Mass effects in the quark-gluon decays of heavy paraquarkonia

Quark-gluon decays of heavy paraquarkonia S0(Q̄Q) → q̄qg are investigated with account of the masses of final quarks. The decay widths and the energy distributions of the final quarks and gluons are calculated in dependence on the relative quark masses. The strong collinear enhancement of the gluon energy distribution at the end of the spectrum is shown to take place in all such decays of ηc and ηb mesons except the decay ηb → c̄cg. The total decay width is shown to have an essential dependence on the final quark masses. The corresponding branching ratios of ηc and ηb mesons are numerically estimated with a good agreement of Br(ηc → s̄sg) with experimental data on ηc decays. To be published in Mod. Phys. Lett. A ∗E-mail: [email protected] Many-partical decays of heavy quarkonia such as S0(Q̄Q) → 3g, q̄qg, S1(Q̄Q) → 4g, q̄qgg depend on the 3gand q̄qg-interactions of the final quarks and gluons and can give the useful information about these interactions. In particular, these interactions give rise to the specific thrust [1, 2] and invariant masses [3, 4] distributions, to the acomplanarity of the fourparticle decays [5], to the collinearization effect of final gluons [6], etc. It should be noted, however, that such effects can be essentially affected by masses of final quarks. For example, the four-particle quark-gluon decays of the orthobottomonium with the production of ūu-, d̄dor s̄s-pairs exhibit a collinear enhancement whereas this effect in such decays with the production of c̄c-pair is absent because of the relatively large mass of c-quark [7]. Hence, the effect of the final quark masses can be essential in the decays of heavy quarkonia with production of q̄q-pairs and it should be taken into account in investigations of such decays. In this paper the quark-gluon decay of heavy paraquarkonium S0(Q̄Q) → q̄qg is investigated with account of the masses of final quarks. The differential and total decay widths are calculated in tree approximation in dependence on the final quark masses and are discussed in comparison with the experimental data on ηc decays. The branching ratios of ηb decay are also predicted and discussed. The decay of a heavy paraquarkonium into a quark-antiquark pair and a gluon S0(Q̄Q) → q̄qg is described by two graphs shown in Fig. 1. The amplitude of this process in the limit of static quarks in the quarkonium can be presented in the form: Mαβ(S0 → q̄qg) = − (ta)αβ 2 √ N c g sψ(0) 2m √ ωε1ε2 (jαβF̃ aP) p2 (Pk) , (1) where ta are the generators of the color group SU(Nc), a = 1, 2, . . . , N 2 c − 1 and α, β = 1, 2, . . . , Nc are the color indices, gs is the strong charge, ψ(~r) is the nonrelativistic wave function of a paraquarkonium in the coordinate space, ω, ε1 and ε2 are the energies of the gluon, quark and antiquark respectively, (jμ)αβ = (ūα(p1)γμuβ(−p2)) is the final quark current, F̃ a μν = εμνρσkρe a σ, e a μ and kμ are the polarization and wave vectors of the gluon, p = p1 + p2, p1μ and p2μ are four-momenta of the final quark and antiquark, Pμ ≃ (2m,~0) is the four-momentum of a paraquarkonium in its rest frame, and m is the mass of the heavy quark in the quarkonium. 2 The differential probability of the decay after summation over colors and polarizations of the final particles can be written in the form: dΓq̄qg = Fc α s|ψ(0)| π2m2 [ (P(p1 − p2)) + (Pk)2 p2(Pk)2 + 4mμ (p2)2 ] × δ(P − p1 − p2 − k) d~ p1d~ p2d~k ε1ε2ω , (2) where Fc = (N 2 c − 1)/(8Nc) is the color factor of the SU(Nc) group (in QCD Nc = 3 and Fc = 1/3), αs = g 2 s/(4π) is the strong coupling constant, and μ = mq/m is the relative mass of a final quark (antiquark). The probability (2) of the three particle decay S0(Q̄Q) → q̄qg of the paraquarkonium depends on two independent variables and as a function of two independent energies can be presented in the form: dΓq̄qg dy1dy2 = dΓq̄qg dy1dx = Fc 2α s|ψ(0)| m2 [ x + (y1 − y2) x2(1− x) + μ (1− x)2 ] , (3) where y1 = ε1/m, y2 = ε2/m and x = ω/m are the relative energies of the quark, antiquark and gluon respectively satisfying the energy conservation low x + y1 + y2 = 2. The expression (3) is rather simple and allows for the relative mass of the final quark. In the particular case of μ = 0 it is consistent with the corresponding result presented in the more complicated form in Ref. [2]. Integrating the Eq. (2) over the momenta of the quark ~p1 and the antiquark ~ p2 or over the momenta of the antiquark ~ p2 and the gluon ~k we obtain the distribution in the energy x of the final gluon or that in the energy y1 of the final quark in the form: dΓq̄qg dx = Fc 8α s|ψ(0)| 3m2 x 1− x (