Relativistic corrections are important in hadronic physics since even for the heavy hadrons there are sizable relativistic corrections. Therefore one should use a relativistic model to describe the higher excited states. This note summarizes our predictions for the decay constants of the S wave and P wave heavy mesons by means of the instantaneous relativistic Bethe Salpeter equation （Salpeter equation）.
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 .
