\[ehha\], we present the most general effective weak Lagrangian for the $b \to c\tau\nu$ quark level transitions in the presence of NP valid at renormalization scale $\mu = m_b$. We also present all the relevant formulas pertinent for our analysis. The results of our analysis are reported in section. \[rd\]. We conclude with a brief summary of our results in section. \[con\].

Theory framework {#ehha}
================

We begin with the most general effective Lagrangian for the $b \to c\,l\nu$ quark level transitions in the presence of NP. That is [@Bhattacharya; @Cirigliano] $$\begin{aligned}
\mathcal L_{\rm eff} &=&
-\frac{4\,G_F}{\sqrt{2}}\,V_{c b}\,\Bigg\{(1 + V_L)\,\bar{l}_L\,\gamma_{\mu}\,\nu_L\,\bar{c}_L\,\gamma^{\mu}\,b_L + 
V_R\,\bar{l}_L\,\gamma_{\mu}\,\nu_L\,\bar{c}_R\,\gamma^{\mu}\,b_R 
+
\widetilde{V}_L\,\bar{l}_R\,\gamma_{\mu}\,\nu_R\,\bar{c}_L\,\gamma^{\mu}\,b_L 
+
\widetilde{V}_R\,\bar{l}_R\,\gamma_{\mu}\,\nu_R\,\bar{c}_R\,\gamma^{\mu}\,b_R + \nonumber \\
&&S_L\,\bar{l}_R\,\nu_L\,\bar{c}_R\,b_L + 
S_R\,\bar{l}_R\,\nu_L\,\bar{c}_L\,b_R +
\widetilde{S}_L\,\bar{l}_L\,\nu_R\,\bar{c}_R\,b_L +
\widetilde{S}_R\,\bar{l}_L\,\nu_R\,\bar{c}_L\,b_R 
+ 
T_L\,\bar{l}_R\,\sigma_{\mu\nu}\,\nu_L\,\bar{q^\prime}_R\,\sigma^{\mu\nu}\,b_L + \nonumber \\
&&\widetilde{T}_L\,\bar{l}_L\,\sigma_{\mu\nu}\,\nu_R\,\bar{q^\prime}_L\,\sigma^{\mu\nu}\,b_R\Bigg\}
 + {\rm H.c.}\,,\end{aligned}$$ where $(V_L,\,V_R,\,S_L,\,S_R,\,T_L)$ represents NP couplings that involve left handed neutrino interactions and $(\widetilde{V}_L,\,\widetilde{V}_R,\,\widetilde{S}_L,\,\widetilde{S}_R,\,\widetilde{T}_L)$ represents NP couplings that involve right handed neutrino interactions. We consider all the NP couplings to be real. In the presence of such NP, the three body differential branching ratios for $B_q \to (P,\,V)\,l\nu$ decays, where $P(V)$ represents pseudoscalar (vector) meson, can be written as [@dutta; @Bardhan:2016uhr; @Watanabe:2017mip]

$$\begin{aligned}
&&\frac{d\Gamma^P}{dq^2}(+) =
\frac{8\,N\,|\overrightarrow{p}_P|\,}{3}\Bigg\{\frac{m_l^2}{q^2}\Big[\widetilde{H}_{T0}^2 + \frac{1}{2}\,H_{0T}^2 + \frac{3}{2}\,
H_{tS}^2\Big]\Bigg\}\,,\qquad
\frac{d\Gamma^P}{dq^2}(-) =
\frac{8\,N\,|\overrightarrow{p}_P|\,}{3}\Bigg\{\frac{m_l^2}{q^2}\Big[H_{T0}^2 + \frac{1}{2}\,\widetilde{H}_{0T}^2 