$D^{\ast }$ polarization vs. $R_{D^{(\ast )}}$ anomalies in the leptoquark models

Polarization measurements in $\bar{B}\to D^{(\ast )}\tau \bar{\nu }$ are useful to check consistency in new physics explanations for the $R_D$ and $R_{D^{\ast }}$ anomalies. In this paper, we investigate the $D^{\ast }$ and $\tau$ polarizations and focus on the new physics contributions to the fraction of a longitudinal $D^{\ast }$ polarization ($F_L^{D^{\ast }}$), which is recently measured by the Belle collaboration $F_{L}^{D^{\ast}} = 0.60 \pm 0.09$,inmodel-independentmannerandineachsingleleptoquarkmodel(${\rm R}_2$,${\rm S}1$and${\rm U}1$)thatcannaturallyexplainthe$R{D^{(\ast)}}$anomalies\mathrm{.}Itisfoundthat$$\mathcal{B}$(B_c^{+} \to \tau^{+} \nu)$severelyrestrictsdeviationfromtheStandardModel(SM)predictionof$F{L, $\text{SM}$}^{D^{\ast}} = 0.46 \pm 0.04$ in the leptoquark models: [0.43, 0.44], [0.42, 0.48], and [0.43, 0.47] are predicted as a range of $F_L^{D^{\ast }}$ for the ${\mathrm{R}}_2$, ${\mathrm{S}}_1$, and ${\mathrm{U}}_1$ leptoquark models, respectively, where the current data of $R_{D^{(\ast )}}$ is satisfied at $1\sigma$ level. It is also shown that the $\tau$ polarization observables can much deviate from the SM predictions. The Belle II experiment, therefore, can check such correlations between $R_{D^{(\ast )}}$ and the polarization observables, and discriminate among the leptoquark models.