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In this paper, we study the OZI-allowed two-body strong decays of $3^-$heavy-light mesons. Experimentally the charmed $D_{3}^{\ast}(2760)$ and thecharm-strange $D_{s3}^{\ast}(2860)$ states with these quantum numbers have beendiscovered. For the bottomed $B(5970)$ state, which was found by the CDFCollaboration recently, its quantum number has not been decided yet and weassume its a $3^-$ meson in this paper. The theoretical prediction for thestrong decays of bottom-strange state $B_{s3}^\ast$ is also given. Therelativistic wave functions of $3^-$ heavy mesons are constructed and theirnumerical values are obtained by solving the corresponding Bethe-Salpeterequation with instantaneous approximation. The transition matrix is calculatedby using the PCAC and low energy theorem, following which, the decay widths areobtained. For $D_{3}^\ast(2760)$ and $D_{s3}^\ast(2860)$, the total strongdecay widths are 72.6 MeV and 47.6 MeV, respectively. For $B_3^\ast$ with$M=5978$ MeV and $B_{s3}^\ast$ with $M=6178$ MeV, their strong decay widths are22.9 MeV and 40.8 MeV, respectively.

Strong decays of $D_{3}^{\ast}(2760)$, $D_{s3}^{\ast}(2860)$, $B_{3}^{\ast}$, and $B_{s3}^{\ast}$