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We show that our QCD calculation of the ' P I charmonium state can be incorporated into a global description of the fine and hyperfine structure of the charmonium and bottomonium systems. We propose a simple renormalization prescription in which all ambiguities are absorbed into a single strong coupling constant a,. We find that the O(a:) perturbative QCD potential leads to a consistent description of all splittings. The results are used to predict the unmeasured bottomonium P-wave hyperfine splittings. PACS number(s): 14.40.G~; 12.38.B~; 12.40.Qq In an earlier paper [I] we demonstrated that the hyperfine splittings of P-wave states in heavy quarkonia can be reliably calculated via perturbative QCD and are consistent with experimental indications [2]. The angular dependence of the wave functions and of the potential is such that the calculation is independent a t the one-loop level of renormalization scheme and scale, leaving the splitting a function of only the quark mass and a,. This independence, and therefore predictive power, does not occur in the calculation of either S-wave hyperfine splittings or the P-wave spin-orbit and tensor splittings. A single choice of the renormalization scale may be made for each quark mass, and despite the uncertainties in the calculation of these splittings, our previous perturbative QCD calculation can be generalized t o all spin splittings with results consistent with experiment. This differs from previous calculations of the same quantities in the greatly simplified approach t o renormalization scheme and correspondingly t o a,. Other authors have fixed I*, and a, using the Grunberg [3] prescription and the experimental data for each state [4], or by minimizing the effect of higher-order terms generated c h a r m 1S t rn, = 1.2 GeV FIG. 1. Hyperfine splitting, A M~S ,~ = M(J/Q) -M ( q c ) , in the charmonium 1 s system, for three values of a,. The large circles give the tree-level results for the same values of a,. They have been placed at p = m, to demonstrate that the one-loop corrections are small for this choice of p. The dotted horizontal lines indicate the experimental 1 u limits. 47 3013 -

We explore the dynamics of mesons composed of spinless quarks connected by a straight flux tube. The mesons are quantized and the constituent motion is relativistic. The methods developed are applied to mesons containing equal mass quarks and heavy-light quarks. For massless quarks a nearly straight leading Regge trajectory with Nambu slope is accompanied by nearly parallel equally spaced daughter trajectories. For heavy-light mesons an analogous structure is found but with double the usual Nambu slope. Comparison with six observed spin-averaged heavy-light meson states yields good agreement. PACS number(s): 12.38.Aw, 12.38.Lg

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