1. Static Polarizability of Mesons in the Quark Model N.V. Maksimenko, S.M. Kuchin

2. Introduction Calculation of the electromagnetic characteristics of the bound systems gives a possibility to study the interaction between the particles of the composite system. Electric polarizability is one of such characteristics. The electromagnetic characteristics of the hadrons taking into account their quark structure may be determined on the basis of the equation of the motion of a composite system in the external electromagnetic field. The modern methods of solutions of equations are not sufficient for accurate calculations, but they give a possibility to acquire the wave function of the basic state of the considered system. An approximate solution turns into finding corrections to the wave function. Finding corrections to the wave functions by the Raley-Schrödinger method requires knowledge of all proper functions of an unperturbed equation. In the case of potential interaction, however, it is possible to suggest method, in which the solutions of an unperturbed equation with the only value of energy are used. The result for the correction to the wave function of an arbitrary discrete state is expressed in the terms of the unperturbed wave function of this state and, in contrast to the Raley- Schrödinger theory of perturbations, it doesn’t require knowledge of the whole spectrum of an unperturbed task.

3. The Y.B. Zeldovich Theory of Perturbation. The Y.B. Zeldovich theory of perturbation method has already been applied for finding corrections to the wave functions when the perturbation is spherical-symmetrical. However, if a bound system is found, for example, in the external electric field, the perturbation in the capacity of which the electric dipole interaction is considered depends not only on r but also on the angle between the radius-vector of the system and external field. Using the Y.B. Zeldovich theory of perturbation we work out a method for finding the correction to the wave function of the system when the perturbation depends also on an angle.

4. In this case we write the Schrödinger equation down in a form of:

5. Now we find an approximate solution of the last equation in a form of series:

6. Where

7. As, introducing the designation and cancelling by we acquire the equation (1.1)

8. The general solution of the equation we find in the form of where and are for the present unknown functions which are determined from the system of equations

9. Relative to, system of equations is the system of two linear inhomogeneous algebraic equations, and the main determiner of this system at that Therefore system of equations has the only solution which we find by the Kramer method and finally we acquire:

10. Whence

11. we present the solution of inhomogeneous equation in the form of (1.2): The linear independence with solution of we find by Abel formula and acquire:

12. Substituting the given expression in (1.2) we acquire (1.3):

13. Let us consider equation (1.1). We write the respective homogeneous equation down (1.4): Joining the terms of the form and designating in equation (1.1) we may rewrite it in the form of (1.5):

14. Equation (1.5) coincides with the equation for an unperturbed task when is substituted for where

15. Determination of the Polarizability of Charged π - Mesons Let us apply the considered above method for finding the correction to the wave function of the bound system which is in the external electric field. The wave function of the basic state has the form of:

16. For the considered case the proper functions in equation (1.3) have the form of: where

17. With taking into account we find that (1.6) Substituting (1.6) in (1.3) we find:

18. Then taking into account rationing factor and the condition of the limitedness of the wave function finally we acquire:

19. We determine the static polarizability from the following expression: where is the dipole electric interaction of the system with the external field E.

21. The analytically results of the calculation of the electric polarizabilities

22. The final results of the calculation of the electric polarizabilities of mesons are presented in the following table

23. Conclusion Thus above-stated methodology makes it possible to find corrections to wave functions of bound systems found in the external electric field that in its turn gives the possibility to calculate numerically the electromagnetic characteristics of mesons like, for example, the electric polarizability. The result for the correction to the wave function of the discrete state is expressed in the terms of the unperturbed wave function of this state and in contrast to the Raley-Schrödinger theory of perturbations it doesn’t require knowing the whole spectrum of an unperturbed task.

24. Thank you for attention!