Comparing the Schrödinger and spinless Salpeter equations for heavy-quark bound states

Steve Jacobs; M. G. Olsson; Casimir Suchyta, III

doi:10.1103/PhysRevD.33.3338

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Comparing the Schrödinger and spinless Salpeter equations for heavy-quark bound states

Steve Jacobs, M. G. Olsson, and Casimir Suchyta, III

Physics Department, University of Wisconsin, Madison, Wisconsin 53706

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Phys. Rev. D 33, 3338 – Published 1 June, 1986Erratum Phys. Rev. D 34, 3536 (1986)

DOI: https://doi.org/10.1103/PhysRevD.33.3338

Abstract

We develop a unified approach to the solution of the Schrödinger and the spinless Salpeter equations. By adjusting the potential and quark masses to best account for the spin-averaged energy levels, we can compare in a realistic way the effect of using a relativistic kinetic energy. Emphasis is given to consideration of relativistic wave-function corrections. We find that the relativistic equation better accounts for the observed energy levels and gives improved results for s- and p-wave annihilation rates and E1 transitions.

Erratum

Erratum: Comparing the Schrödinger and spinless Salpeter equations for heavy-quark bound states

Steve Jacobs, M. G. Olsson, and Casimir Suchyta, III

Phys. Rev. D 34, 3536 (1986)

References (27)

T. Appelquist, R. M. Barnett and K. D. Lane, Annu. Rev. Nucl. Part. Sci. 28, 387 (1978); J. L. Richardson, Phys. Lett. 82 B, 272 (1979); W. Buchmüller and S.-H. H. Tye, Phys. Rev. D 24, 132 (1981).
E. Eichten, K. Gottfried, T. Kinoshita, K. D. Lane and T. M. Yan, Phys. Rev. D 17, 3090 (1978); ibid. 21, 203 (1980).
A. Martin, Phys. Lett. 93 B, 338 (1980).
K. J. Miller and M. G. Olsson, Phys. Rev. D 25, 2383 (1982).
Kaoru Hagiwara, Steve Jacobs, M. G. Olsson and K. J. Miller, Phys. Lett. 131 B, 455 (1983).
D. P. Stanley and D. Robson, Phys. Rev. D 21, 3180 (1980); L. Hostler and W. Repko, Ann. Phys. (N.Y.) 130, 329 (1980); Stephen Godfrey and Nathan Isgur, Phys. Rev. D 32, 189 (1985) L. J. Nickisch, L. Durand and B. Durand, 30, 660 (1984).
Bernice Durand and Loyal Durand, Phys. Rev. D 25, 2312 (1982).
Bernice Durand and Loyal Durand, Phys. Rev. D 30, 1904 (1984).
Loyal Durand, Phys. Rev. D 32, 1257 (1985).
Godfrey and Isgur (Ref. 6).
C. Quigg and J. L. Rosner, Phys. Rep. 56, 167 (1979); Avinash Khare, Nucl. Phys. B152, 533 (1979).
Richard McClary and Nina Byers, Phys. Rev. D 28, 1692 (1983).
T. Appelquist and H. D. Politzer, Phys. Rev. Lett. 34, 43 (1975); A. DeRújula and S. L. Glashow, ibid. 34, 46 (1975); R. Barbieri, R. Gatto and R. Kögerler, Phys. Lett. 60 B, 1976 (1976); V. A. Novikov, L. B. Okun, M. A. Shifman, A. I. Vainshtein, M. B. Voloshin and V. I. Zakharov, Phys. Rep. 41 C, 1 (1978); E. Remidi, in Theory of Fundamental Interactions, proceedings of the International School of Physics, ``Enrico Fermi,'' Course 81, Varenna, 1980, edited by G. Costa and R. R. Gatto (North-Holland, Amsterdam, 1980).
G. P. Lepage, in Proceedings of the 1983 International Symposium on Lepton and Photon Interactions at High Energy, Ithaca, New York, edited by D. G. Cassel and D. L. Kreinick (Newman Laboratory of Nuclear Studies, Cornell University, 1983).
M. G. Olsson, A. D. Martin and A. W. Peacock, Phys. Rev. D 31, 81 (1985).
D. Hitlin, in Proceedings of the 1983 International Symposium on Lepton and Photon Interactions at High Energy, Ithaca, New York (Ref. 14).
C. Baglin et al.., contribution to the VII European Symposium on Antiproton Interactions, Durham, England, 1984 (unpublished); A. Silverman, in Proceedings of the XXII International Conference on High Energy Physics, Leipzig, 1984, edited by A. Meyer and E. Wieczorek (Akademie der Wissenschaften der DDR, Zeuthen, DDR, 1984); M. Macri and A. Lundby (private communication).
M. Oreglia (private communication); a similar width is obtained from the $χ_{2}$ -> ψ gamma branching ratio and a calculation of the radiative decay rate (see Ref. 15).
Juliet Lee-Franzini, in Physics in Collision V, proceedings of the Fifth International Conference, Autun, France, 1985, edited by B. Aubert and L. Montanet (Editions Frontie´res, Gif-sur-Yvette, France, 1985).
R. Barbieri, M. Caffo, R. Gatto and E. Remiddi, Phys. Lett. 95 B, 93 (1980).
I. Stakgold, Green's Functions and Boundary Value Problems (Wiley, New York, 1984).
Stanley and Robson (Ref. 6); Godfrey and Isgur (Ref. 6); Durand, Durand, and Nickish (Ref. 6).
Omitted end note.
M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).
E. Joachin Weniger, J. Math. Phys. 26, 276 (1985).
A. Ralston and P. Rabinowitz, A First Course in Numerical Analysis (McGraw-Hill, New York, 1978).
J. H. Wilkinson and C. Reinsch, Linear Algebra: A Handbook for Automatic Computation (Springer, Berlin, 1971).

[c1] T. Appelquist, R. M. Barnett and K. D. Lane, Annu. Rev. Nucl. Part. Sci. 28, 387 (1978); J. L. Richardson, Phys. Lett. 82 B, 272 (1979); W. Buchmüller and S.-H. H. Tye, Phys. Rev. D 24, 132 (1981).

[c2] E. Eichten, K. Gottfried, T. Kinoshita, K. D. Lane and T. M. Yan, Phys. Rev. D 17, 3090 (1978); ibid. 21, 203 (1980).

[c3] A. Martin, Phys. Lett. 93 B, 338 (1980).

[c4] K. J. Miller and M. G. Olsson, Phys. Rev. D 25, 2383 (1982).

[c5] Kaoru Hagiwara, Steve Jacobs, M. G. Olsson and K. J. Miller, Phys. Lett. 131 B, 455 (1983).

[c6] D. P. Stanley and D. Robson, Phys. Rev. D 21, 3180 (1980); L. Hostler and W. Repko, Ann. Phys. (N.Y.) 130, 329 (1980); Stephen Godfrey and Nathan Isgur, Phys. Rev. D 32, 189 (1985) L. J. Nickisch, L. Durand and B. Durand, 30, 660 (1984).

[c7] Bernice Durand and Loyal Durand, Phys. Rev. D 25, 2312 (1982).

[c8] Bernice Durand and Loyal Durand, Phys. Rev. D 30, 1904 (1984).

[c9] Loyal Durand, Phys. Rev. D 32, 1257 (1985).

[c10] Godfrey and Isgur (Ref. 6).

[c11] C. Quigg and J. L. Rosner, Phys. Rep. 56, 167 (1979); Avinash Khare, Nucl. Phys. B152, 533 (1979).

[c12] Richard McClary and Nina Byers, Phys. Rev. D 28, 1692 (1983).

[c13] T. Appelquist and H. D. Politzer, Phys. Rev. Lett. 34, 43 (1975); A. DeRújula and S. L. Glashow, ibid. 34, 46 (1975); R. Barbieri, R. Gatto and R. Kögerler, Phys. Lett. 60 B, 1976 (1976); V. A. Novikov, L. B. Okun, M. A. Shifman, A. I. Vainshtein, M. B. Voloshin and V. I. Zakharov, Phys. Rep. 41 C, 1 (1978); E. Remidi, in Theory of Fundamental Interactions, proceedings of the International School of Physics, ``Enrico Fermi,'' Course 81, Varenna, 1980, edited by G. Costa and R. R. Gatto (North-Holland, Amsterdam, 1980).

[c14] G. P. Lepage, in Proceedings of the 1983 International Symposium on Lepton and Photon Interactions at High Energy, Ithaca, New York, edited by D. G. Cassel and D. L. Kreinick (Newman Laboratory of Nuclear Studies, Cornell University, 1983).

[c15] M. G. Olsson, A. D. Martin and A. W. Peacock, Phys. Rev. D 31, 81 (1985).

[c16] D. Hitlin, in Proceedings of the 1983 International Symposium on Lepton and Photon Interactions at High Energy, Ithaca, New York (Ref. 14).

[c17] C. Baglin et al.., contribution to the VII European Symposium on Antiproton Interactions, Durham, England, 1984 (unpublished); A. Silverman, in Proceedings of the XXII International Conference on High Energy Physics, Leipzig, 1984, edited by A. Meyer and E. Wieczorek (Akademie der Wissenschaften der DDR, Zeuthen, DDR, 1984); M. Macri and A. Lundby (private communication).

[c18] M. Oreglia (private communication); a similar width is obtained from the $χ_{2}$ -> ψ gamma branching ratio and a calculation of the radiative decay rate (see Ref. 15).

[c19] Juliet Lee-Franzini, in Physics in Collision V, proceedings of the Fifth International Conference, Autun, France, 1985, edited by B. Aubert and L. Montanet (Editions Frontie´res, Gif-sur-Yvette, France, 1985).

[c20] R. Barbieri, M. Caffo, R. Gatto and E. Remiddi, Phys. Lett. 95 B, 93 (1980).

[c21] I. Stakgold, Green's Functions and Boundary Value Problems (Wiley, New York, 1984).

[c22] Stanley and Robson (Ref. 6); Godfrey and Isgur (Ref. 6); Durand, Durand, and Nickish (Ref. 6).

[c23] Omitted end note.

[c24] M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

[c25] E. Joachin Weniger, J. Math. Phys. 26, 276 (1985).

[c26] A. Ralston and P. Rabinowitz, A First Course in Numerical Analysis (McGraw-Hill, New York, 1978).

[c27] J. H. Wilkinson and C. Reinsch, Linear Algebra: A Handbook for Automatic Computation (Springer, Berlin, 1971).

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Comparing the Schrödinger and spinless Salpeter equations for heavy-quark bound states

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