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Search for $B_c(ns)$ via the $B_c(ns)\to B_c(ms)π^+π^-$ transition at LHCb and $Z_0$ factory

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, $B_c^*(ns)\to B_c+γ$ is not easy to be clearly measured, instead, the trajectories of $π^+$ and $π^-$ occurring in the decay of $B_c(ns)\to B_c(ms)+π^+π^-$ ($n>m$) 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 $B_c(ns)\to B_c(ms)+π^+π^-$ 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.