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How accurate is the local-duality model for the pion elastic form factor?
Irina Balakireva
SINP, Moscow State University
We study the accuracy of the pion form factor, obtained with a local-duality (LD) version of dispersive sum rules. To probe this accuracy, we make use of a potential model, where the exact form factor may be calculated from the solution of the SchrÆdinger equation and confronted with the local-duality form factor. The deviation between these quantities is found to be below 20% in the region Q>2-3 GeV, independently of the specific form of the confining potential. We argue that the LD model for elastic form factors in QCD has at least this level of accuracy.

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Motivation
The pion elastic form factor at Q = 3 - 8 GeV is sensitive to the specific features of the onset of the perturbative regime and opens the possibility to study the subtle details of the pion structure. The experimental study of the pion form factor in this region will become available with an upgrade of the JLab in the next few years. The theoretical description of the pion form factor at large Q directly from QCD is a complicated problem. Several versions of the method of QCD sum rules have been applied to this problem. We still have a strong discrepancy between the results from various versions. Our goal: to study the accuracy of the pion form factor obtained from socalled local-duality (LD) version of QCD sum rules.

2


Two- and three-point correlators
· Two-point correlator

p2 =


()

Tj ( x ) j + (0 ) eipx dx

is the physical vacuum; here j(x) is a short-hand notation for the interpolating current j5(x) of the positively charged pion j5 (0 ) ( p ) = ip f ; for brevity, we omit Lorentz indices.

·

Three-point correlator

2 p12 , p2 , q 2 =
Here q = p1 - p

(

)

Tj ( x1 )J (0 ) j + ( x2 ) e

ip1 x1 -ip 2 x

2

dx1dx

2

2

; J(0) denotes the electromagnetic current J(0).

We consider the case of space-like momentum transfers:

Q 2 = -q 2 > 0
3


The Borel transform for two-point correlator
Next: Borel transform of the correlators; Green functions in Minkowski z evolution operators in Euclidean The · · · Borel transform leads to several improvements: Suppresses the contributions of the excited states; Improves the convergence of the perturbative expansion; Provides the necessary smearing required by quark-hadron duality.

The Borel transform

p 2 for two-point correlator has the form:

OPE

( ) =
0



pert

(s )

e

- s

ds +

cond 2 s

( ),



pert

(s ) = 0 (s ) + s 1 (s )

+O

()

r are spectral densities of two-point diagrams of perturbation theory:

Making use of hadron intermediate states for two-point correlator gives:

( ) = f e

2 2 - m

+ excited states
4


The double Borel transform for three-point correlator
·
2 2 The double Borel transform p1 ; p2 for the three-point 2 2





correlator has the form:

( , Q ) =
·
pert




00



pert

(

s1 , s2 , Q ) e

-

s1 2

e

-

s2 2

ds1ds2 + cond ( , Q

)

(

s1 , s2 , Q ) = 0 (s1 , s2 , Q ) + s 1 (s1 , s2 , Q ) + O

()
2 s

i are the double spectral densities of three-point diagrams of the perturbation theory:



Making use of hadron states for three-point correlator we obtained:

( , Q ) = F (Q ) f e

2 2 - m

+ excited states
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Duality assumption
The contribution of the excited states is dual to the highenergy region of the diagrams of the perturbation theory.
Using this assumption the sum rules take the form: for the two-point correlator:
s
eff

( )

f2 e

- m
2

=


0



pert

(s )e

- s

ds +

sG

2

12

+

176 s q q 2 + 81

2

for the three-point correlator:
seff (Q , ) seff (Q ,

F (Q ) f e

)

2 2 -m

=


0


0



pert

(s1

, s2 , Q )e

s -1 2

e

s -2 2

ds2ds1 +

s G
24

2

4s q q 2 + (13 + Q 2 ) + 81

2

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LD sum rules
F (Q ) f2 e
s - m
2 eff

(Q , )

s

eff

(Q , )

=


0


0



pert

(s1

, s2 , Q )e

s -1 2

e

s -2 2

ds2 ds1 +

s G
24

2

4 s q q 2 + (13 + Q 2 ) + 81

2

There are two ways for considering the region of large Q : · Resummation of power corrections: non-local condensates; · Set = 0 : Local-duality (LD) sum rules. For massless quarks: ­ for two-point correlator:
s
eff

f

2



=


0



pert

s (s )ds = 2 1 + + O ( s2 ); 4
s
eff
s
eff

­ for three-point correlator:

F (Q ) f2 =

s

eff

(Q )
0

(Q )


0



pert

(s1

, s2 , Q )ds1ds2
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Properties of the spectral densities
· At Q 0 the spectral densities of two- and three-point functions are related to each other by the Ward identity:

lim i (s1 , s2 , Q ) = i (s1 ) (s1 - s Q0
lim Q 0 (s1 , s2 , Q ) lim Q 1 (s1 , s2 , Q ) 1 ; Q4 8 = 2 0 (s1 ) 0 (s2 ). Q ~

2

)

·

For Q explicit calculations give:

Properties of the pion form factor in QCD
· Normalization:

F (0) = 1

8 f2 s + · Factorization theorem: F (Q ) = Q2 4 2 f 2 s eff (Q 0 ) = ; s eff (Q ) = 4 If we set s 1+

2

f 2 .

the form factor obtained from the LD sum rule satisfies the correct normalization at Q = 0 and reproduces the asymptotic behavior according to the factorization theorem for the form factor at large Q . These values are not far from each other! So, it is easy to construct an interpolation function seff (Q ) for all Q .
Notice: the model is not expected to work well at small Q because the OPE is not applicable here.



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The local-duality model for hadron elastic form factors
a. Based on a dispersive three-point sum rule at = 0 (i.e. infinitely large Borel mass parameter). In this case all power corrections vanish and the details of the nonperturbative dynamics are hidden in a single quantity - the effective threshold seff (Q ) b. Makes use of a model for seff (Q ) based on a smooth interpolation between its values at Q = 0 determined by the Ward identity and at Q determined by factorization. Since these values are not far from each other, one believes that the details of the interpolation are not essential. For instance, a self-consistent expression may be used:

s

eff

(Q )

=

4 2 f 1+

s (Q



2

)
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Where the accuracy of this model may be tested?
· · · · · The model may be tested in quantum mechanics for the case of the potential containing the Coulomb and Confining interactions. The SchrÆdinger equation may be solved. The exact form factors may be calculated. The LD form factor may be constructed. The results may be compared. The accuracy may be tested. We consider two different confining potentials:

V (r ) = -


r

+ Vconf (r ), Vconf (r ) = r

(1)

Vconf

m 2 r (r ) = 2

2

(2)

Parameters relevant for hadron physics are used.
m = 0.35 GeV , = 0.5 GeV , = 0.168 GeV , = 0.3.

The parameters are chosen such that for both confining potentials the same value of decay constant is obtained. Therefore the LD form factors is the same for both cases.
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Numerical results
· Results for quantum-mechanical potential model for various confining potentials: (a) the exact vs LD form factors; (b) the exact vs LD effective thresholds;
Q FHQL 0.12
4

k ef f HQL 0.8

0.1 0.08 0.06 0.04 0.02
Factoriz ation Q @GeV D LD Sum Rule Vconf ~ r
2

0.75 0.7

Vconf ~ r LD
Fac torization

0.65
Normalization

Vconf ~ r

0.6 0.55 3 3.5 4

Vconf ~ r

2

0.5

1

1.5

2

2.5

Q @GeV D

1

2

3

4

5

6

7

Blue and red lines. Left: the exact form factors obtained from the solution of the Schroedinger equation vs Q for the harmonic oscillator and linear potentials. Right: the exact thresholds which reproduce these form factors by LD expression:

1 F (Q ) = 2 f

k

eff

(Q )
0

k

eff

(Q )




0



pert

(k1

, k 2 , Q )dk1dk

2

(The variable k is related to the variable s, used above by: s = 4(k + m ) ) . Black lines: LD model. The LD model provides a good approximation to the exact thresholds and the form factors at Q bigger than 2-3 GeV. Notice: the exact threshold never exceeds the asymptotic value by more than 10%; this accuracy improves with Q. 11
2 2


Numerical results
· Results for the pion form factor in QCD: (a) various models for the effective threshold;
seffHQ2L 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 2.5 5 7.5 10 12.5 15 17.5 20 Q2 @ GeV
2D

(b) the corresponding pion form factor.
Q2FpHQ2L

Bakulev et al Upper boundary

0.6 0.5 0.4

Upper boundary Bakulev et al

LD

0.3
LD

0.2 0.1 2.5 5 7.5 10
pQCD asymptotics

Q2 @GeV 2D 12.5 15 17.5 20

· ·

·

Black lines: predictions of the LD model for the effective threshold and for the form factor. Red lines: the upper boundary for the threshold and for the form factor (According to the results for the potential model we set the upper boundary for the threshold by 10% higher than the asymptotic threshold). Green lines: the result of the sum rule with non-local condensates. The green threshold exceeds our upper boundary by more than 10% which seems quite improbable taking into account our experience from potential model. Our point of view: the problem here lies in the procedure of extracting the form factor from the correlator with nonlocal condensats, but this should be tested.

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Summary and conclusions
We studied the LD model which may be formulated in any theory where the form factor at large momentum transfers satisfies the factorization theorem (i.e., any theory containing both Coloumb and Confining interactions).

The LD model for the elastic form factor:
·
­ ­

is based on the dispersive three-point sum rule at t=0:
In this case power corrections vanish; the details of the non-perturbative dynamics are hidden in one quantity ­ effective threshold.

·

s makes use of a specific model foreff two close values:
(Q ) Our main conclusions are:
s
eff

based on a smooth interpolation between the

­ ­

s

eff

(Q

0

)

fixed by the Ward identity; determined by factorization.

1.

The LD model provides a good description of the elastic form factor in the region Q>2-3 GeV. The estimate for the upper boundary of the form factor may be obtained by setting:

s
2.

eff

(Q )

1.1 4 2 f



2

3.

In the region Q=(1-2)GeV, the exact effective threshold exhibits rapid variation with Q. The error of the LD form factor in this region may reach 30-40% level. For a relativistic theory a smaller error than for a non-relativistic theory is expected. For the pion form factor, we confirm the accuracy of the LD model at the level better than 20% for Q> 3-4 GeV. We point out that our prediction is considerably lower than the prediction of the approach based on the sum rule with non-local condensates. 13