  Recently, the experimental measurements of the branching ratios and different
polarization asymmetries for the processes occurring through
flavor-changing-charged current $b\rightarrow c\tau\overline{\nu}_{\tau}$
transitions by BABAR, Belle, and LHCb show some sparkling differences with the
corresponding SM predictions. Assuming the left handed neutrinos, we add the
dimension-six vector, (pseudo-)scalar, and tensor operators with complex WCs to
the SM WEH. Together with 60%, 30% and 10% constraints coming from the
branching ratio of $B_{c}\to\tau\bar{\nu}_{\tau}$, we analyze the parametric
space of these new physics WCs accommodating the current anomalies in the
purview of the most recent HFLAV data of $R_{\tau/{\mu,e}}\left(D\right)$,
$R_{\tau/{\mu,e}}\left(D^*\right)$ and Belle data of $F_{L}\left(D^*\right)$
and $P_{\tau}\left(D^*\right)$. Furthermore, we derive the sum rules which
correlate these observables with $R_{\tau/{\mu,e}}\left(D\right)$ and
$R_{\tau/{\mu,e}}\left(D^*\right)$. Using the best-fit points of the new
complex WCs along with the latest measurements of
$R_{\tau/{\mu,e}}\left(D^{(*)}\right)$, we predict the numerical values of the
observable $R_{\tau/\ell}\left(\Lambda_c\right)$,
$R_{\tau/\mu}\left(J/\psi\right)$ and $R_{\tau/\ell}\left(X_c\right)$ from the
sum rules. Apart from finding the correlation matrix among the observables
under consideration, we plot them graphically which is useful to discriminate
different NP scenarios. Finally, we study the impact of these NP couplings on
various angular and the CP triple product asymmetries, that could be measured
in some ongoing and future experiments. The precise measurements of these
observables are important to check the SM and extract the possible NP.
