Closing the Gap on $R_{D^{\ast }}$ by including longitudinal effects

Measurements of the $R_{D^}\equiv$\mathrm{B}\mathrm{r}$(B\rightarrow \tau\nu D^)/$\mathrm{B}\mathrm{r}$(B\rightarrow e\nu D^*)$ parameter remain in tension with the standard model prediction, despite recent results helping to close the gap. The standard model prediction it is compared with considers the $D^{\ast }$ as an external particle, even though what is detected in experiments is a $D\pi$ pair it decays into, from which it is reconstructed. We argue that the experimental result must be compared with the theoretical prediction considering the full 4-body decay $(B\to l\nu D^{\ast }\to l\nu D\pi )$. We show that the longitudinal degree of freedom of the off-shell $D^{\ast }$ helps to further close the disagreement gap with experimental data. We find values for the ratio $R_{D\pi}^l \equiv {$\mathrm{B}\mathrm{r}$(B\rightarrow \tau\nu_\tau D \pi)}/$\mathrm{B}\mathrm{r}$(B\rightarrow l\nu_l D\pi)$of$R_{D\pi}^e=0.274\pm0.003$and$R_{D\pi}^\mu=0.275\pm0.003$, where the uncertainty comes from the uncertainty of the form factors parameters. Comparing against $R_{D\pi }$ reduces the gap with the latest LHCb result from $1.1\sigma$ to $0.48\sigma$, while the gap with the latest Belle result is reduced from $0.42\sigma$ to just $0.10\sigma$ and with the world average results from $3.7\sigma$ to $2.1\sigma$. Erratum added at the end of the file.