  The measurement of $R_D$ ($R_{D^*}$), the ratio of the branching fraction of
$\overline{B} \to D \tau \bar{\nu}_\tau (\overline{B} \to D^* \tau
\bar{\nu}_\tau)$ to that of $\overline{B} \to D l \bar{\nu}_l (\overline{B} \to
D^* l \bar{\nu}_l)$, 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 $\rm SU(3)_{C} \times
SU(2)_{L} \times U(1)_{Y}$ gauge invariance. We first show that, in general,
$R_D$ 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 $R_{D^*}$ individually and
also together. Motivated by the recent measurement of the $\tau$ polarisation
in $\overline{B} \to D^* \tau \bar{\nu}_\tau$ decay, $P_\tau (D^*)$ by the
Belle collaboration, we study the impact of a more precise measurement of
$P_\tau (D^*)$ (and a measurement of $P_\tau (D)$) on the various possible NP
explanations. Furthermore, we show that the measurement of $R_{D^*}$ 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.
