Analysis of $S$-wave $ωπ$ dynamics behavior in multi-body $B_{\ell4}$ and $B_{\ell6}$ decays

The $b_1(1235)$ state has a sizable decay width, which have longstanding puzzle in treating as a final state in theoretical calculations may lead to systematic deviations in the predicted observables. To address this issue and further explore the internal structure of this resonance, we study the multi-body semileptonic decays $B_{\ell4}$ and $B_{\ell6}$. Within the approximation where the $b_1(1235)$ resonance dominates, we study the $B_{\ell4}$ decay $B\to b_1(1235)(\toωπ)\ell^+ν_\ell$ in the QCD light-cone sum rules framework. The corresponding transition form factors (TFFs) are calculated by constructing a correlation function with a right-handed chiral current. As the main nonperturbative input for the TFFs calculation, the transverse twist-2 light-cone distribution amplitude (LCDA) of the $b_1(1235)$ resonance is constructed using the light-cone harmonic oscillator model, and the resulting LCDA exhibits a single-peak behavior. At the large-recoil region, we obtain the TFFs $V_{0, 1, 2}(0)$ and $A(0)$, together with the corresponding TFFs ratios $r_V = 1.530_{-0.016}^{+0.013}$ and $r_2 = 0.553_{-0.015}^{+0.012}$. We then further study the complete $B_{\ell4}$ decay by incorporating the $b_1(1235)\toωπ$ resonant line shape and obtain the corresponding observables. The results show that the branching ratio of the $B_{\ell4}$ decay is about $1/3$ of that obtained when $b_1(1235)$ is treated as a final state. We also account for the subsequent decay $ω\to π^+π^-π^0$ and further predicted the branching ratio of the $B_{\ell6}$ decay $B\to b_1(1235)(\toω(\to π^+π^-π^0)π)\ell^+ν_\ell$. These results may provide useful references for future experimental measurements and further theoretical studies.