Finding $B_c(3S)$ States via Their Strong Decays

The experimentally known $B_c$ states are all below open bottom-charm threshold, which experience three main decay modes, and all induced by weak interaction. In this work, we investigate the mass spectrum and strong decays of the $B_c(3S)$ states, which just above the threshold, in the Bethe-Salpeter formalism and $^3P_0$ model. The numerical estimation gives $M(B_c(3^1S_0))=7273\ {\rm MeV}$, $M(B_c^*(3^3S_1))=7304\ {\rm MeV}$, $\Gamma\left(B_c(3^1S_0)\to B^*D\right)=26.02^{+2.33}_{-2.21}\ {\rm MeV}$, $\Gamma\left(B_c^*(3^3S_1)\to BD\right)=3.39^{+0.27}_{-0.26}\ {\rm MeV}$, $\Gamma\left(B_c^*(3^3S_1)\to B^*D\right)=14.77^{+1.40}_{-1.33}\ {\rm MeV}$ and $\Gamma\left(B_c^*(3^3S_1)\to BD^*\right)=6.14^{+0.58}_{-0.54}\ {\rm MeV}$. Compared with previous studies in non-relativistic approximation, our results indicate that the relativistic effects are notable in $B_c(3S)$ exclusive strong decays. According to the results, we suggest to find the $B_c(3S)$ states in their hadronic decays to $B$ and $D$ mesons in experiment, like the LHCb.