Relating Quarkonium Wave Functions at the Origin

Within the context of nonrelativistic potential models, we obtain several formulas (with varying degrees of rigor) relating the wave functions at the origin of the \(c{\bar c}\), \(b{\bar c}\) and \(b{\bar b}\) S-wave quarkonium systems. One of our main results is a model-independent relation which seems to hold to within 3% for any reasonable choice of interquark potential and any choice of radial quantum number -- namely, \({|\Psi_{b{\bar c}}(0)|^2 \simeq |\Psi_{c{\bar c}}(0)|^{1.3}|\Psi_{b{\bar b}}(0)|^{0.7}}\) (the exponents are motivated in the text). One of the physical consequences of this result is the following relationship between heavy meson masses which we expect to hold at about the 10% level: \({M_{B_c^*}-M_{B_c}\simeq (0.7)(M_{J/\psi} -M_{\eta_c})^{0.65}(M_{\Upsilon}-M_{\eta_b})^{0.35}}\).