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

Abstract

Measurements of the $R_{D^{\ast }}\equiv \mathrm{Br}(B\to \tau \nu D^{\ast })/\mathrm{Br}(B\to e\nu D^{\ast })$ 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 $R_{D\pi }^l\equiv \mathrm{Br}(B\to \tau {\nu }_{\tau }D\pi )/\mathrm{Br}(B\to l{\nu }_{l}D\pi )$ of $R_{D\pi }^e=0.274\pm 0.003$ and $R_{D\pi }^{\mu }=0.275\pm 0.003$, 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.

arXiv commentTable 1 updated. After publication we found a bug in the transcription code for the numerical integration. Using the corrected values, the results on $R_{D\pi }$ are consistent with $R_{D^{\ast }}$ at the current precision. This modifies our previous finding that lead us to a different conclusion

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