GLUONIC AND LEPTONIC DECAYS OF HEAVY QUARKONIA AND THE DETERMINATION OF ALPHA(S)(M(C)) AND ALPHA(S)(M(B))

The determination of the QCD running coupling constants alpha(s)(m(c)) and alpha(s)(m(b)) is studied with heavy quarkonia c (c) over bar and b (b) over bar decays. The decay rates of V --> 3g and V --> e(+)e(-) for V = J/psi and Upsilon are given in terms of the Bethe-Salpeter am plitudes. To the first-order relativistic correction of on-shell quark s, for the leptonic decay, we have Gamma(V --> e(+)e(-)) = (4 pi alpha (2)e(Q)(2)/m(Q)(2))\integral d(3)q[1-(2 (q) over right arrow(2)/3m(Q)( 2))]psi(Sch)((q) over right arrow\(2), which agrees with the NRQCD res ult, while for the gluonic decay we find Gamma(V --> 3g) = [40(pi(2)-9 )alpha(s)(3)(m(Q))/81m(Q)(2)]\ integral d(3)q[1-(2.95 (q) over right a rrow(2)/m(Q)(2))]psi(Sch)((q) over right arrow)\(2). Here psi(Sch)((q) over right arrow is the Q (Q) over bar bound-state wave function in m omentum space, and m(Q) is the heavy quark mass. This result clearly s hows that the relativistic correction (due to the (q) over right arrow (2)/m(Q)(2) term in the decay widths) suppresses the gluonic decay mor e severely than the leptonic decay. We then estimate these decay width s by further including the first-order QCD radiative corrections (give n in the <(MS)over bar> scheme and at the heavy quark mass scale) on t he basis of the factorization assumption, and using the meson wave fun ctions which are obtained with a QCD-inspired interquark potential. Us ing the experimental values of the ratio R(g) = Gamma(V --> 3g)/Gamma( V --> e(+)e(-)) approximate to 10, 32 for V = J/psi, Upsilon, respecti vely, and the calculated widths, we find alpha(s)(m(c)) = 0.26-0.29 an d alpha(s)(m(b)) = 0.19-0.21 at m(c) = 1.5 GeV and m(b) = 4.9 GeV. The se values for the QCD running coupling constant are substantially enha nced, as compared with the ones obtained without relativistic correcti ons, and are potentially consistent with the QCD scale parameter Lambd a(<(MS)over bar>)((4)) approximate to 200 MeV. We emphasize, however, that our numerical results of the running coupling constant mainly ser ve as an improved estimate rather than a precise determination for whi ch the existing theoretical uncertainties due to higher order relativi stic corrections and the scheme dependence of the radiative correction s should be further clarified, and a first principles estimate of the nonperturbative bound-state effects should be further studied.