Relativistic treatment of quarkonium in QCD

Abstract We have formulated a relativistic two-body equation for quark-antiquark bound states using the expansion in intermediate states. We have used the asymptotically free QCD gluon exchange with either a scalar or equal vector and scalar linear potential. Using these we have fitted the upsilon, charmonium, D, F, s s , s -u, d and u, d quark mesons for the rho and above. We have also calculated relativistically their widths, singlet-triplet and P-state splittings.

We have formulated a relativistic two-body equation for quark-antiquark systems using the multiquantum intermediate state expansion of the field equations used by Greenberg [1], Gross [2], and Johnson [3]. We will summarize the formulation below. A more detailed analysis will be published elsewhere [4]. We have applied this to quark-antiquark systems and have compared the results with the T, ~, B, F, and D families as well as lower mass mesons which are not affected by chiral symmetry breaking or large mixings. We have used an asymptotically free QCD vector exchange and a linear part which is either totally a scalar exchange or an equal combination of vector and scalar exchanges. The scalar results are presented in this letter; see ref. [4] for the other case.
We will present our fits to the spectra and leptonic decay widths for the mesonic systems excluding the lightest pion, kaon, and etas. The parameters used are the linear slope, the QCD potential scale AR, the quark masses, and a cutoff on the slowly diverging integral equations which represents the effects of ¢~ Supported in part by the National Science Foundation and the Israeli Academy of Science and Humanities Basic Research Foundation.
higher mass intermediate states, and is reset for each qq system. The derivation starts with the field theory equations for the quark field and for the vector and scalar potential fields. We use the asymptotic freedom modified vector gluon exchange plus effective linear exchange as our best knowledge of the solution to the potential field equations. We then proceed with the equation for the quark field in the potentials produced by the antiquark: where m 1 is the quark mass. The equation may be solved in terms of all of its matrix elements between the bound state and the quark-gluon basis. To do this we insert a complete set of states on the right-hand side between a potential field and the quark field. The valence quark approximation which we use is obtained by limiting the intermediate states to the single antiquark states, which gives an integral equation for the matrix element of the quark field between the bound state and the on-shell antiquark state. The resulting integral equation has the following properties: (a) Upon performing a partial wave analysis it becomes a one-variable integral equation.

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(b) Since one leg is on-shell it is current conserving and even the one-gluon exchange is gauge invariant. (c) In the nonrelativistic limit it reduces to the Darwin-Breit hamiltonian, which includes the U2/c 2 effects.
(d) It has the Dirac equation as a limit as the on-shell antiquark becomes extremely heavy relative to the off-shell quark. (e) It includes retardation effects which cannot be reproduced by spatial potentials. For another application to deeply bound composite models of fermions, see ref. [5].
We define the matrix element of the quark field between the bound state and the on-shell antiquark state as (2) where 60B = (~2 + M2)1/2, 60 = ([52 + m2)1/2, m2 is the antiquark mass and M is the unknown bound state mass. We transform this to a 4 × 4 component wave function ~(p), x with subsidiary condition that ,I,@ + m2) = 0. The equation for q~ is then We then define large and small Dirac components for the quark field, labeled by G, F, and for the antiquark field, labeled by u, d, as parts of where (;d is the large component in the non relativistic limit. The subsidiary condition on • leaves only Gd and Fd as independent: In order to perform an angular momentum decomposition we go to a direct product representation with Gd = GdOy and Fd = FdOy. We then expand in an an- The integral equation diverges slowly at large momentum no worse than logarithmically. Before that becomes important, however, the neglected multiquanta states become important in competing for probability or amplitude with the valence quark states. This is put in as a cutoff which can vary to describe the differing physical cutoffs in each channel and is of the form: S(Ac, p) = A~/(A~ + p2).
Other forms of the cutoff were also tried but do not affect the success of the fits. For the potential we start with the one-gluon exchange modified by the renormalization group asymptotic freedom corrections at short distances in a form • ~ 2 [6] that avoids ~seudopoles at q = A 2 and behaves as [~2 ln(1 + c72/A~)]-1. As a gluon exchange this must occur in the vector potential, but it also contains a q-4 limit at small ~2 corresponding to a linear potential at large r. In a relativistic equation, however, this purely vector linear potential would lead to a Klein paradox. To avoid this we subtract off the linear part and use for our gluon exchange the relativistic form: #1 For spectroscopy with a more exact asymptotic freedom form see ref.
We then add in the linear potentials separately in the scalar part for the fits of this paper (see also ref. [4] for a combination of vector and scalar linear parts). We first present our fit to the charmonium system. We expect that for the centers of the levels before splitting, the parameters used in previous nonrelativistic fits [6] will be most useful. This is indeed the case, and we take AR = 0.4 GeV and K s = 0.15 GeV 2. The P state splittings, the leptonic decay widths and the ~-r/c splittings are then fitted by choosing the cutoff Ac appropriately. The quark mass is then fixed to obtain the correct value for the 4(3097). The fits are thus presented in table 1 in terms of the excitation energy above the ~b. The location of the levels are in general of the same accuracy as other fits [8]. We take due caution in the accuracy expected since studies of the mixing to D-D channels have shown shifts from the bound state equation masses of the order of 100 MeV [9].
Besides the spectral levels, in table 1 we include the leptonic decay width of the ~ and the ratios of the leptonic decay widths of the higher S states; the splitting of the 3P 2 3P 0 levels; the ~-r/c and ~'-r/' c splittings and the ratio Rp = (3P2-3P1)/(3P 1 3P0). In further words of caution, the QCD corrections to the leptonic widths are not firmly established [10] and the ~ width was only constrained to be no more than a factor of two greater than the observed width, in agreement with many other fits. The ratios of widths are considered to be less ambiguous and are consistent with the data. The ratio Rp is unique, independent of matrix elements, if the potential is a single pure power law [11]. Here, however, two powers are involved in the potential in even the simplest approximation and Rp depends on the details of the wave functions and is of less fundamental importance. A level shift from multimeson channel mixing of even 25 MeV can change this from 1.0 to 0.5, and such mixing calculations are called for. For the upsilonium spectra and widths, we use the same value of • s and of A R found in charmonium and vary Ac to fit the splittings and widths, and then set mb to set the upsilon (9460) correctly. As before we measure the excitation energy from the upsilon, and give the results in table 1. The widths are closer to the experimental ones than in the charmonium case, indicating the expected decrease in the QCD corrections at larger Q2. One tentative discrepancy is that the newest results for the P states [ 12] are showing near equality for the 23P2-23p 0 splitting with the 13P2 -13P 0 splitting, being 38 MeV and 41 MeV, respectively. The calculations yield a ratio of about 0.65 between these splittings.
Since the relativistic equation which we use reduces to the Dirac equation for a heavy antiquark and a light quark, we should be able to describe such mesons as the D, D*, F, F*, and B [13]. We do fit the D and D* states but it requires using two parameters, the light (u or d) quark mass and the cutoff for that system. Until more states are found in that system, the approach cannot really be tested there. The same is true of the F and F* which are fit by setting the strange quark mass and the cutoff for the system. For the newly found B meson, since the b and u, d quark masses are now fixed, we only need to fit the cutoff and then predict the B* mass. The D and D* determine the u and d quark masses to be 250 MeV and the F and F* determine the strange quark mass to be 410 MeV. The mass difference of the B* and B is predicted to be about 50 MeV, as in other calculations.
The (u, d) or light quarks, and the s quark mesons require a description by a relativistic equation. In this work we must neglect the pion, kaon, and r/mesons, i.e. the 0-nonet, since we do not include chiral symmetry breaking or dynamical mixing calculations. We also leave out the ~i 0+(983) and the S* 0+(975)-e 0+(1300) since these show strong couplings to mesonic channels [14] and are not well described by the valence quark approximation. Thus in table 2 we present the L = 1 light and strange quark mesons and the 1--L = 0 mesons.
In this work [4] we have used a relativistic bound state expansion and equation to simultaneously study the heavy quark, heavy-light and light quark mesons. Our relativistic treatment included spin-orbit, spinspin, and retardation effects automatically without recourse to a u/c expansion ,2. Multiquark state effects were included approximately by using a high momentum cutoff. The approach shows successful spectra and leptonic decay rate calculations but does not explain all discrepancies of the physical spectra from the nonrelativistic calculations. A more detailed presentation including results with equal vector and scalar linear potentials will be published elsewhere [4]. Finally we note that the strange or light (u or d) quark mass values increase as the energy scale decreases. This is qualitatively consistent with the mass 4-2 For a recent review of results of v2]¢ 2 expansions see ref. [151.