Prog. Theor. Phys. Vol. 84 No. 1 (1990) pp. 94-107

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Relativistic Quarkonium Model with Retardation Effect. I

— Mass Level and Lepton-Pair Decay —

Hitoshi Ito

Department of Physics, Faculty of Science and Technology
Kinki University, Higashi-Osaka 577

(Received November 9, 1989)

Abstract:

A new relativistic two-body equation is proposed which has the charge-conjugation symmetry. The renormalization of the wave function at the origin (WFO) is done by incorporating the corresponding vertex equation. By using this model, the heavy-quarkonium phenomenology is developed putting emphasis on the short-distance interaction. The typical scale of the distance restricting the applicability of the ladder model for the mass spectra is found to be 0.13 fm : By assuming the equivalent high-momentum cutoff for the gluonic correction, good results are obtained for the charmonium masses. The improved fine-splittings of the b b states are obtained by inclusion of the retardation. Leptonic decay rates are predicted by assuming the renormalized WFO reduced by another high-momentum cutoff.

URL : http://ptp.ipap.jp/link?PTP/84/94/
DOI : 10.1143/PTP.84.94

References:

J. Morishita et al., Z. Phys. C19 (1983), 167.
R. McClary and N. Byers, Phys. Rev. D28 (1983), 1692[APS].
H. Ito, Prog. Theor. Phys. 75 (1986), 1416[PTP].
H. Ito, Prog. Theor. Phys. 74 (1985), 1092[PTP].
G. Feldman, T. Fulton and J. Townsend, Phys. Rev. A8 (1973), 1149[APS].
R. Barbieri et al., Phys. Lett. 57B (1975), 455.
W. Celmaster, Phys. Rev. D19 (1979), 1517[APS].
H. Ito, Prog. Theor. Phys. 77 (1987), 681[PTP].
We have reexamined the comparison with the nonrelativistic WFO (or width) made in Table I of this paper We obtain Z₁(c c)=0.512 and Z₁(b b)=0.621 if we assume the new renormalization point. The widths shown in the columns Γ^R and Γ^R(G_t=0) should be changed accordingly.
H. Ito, Prog. Theor. Phys. 67 (1982), 1553[PTP].
The relation between the WFO and the vertex renormalization was recognized in some literatures, W. E. Caswell and G. P. Lepage, Phys. Rev. A18 (1978), 810[APS].
K. Higashijima, Phys. Rev. D18 (1978), 2128[APS].
E. E. Salpeter, Phys. Rev. 87 (1952), 328[APS].
H. Ito, Prog. Theor. Phys. 80 (1988), 874[PTP].
We have reexamined the small coupling limit of the WFO by assuming the new renormalization point. We get
Ψ₁^R(0) ≈ψ₁(0)^NR(1-0.594^0.96)
for the vector state and
Ψ₁^R(0) ≈ψ₁(0)^NR(1-0.825 α^0.96)
for the ps-scalar state. The numerical errors in the correction terms may be 2 ∼3%.
The WFO of the (one body) Dirac equation is renormalized by Z₁ for the weak vertex, H. Ito, Prog. Theor. Phys. 78 (1987), 978[PTP]; ibid. 83 (1990), 1064[PTP].
See Appendix for the renormalization point of this model.
H. Ito, Prog. Theor. Phys. 76 (1986), 567[PTP].
The retardation effect was estimated in this paper. However, there was an error in treating the retardation and the results in the last column of Table I of this paper are incorrect.
N. Fukuda, K. Sawada and M. Taketani, Prog. Theor. Phys. 12 (1954), 156[PTP].
S. Okubo, ibid., 603.
M. Shintani, Preprint YWU-PHYS-8901 (1989).
E. E. Salpeter and H. A. Bethe, Phys. Rev. 84 (1951), 1232[APS].
W. Glöckle and Y. Nogami, Phys. Rev. D35 (1987), 3840[APS].
H. Ito, Prog. Theor. Phys. 63 (1980), 2140[PTP]. See also Ref. 8) for an erratum.
See for example,
J. M. Jauch and F. Rohrlich, The Theory of Photons and Electrons (Springer, New York, 1976), p. 246.
H. Ito, Proc. of BNL Workshop on Glueballs Hybrids and Exotic Hadrons, August 1988, Upton, ed. S. U. Chung (AIP Conf. Proc. No. 185, 1989), p. 586. Inclusion of the confining potential in the vertex equation, in this paper, is not adequate to the renormalization condition stated in § 3.
Particle Data Group, Phys. Lett. 204B (1988), 1.
K. Königsmann, Preprint DESY 87-151 (1987).
K. Königsmann, Proc. of BNL Workshop on Glueballs Hybrids and Exotic Hadrons, August 1988, Upton, ed. S. U. Chung (AIP Conf. Proc. No. 185, 1989), p. 36.
K. Königsmann, Proc. of the 6th International Conference on Physics in Collision, September 1986, Chicago, ed. M. Derrick (World Scientific), p. 147.
J. P. Alexander et al., SLAC-PUB-4501 (1988).
C. Baglin, et al., Phys. Lett. 171B (1986), 135.

Citing Article(s) :

Progress of Theoretical Physics Vol. 89 No. 3 (1993) pp. 763-765 :
On the Renormalization Condition of the Relativistic Wave Function at the Origin

Hitoshi Ito