Semirelativistic study on the semileptonic decays of $B_q$ mesons to orbital excited heavy Tensors

Based on the method of solving the complete Salpeter equation, we study the semileptonic decays of a $0^-$ heavy meson to $1P$, $2P$, or $3P$ heavy tensor mesons, $B_q \to (\bar c q)(nP) \ell^+ \nu_\ell$ $(q=u,d,s,c;n=1,2,3)$. The obtained branching ratio of $\mathcal{B} (B \rightarrow D_2^{\star}(2460)\ell^{+} \nu_{\ell})$ agrees with the experimental data. We predict $\mathcal{B}\left(B_s^{0} \rightarrow D_{s2}^{\star-}(1P) \ell^{+} \nu_{\ell}\right)$$=$$3.76\times 10^{-3}$ and $\mathcal{B}\left(B_c^+ \rightarrow \chi_{c2}(1P)\ell^{+} \nu_{\ell}\right)$$=$$1.82\times 10^{-3}$. The branching ratios of decays to $2P$ and $3P$ final states are found to be very small. The ratios $\mathcal{R}(\bar{D}_{2}^{\star 0})=0.045$, $\mathcal{R}({D}_{s2}^{\star})=0.048$ and $\mathcal{R}({\chi}_{c2})=0.059$ are also obtained. This study focuses on the contribution of relativistic corrections. The wave function of the pseudoscalar includes non-relativistic $S$-wave and relativistic $P$-wave. While for a tensor, it contains non-relativistic $P$-wave and relativistic $P$, $D$ and $F$ waves in its wave function. We find the individual contributions of relativistic partial waves are significant in the decay $B \to D_2^{\star }(2460)\ell^{+} \nu_{\ell}$, but the overall contribution of the relativistic effect is $24.4\%$, which is small due to cancellation. Similarly, for the decay $B_s^{0} \rightarrow D_{s2}^{\star-}(1P) \ell^{+} \nu_{\ell}$, the contribution of the relativistic effect is $28.8\%$. While for $B_c^+ \rightarrow \chi_{c2}(1P)\ell^{+} \nu_{\ell}$, the individual contributions of relativistic partial waves and the overall relativistic correction are both small, the later of which is $22.1\%$.