Longitudinal leading-twist distribution amplitude of the $^1P_1$-state $b_1(1235)$ meson and its implications on $B\to b_1(1235)\ell^+\nu_\ell$ decays

In the paper, we derive the $\xi$-moments $\langle\xi_{2;b_1}^{n;\|}\rangle$ of the longitudinal leading-twist distribution amplitude $\phi_{2;b_1}^{\|}$ for $^1P_1$-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 $\langle\xi_{2;b_1}^{1;\|}\rangle= -0.647^{+0.118}_{-0.113}$ and $\langle\xi_{2;b_1}^{3;\|}\rangle = -0.328^{+0.055}_{-0.052}$, respectively. Using those moments, we then fix the Gegenbauer expansion series of $\phi_{2;b_1}^{\|}$ 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 ${\cal B}(\bar B^0 \to b_1^+(1235)e^- \bar\nu_e) = 2.179^{+0.553}_{-0.422}\times 10^{-4}$, ${\cal B}(B^0 \to b_1^-(1235)\mu^+\nu_\mu) = 2.166^{+0.544}_{-0.415}\times 10^{-4}$, ${\cal B}(B^+ \to b_1^0(1235)e^+\nu_e) = 2.353^{+0.597}_{-0.456}\times 10^{-4}$, and ${\cal B}(B^+ \to b_1^0(1235)\mu^+\nu_\mu) = 2.339^{+0.587}_{-0.448}\times 10^{-4}$, respectively.