Longitudinal leading-twist distribution amplitude of the ${}^1{P}_1$-state $b_1(1235)$ meson and its implications on $B\to b_1(1235){\mathrm{\ell }}^{+}{\nu }_{\mathrm{\ell }}$ decays

In the paper, we derive the $\xi$-moments $⟨{\xi }_{2;b_1}^{n;\mathrm{\parallel }}⟩$ of the longitudinal leading-twist distribution amplitude ${\phi }_{2;b_1}^{\mathrm{\parallel }}$ for ${}^1{P}_1$-state $b_1(1235)$-meson by using the QCD sum rules under the background field theory. Considering the contributions from the vacuum condensates up to dimension-six, its first two non-zero $\xi$-moments at the scale 1 GeV are $⟨{\xi }_{2;b_1}^{1;\mathrm{\parallel }}⟩=-0.647_{-0.113}^{+0.118}$ and $⟨{\xi }_{2;b_1}^{3;\mathrm{\parallel }}⟩=-0.328_{-0.052}^{+0.055}$, respectively. Using those moments, we then fix the Gegenbauer expansion series of ${\phi }_{2;b_1}^{\mathrm{\parallel }}$ and apply it to calculate $B\to b_1(1235)$ transition form factors (TFFs) that are derived by using the QCD light-cone sum rules. Those TFFs are then extrapolated to the physically allowable $q^2$-range via the simplified series expansion. As for the branching fractions, we obtain $\mathcal{B}({\bar{B}}^0\to b_1^{+}(1235)e^{-}{\bar{\nu }}_e)=2.179_{-0.422}^{+0.553}\times {10}^{-4}$, $\mathcal{B}(B^0\to b_1^{-}(1235){\mu }^{+}{\nu }_{\mu })=2.166_{-0.415}^{+0.544}\times {10}^{-4}$, $\mathcal{B}(B^{+}\to b_1^0(1235)e^{+}{\nu }_e)=2.353_{-0.456}^{+0.597}\times {10}^{-4}$, and $\mathcal{B}(B^{+}\to b_1^0(1235){\mu }^{+}{\nu }_{\mu })=2.339_{-0.448}^{+0.587}\times {10}^{-4}$, respectively.