1. Maksimenko N.V., Vakulina E.V., Deryuzkova О.М. Kuchin S.М. GSU, GOMEL The Amplitude of the Сompton Scattering of the Low-Energy Photons at Quark-Antiquark System on the Basis of the Bethe-Salpeter Equation with Instantaneous Interaction.

2. In the basis of the low-energy theorems for interaction of the photons with hadrons lay common relativistic principles of quantum field theory. Therefore these theorems concern to the fundamental theorems. At the same time hadrons are composite quark systems. Therefore the research of the process of the photons interaction with quark systems allows to obtain new information on structure of hadrons in the region of energies, where methods of the theories of perturbation quantum chromodynamics can not be sequentially used. For the description of properties of bound systems in quantum field theory at present time the covariant relativistic Bethe-salpeter equation, is mainly applied. However its practical use is associated with a lot of difficulties. Therefore, solution of the concrete tasks and physical intrpretation of the obtained solutions are limited to reduction of BSE three-dimensional form, in which the vertex function of the system of two particles is projected on the equal times surface. Such approach has shown to be effective and for the description of bound states of particles in an external electromagnetic field.

3. In the given work the Сompton scattering amplitude on the two quark system at low energies is obtained on the basis of the diagram approach in the simultaneous approximations of the meson-quark transition vertex functions, that allows to express factors of low- energy decomposition of the amplitude in frequence of radiation trough wave functions of compound system.

4. In the given work, the tensor of the Сompton scattering at quark-antiquark system in the relativistic impulse approximation can be presented as follows (1) In this expression the following denotations are entered: is the quark propagator, is the quark charge with which a photon interacts, is the Dirac matrix, are vertex functions of meson-quark transition at the initial and terminal states, и are impulses of quark, antiquark, falling and dispersed accordingly.

5. On the basis of conservation laws of four impulses also the following denotations are entered. Let us introduce variables from which it follows, that. Let us take into consideration that the vertex function doesn’t depend on Such approximation is valid in the case when the potential of iteraction between quarks doesn’t depend on.

6. Let us select an integral in (1) where the indexes a, c, d, e, f, g and 1 take values 1, 2, 3, and 4 and the matrix. In fact, the function is a Green two particles function in the two-times approximation of the photon scattering on one of the quarks. Let us put down the equation (2) in a more compact form. (2), defined in the expression (2), (3)

7. Now let us calculate the tensor that is we take into consideration that the impulse of the photon k is a little value in comparison with rest energy of a quark. Then with the help of the relation (4) in the low energy approximation, we can transform the propagators and in the expression (3) on the four-measured impulse k. First let us inspect a more simple expression of the form (5) Being limited by the second order of transformation on k with the help of the rellation (4) we obtain (6)

8. In this case the equation (5) has the form (7) To calculate let us present the propagators and can be presented as follows: (8) (9)

9. , The greatest contributions to the integrals of the terms of propagators of the forms (8) and (9): (10) (11) in (7) are given by the products In the right part of the expression (7) we use the Gordon identity where. (12) Using (12), (10) and (11) the expression can obtain the form (13) (7)

10. where. In the transformation (13) in brackets there is the sum of the geometric progression (14) Using (14), the Green function (13) can be presented in the form of (15)

11. With the help of the Green function (15)let us set the relation of the vertex function - with the simultaneous wave function of the system. where (16) Let us take into consideration that the vertex function doesn’t depend on In this case according to (15) we obtain (17) Let the vertex have the form (18)

12. Then The function of the expression (17) can be presented as follows In the equation (19) the function is defined (20) (19) With the help of the bond of the scalar part of the vertex function and the wave function of the compound system in the relativistic impulse approximation and define normalization of the function. we can calculate a form-factor Indeed, if the with the compound system ( is a vector of polarization of a photon) is the amplitude of interaction of a photon

13. can be presented as follows In the equation (21) is a formfactor of the system, and the vertex have the following form (22) (23) (21) functions g and If in the equation (21) we take into consideration the evident form of the,(22) and (23) accordingly, in such case functions g and

14. In the system of rest target the following cinematic relations are valid Let us redefine in (24) the wave functions Then the equation (24) can be presented as follows From the relation (26) is seen that under the condition of the normalization of function (27) (24) (25) (26)

15. Now with an analogous method we can calculate green function of tensor of the Compton scattering (1) The two-times Green function for tensor of the expression has the form In the expression (28) the following indications are introduced: and falling photons accordingly. (28) are the vectors of polarisation and the impulses of the dispersed

16. In the system of rest target the amplitude of the Compton scattering In formula (29) and are the quark charges,. Thus, knowing the wave functions Compton scattering in the field of low energies ( with the definite electromagnetic characteristics of the bound system. in the impulse approximation has the form (29) we can define the amplitude of the ), and therefore also the coefficients of the low-energy transformation,which are bound