  In this article, we investigate the structures of the pseudoscalar charmonium
and bottomonium in the framework of the coupled rainbow Schwinger-Dyson
equation and ladder Bethe-Salpeter equation with the confining effective
potential (infrared modified flat bottom potential).
  As the current masses are very large, the dressing or renormalization for the
$c$ and $b$ quarks are tender, however, mass poles in the timelike region are
absent. The Euclidean time fourier transformed quark propagator has no mass
poles in the timelike region which naturally implements confinement. The
Bethe-Salpeter wavefunctions for those mesons have the same type (Gaussian
type) momentum dependence and center around zero momentum with spatial
extension to about $q^2=1GeV^2$ which happen to be the energy scale for Chiral
symmetry breaking, the strong interactions in the infrared region result in
bound states. The decay constants for those pseudoscalar heavy quarkonia are
compatible with the values of experimental extractions and theoretical
calculations.
