3\,m_V^2-q^2\Big)\,T_2(q^2) + \frac{2\,m_{B_q}\,|\overrightarrow{p}_V|}
{m_{B_q}^2 - m_V^2}\,T_3(q^2)\Big]\,.\end{aligned}$$

Here we denote $G_V = 1+V_L+V_R$, $G_A =1+ V_L - V_R$, $G_S = S_L+S_R$, $G_P=S_L-S_R$, $\widetilde{G}_V=\widetilde{V}_L + \widetilde{V}_R$, $\widetilde{G}_A=\widetilde{V}_L - \widetilde{V}_R$, $\widetilde{G}_S=\widetilde{S}_L + \widetilde{S}_R$, and $\widetilde{G}_P=\widetilde{S}_L - \widetilde{S}_R$. Again, $|\overrightarrow{p}_{P(V)}| = 
\sqrt{\lambda(m_{B_q}^2,\,m_{P(V)}^2,\,q^2)}/2\,m_{B_q}$ denotes the three momentum vector of the outgoing meson. The ratio of branching ratios and $\tau$ polarization fractions for these decay modes are

$$\begin{aligned}
&&R_{M} = \frac{\mathcal B(B_q \to M\tau\nu)}{\mathcal B(B_q \to M\,l\nu)}\,,\qquad
P_{\tau}^{M} = \frac{\Gamma^{M}(+) - \Gamma^{M}(-)}{\Gamma^{M}(+) + \Gamma^{M}(-)}\,, \end{aligned}$$

where, $l$ is either an electron or a muon and $B_q$ is either a $B$ meson or a $B_c$ meson. Similarly, $M$ refers to the outgoing pseudoscalar or vector meson. Again, $\Gamma(+)$ and $\Gamma(-)$ denote the decay widths of positive and negative helicity $\tau$ lepton, respectively.

Numerical analyses {#rd}
==================

For the numerical estimates of all the observables we first report all the input parameters. For the quark, lepton, and meson masses, we use $m_b(m_b)=4.18\,{\rm GeV}$, $m_c(m_b)=0.91\,{\rm GeV}$, $m_e= 0.510998928\times 10^{-3}
\,{\rm GeV}$, $m_{\mu}= 0.10565837151\,{\rm GeV}$, $m_{\tau}= 1.77682\,{\rm GeV}$, $m_{J/\Psi}= 3.0969{\rm GeV}$, $m_{B^-}= 5.27931\,{\rm GeV}$, $m_{B_c}= 6.2751\,{\rm GeV}$, $m_{D^0}= 1.86483\,{\rm GeV}$, and $m_{{D^{\ast}}^0}= 2.00685\,{\rm GeV}$ [@pdg]. Similarly, for the mean lifetime of $B^-$ and $B_c$ meson, we use $\tau_{B^-} = 1.638 