A Comment on the Letter by Y. J. Ng et al., Phys. Rev. Lett. 55, 916 (1985)

Gupta, Suraj N.

Radford, Stanley F.

Repko, Wayne W.

English

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Physics School of Science and Mathematics
12-1985
Comment on Spin-Dependent Forces in Heavy-
Quark Systems
Suraj N. Gupta
Wayne State University
Stanley F. Radford
The College at Brockport, sradford@brockport.edu
Wayne W. Repko
Michigan State University, repko@pa.msu.edu
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© 1985 The American Physical Society
URL:http://link.aps.org/doi/10.1103/PhysRevLett.55.3006 DOI:10.1103/PhysRevLett.55.3006
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Gupta, Suraj N.; Radford, Stanley F.; and Repko, Wayne W., "Comment on Spin-Dependent Forces in Heavy-Quark Systems" (1985).
Physics. 6.
https://digitalcommons.brockport.edu/phs_facpub/6
VOLUME 55, NUMBER 27 PHYSICAL REVIEW LETTERS 30 DECEMBER 1985
Comment on "Spin-Dependent Forces in
Heavy-Quark Systems"
A few years ago, the bb and cc spectra were investi-
gated by us' with the use of a perturbative quantum-
chromodynamic potential supplemented by a scalar-
exchange linear confining potential. An essential
feature of our model was the inclusion of the complete
one-loop radiative corrections in the perturbative part
of the quark-antiquark potential, and our results have
been found to be in excellent agreement with experi-
ments. 2 Recently this problem has been reexamined
by Ng, Pantaleone, and Tye. 3 Since spectroscopy of
heavy quarkonia is particularly suitable for a confron-
tation of quantum chromodynamics with the experi-
mental data, we shall comment on the similarities as
well as the differences between our treatment and that
of Ng, Pantaleone, and Tye.
QCD radiative corrections The.—Bethe-Salpeter ap-
proach has been extensively used for the treatment of
bound states for many years. More recently, an alter-
native and simpler treattnent has been developed by
Gupta and Radford by deriving the two-particle poten-
tial from the scattering operator, 4 and they have used
it to obtain the complete one-loop QCD radiative
corrections to the quark-antiquark potential. 5
As in our earlier treatment, Ng, Pantaleone, and
Tye have included the radiative corrections, obtained
by Gupta and Radford, in the perturbative potential
for the quark-antiquark system. 6 They have, however,
used the tnodified minimal-subtraction (MS) renor-
malization scheme7 for this purpose. On the other
hand, we used the Gupta-Radford (GR) renormaliza-
tion scheme, a which is more physical because it is a
simplified momentum-space subtraction scheme.
Since the results in the GR and MS schemes become
equivalent if we set
49 —10nft'3
pGR= p sp-
nf
the difference between the two schemes is equivalent
to a difference in the choice of the renormalization
scale p, .
Determination ofp, and a, . The choice of ren—ormal-
ization scale p, has been widely discussed in the litera-
ture, but there is no consensus as to which prescription
is to be preferred. Ng, Pantaleone, and Tye have
chosen Grunberg's prescription, in which p, is chosen
so as to make the next-to-leading order coefficient
vanish. We proposed and used another prescription'
with a view to minimizing the effect of higher-order
terms which would be generated by renormalization-
group improvement of our potential.
Ng, Pantaleone, and Tye point out that different
physical processes in general have different values of
o.„and they treat contributions to quarkonium energy
levels resulting from various types of terms in the
quark-antiquark potential as different physical
processes. Moreover, their wave functions have been
determined with the use of a potential which yields the
rather large value of 300 MeV for AMs. On the other
hand, we used the same value of n, for all potential
terms, and our value of A agrees with the generally ac-
cepted value of this parameter.
Long-range quark-antiquark potential. —Ng, Pantale-
one, and Tye have attempted to deduce the long-range
spin-dependent potential with use of a general treat-
ment, which has been applied by them to four types of
potentials. They have thus arrived at the conclusion
that "better data will be able to distinguish the model
of short-range spin forces from the model of scalar
long-range force and other models. "
We have ourselves investigated several types of con-
fining potentials during the past several years, but we
found only the results obtained with a scalar-exchange
linear confining potential worthy of publication. ' lt is
also interesting that the spin dependence in such a
confining potential appears in the form of a spin-orbit
term, which plays only a minor role in the P-state split-
tings of bb, while its effect is considerably larger for cc.
Considering both the bb and cc results, we believe that
the confining quark-antiquark potential for heavy
quarkonia behaves predominantly as a scalar-exchange
linear potential.
This work was supported in part by the U.S. Depart-
ment of Energy and the National Science Foundation.
Suraj N. Gupta and Stanley F. Radford
Department of Physics
Wayne State University
Detroit, Michigan 48202
Wayne W. Repko
Department of Physics
Michigan State University
East Lansing, Michigan 48824
Received 23 October 1985
PACS numbers: 12.40.Qq, 12.35.Eq, 14.40.Gx
~S. N. Gupta, S. F. Radford, and W. W. Repko, Phys.
Rev. D 26, 3305 (1982).
2K. Berkelman, Cornell University Report No. CLNS-
85/649, 1985 (unpublished); S. Cooper, SLAC Report No.
SLAC-PUB-3555, 1985 (unpublished) .
3Y. J. Ng, J. Pantaleone, and S.-H. H. Tye, Phys. Rev.
Lett. 55, 916 (1985).
&S. N. Gupta, Nucl. Phys. 57, 19 (1964); S. N. Gupta and
S. F. Radford, Phys. Rev. D 21, 2213 (1980).
5S. N. Gupta and S. F. Radford, Phys. Rev. D 24, 2309
(1981),and 25, 3430 (1982).
Note that Ref. 3 contains misprints, which can be correct-
ed with the help of the results in Refs. 5 and 1.
7W. A. Bardeen et al. , Phys. Rev. D 18, 3998 (1978).
S. N. Gupta and S. F. Radford, Phys. Rev. D 25, 2690
(1982) .
3006 1985 The American Physical Society


Comment on Spin-Dependent Forces in Heavy-Quark Systems

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