

)

15
p

20
N (1 5 3 5 )

0 .2

F
1

p

p



N (1 4 4 0 )

0 .4

0 .1 0 .0 - 0 .1 - 0 .2 0

F
1

p



0 .3 0 .2 0 .1 0 .0 0
2 p

PDG A z n a u ry a n 2 0 0 9

F
2

p

1

Q
2

(G e V
2

2

PDG A z n a u ry a n 2 0 0 9 A z n a u ry a n 2 0 0 5

F

1

p

,F

F
2

p

)

3

4

1

Q
2

(G e V
2

2

)

3

4

Helicity amplitudes and FFs for the transition p N+ (1440) ( /D O F = 0.97)
2

Helicity amplitudes and FFs for the transition p N+ (1535) (2 /D O F = 0.65)


Introduction Symmetries of the RS field Symmetries of the free RS field Consequences of symmetry breaking Point and gauge invariant N R-interactions N R-Lagrangian for J = 3/2 N R-Lagrangians for J 3/2 Multi-pole vector-mesonídominance model FFs as dispersionlike expansions pQCD constraints on the model Data analysis "Scaling" of the N (1232)-FF ratios Conclusion


N (1232)-transition
+ (1232): J P = 3 2

1 .0 0 .8

p



( 1 2 3 2 )

) /(3 G
0 .6 0 .4 0 .2 0 -1 0 -2 0 -3 0
S F ro p u z l lo a rv J lia A na E sne v1 er D ur r2 9 is ia z ya 0
D

Magnetic FF:
3 MN 2 (MR 2 - MN ) MN )2 Q2 - 1/2

GM = -



2 (MR + ½ A1/2 + 3A3/2 ,

½

F ro Sp Az Sp

lo a rv nau a rv

v1 er r er

9 is ya is

99 2005 n 2009 2008

K a m a lo v 2 0 0 1 S ta v e 2 0 0 8 V illa n o 2 0 0 9

G

*

M

R R
SM

EM

The ratio of the electric and amplitudes to the magnetic one: A1/2 - A3/2 / 3 REM = , A1/2 + 3A3/2 2S1/2 RSM = . A1/2 + 3A3/2

Coulomb

(% )

0 .1

99 2005 2008 n 2009 06

K Ke St V

am l a il

al ly 2 ve 2 la n o

ov 2001 005 008 2009

R

EM

,R

SM

Q
2

(G e V
2

1

)

10

Magnetic FF and the amplitude ratios for the transition p (1232) (2 /D O F = 1.51)


"Scaling" of the ratios of the elastic form factors

0.3 (Q2/Log2(Q2/2))F2/F
1

0.2

=400MeV =300MeV =200MeV [2] [3]

F2 (Q2 ) 1 Q2 2 ln2 2 F1 (Q2 ) Q

0.1

0 0 1 2 Q
2

3 (GeV2)

4

5

6

A. V. Belitsky, X. Ji, F. Yuan, Phys. Rev. Lett. 91, 092003 (2003)


"Scaling" of the ratios of the N (1232) FFs
1

F 1/F
2

F 1/F 2, F 3/F

0

F 3/F
2

2

-1

F ro Sp Ju Vi

1

l ar lia ll

ov v an

1 e ri Di o

99 s2 az 20
2

005 2008 09
2

9

K a m a lo v 2 0 0 1 K e lly 2 0 0 5 A z n a u ry a n 2 0 0 9

Q

(G e V

)

10

Ff (Q2 ) 1 Q2 2 lnNf 2 , F2 (Q2 ) Q for Q2

f = 1, 3

0.4 GeV2 , = 0.29 GeV, N3 = 2, N1 = 2.7.


N (1232)-transition
+ (1232): J P = 3 2

1 .0 0 .8

p



( 1 2 3 2 )

) /(3 G
0 .6 0 .4 0 .2 0 -1 0 -2 0 -3 0
S F ro p u z l lo a rv J lia A na E sne v1 er D ur r2 9 is ia z ya 0
D

Magnetic FF:
3 MN 2 (MR 2 - MN ) MN )2 Q2 - 1/2

GM = -



2 (MR + ½ A1/2 + 3A3/2 ,

½

F ro Sp Az Sp

lo a rv nau a rv

v1 er r er

9 is ya is

99 2005 n 2009 2008

K a m a lo v 2 0 0 1 S ta v e 2 0 0 8 V illa n o 2 0 0 9

G

*

M

R R
SM

EM

The ratio of the electric and amplitudes to the magnetic one: A1/2 - A3/2 / 3 REM = , A1/2 + 3A3/2 2S1/2 RSM = . A1/2 + 3A3/2

Coulomb

(% )

0 .1

99 2005 2008 n 2009 06

K Ke St V

am l a il

al ly 2 ve 2 la n o

ov 2001 005 008 2009

R

EM

,R

SM

Q
2

(G e V
2

1

)

10

Magnetic FF and the amplitude ratios for the transition p (1232) (2 /D O F = 1.51)


Introduction Symmetries of the RS field Symmetries of the free RS field Consequences of symmetry breaking Point and gauge invariant N R-interactions N R-Lagrangian for J = 3/2 N R-Lagrangians for J 3/2 Multi-pole vector-mesonídominance model FFs as dispersionlike expansions pQCD constraints on the model Data analysis "Scaling" of the N (1232)-FF ratios Conclusion


Results and conclusions

We have constructed effective Lagrangians for the N RV -interactions that possess the gauge and point invariance of the RS field. The symmetry ensures mathematical coherence of the theory and fixes all three terms of the minimally local Lagrangian. The point and gauge invariance unifies the structure and properties of the Lagrangians for arbitrarily high spins. The multi-pole vector-mesonídominance model constrained by high-Q2 pQCD predictions is in a good agreement with the available data on the transitions to the resonances (1232), N (1440), N (1535), N (1520), and N (1680).


Results and conclusions

The ratios of the point and gauge invariant form factors N (1232) exhibit asymptotic scaling behavior at momentum transfers as low as 0.4 GeV2 . While the high-Q2 scaling of the FFs is well understood as a consequence of the asymptotic freedom, the dynamics leading to the low-Q2 scaling of the FF ratios in the nonperturbative domain of QCD is still to be established both qualitatively and quantitatively.


G. Vereshkov, N. Volchanskiy. Q2 -evolution of nucleon-to-resonance transition form factors in a QCD-inspired vector-mesonídominance model, Phys. Rev. D 76, 073007 (2007). G. Vereshkov, N. Volchanskiy. Low-Q2 scaling behavior of the form-factor ratios for the N (1232)-transition, Phys. Lett. B 688, 168 173 (2010). G. Vereshkov, N. Volchanskiy. Symmetries of higher-spin fields and the electromagnetic N N (1680) form factors, Phys. Rev. C 82, 045204 (2010). V.I. Kuksa, N.I. Volchanskiy. Factorization effects in a model of unstable particles, Int. J. Mod. Phys. A 25, 2049 2062 (2010).