1. NUMERICAL INVESTIGATION OF BOUND RELATIVISTIC TWO-PARTICLE SYSTEMS WITH ONE-BOSON EXCHANGE POTENTIAL Yu.A. Grishechkin and V.N. Kapshai Francisk Skorina Gomel State University

2. Two-particle equations of quasipotential type Integral equations for bound s-states in the relativistic configurational representation (RCR) One-boson exchange potentials and their superposition Numerical method of solving Spectrum of the orthopositronium Decay width of the parapositronium Plan of the talk

3. Two-particle equations of quasipotential type − Logunov-Tavkhelidze equation − Kadyshevsky equation − Logunov-Tavkhelidze modified equation − Kadyshevsky modified equation 2E – two-particle system energy Integral equations for bound states in the momentum representation

4. Relativistic configurational representation RCR is as expansion over functions Transformations of the two-particle wave function of relative motion – from the momentum representation to the RCR – from the RCR to the momentum representation For s-states this transformation analogous the Fourier transformation where χ – is the rapidity, connected with momentum by relation

5. Integral two-particle equations for bound s-states in the RCR j = 1 Green functions in the RCR: j = 2 j = 3 j = 4 − Logunov-Tavkhelidze equation − Kadyshevsky equation − Logunov-Tavkhelidze modified equation − Kadyshevsky modified equation r – is radius-vector modulus in the RCR – normalization condition of wave function

6. Two-particle equations for bound s-states in the momentum representation Logunov-Tavkhelidze equation: Kadyshevsky equation: Normalization conditions of wave functions:

7. One-boson exchange potentials in the momentum representation One of the first one-boson exchange potentials for two scalar particles obtained in framework quasipotential approach is This potential was obtained on the basis of diagram technique of the quantum field theory Hamiltonian formulation [1]. Mass of exchange boson is equal to zero. For s-states this potential has the form [1] Kadyshevsky V.G. Quasipotential type equation for the relativistic scattering amplitude / Nucl. Phys. – 1968.– V.B6, №1. – P.125-148.

8. In article [2] one-boson exchange potentials were obtained on the basis of retarded and causal Green functions calculation. where [2] Капшай В.Н., Саврин В.И., Скачков Н.Б. О зависимости квазипотенциала от полной энергии двухчастичной системы / – ТМФ, 1986. – Т.69, №3. – С. 400-410.

9. Potentials of two fermions interaction in the cases of different total spin and total angular momentum values were obtained in article [3] as coefficient of general three-dimensional potential for two fermions [4] expansion into the spherical spinors. Total spin of system is equal to zero: [3] Двоеглазов В.В, Скачков Н.Б., Тюхтяев Ю.Н., Худяков С.В. Релятивистские парциальные интегральные уравнения для волновой функции системы двух фермионов / – ЯФ, 1991. – Т.54, №3. – С. 658-668. [4] Архипов А.А. Приближение одноглюонного обмена для квазипотенциала взаимодействия двух кварков в квантовой хромодинамике / – ТМФ, 1990. – Т.83, №3. – С. 358-373.

10. Total spin of system is equal to one: where

11. Potentials in the RCR One-boson exchange potential Parameter α is associated with mass of exchange boson µ Mass of exchange boson is equal to zero Superposition of two one-boson exchange potentials with masses are equal to zero and 2m Coulomb potential in the RCR

12. Numerical method of equations solving in the momentum representation For solving of equations the rectangles quadrature method was used after replaced the variable Nonlinear energy eigenvalue problem for algebraic equation systems. For solving this problem the iteration method was used [5]. Initial value energy 2E (0) [5] T.M. Solov’eva Numerical Calculation of the Energy Spectrum of a Two-Fermion System / Comp. Phys. Comm., 136 (2001), p. 208-2011. To find the eigenvalues of the matrices one can use standard methods. Process has to be continued until the condition holds, ε – accuracy. Then, the Richardson extrapolation process is applied to energy values 2E, and normalized wave functions, obtained on two grids N and 2N.

13. Numerical method of equations solving in the RCR To find solutions of integral equations in the RCR we used the composite Gauss quadrature method. Nonlinear energy eigenvalue problem for algebraic equation systems. For solving this problem the iteration method was used too. Initial value w (0) (energy 2E (0) =2m cos(w (0) ) ) Process has to be continued until the condition holds, ε – accuracy.

14. Results of equations solving in the momentum representation Wave functions for Logunov-Tavkhelidze equation for potential Coupling constant is equal to the fine structure constant

15. Results for spectrum

16. Convergence of the energy spectrum for scalar particles

17. Experimental date for frequency of transition from the ground state to the first excited for orthopositronium: 1233607216.4(3.2)МHz.

18. Results for spectrum Method was tested for solving modified Logunov-Tavkhelidse equation in the case of potential Exact expression for quantization condition of energy Results of equations solving in the RCR

19. Experimental date for frequency of transition from the ground state to the first excited for orthopositronium: 1233607216.4(3.2)МHz. Frequency of transition obtained by solving the Schrödinger equation with the Coulomb potential: 1233690736MHz. Frequency of transition obtained from the quantization condition 2E n =(4m 2 -λ 2 /n 2 ) 1/2 : 1233695868MHz. The solution of the Kadyshevsky equation with the potential V(r)=tanh(πmr/2)/r gives the best agreement with experimental data.

20. Decay width for two-particle systems The decay width for systems of two scalar particles into two photons [6]: [6] Г.А. Козлов О распаде связанного состояния µ + µ - -пары в e + e - далитц-пару и γ - квант / ТМФ, 60 (1984), №1, с. 24-36.

21. The substitution of non-relativistic wave function for Coulomb potential into this formula gives value: The substitution of exact wave function for potential gives value: The experimental value for parapositronium:

22. Conclusions Our methods allows to solve effectively two-particle integral equations in the momentum representation and in the RCR for slowly decreasing potentials like Coulomb potential The wave functions in the RCR allow to calculate decay width simply The results of numerical solution for energy spectrum and decay width in the case of one-boson exchange potential and superposition of such potentials give good agreement with the experimental values for positronium. Herewith results for phenomenological potentials coincide with experimental values better then results for potentials which derived strongly.