A closer look at the $R_D$ and $R_{D^{\ast }}$ anomalies

The measurement of $R_D$ ($R_{D^{\ast }}$), the ratio of the branching fraction of $\overline{B} \to D \tau \bar{\nu}\tau (\overline{B} \to D^* \tau \bar{\nu}\tau)$tothatof$\overline{B} \to D l \bar{\nu}l (\overline{B} \to D^* l \bar{\nu}l)$,shows$1.9 \sigma$ $(3.3\sigma )$ deviation from its Standard Model (SM) prediction. The combined deviation is at the level of $4 \sigma$ according to the Heavy Flavour Averaging Group (HFAG). In this paper, we perform an effective field theory analysis (at the dimension-6 level) of these potential New Physics (NP) signals assuming $\rm SU(3){C} \times SU(2){L} \times U(1){Y}$gaugeinvariance\mathrm{.}Wefirstshowthat,ingeneral,$R_D$and$R{D^*}$ are theoretically independent observables and hence, their theoretical predictions are not correlated. We identify the operators that can explain the experimental measurements of $R_D$ and $R_{D^{\ast }}$ individually and also together. Motivated by the recent measurement of the $\tau$ polarisation in $\bar{B}\to D^{\ast }\tau {\bar{\nu }}_{\tau }$ decay, $P_{\tau }(D^{\ast })$ by the Belle collaboration, we study the impact of a more precise measurement of $P_{\tau }(D^{\ast })$ (and a measurement of $P_{\tau }(D)$) on the various possible NP explanations. Furthermore, we show that the measurement of $R_{D^{\ast }}$ in bins of $q^2$, the square of the invariant mass of the lepton-neutrino system, along with the information on $\tau$ polarisation and the forward-backward asymmetry of the $\tau$ lepton, can completely distinguish the various operator structures. We also provide the full expressions of the double differential decay widths for the individual $\tau$ helicities in the presence of all the 10 dimension-6 operators that can contribute to these decays.