Measurement of the ratio of branching fractions $\mathscr{B} (B^0 \to D^{*-} \tau^+ \nu_\tau) / \mathscr{B} (B^0 \to D^{*-} \mu^+ \nu_\mu)$ with hadronic $\tau$ three-prong decays

A measurement of the ratio of semileptonic branching fractions $\mathscr{R}(D^{*-}) \equiv \mathscr{B}(B^0\to D^{*-}\tau^+ \nu_\tau )/\mathscr{B}(B^0\to D^{*-}\mu^+ \nu_\mu )$ is performed using a data sample of proton-proton collisions, corresponding to an integrated luminosity of 3 fb$^{-1}$, collected by the LHCb experiment at center-of-mass energies of 7 and 8 TeV. For the first time, $\mathscr{R}(D^{*-})$ is determined using the $\tau$ lepton decay with three charged pions in the final state. The $B^0\to D^{*-}\tau^+ \nu_\tau$ yield is normalized to that of the $B^0\to D^{*-}\pi^+ \pi^- \pi^+$ mode. A measurement of $\mathscr{B}(B^0\to D^{*-}\tau^+ \nu_\tau )/\mathscr{B}(B^0\to D^{*-}\pi^+ \pi^- \pi^+ ) =1.93\pm 0.13\pm 0.17$ is obtained, where the first uncertainty is statistical and the second systematic. The value of $\mathscr{B}(B^0\to D^{*-}\tau^+ \nu_\mu ) = (1.39\pm 0.09\pm 0.12\pm 0.06)$\% is obtained, where the third uncertainty is due to the limited knowledge of the branching fraction of the normalization mode. Using the well-measured branching fraction of the $B^0\to D^{*-}\mu^+ \nu_\mu$ decay, a value of $\mathscr{R}(D^{*-}) = 0.285\pm 0.019\pm 0.025\pm 0.013$ is established, where the first uncertainty is statistical, the second systematic, and the third is due to the limited knowledge of the branching fractions of the normalization and of the $B^0\to D^{*-}\mu^+ \nu_\mu$ modes. This measurement is in agreement with the Standard Model prediction and with previous results. The present systematic uncertainty can be reduced by joint efforts of the LHCb, BABAR, BELLE and BES collaborations. This novel analysis technique will also enable the search for SM deviations in the event distributions, in addition to the event yield, thanks to its unique capability to select high statistics (a few thousands events) highly enriched ( 50%) in semitauonic decays. LHCb will also use the exact same method to perform the measurement of all other B hadrons semitauonic decays, including those coming from Λ0b and B+c hadrons.