Large-$j$ Expansion Method for Two-Body Dirac Equation

A. Duviryak

Symmetry, integrability and geometry: methods and applications, Tome 2 (2006)

Résumé

By using symmetry properties, the two-body Dirac equation in coordinate representation is reduced to the coupled pair of radial second-order differential equations. Then the large-$j$ expansion technique is used to solve a bound state problem. Linear-plus-Coulomb potentials of different spin structure are examined in order to describe the asymptotic degeneracy and fine splitting of light meson spectra.

Détail

Keywords: Breit equation, two body Dirac equation, large-$N$ expansion, Regge trajectories.

Bibliographie

[1] Breit G., “The effect of retardation on the interaction of two electrons”, Phys. Rev., 34 (1929), 553–573 | DOI | MR | Zbl

[2] Barut A. O., Komy S., “Derivation of nonperturbative reativistic two-body equation from the action principle in quantum electrodynamics”, Fortschr. Phys., 33 (1985), 309–318 | DOI | MR

[3] Barut A. O., Ünal N., “A new approach to bound-state quantum electrodynamics”, Phys. A, 142 (1987), 467–487 | DOI

[4] Grandy W. T. Jr., Relativistic quantum mechanics of leptons and fields, Kluwer Academic Publishers, Dordrecht–Boston–London, 1991 | MR

[5] Darewych J. W., Di Leo L., “Two-fermion Dirac-like eigenstates of the Coulomb QED Hamiltonian”, J. Phys. A: Math. Gen., 29 (1996), 6817–6841 | DOI | MR | Zbl

[6] Darewych J. W., “Few-particle eigenstates in the Yukawa model”, Condensed Matter Physics, 1:3(15) (1998), 593–604

[7] Darewych J. W., Duviryak A., “Exact few-particle eigenstates in partially reduced QED”, Phys. Rev. A, 66 (2002), 032102, 20 pp., ages ; arXiv:nucl-th/0204006 | DOI

[8] Duviryak A., Darewych J. W., “Variational wave equations of two fermions interacting via scalar, pseudoscalar, vector, pseudovector and tensor fields”, Cent. Eur. J. Phys., 3:3 (2005), 1–17 | MR

[9] Fushchich W. I., Nikitin A. G., “On the new constants of motion for two- and three-particle equations”, J. Phys. A: Math. Gen., 23 (1990), L533–L535 | DOI | MR

[10] Theor. Math. Phys., 88:3 (1991), 960–967 | DOI | MR

[11] Simenog I. V., Turovsky A. I., “A relativistic model of the two-nucleon problem with direct interaction”, Ukraïn. Fiz. Zh., 46 (2001), 391–401 (in Ukrainian)

[12] Simenog I. V., Turovsky A. I., “The model of deuteron in Dirac–Breit approach with direct interaction”, J. Phys. Studies, 8 (2004), 23–34 (in Ukrainian)

[13] Krolikowski W., “Relativistic radial equations for 2 spin-1/2 particles with a static interaction”, Acta Phys. Polon. B, 7 (1976), 485–496

[14] Childers R. W., “Effective Hamiltonians for generalized Breit interactions in QCD”, Phys. Rev. D, 36 (1987), 606–614 | DOI

[15] Brayshaw D. D., “Relativistic description of quarkonium”, Phys. Rev. D, 36 (1987), 1465–1478 | DOI

[16] Tsibidis G. D., “Quark-antiquark bound states and the Breit equation”, Acta Phys. Polon. B, 35 (2004), 2329–2366 ; arXiv:hep-ph/0007143

[17] Theor. Math. Phys., 51:2 (1982), 447–453 | DOI

[18] Crater H. W., Wong C. W. and Wong C.-Y., “Singularity-free Breit equation from constraint two-body Dirac equations”, Internat. J. Modern Phys. E, 5 (1996), 589–615 ; arXiv:hep-ph/9603402 | DOI

[19] Mlodinov L. D., Shatz M. P., “Solving the Schrödinger equation with use of $1/N$ perturbation theory”, J. Math. Phys., 25 (1984), 943–950 | DOI | MR

[20] Imbo T., Pagnamenta A., Sukhatme U., “Energy eigenstates of spherically symmetric potentials using the shifted $1/N$ expansion”, Phys. Rev. D, 29 (1984), 1669–1681 | DOI

[21] Vakarchuk I. O., “The $1/N$-expansion in quantum mechanics. High-order approximations”, J. Phys. Studies, 6 (2002), 46–54 | MR | Zbl

[22] Mustafa O., Barakat T., “Nonrelativistic shifted-l expansion technique for three- and two-dimensional Schrödinger equation”, Commun. Theor. Phys., 28 (1997), 257–264 ; arXiv:math-ph/9910040

[23] Mustafa O., Barakat T., “Relativistic shifted-l expansion technique for Dirac and Klein–Gordon equations”, Commun. Theor. Phys., 29 (1998), 587–594 ; arXiv:math-ph/9910039

[24] Todorov I. T., “Quasipotential equation correspondong to the relativistic eiconal approximation”, Phys. Rev. D, 3 (1971), 2351–2356 | DOI

[25] Rizov V. A., Sazdian H., Todorov I. T., “On the relativistic quantum mechanics of two interacting spinless particles”, Ann. of Phys. (NY), 165 (1985), 59–97 | DOI | MR

[26] Duviryak A., “Heuristic models of two-fermion relativistic systems with field-type interaction”, J. Phys. G, 28 (2002), 2795–2809 ; arXiv:nucl-th/0206048 | DOI

[27] Lucha W., Schoberl F. F., Gromes D., “Bound states of quarks”, Phys. Rep., 200:4 (1991), 127–240 | DOI

[28] Eddington A. S., “The charge of an electron”, R. Soc. Lond. Proc. Ser. A, 122:789 (1929), 358–369 | DOI | Zbl

[29] Gaunt J. A., “The triplets of Helium”, Philos. Trans. R. Soc. Lond. Ser. A, 228 (1929), 151–196 | DOI | Zbl

[30] Salpeter E. E., “Mass corrections to the fine structure of hydrogen-like atoms”, Phys. Rev., 87 (1952), 328–343 | DOI | Zbl

[31] Faustov R. N., “The proton structure and hyperfine splitting of hydrogen energy levels”, Nucl. Phys., 75 (1966), 669–681 | DOI

[32] Khelashvili A. A., Quasipotential equation for the system of two particles with spin 1/2, Communications of the Joint Institute for Nuclear Physics, P2–4327, Dubna, 1969 (in Russian)

[33] Long C., Robson D., “Bound states of a relativistic quark confined by a vector potential”, Phys. Rev. D, 27 (1983), 644–646 | DOI

[34] Baric N., Jena S. N., “Lorentz structure vs relativistic consistency of an effective power-law potential model for quark-antiquark systems”, Phys. Rev. D, 26 (1982), 2420–2429 | DOI

[35] Haysak I., Lengyel V., Shpenik A., Challupka S., Salak M., “Quark masses in the relativistic analytic model”, Ukraïn. Fiz. Zh., 41 (1996), 370–372 (in Ukrainian)