Semirelativistic potential model for heavy quarkonia

Suraj N. Gupta; Stanley F. Radford; Wayne W. Repko

doi:10.1103/PhysRevD.34.201

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Semirelativistic potential model for heavy quarkonia

Suraj N. Gupta and Stanley F. Radford

Wayne W. Repko

Department of Physics, Wayne State University, Detroit, Michigan 48202

Department of Physics, Michigan State University, East Lansing, Michigan 48824

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Phys. Rev. D 34, 201 – Published 1 July, 1986

DOI: https://doi.org/10.1103/PhysRevD.34.201

Abstract

The cc¯, bb¯, and t t¯ spectra are investigated with the use of a semirelativistic potential model described in an earlier paper. Results for the energy levels, leptonic widths, and E1 transition widths are compared with the experimental data for cc¯ and bb¯ and predicted for t t¯. We also find that the quark-antiquark interaction can best be described by a quasistatic rather than a momentum-dependent potential, and propose a theoretical justification for this surprising conclusion.

References (21)

S. N. Gupta, S. F. Radford and W. W. Repko, Phys. Rev. D 31, 160 (1985).
S. N. Gupta, S. F. Radford and W. W. Repko, Phys. Rev. D 26, 3305 (1982); ibid. 30, 2424 (1984).
G. Arnison et al., Phys. Lett. 147B, 493 (1984).
These results for the $c $\bar{c}$ energy levels slightly differ from those in Ref. 1 because we have eliminated a computational error in our earlier work and also carried out a more rigorous search for the optimum values of the input parameters.
In accordance with the standard spectroscopic notation, we have denoted the lowest P states as 2P.
Particle Data Group, Rev. Mod. Phys. 56, S1 (1984). For more recent data on heavy-quark physics, see K. Berkelman, Cornell University Report No. CLNS-85/649, 1985 (unpublished); S. Cooper, SLAC Report No. SLAC-PUB-3819, 1985 (unpublished).
R. Van Royen and V. F. Weisskopf, Nuovo Cimento 50, 617 (1967).
R. Barbieri et al., Phys. Lett. 57B, 455 (1975).
Note that as a result of the mixing of the 2 ${}^{3}S_{1}$ state of c $\bar{c}$ with the lowest $^{3}$ $D_{1}$ state, our theoretical value for $Γ_{ee}$ (2S) corresponds to the experimental result for $Γ_{ee}$ ( ψ prime )+ $Γ_{ee}$ ( ψ prime prime ).
F. C. Porter, in Proceedings of the 9th SLAC Summer Institute on Particle Physics, 1981, edited by A. Mosher (Stanford University, Stanford, California, 1981), p. 355.
The importance of perturbed wave functions in the treatment of E1 transitions has been emphasized by several authors. See R. McClary and N. Byers, Phys. Rev. D 28, 1692 (1983), and references therein. See also the recent relativized quark-model treatment of S. Godfrey and N. Isgur, ibid. 32, 189 (1985), which yields satisfactory results for the E1 transitions in $c $\bar{c}$ .
For recent experimental work on the P states of $b $\bar{b}$ , which has aroused much interest, see K. Han et al., Phys. Rev. Lett. 49, 1612 (1982); C. Klopfenstein et al., ibid. 51, 160 (1983); P. Haas et al., ibid. 52, 799 (1984); R. Nernst et al., ibid. 54, 2195 (1985).
P. Moxhay and J. L. Rosner, Phys. Rev. D 31, 1762 (1985).
W. Buchmüller and S.-H. H. Tye, Phys. Rev. D 24, 132 (1981).
It is interesting that our values of | p vec |/ mu for the ground states of c $\bar{c}$ , $b $\bar{b}$ , and $t $\bar{t}$ are 0.36, 0.36, and 0.32, respectively, which indicates that $mu is closely related to the quark momentum | p vec | rather than its mass m.
S. N. Gupta and S. F. Radford, Phys. Rev. D 25, 2690 (1982).
For the derivation of nonrelativistic potentials from the scattering operator, we have followed the treatment of S. N. Gupta, Nucl. Phys. 57, 19 (1964). The linear scalar-exchange potential takes the form (6.1) when the on-shell quark-antiquark scattering matrix element in the center-of-mass frame is expressed in the simplest possible form, while other forms can be obtained by adding on-shell vanishing terms. See, for instance, T. Barnes and G. I. Ghandour, Phys. Lett. 118B, 411 (1982), and references therein.
We obtained unsatisfactory results with the momentum-dependent scalar-exchange confining potential when used either in the form (6.1) or in the Barnes-Ghandour form cited in Ref. 17.
We have also looked at the $c $\bar{c}$ and $b $\bar{b}$ spectra with the use of the quasistatic and the momentum-dependent forms of a linear vector-exchange confining potential. We found unacceptably large spin splittings of energy levels, and concluded that vector-exchange component of the confining potential, if any, is quite small compared with the scalar-exchange component.
S. N. Gupta and S. F. Radford, Phys. Rev. D 32, 781 (1985).
Our values of p ${vec}^{2}$ / $m^{2}$ for the ground states of $c $\bar{c}$ , $b $\bar{b}$ , and $t^ $\bar{t}$ are 0.284, 0.076, and 0.018, respectively.

[c1] S. N. Gupta, S. F. Radford and W. W. Repko, Phys. Rev. D 31, 160 (1985).

[c2] S. N. Gupta, S. F. Radford and W. W. Repko, Phys. Rev. D 26, 3305 (1982); ibid. 30, 2424 (1984).

[c3] G. Arnison et al., Phys. Lett. 147B, 493 (1984).

[c4] These results for the $c $\bar{c}$ energy levels slightly differ from those in Ref. 1 because we have eliminated a computational error in our earlier work and also carried out a more rigorous search for the optimum values of the input parameters.

[c5] In accordance with the standard spectroscopic notation, we have denoted the lowest P states as 2P.

[c6] Particle Data Group, Rev. Mod. Phys. 56, S1 (1984). For more recent data on heavy-quark physics, see K. Berkelman, Cornell University Report No. CLNS-85/649, 1985 (unpublished); S. Cooper, SLAC Report No. SLAC-PUB-3819, 1985 (unpublished).

[c7] R. Van Royen and V. F. Weisskopf, Nuovo Cimento 50, 617 (1967).

[c8] R. Barbieri et al., Phys. Lett. 57B, 455 (1975).

[c9] Note that as a result of the mixing of the 2 ${}^{3}S_{1}$ state of c $\bar{c}$ with the lowest $^{3}$ $D_{1}$ state, our theoretical value for $Γ_{ee}$ (2S) corresponds to the experimental result for $Γ_{ee}$ ( ψ prime )+ $Γ_{ee}$ ( ψ prime prime ).

[c10] F. C. Porter, in Proceedings of the 9th SLAC Summer Institute on Particle Physics, 1981, edited by A. Mosher (Stanford University, Stanford, California, 1981), p. 355.

[c11] The importance of perturbed wave functions in the treatment of E1 transitions has been emphasized by several authors. See R. McClary and N. Byers, Phys. Rev. D 28, 1692 (1983), and references therein. See also the recent relativized quark-model treatment of S. Godfrey and N. Isgur, ibid. 32, 189 (1985), which yields satisfactory results for the E1 transitions in $c $\bar{c}$ .

[c12] For recent experimental work on the P states of $b $\bar{b}$ , which has aroused much interest, see K. Han et al., Phys. Rev. Lett. 49, 1612 (1982); C. Klopfenstein et al., ibid. 51, 160 (1983); P. Haas et al., ibid. 52, 799 (1984); R. Nernst et al., ibid. 54, 2195 (1985).

[c13] P. Moxhay and J. L. Rosner, Phys. Rev. D 31, 1762 (1985).

[c14] W. Buchmüller and S.-H. H. Tye, Phys. Rev. D 24, 132 (1981).

[c15] It is interesting that our values of | p vec |/ mu for the ground states of c $\bar{c}$ , $b $\bar{b}$ , and $t $\bar{t}$ are 0.36, 0.36, and 0.32, respectively, which indicates that $mu is closely related to the quark momentum | p vec | rather than its mass m.

[c16] S. N. Gupta and S. F. Radford, Phys. Rev. D 25, 2690 (1982).

[c17] For the derivation of nonrelativistic potentials from the scattering operator, we have followed the treatment of S. N. Gupta, Nucl. Phys. 57, 19 (1964). The linear scalar-exchange potential takes the form (6.1) when the on-shell quark-antiquark scattering matrix element in the center-of-mass frame is expressed in the simplest possible form, while other forms can be obtained by adding on-shell vanishing terms. See, for instance, T. Barnes and G. I. Ghandour, Phys. Lett. 118B, 411 (1982), and references therein.

[c18] We obtained unsatisfactory results with the momentum-dependent scalar-exchange confining potential when used either in the form (6.1) or in the Barnes-Ghandour form cited in Ref. 17.

[c19] We have also looked at the $c $\bar{c}$ and $b $\bar{b}$ spectra with the use of the quasistatic and the momentum-dependent forms of a linear vector-exchange confining potential. We found unacceptably large spin splittings of energy levels, and concluded that vector-exchange component of the confining potential, if any, is quite small compared with the scalar-exchange component.

[c20] S. N. Gupta and S. F. Radford, Phys. Rev. D 32, 781 (1985).

[c21] Our values of p ${vec}^{2}$ / $m^{2}$ for the ground states of $c $\bar{c}$ , $b $\bar{b}$ , and $t^ $\bar{t}$ are 0.284, 0.076, and 0.018, respectively.

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Semirelativistic potential model for heavy quarkonia

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