The value of the quarkonium wave function at the origin is an important quantity for studying many physical problems concerning a heavy quarkonium. This is because it is widely used to evaluate the production and decay amplitudes of the heavy quarkonium within the effective field theory framework, e.g., the nonrelativistic QCD (NRQCD). In this paper, the values of the Schrödinger radial wave function or its first nonvanishing derivative at zero quark-antiquark separation, i.e., |(|cc) [

The programming new e + e − collider with high luminosity shall provide another useful platform to study the properties of the doubly heavy Bc meson in addition to the hadronic colliders as LHC and TEVATRON. Under the 'New Trace Amplitude Approach', we calculate the production of the spin-singlet Bc and the spin-triplet B * c mesons through the Z 0 boson decays, where uncertainties for the production are also discussed. Our results show Γ ( PACS numbers: 12.38. Bx, 12.39.Jh, 14.40Lb, 14.40.Nd The B c meson is a double heavy quark-antiquark bound state and carries flavors explicitly. Since its first discovery at TEVATRON [1], B c physics is attracting more and wide interests. Recently, many progresses have been made for the hadronic production of B c meson at high energy colliders as LHC and TEVATRON. A computer program BCVEGPY for the direct hadronic production of B c meson has been presented in Refs. [2,3]. And it has been found that the indirect production of B c via top quark decays can also provide useful information on B c meson [4][5][6][7].Comparing with the hadronic colliders, an e + e − collider has its own advantages, mainly because of its lower background. As for the previous LEP-I experiment, no B c events have been found due to its lower collision energy and low luminosity [8,9]. However, if the luminosity of the e + e − collider can be raised up to L ∝ 10 34 cm −2 s −1 or even higher as programmed by the Internal Linear Collider (ILC) [10], then there might have enough events. Moreover, if the e + e − collider further runs at the Z 0 -boson energy, the resonance effects at the Z 0 peak may raise the production rate up to several orders. It has been estimated by Ref.[11] that more than 10 9∼10 Z 0 -events can be produced at ILC per year, which is about 3 ∼ 4 orders higher than that collected by LEP-I. Such a high luminosity collider is called as GigaZ [11] or a Z-factory [12]. Then it will open new opportunities not only for high precision physics in the electro-weak sector, but also for the hadron physics.The production of B c through Z 0 decays has been studied in Refs [8,9,13] with various methods. Since the process is very complicated, it would be helpful to have a cross check of these results. Furthermore, considering the forthcoming Z-factory, it may be interesting to know the theoretical uncertainties in estimating of B c production.For the purpose, we need to calculate the processc +b+c, whose Feynman diagrams are shown in Fig.(1). According to the NRQCD factorization formula * e-mail:wuxg@cqu.edu.cnc[14], the decay width for the process Z 0 → B ( * ) c + b +c can be written in the following factorization form:where the matrix element O H (n) is proportional to the inclusive transition probability of the perturbative state cb[n] into the bound states of B c . As for the two colorsinglet S-wave states cb[are related with the Bethe-Salpeter wave function at the origin that can be determined by the potential model [15][16][17][18][19][20]. dΓ(Z 0 → cb[n] + b +c) stands for the short-distan...

The production of the heavy (cc)-quarkonium, (cb)-quarkonium and (bb)-quarkonium states [(QQ ′ ) quarkonium for short], via the W + semi-inclusive decays, has been systematically studied within the framework of the non-relativistic QCD. In addition to the two color-singlet S-wave states, we also discuss the production of the four color-singlet P -wave states |(QQ ′ )( 1 P1)1 and |(QQ ′ )( 3 PJ )1 [with J = (0, 1, 2)] together with the two color-octet components |(QQ ′ )( 1 S0)8 and |(QQ ′ )( 3 S1)8 . Improved trace technology is adopted to derive the simplified analytic expressions at the amplitude level, which shall be useful for dealing with the following cascade decay channels. At the LHC with the luminosity L ∝ 10 34 cm −2 s −1 and the center-of-mass energy √ S = 14 TeV, sizable heavy-quarkonium events can be produced through the W + boson decays, i.e. 2.57 × 10 6 ηc, 2.65 × 10 6 J/Ψ and 2.40 × 10 6 P -wave charmonium events per year can be obtained; and 1.01 × 10 5 Bc, 9.11 × 10 4 B * c and 3.16 × 10 4 P -wave (cb)-quarkonium events per year can be obtained. Main theoretical uncertainties have also been discussed. By adding the uncertainties caused by the quark masses in quadrature, we obtain Γ W + →(cc)+cs = 524.8 +396.3 −258.4 KeV, Γ W + →(cb)+bs = 13.5 +4.73 −3.29 KeV, Γ W + →(cb)+cc = 1.74 +1.98 −0.73 KeV and Γ W + →(bb)+cb = 38.6 +13.4 −9.69 eV.

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