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We calculate the fully-differential rate of the decays $B\to P\tau(\to\mu\bar\nu\nu)\bar\nu$ where $P = D,\pi$, which is a background to thesemimuonic decays $B\to P\mu\bar\nu$. The decays with a $3\nu$ final state canhave a sizeable impact on the experimental analyses of the ratios $R_D$ and$R_\pi$, depending on the event selection in the analysis. We outline astrategy which permits the extraction of $R_P \mathcal{B}(\tau \to\mu\bar\nu\nu)$ from the neutrino-inclusive rate. Our analytic results can alsobe used to test both existing and upcoming experimental analyses. We furtherprovide Monte Carlo samples of the 5D rate of the neutrino-inclusive decays$B\to P\mu X_{\bar\nu}$.

Impact of leptonic $\tau$ decays on the distribution of $B \to P \mu\bar\nu$ decays