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- isovector axial form factors in QCD

- isovector axial form factors in QCD
Ulas Ÿzdem ?
(in collaboration with A. Küçükarslan and A. Ÿzpineci , PRD 90, 2014)

Bogoliubov Laborator y of Theoretical Physics, JINR, RUSSIA and 'anakkale Onsekiz Mar t University, TURKEY

The XXII International Workshop High Energy Physics and Quantum Field Theor y June 24 ? July 1, 2015 Samara, Russia

Ulas Ÿzdem ?

- isovector axial form factors in QCD


- isovector axial form factors in QCD

Introduction Light-cone QCD sum rules - Transition in QCD Results Conclusion

Outline

Introduction Light-cone QCD sum rules (LCSR) - Isovector axial vector transition in QCD Results Conclusions

Ulas Ÿzdem ?

- isovector axial form factors in QCD


- isovector axial form factors in QCD

Introduction Light-cone QCD sum rules - Transition in QCD Results Conclusion

Motivation

Understanding the structure of the resonance is great relevance to nuclear phenomenolgy. The is a rather broad resonance close to the N theroshold. it therefore couples strongly to nucleons and pions making it impor tant ingredient in chiral expansions. The bar yon resists the experimental probing due to its shor t lifetime ( 10-23 )

Ulas Ÿzdem ?

- isovector axial form factors in QCD


- isovector axial form factors in QCD

Introduction Light-cone QCD sum rules - Transition in QCD Results Conclusion

Introduction

Form Factors ? Why are they impor tant ? FFs describe how hadrons interact with each other and give information about the internal structure of the hadrons. FFs are well known that many fundamental proper ties hadrons, e.g., the distribution of their charge, the origin and strength of their magnetization, and their (possibly deformed) shape, can be studied on the basis of hadron form factors. Specifically, many of the classical hadron structure obser vables can be directly defined from form factors, including the charges or coupling constants, par ticularly the electromagnetic, axial and tensor charge of hadrons.

Ulas Ÿzdem ?

- isovector axial form factors in QCD


- isovector axial form factors in QCD

Introduction Light-cone QCD sum rules - Transition in QCD Results Conclusion

Light-cone QCD sum rules

Form factors are non-per turbative objects. To study these processes, a reliable non per turbative method is needed. One of these methods is the LCSR* approach. LCSR method is one of the most powerful and applicable non-per turbative tools to hadron physics. In the LCSR, the hadronic parameters are expressed in terms of the proper ties of the vacuum and the distribution amplitudes. In the LCSR, OPE is carried out near light cone (x One star ts with a correlation function of the form,
÷ 2

0).

(p, q ) = i

d 4 xe

iqx

0|T {J÷ (x )J (0)} |h(p, s)

*Braun et al. Z. Phys C 1989, Braun et al., Nucl.Phys. B 1989, Chernyak et al. Nucl.Phys. B 1990
Ulas Ÿzdem ? - isovector axial form factors in QCD


- isovector axial form factors in QCD

Introduction Light-cone QCD sum rules - Transition in QCD Results Conclusion

Hadronic Representation

The correlation functions can be expressed in terms of the proper ties of hadrons and also in terms of the proper ties of the vacuum. By inser ting complete sets of hadronic states, the correlation function can be written as: ÷ (p, q ) =
p

0|J÷ |h (p ) h (p )|J |h(p) + ... m2 - p 2
h

matrix elements 0|J÷ (0)|h (p ) = h u ÷ (p ) where u ÷ (p ) spinor and
h

residue.

The matrix elements h (p )|J |h(p) can be expressed in terms of coupling constants or form factors.

Ulas Ÿzdem ?

- isovector axial form factors in QCD


- isovector axial form factors in QCD

Introduction Light-cone QCD sum rules - Transition in QCD Results Conclusion

QCD Representation

The correlation function can also be calculated in the deep Euclidean region using OPE: =
d

Cd (x 2 )Od (x )

In the case of mass sum rules or traditional sum rules, Od (x ) are local operators. After Fourier transform, the correlation function becomes: =
d

Cd (p2 ) h (p )|Od |h(p)

In case of light cone sum rules, matrix elements of the form h (p )|Od |h(p) are needed. The matrix elements are expanded around x amplitudes.
2

0 in terms of distribution

Ulas Ÿzdem ?

- isovector axial form factors in QCD


- isovector axial form factors in QCD

Introduction Light-cone QCD sum rules - Transition in QCD Results Conclusion

QCD sum rules

Two expressions for the correlation function is matched using spectral representation. (p2 ) = ds
(s ) s -p 2

+ p2 (polynomial )

To subtract the contributions of higher states and continuum, quark hadron duality is assumed:
hadron s0 - s2
M



=
0

ds(s)e

For p2 > 0, the correlation function is calculated in terms of hadronic parameters. In the deep Euclidean region, p2 << 0, the correlation function is calculated using the OPE in terms of QCD degrees of freedom. Sum rules are obtained by matching the two representation using spectral representation..

Ulas Ÿzdem ?

- isovector axial form factors in QCD


- isovector axial form factors in QCD

Introduction Light-cone QCD sum rules - Transition in QCD Results Conclusion

Hadronic Representation

÷

(p, q ) = i

d 4x e

iqx

A 0|T J÷ (0)J (x ) |(p)

In this par t, we show how to correlation function is related to the physically obser ved hadrons. Inser ting a complete set of intermediate hadronic states with the same quantum number as inter polating currents, (p, q ) =
s

÷

0|J÷ |(p , s ) (p , s )|J |(p, s) + ... 2 m - p 2

where; 0|J÷ (0)| = ÷ (s , p ) -i (p , s ) g 2 + qq 2 4M q 5 2M

(p , s )|A (x )|(p, s) =



A A g1 (q 2 ) 5 + g3 (q 2 )

A A h1 (q 2 ) 5 + h3 (q 2 )

q 5 2M

(p, s)

Ulas Ÿzdem ?

- isovector axial form factors in QCD


- isovector axial form factors in QCD

Introduction Light-cone QCD sum rules - Transition in QCD Results Conclusion

Hadronic Representation
Hadronic side of the correlation function;
÷

=i


2 2(M



- p 2)

-

2 (3M + 4q 2 )

24

5 M

A h3 (q 2 ) -

A g3 (q 2 ) 3 3M

q÷ q q 5 q /




4 2 A + g1 (q 2 ) - q÷ 5 + p 5 ÷ + q 5 ÷ - / q÷ q 5 / 3 3M
A A - g3 (q 2 ) + 2g1 (q 2 ) q 5 ÷ + 2 (3M - 4q 2 ) 4 6M A g3 (q 2 ) A 2g1 (q 2 ) 2 (M + 2q 2 )

6

4 M

A h1 (q 2 ) +

A 2g1 (q 2 ) 2 3M

q÷ q 5 q /



+

A h1 (q 2 ) +

3

2 M

q÷ p 5 q -
A h1 (q 2 )

A h1 (q 2 ) 3 M

q÷ p q 5 q /



-

2M -

q q 5 ÷ + /

-

2 (3M + 2q 2 ) 3 12M

q÷ 5 q

+ -

2 (-3M + 5q 2 ) 4 12M

A h3 (q 2 ) -

2 (5M + 2q 2 ) 4 6M

A h1 (q 2 ) -

A 4g1 (q 2 ) 2 3M

q÷ q 5 q



2g3 (q 2 ) q÷ q 5 q 2 3M



Ulas Ÿzdem ?

- isovector axial form factors in QCD


- isovector axial form factors in QCD

Introduction Light-cone QCD sum rules - Transition in QCD Results Conclusion

QCD Representation

Correlation Function
÷

(p, q ) = i

d 4x e

iqx

A 0|T J÷ (0)J (x ) |(p)

where;
J÷ (0)= 1 abc 3 1 2

[2(u

aT

(0)C ÷ d b (0))u c (x ) + (u

aT

(0)C ÷ u b (0))d c (0)]

A J (x ) =

? ? u d (x ) 5 u d (x ) - d e (x ) 5 d e (x )

Ulas Ÿzdem ?

- isovector axial form factors in QCD


- isovector axial form factors in QCD

Introduction Light-cone QCD sum rules - Transition in QCD Results Conclusion

QCD Representation
QCD side of the correlation function: i 16 3
4 iqx abc a b c



÷

=

d xe


( C ÷ )
b



( 5 )


4

0|u (x 1)u (x 2)d (x 3)|(p, s) + S (-x )


2 S (-x ) -4 Where; S (x ) = ? qq ix / - 2 2 x 4 12
abc a b abc a

+ 2 S (-x )
c



+ S (-x )








0|u (x 1)u (x 2)d (x 3)|(p, s)

2 S (-x ) + S (-x )



1+

2 m0 x 16

2

1

- igs
0

d

x / G÷ 16 2 x 4

÷

- x G

÷

÷





i 4 2 x

2

.

0|

u (x 1)u (x 2)d (x 3)|(p) =

c

f ÷ ÷ V (xi )M (÷ C ) + A(xi )M (÷ 5 C )(5 ) 4
÷

+ T (xi )(i ÷ p C )

*C.E. Carlson and J. L. Poor., PRD 38, 1988
Ulas Ÿzdem ? - isovector axial form factors in QCD


- isovector axial form factors in QCD

Introduction Light-cone QCD sum rules - Transition in QCD Results Conclusion

Results
The QCD sum rules are obtained by matching the shor t-distance (QCD side) of the correlation function with the long-distance (phenomenological side) calculation and taking the fourier transformations, we obtain;
For the - axial transition
A 2

g1 (q )

M - p

2

f M =- 3
1

1

dx2
0

1 (q - px2 )2
1-x3 0

1-x2 0

dx1 4V (x1 , x2 , 1 - x1 - x2 )

-
0

dx3

1 (q - px3 )2
1

dx1 [-T + A - 2V ](x1 , 1 - x1 - x3 , x3 )

g3 (q )

A

2

M - p

2

f M =- 3
1

dx2
0

1 (q - px2 )2
1-x3 0

1-x2 0

dx1 [2T + 4A + 8V ](x1 , x2 , 1 - x1 - x2 )

+
0

dx3

1 (q - px3 )

2

dx1 [-3T + 3A + 2V ](x1 , 1 - x1 - x3 , x3 ) ,

Ulas Ÿzdem ?

- isovector axial form factors in QCD


- isovector axial form factors in QCD

Introduction Light-cone QCD sum rules - Transition in QCD Results Conclusion

Results
0.8 0.7 g1 (Q = 2.0 GeV ) 0.6 0.5 0.4 0.3 0.2 0.1 2 4
2
2

s0 = 2.0 GeV s0 = 2.5 GeV s0 = 3.0 GeV s0 = 3.5 GeV s0 = 4.0 GeV

2 2 2

0.4 0.35 g1 (Q = 4.0 GeV ) 0.3 0.25 0.2 0.15 0.1 0.05
2

s0 = 2.0 GeV s0 = 2.5 GeV s0 = 3.0 GeV s0 = 3.5 GeV s0 = 4.0 GeV

2 2 2 2 2

2 2

2

2

6

8
2

10

0

2

4
2

6

8
2

10

M (GeV )
2.4 2.2 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2
s0 = 2.0 GeV s0 = 2.5 GeV s0 = 3.0 GeV s0 = 3.5 GeV s0 = 4.0 GeV
2 2 2 2 2

M (GeV )
0.72
s0 = 2.0 GeV s0 = 2.5 GeV s0 = 3.0 GeV s0 = 3.5 GeV s0 = 4.0 GeV
2 2 2 2 2

g3(Q = 2.0 GeV )

g3(Q = 4.0 GeV )

2

0.63 0.54 0.45 0.36 0.27 0.18 0.09

2

2

4
2

6

8
2

10

2

2

2

4
2

6

8
2

10

M (GeV )

M (GeV )

2 Figure: The dependence of the form factors; on the Borel parameter squared MB for the values of the continuum threshold s0 = 2.0 GeV 2 , s0 = 2.5 GeV 2 s0 = 3.0 GeV 2 , s0 = 3.5 GeV 2 and A A s0 = 4.0 GeV 2 and Q 2 = 2.0 and 4.0 GeV 2 , for g1 and g3 axial form factors.

Ulas Ÿzdem ?

- isovector axial form factors in QCD


- isovector axial form factors in QCD

Introduction Light-cone QCD sum rules - Transition in QCD Results Conclusion

Results
0.8 0.7 0.6 0.5

s0 = 2.0 GeV s0 = 2.5 GeV s0 = 3.0 GeV

2 2 2

g1(Q )

2

0.4 0.3 0.2 0.1 0 0 2 4
2 2

6

8

10

Q (GeV )
2.5 2 1.5 1 0.5 0 0 2 4
2

s0 = 2.0 GeV s0 = 2.5 GeV s0 = 3.0 GeV

2 2 2

g3 (Q )

2

6
2

8

10

Q (GeV )

Figure: The dependence of the form factors for the values of the continuum threshold A A s0 = 2.0 GeV 2 , s0 = 2.5 GeV 2 , s0 = 3.0 GeV 2 and M 2 = 3.0 for g1 and g3 axial form factors.

Ulas Ÿzdem ?

- isovector axial form factors in QCD


- isovector axial form factors in QCD

Introduction Light-cone QCD sum rules - Transition in QCD Results Conclusion

Results

Axial Charge; 2 Fit function; gA (q 2 ) = gA (0) exp[-Q 2 /MA ] Fit Region [1.0 - [1.5 - [2.0 - (GeV2 ) 10] 10] 10] gA (0) 3.48 2.64 2.10 MA (GeV) 1.15 1.24 1.32

Table: The values of exponential fit parameters, gA and MA for axial form factors.

Ulas Ÿzdem ?

- isovector axial form factors in QCD


- isovector axial form factors in QCD

Introduction Light-cone QCD sum rules - Transition in QCD Results Conclusion

Results

[1] gA -1.9 + 0.1

[2] -4.50

[3] -4.47

[4] -4.48

[5] -4.30

This Work -2.70 + 0.6

Table: Different results from theoretical models which are Lattice QCD, ChPT, quark models and also our model.

12345-

Alexandrou et al., PRD 87, 2013. Jiang et al.,PRD 78, 2008. Theussl et al., EPJC 12, 2001. Glozman et al., PRD 58, 1998. Glantshnig et al., EPJA 23, 2005.
Ulas Ÿzdem ? - isovector axial form factors in QCD


- isovector axial form factors in QCD

Introduction Light-cone QCD sum rules - Transition in QCD Results Conclusion

Conclusion

In this study, we have calculated the isovector axial vector form factors for bar yon within the LCSR method.
A A AA We extract axial g1 , g3 form factors. The form factors h1 , h3 could not obtain from our approach since the necessar y DAs have not been known yet.

We compared our fit results with the lattice QCD, ChPT and quark models. Unfor tunately, there is no experimental data yet to compare our results within this region.

Ulas Ÿzdem ?

- isovector axial form factors in QCD


- isovector axial form factors in QCD

Introduction Light-cone QCD sum rules - Transition in QCD Results Conclusion

Thank you......

Ulas Ÿzdem ?

- isovector axial form factors in QCD