Spectrum for Heavy Quankonia and Mixture of the Relevant Wave Functions within the Framework of Bethe-Salpeter Equation

Considering the fact that some excited states of the heavy quarkonia (charmonium and bottomonium) 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 QCD-inspired kernel. Such a kernel for the heavy quarkonia, relating to potential of 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 and 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).
26 pages, 8 figures