Abstract
A covariant shell model of three Dirac fermions in the center-of-mass frame is analytically solved for a particular spin-dependent harmonic-oscillator potential. Three equal-mass quarks are all assumed to be in states with identical quantum numbers ofj π=1/2+ for the shell-model ground state. The sixtyfour-component composite wave function for three Dirac fermions is reduced to a set of four coupled differential equations for the four independent composite radial components. Hyperspherical coordinates are used. The two-body potential used vanish in the small component of the composite wave function. The various components of the composite wave function are found to have aρ l exp(−κρ 2) hyperradial dependence, where ρ is the hyperradius, andl is the sum of the orbital angular momentum acting in the various components of the composite wave function. For an eigenergy equal to the proteon rest-mass energy, the root-mean-square radius calculated is about 0.84 fermi, about 80% of that calculated in an independent-particle harmonic-oscillator model with the same rest-mass energy for the system. A sixth-order polynomial equation is found for the rest-mass energy of the system in terms of the potential-strength parameter, and the quark mass, For a quark mass of zero, a quadratic equation inE 3 is obtained. One solution agrees with earlier work, and the other can be rejected on physical grounds. The various components of the composite three-body wave function are dominated by thel=0s-state component as the quark mass approaches one third of the system rest-mass energy. For zero quark mass thes-wave component is about 29% of the normalization, thep wave about 44%, thed wave about 22%, and thef wave 4% of the normalization.
Similar content being viewed by others
References
Barut, A. O., Bo Wei Xu: Physica Scripta26, 129 (1982)
Barut, A. O., Strobel, G. L.: Few-Body Systems1, 167 (1986)
Barut, A. O., Komy, S.: Fortschr. Phys.33, 309 (1985)
Weise, W.: Int. Rev. of Nucl. Physics1, 1 (1984)
Feynman, R. P., et al.: Phys. Rev.D3, 2706 (1971)
Isgur, N., Karl, G.: Phys. Rev.D21, 3175 (1980)
Gartenhaus, S., Schwartz, C.: Phys. Rev.108, 482 (1957)
Friar, J. L., Gibson, B. F., Payne, G. L.: Phys. Rev.C22, 284 (1980)
Tegen, R., Brockmann, R., Weise, W.: Z. Phys.A307, 339 (1982)
Tegen, R., Weise, W.: Z. Phys.A314, 357 (1983)
Barik, N., Dash, B. K.: Phys. Rev.D34, 2092 (1986)
Palladino, B. E., Ferrelka, P. L.: Phys. Rev.34, 2168 (1986)
Basdevant, J. L., Boukrae, S.: Z. Phys.C36, 103 (1986)
Gattone, A. O., Hwang, W. R. P.: Phys. Rev.D31, 2874 (1985)
Chizhov, A. V., Dorokhov, A. E.: Phys. Lett.157B, 85 (1985)
de Forest, T.: Phys. Rev.C22, 2222 (1980)
Donoghue, J. F., Johnson, K.: Phys. Rev.D21, 1975 (1980)
Wong, C. W.: Phys. Rev.D24, 1416 (1981)
Foldy, L. L.: Phys. Rev.D10, 1777 (1974)
Baz, A. I., Zhukov, M. V.: Sov. J. Nucl. Physics11, 435 (1970)
Halzen, F., Martin, A. D.: Quarks and Leptons, p. 54. New York: Wiley 1984
Strobel, G. L.: Hadronic Journal9, 181 (1986)
Isgur, N., Karl, G.: Phys. Rev.D18, 4187 (1978)
De Tar, C.: Phys. Rev.D17, 323 (1978)
Oka, M., Yazaki, K.: Prog. Theor. Phys.66, 551 (1981)
Furui, S., Faessler, A.: Nucl. Phys.A357, 413 (1983)
Zeitnitz, B.: Nucl. Phys.A416, chapt. I–III (1984)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Strobel, G.L., Hughes, C.A. Analytic solution of the covariant shell model of three dirac fermions with harmonic forces. Few-Body Systems 2, 155–168 (1987). https://doi.org/10.1007/BF01113296
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF01113296