On Relativistic Corrections to Mass Spectra in Heavy Quarkonia

Abstract

It is shown that the relativistic gluon corrections for the states with j'O are consistently included in the instantaneous ladder BS equation but the higher-order kernels may contribute to the ground states with j=O. The retardation correction is examined. The relativistic effect is shown to be essential to get the reasonable value for M(T/e). The relativistic effects are important for the study of the charmonium and are not negligible even in the b b system. It is found, in a recent paper,') that the experimental mass spectrum of the charmonium is very well reproduced in the relativistic model based on the instantaneous Bethe-Salpeter equation if the scalar (exchange) potential is assumed for the confining force. The gluonic correction (transverse-polarization part in the Coulomb gauge) restricted by the cutoff for the short-distance interaction (for the momenta p > 1.5 mq) plays an essential role in adjusting the fine and hyperone splittings, and the ladder itera­ tion of the gluonic correction is included in the calculations. That is, some part of the higher­ order corrections is formally included in the model of Ref. 1). If this iteration really has non-negligible effects, we should infer that the other higher order (in as) kernels which are neglected in the model also have significant effects in the short-distance region (r ~ 1/ mq).*) This means that the ladder approximation is not sufficient. If, on the other hand, the effects of the iteration are negligible, we can say that the ladder model provides a selfconsistent framework for the gluonic correction. It turns out to be equiva­ lent to the model based on the first-order perturba­ tion theory. The first purpose of the present paper is to examine the consistency of the ladder approxima­ tion. It will be shown that the ladder approxima­ tion is satisfactory for the states with j =\= 0 but the corrections from the second-order kernels may be *> We know, for example, that the second-order (in as) kernels with the pair-creation intermediate states have some effects in the short-distance region (see the Appendix of Ref. 1)). considerably large for the ground states with j = O. In other words, the masses of the 11 So and the 1 3 Po states are important for the study of the higher-order gluonic interaction where the non­ Abelian character of the QCD appears. The second purpose is to examine the retardation correction. We lastly show that the highel-order p2/m2 corrections are important for the char­ monium, especially for obtaining the reasonable value of M(T/e). We first describe the model briefly. The in­ stantaneous BS equation is written as {M - Hl(p) - H2( - p) }x(p)