Quarkonium spectroscopy in a potential model with vacuum-polarization corrections

We consider a potential model taking long-distance vacuum-polarization corrections as suggested by Poggio and Schnitzer, which enables one to interpolate between $c\overline{c}$, $b\overline{b}$, and $t\overline{t}$ systems. Taking special care for the accuracy of the numerical integration near the origin, we have developed a numerical method to obtain the heavy-quark-antiquark bound states along with their leptonic widths. We obtain the above flavor-independent potential giving good agreement with the so-called experimental mass splitting of the $1S-2S$ states of the $\psi$ and $\Upsilon$ family with reasonable values of the quark-gluon coupling constant ${\alpha }_s$, which do not deviate very much from the quantum-chromodynamics value. We obtain some of the bound states of the hypothetical $t\overline{t}$ family and observe that the effect of screening of the potential due to the vacuum-polarization could decreases with increase of the mass of the heavy quark forming the quarkonium.