Considering the fact that some excited states of the heavy quarkonia (charmonium and bottomonium) are still missing in experimental observations and potential applications of the relevant wave functions of the bound states,we re-analyze the spectrum and the relevant wave functions of the heavy quarkonia within the framework of Bethe-Salpeter (B.S.) equation with a proper QCDinspired kernel.Such a kernel for the heavy quarkonia,relating to potential of the non-relativistic quark model,is instantaneous,so we call the corresponding B.S.equation as BS-In equation throughout the paper.Particularly,a new way to solve the B.S.equation,which is different from the traditional ones,is proposed here,and with it not only the known spectrum for the heavy quarkonia is re-generated,but also an important issue is brought in,i.e.,the obtained solutions of the equation ‘automatically’ include the ‘fine’,‘hyperfine’ splittings and the wave function mixture,such as S-D wave mixing in J PC = 1-states,P-F wave mixing in J PC = 2 ++ states for charmonium,bottomonium etc.It is pointed out that the best place to test the wave mixture probably is at Z-factory (e + e-collider running at Z-boson pole with extremely high luminosity).
The magnetic dipole transitions between the vector mesons B-c and their relevant pseudoscalar mesons B c (B c ,B-c ,B c (2S ),B-c (2S ),B c (3S ),B-c (3S ) etc.,the binding states of (c) system) of the B c family are interesting.The ‘hyperfine’ splitting due to spin-spin interaction is an important topic for understanding the spin-spin interaction and the spectrum of the the (c) binding system.The knowledge about the magnetic dipole transitions is also very useful for identifying the vector boson B-c mesons experimentally,whose masses are just slightly above the masses of their relevant pseudoscalar mesons B c .Considering the possibility to observe the vector mesons via the transitions at Z 0 factory and the potential use of the theoretical estimate on the transitions,we fucus our efforts on calculating the magnetic dipole transitions,i.e.a precise calculation of the rates for the transitions such as decays B-c → B c γ and B-c → B c e + e-,and particularly work in the Bethe-Salpeter framework.As a typical example,we carefully investigate the dependence of the rate Γ(B-c → B c γ) on the mass difference ΔM = M B-c-M B c .
Considering the possibility to build an e + e-collider at the energies around Z-boson resonance with a luminosity so high as L ∝ 10 34 cm-2 s-1 (even higher) and the abilities of a modern synthesis detector,we systematically calculate the exclusive two body processes of the heavy quarkonium production: e + eannihilates into a heavy quarkonium and a photon,involving the initial state radiation (i.e.ISR) cases,at the energies around the Z-boson resonance.Since the couplings of Z-boson to quarks contain an axial vector term as well as a vector one,a charmonium such as J/ψ or η c or h c or χ cJ ···,or a bottomonium such as Υ or η b or h b or χ bJ ···,associating with a photon,may be produced respectively via Z-boson annihilation.If we call such a collider with so high luminosity and running around the Z-boson resonance as a Z-factory,then our results obtained here indicate that experimental studies of the various heavy quarkona (their ground and excited states) via the two-body processes at a Z-factory have outstanding advantages,especially,the production of the possible states with quantum numbers J PC = 1-via ISR.
