  It is interesting to study the characteristics of the whole family of $B_c$
which contains two different heavy flavors. LHC and the proposed $Z^0$ factory
provide an opportunity because a large database on the $B_c$ family will be
achieved. $B_c$ and its excited states can be identified via their decay modes.
As suggested by experimentalists, $B_c^*(ns)\to B_c+\gamma$ is not easy to be
clearly measured, instead, the trajectories of $\pi^+$ and $\pi^-$ occurring in
the decay of $B_c(ns)\to B_c(ms)+\pi^+\pi^-$ ($n>m$) can be unambiguously
identified, thus the measurement seems easier and more reliable, therefore this
mode is more favorable at early running stage of LHCb and the proposed $Z^0$
factory. In this work, we calculate the rate of $B_c(ns)\to B_c(ms)+\pi^+\pi^-$
in terms of the QCD multipole-expansion and the numerical results indicate that
the experimental measurements with the luminosity of LHC and $Z^0$ factory are
feasible.
