Semirelativistic study of the semileptonic decays of Bq 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, Bq→(c̄q)(nP)⁺ν_ $(q=u,d,s,c;n=1,2,3)$. The obtained branching ratio of B(B→D₂^(2460)⁺ν_) agrees with the experimental data. We predict $B({B}ₛ⁰{→}{D}ₛ₂^{{}{-}}(1P){{}}⁺{{ν}}_{{}})=3.76×{}{10}^{{-}3}$ and $B({B}c⁺{→}{{χ}}c2(1P){{}}⁺{{ν}}_{{}})=1.82×{}{10}^{{-}3}$. The branching ratios of decays to $2P$ and $3P$ final states are found to be very small. The ratios $R({D̄}₂^{{}0})=0.045$, $R({D}ₛ₂^{{}})=0.048$ and $R({{χ}}c2)=0.059$ are also obtained. This study focuses on the contribution of relativistic corrections. The wave function of the pseudoscalar includes nonrelativistic $S$-wave and relativistic $P$-wave. While for a tensor, it contains nonrelativistic $P$-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 $B{→}{D}₂^{{}}(2460){{}}⁺{{ν}}_{{}}$, but the overall contribution of the relativistic effect is 24.4%, which is small due to cancellation. Similarly, for the decay ${B}ₛ⁰{→}{D}ₛ₂^{{}{-}}(1P){{}}⁺{{ν}}_{{}}$, the contribution of the relativistic effect is 28.8%. While for ${B}c⁺{→}{{χ}}c2(1P){{}}⁺{{ν}}_{{}}$, the individual contributions of relativistic partial waves and the overall relativistic correction are both small, the later of which is 22.1%.