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FOURTEENTH LOMONOSOV CONFERENCE ON ELEMENTARY PARTICLE PHYSICS Moscow, 19 ­ 25 August, 2009

Electromagnetic structure functions of nucleons in the region of very small x
E.V. Bugaev (INR, Moscow) B.V. Mangazeev (Irkutsk State Univ.)
21 August, 2009


Electromagnetic structure functions of
2 2 ) The electromagnetic structure functions of nucleons 1 2 and are connected with the total absorption cross sections of a virtual photon with transverse or longitudinal polarization by the simple relations:

nucleons and the total photoabsorption cross sections F (Q , ) F (Q ,

Where is squared transferred momentum, of the photon,



is lab system energy

In turn the absorption of the virtual photon by the nucleon by the optical theorem is connected with the Compton forward scattering of a virtual photon 2


Compton scattering amplitude

This is GVDM (General Vector Dominance Model ) approach for Compton scattering amplitude.

And this diagram represents schematically q of the contribution q - channel in Compton scattering amplitude.

Perturbative QCD models are not sufficient for a description of electromagnetic structure We know that GVDM is functions in the region of small Bjorken x and inefficient for large Q2. Q2. Therefore two component approach is needed (GVDM /soft component/+QCD /hard component/). 3


For GVDM approach (which is developed below) we need some model of vector mesons. The following two modern approaches for a description of hadrons are typical:

Modern approaches for a description of hadrons

1) holographic dual of QCD ("hQCD"),
e.g., T.Sakai, S.Sugimoto, hep-th/0507073; J.Hirn, V.Sanz, hep-ph/0507049; A.Karch et al, hep-ph/0602229.

2) dimensionally deconstructed QCD ("ddQCD"),
e.g., D.Son, M.Stephanov, Phys.Rev.D69:065020,2004. The important predictions of the modern QCD models are followings:

1)The family of vector mesons with infinite numbers of particles: , 1, 2, ..., n, ... , etc. 2)The concrete mass spectrum of vector mesons. In the models with "soft-wall" square of meson mass is proportional to the number of meson .
4


Mass spectrum of vector mesons
In models with "soft-wall" square of meson mass is proportional to number of meson

This prediction is satisfactorily agree with experiment (A.Karch et al, hepph/0602229)

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Model of the hadronic amplitudes (Bugaev at al,hepph/9912384)

(3D-reduction of BS equation)
We used the simplest model of VN-scattering: two-gluon exchange approximation. For a calculation of the corresponding diagrams one must know, in particular, - qq -wave functions of the mesons. In a relativistic constituent quark model these wave functions are obtained from the Bethe-Salpeter (BS) equation (we consider the case of scalar identical quarks):

For the utilization of this equation it is convenient to use the quasipotential formalism in a light-front form (see, e.g., W.Jaus, Phys.Rev. D41(1990)3394, S.Chakrabarty et al., Progr.Part.Nucl.Phys. 22(1989)43.). The corresponding reduction of the BS-equation leads to the threedimensional equation

Where momentum".

is three-dimensional so called "inner

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Model of the hadronic amplitudes
(harmonic oscillator kernel)
To solve this equation we assume that the kernel K has only the long range confining term of the hadronic oscillator type (S.Chakrabarty et al., Progr.Part.Nucl.Phys. 22(1989)43):

with two parameters: (a "spring constant") and a zero-point energy 0. Then the equation formally coincides with the equation for a quantum-mechanical 3D-oscillator:

Solutions of this equation, its eigenfunctions and eigenvalues are well known. We will use them for a description of the -family. 7


Model of the hadronic amplitudes
(mass spectrum)
The mass spectrum of radial excitations is given by the ratio :

and has the linear form

2 mn = a + bn

For

m ' = 1.33GeV

we obtain

(We assume that 2 qq / 2 M is a constant (i.e., is independent on M). In this case the meson mass spectrum has this form.) With the numerical value m ' = 1.33 GeV mq = 0.3 GeV one has

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Model of the hadronic amplitudes
(wave functions and amplitudes)
This is -meson wave function From the transverse momentum distance between quarks ). the expression for wave function The VnN-scattering amplitude (diagonal amplitude ) has the form , we turn to the dual variable (transverse In this representation one has

Here,

is an amplitude for the scattering of the qq -pair with a fixed

on the nucleon (in the two-gluon exchange approximation). The V-factor describes the ggNN-vertex.

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Nondiagonal contributions and cross sections
Similarly nondiagonal amplitude is given by the expression According to the GVDM total absorption cross sections of a virtual photon with transverse or longitudinal polarization has the form

We assume, as usual, that the coupling constants fn are proportional to the masses of mesons

Now we can calculate the contributions of diagonal and nondiagonal transitions in cross-sections. 10


The contributions of the diagonal and nondiagonal transitions in a cross-section of the real photon Vn + N Vn ' + N (mkbn)

n
0 1 2 3 4 5 6 7 8

n'

0

1

2

3

4

5

6

7

8
0.2 0.2 0.8 2.4

70.9 20.3 -6.7 -6.7 10.0 29.8 2.4 -4.2 -1.5 -0.5 -0.2 0.7 -1.1 0.3 -0.4 1.4 -3.2 0.4 -0.9 0.2 -0.4

2.4 -1.5 6.8 -3.2 5.2 19.9 1.0 -2.3 0.2 -0.8

0.7 -0.5 1.2 -0.9 4.1 -2.3 3.4 15.2 0.8 -1.8

0.3 -0.2 0.3 -0.4 0.9 -0.8 2.8 -1.8 2.4 12.3

20.3 40.8 10.0 -4.2 6.8 23.9 1.2 -2.6 0.3 -0.8

1.4 -1.1 5.2 -2.6 4.1 17.2 0.9 -2.0

0.4 -0.4 1.0 -0.8 3.4 -2.0 2.8 13.6

T T

= 317 mkbn Experimental 114 mkbn
Calculated

The results of calculations for real photon are presented in this table. As the table shows the calculated cross section significantly exceeds the values known from experiment.

We see that the destructive interference effects and corresponding cancellations of Vn + N Vn ' + N amplitudes inside of VDM sums are small. In other words the approach of non-diagonal VDM alone cannot describe the data. As a result some modification of the standard VDM scheme is needed: cut-off factors reducing the probability of initial -V transitions must be introduced. (Bugaev, Mangazeev, Shlepin, hep-ph/9912384, Bugaev, Shlepin, Phys.Rev.D67:034027,2003)


Cut-off factors
The first stage of the photoabsorption process is the gamma-qq -transition. The differential probability of this transition is given by this expression

x is the fraction of the photon 3-momentum carried by the quark. An invariant mass of the qq -pair is this Is the transverse momentum. It follows from here that, at fixed invariant mass of the qq -pair of pair's phase volume having smaller then is given by this expression the relative part

Where is cut-off factor, which is proportional to the part of pair's phase volume More accurate expressions are given in the paper Bugaev, Shlepin, Phys.Rev.D67:034027,2003) We assume that vector meson forms if (and only if) the value of the quark is smaller than .. The value of is the model parameter. 12


GVDM formulas modification
To take the cut-off into account in GVDM formulas the cut-off factors must be introduced: (Bugaev, Mangazeev, Shlepin, hep-ph/9912384)

Here

are cut-off factors. (Bugaev, Shlepin, Phys.Rev.D67:034027,2003) we have chosen the value -meson mass divided

For the magnitude of by two

p

max

= M / 2 = 0.385 GeV.
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The contributions transitions in a crossVn + N Vn ' + N section of the real photon (mkbn) with accounting of cut-off factors

n

n'
0 0 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8
70.94 10.16 -2.61 0.78 -0.44 0.18 -0.13 0.07 -0.04 10.16 10.27 1.94 -0.68 0.21 -0.14 0.05 -0.04 0.02 -2.61 1.94 4.48 0.86 -0.36 0.12 -0.09 0.03 -0.03 0.78 -0.68 0.86 2.56 0.49 -0.23 0.08 -0.06 0.02 -0.44 0.21 -0.36 0.49 1.66 0.31 -0.16 0.06 -0.05 0.18 -0.14 0.12 -0.23 0.31 1.17 0.21 -0.12 0.04 -0.13 0.05 -0.09 0.08 -0.16 0.21 0.88 0.15 -0.09 0.07 -0.04 0.03 -0.06 0.06 -0.12 0.15 0.68 0.11 -0.04 0.02 -0.03 0.02 -0.05 0.04 -0.09 0.11 0.54

T T

= 114 mkbn Experimental 114 mkbn
Calculated

We see that the introduction of cut-offs motivated by QCD leads to the correct typical value of the calculated photoabsorption cross section. 14


Energy dependence of cross section (soft component) and hard component
For GVDM (soft) component the energy dependence was chosen in the Regge-type form

1.29 s p ( s ) = 114 * + s 1600

0.06

.

( in bn, s in GeV2)

For the hard component, we used the color dipole model and the parameterization of the dipole cross section (p(QCD part) from the work by J.Forshaw, G.Kerley and G.Shaw, Phys.Rev.D60, 074012 (1999).

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Result of the calculations for the photoabsorption cross section of real photon The result of the calculations for the photoabsorption cross section of
real photon is shown on the picture:

Blue line is the soft (GVDM) component. Red line is the total (soft + hard).

The sum of two components agrees with the experimental data.

s , GeV
16


Result of the calculations for the structurefunction F2 This picture presents our predictions
for the structure function F2.

F2 ( x, Q 2 )

Particle Data Group, 2007 Blue line is the soft (GVDM) component. Red line is the total (soft + hard). To achieve a good agreement with the experimental data accounting for the hard contribution is required.

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Finally, here only the sum of soft and hard parts is represented.

Result of the calculations for the structure function F2

F2 ( x, Q 2 )

Particle Data Group, 2007 Red line is the total (soft + hard). The good agreement with the available data in the region of small x (x < 0.003, Q2 < 100 GeV2) is obtained. 18


CONCLUSIONS
1. If no cut-offs are introduced, GVDM is not able to describe photoabsorption data. 2. The introducing of the cut-off factors motivated by QCD can give the correct predictions. 3. To achieve a good agreement with the experimental data accounting for the hard contribution is required. 4. The present model has, for a description of the soft component, the minimum number of parameters, in fact, only the parameter . 5. The good agreement with the available data in the region of small x (x < 0.003, Q2 < 100 GeV2) is obtained.