  In this article, we investigate the under-structures of the $\pi$ and B
mesons in the framework of the Bethe-Salpeter equation with the bare
quark-gluon vertex, bare quark propagator and the confining effective potential
(infrared modified flat bottom potential) firstly.
  Although the bare quark propagator can not embody dynamical chiral symmetry
breaking and has a mass pole in the time-like region, it can give reasonable
results for the values of decay constants $f_\pi$ and $f_B$ compared with the
values of experimental data and other theoretical calculations, such as lattice
simulations and QCD sum rules. Secondly, we explore those mesons within the
framework of the coupled rain-bow Schwinger-Dyson equation and ladder
Bethe-Salpeter equation. The Schwinger-Dyson functions for the $u$ and $d$
quarks are greatly renormalized at small momentum region and the curves are
steep which indicates an explicitly dynamical symmetry breaking. The Euclidean
time fourier transformed quark propagator has no mass poles in the time-like
region which naturally implements confinement. As for the $b$ quark, the
current mass is very large, the renormalization is more tender, however, mass
pole in the time-like region is also absent. The Bethe-Salpeter wavefunctions
for both the $\pi$ and B mesons have the same type (Gaussian type) momentum
dependence as the corresponding wavefunctions with the bare quark propagator,
however, the quantitative values are changed and the values for the decay
constants $f_\pi$ and $f_B$ are changed correspondingly.
